When I turned 21, armed with basic knowledge from Edward Thorp’s Beat the Dealer, I decided to try my luck at blackjack. Beat the Dealer was the first book to accurately describe “Basic Strategy”—the best way to play each hand given casino rules and no knowledge of cards removed from play—and the first book to describe card counting systems that could, in principle, give players an advantage. I was surprised that the players and dealers who seemed most knowledgeable about the game “corrected” my play in ways that consistently deviated from Basic. Why would the most experienced players consistently deviate from a strategy I knew to be optimal?

A few years later, after moving to Prague, I was puzzled for a different reason. Experienced players there also shared false beliefs about how to play, but their beliefs were different from those in Las Vegas. Czech players were noticeably less likely than their Las Vegas counterparts to hit (take additional cards) or to double the size of their bets when Basic said they should, and—unlike in Las Vegas, where corrections to my “mistakes” felt intended as helpful advice—Czech players were often upset, sometimes leaving the table in a display of anger. Why should groups of experienced players in two places converge on different false beliefs? And why should the Czech—but not Las Vegas—players get angry about my misperceived poor choices, which I knew to have an unpredictable impact on their own success?

Those two paragraphs introduce one of two articles just accepted for publication (hooray) that challenge the way those questions usually get answered. They get at the heart of what this Substack is about: decision-making in the wild isn’t what lab experiments about heuristics, biases, and human rationality often imply.

Both papers explore the puzzles described above through the lens of what I refer to as culture-bound heuristics: strategies for solving problems, making judgments, or coming to a decision that emerge from, and only make sense within, specific cultural contexts. The first paper takes a deep dive into one particular strategy often taught by dealers and other players to beginner blackjack players to help them decide whether to hit (take another card) or stand (stop taking cards and end one’s turn), two of the most frequent blackjack decisions. One step in the heuristic is to imagine that all following cards will be 10-value cards (10s, jacks, queens, and kings). That belief is wrong more than twice as often as it’s right. Players often use the heuristic to imagine the next three cards in a row. In those cases the heuristic’s prediction will be wrong more than 97 times out of 100. Perhaps not surprisingly, that assumption is also recognized as transparently false by the dealers and players who teach it, and by the beginners who learn and use it. Anyone familiar with a standard 52-card deck of playing cards, or who has played blackjack for more than a few minutes, will know it to be false.

Why would blackjack players use a strategy based on transparently false likelihood judgments? The surprising answer is because it works. It gets 58 of the 60 hit–stand decisions that are remotely ambiguous correct; it is easier to teach and learn than the optimal strategy; and the two choices it gets wrong have a trivial impact on the players’ expected losses (players using this strategy do 0.04% worse than perfect Basic Strategy users, losing 4 more cents for every $100 bet). Remarkably, it outperforms the hit–stand playing strategies used by even the most experienced blackjack players, despite the fact that players systematically improve with experience.

It better explains previous findings about putative cognitive biases: likelihood miscalibration ascribed to experienced blackjack dealers (Keren & Wagenaar, 1985) and risk aversion (Wagenaar & Keren, 1985) or omission bias (Carlin & Robinson, 2009) ascribed to blackjack players. It also predicts the only two hit–stand choices that blackjack players increasingly get wrong as they gain playing experience: the two hit–stand choices the heuristic gets wrong (for you blackjack aficionados, that’s whether or not to hit a hand total of 12 against the dealer’s exposed 2 or 3).

The second paper steps back to consider characteristics of culture-bound heuristics more generally, not just in casinos. Along with three cases from blackjack, it considers the example of traditional Micronesian canoe navigation techniques (Hutchins & Hinton, 1984). These navigators, using techniques now lost to most Micronesian cultures, imagine that the islands move while their canoes stand still. Similar to the blackjack heuristic just described, they know that belief to be false, yet they imagine it to be true. By doing so, they are able to navigate more efficiently and reliably across long distances, and often over several days.

Below are the titles and abstracts of both papers. If you’d like copies of the full manuscripts, just reach out—they haven’t been published yet, but I’d be happy to share pre-publication versions.

Paper 1 (the 10-heuristic in blackjack): “A heuristic based on transparently false likelihoods improves gamblers’ expected value in the wild”

ABSTRACT: Casino blackjack players learn a simple heuristic that helps them decide when to take additional cards, the most common blackjack decision. The heuristic assumes that all upcoming cards will be 10-value cards, even though that assumption is true fewer than one in three times. The heuristic results in systematic error, but it is also adaptive: it is easier to learn than the optimal strategy, the cost of using it is trivial, and its use is associated with better expected returns. The heuristic helps explain inconsistent previous findings about blackjack likelihood judgments and decision biases. The research relies on mixed methods including qualitative data from 1.5 years of ethnographic fieldwork as a blackjack dealer and player, and quantitative data from interviews with players about how they play each blackjack hand. The heuristic is used as a case to support several theoretical contentions: (a) despite established precedent, gambling is not a good domain-general metaphor for decision making under risk or uncertainty; (b) even in a small-world domain where outcome likelihoods can be calculated and monetary outcomes are unambiguous, using subjective probability to infer expected value may be both uncommon and non-normative; and (c) a focus on narrow, domain- and culture-specific heuristics and biases—despite their limited scope—offers valuable lessons about how, and how well, people make decisions.

Paper 2: “Culture-bound heuristics”

ABSTRACT: Drawing on theoretical insights regarding the interdependence of culture and cognition, this paper argues that culture fundamentally shapes both decision processes (heuristics) and how well they work (e.g., biases). It further argues that the importance of culturally interdependent heuristics (“culture-bound heuristics”) has been underappreciated because of theoretical and methodological norms that tend to remove culture from consideration. The paper first considers theoretical background: content- and context-impoverished norms in how judgments and decisions under uncertainty have been modeled and empirically studied, followed by critical responses to those norms. Second, it uses four case studies to support an argument that decisions in the wild are often cultural-domain-specific (e.g., only used for blackjack or for Micronesian canoe navigation) and inseparable from systems of beliefs, values, practices, task environments, and people. Furthermore, these cases suggest that by removing cultural content and context from experimental stimuli to get at putatively basic cognitive processes, researchers may have systematically overgeneralized and misidentified both heuristics and biases. The paper concludes by recommending expanded methodological and theoretical approaches for identifying and evaluating judgment and decision making that take culture’s importance to cognition seriously. This expansion emphasizes the value of (1) ethnographic methods that richly explore the decision task environment and folk conceptions of decision processes as part of the hypothesis generation stage; and (2) cross-cultural and longitudinal comparative research to test those hypotheses and further explore how decision processes and their effectiveness vary over time and place “in the wild.”

The term chasing is shorthand for chasing losses, the act of increasing the persistence and, usually, the size of one’s bets after extensive or improbable losses in the—often desperate—hope of recouping those losses. It tends to correspond with a manic mixture of strong emotions and thoughts related to having been very unlucky already, having made and lost bets that the gambler started the day committed not to make, and finally being in a state of desperation with near total loss of self-control. Chasing can look like belief in the gambler’s fallacy because bets are increased after an improbable series of losses, and then they tend to decrease when those losses finally result in a big win, as they often eventually do.

Mathematically, chasing works much like the martingale system described in the previous essay. The blessing and the curse of chasing, like the blessing and the curse of martingale, is that it so often works. If my only goal is to get back to even and I keep increasing the size of my bet hoping for one good win that can make up for all the losses, chasers will often get that win, sometimes after going into debt that they can ill afford, reinforcing subsequent reliance on the dangerous behavior.

It is worth note that just as there are many species of martingale-like betting systems, there are many ways to chase, with different relationships to the likelihood one will eventually getting back to even or one will return to the chase. One way to chase is to bet on unlikely outcomes with large payoffs (such as parlays in sports betting, betting large amounts on single numbers in roulette, or hoping for a jackpot in slot machines). That kind of chasing is less likely to work. Another way is more directly akin to martingale: playing even-money bets with close to 50:50 odds and simply betting more and more after the losses. Chasing rarely looks just like martingale because the bets are less systematic, and so it also does not work as often.

Why is a system that usually works a curse? Because, like martingale, when it does not work, it results in losses far larger than the gambler had bargained for. If the impetus for chasing one’s bets is a large loss that leads the gambler to lose their good sense, then the result of an unsuccessful chase can be disastrous: debt that cannot be repaid, lying to one’s spouse and children about the extent of the losses, theft from employers or company accounts, suicide. Because chasing so often work and is associated with pathological gambling, those who chase often do so again and again over time, almost ensuring that eventually the consequences will be severe.

It would be nice to conclude that chasing is simply irrational with the idea that if gamblers could be taught the consequences of the activity, they would see how foolish it is. There are two problems with this picture. First, gamblers rarely chase because they believe it is rational. Nearly every experienced gambler will advise less experienced gamblers not to chase. Yes nearly every one of those gamblers will be able to tell you about a time—knowing full well what a bad idea chasing is—when they did it against their better judgment. Chasing is an act of desperation, done when it subjectively feels to the gambler like it is their only hope out of an already terrible situation. That belief may not be true, but the miserable feeling in the moment and the recognition that a successful chase could erase that feeling is often enough to drive gamblers to chase even while they consciously tell themselves what a terrible idea it is.

Second, like the person who rationally uses the martingale system when desperate to win just $50 more dollars that they need for an operation, chasing can often be a rational response to a desperate situation. If I have already lost more than I can afford to lose, to the point where my bank account is empty and I do not know how I will pay rent or pay back a debt, the marginal disutility of the extra losses (in the form of debt I accrue by maxing out a credit card, for example) may be less that the disutility of the original loss, even if objectively the additional loss is much higher. Consider someone who has lost $10,000, and—in the process—gone from having $5,000 in the bank to having credit card debt of $5,000. How much debt after that point is equivalently bad to that initial loss of $10,000? Depending on the person’s situation: their capacity to earn back the lost money through work, the possibility of losing one’s spouse or house or car as a result, etc., there may be no amount of additional debt that may seem worse than the initial $10,000. This is part of the reason that casino gambling is often referred to as a tax on the poor: in desperation where the only possible way to get out of poverty is gambling, it can be rational—for those who can afford it least—to play a game with a negative expected return.

This is the inverse of how it can be rational for wealthy people to take insurance (a negative expected value bet), even while it is irrational for those in poverty to take insurance: the wealthy person is managing their risk, the marginal cost for the insurance will not affect their lifestyle. The potential, low probability cost of losing everything, however, can change that lifestyle. If a bad insurance bet, from the perspective of expected value, can safeguard against that risk, then wonderful. But for those in poverty, the cost of insurance itself is significant, whereas there may be essentially nothing to lose to insure against. Thus just as a relatively wealthy person can be rational to take a negative-expected return gamble (that is, insurance), a person in poverty can be rational not to take insurance even if they were lucky enough to find an insurance policy with a positive expected return. Similarly, it can be rational for a gambler who has lost everything to “manage their risk” in the opposite direction from how a wealthy person who takes insurance manages risk. If they have relatively little to lose by going further into debt than they already are, but a great deal to win if they get improbably lucky, then playing a game with a negative expected return can have positive expected utility (despite having negative expected value).

This source of rationality, however, is also central to the curse. For gamblers are rarely in such abject states before they have already chased many times, the potential wins are rarely enough to recoup the sustained losses such gamblers have already faced, and even when such wins occur, it is rare that the gambler will then walk away from gambling, thankful for their final luck (even if they swear to themselves ahead of time that they will). The grandmother in Dostoevsky’s The Gambler, quoted extensively in this early post, provides a vivid case of chasing, of the repeated belief that one will walk away after winning, and of the desperate, irrational fever that commonly goes hand-in-hand with the failure to walk away and the eventual return to the chase. Ironically, there often comes a point when the gambler intentionally seeks to lose everything, recognizing that their own will power will not be enough and the only way to escape the cycle is to lose everything.

For those interested in learning more, the clinical psychologist Henry Lesieur—who worked extensively with problem and pathological gamblers—has written about the psychology and horrific experiences underly chasing in his 1984 book, The Chase. The actor Philip Seymour Hoffman, who died before his time, has an award-worthy performance portraying the real-life Brian Molony who embezzled more than $10 million caught up in chasing (based on the book Stung by journalist Gary Ross).

Although many comparisons have been made to the martingale system described in the previous essay, it should be clear by now that chasing is also quite different from martingale. Players rarely endorse chasing, sometimes acknowledging its pending harm even when they are in the midst of doing it. It is a wonderful example of the inner conflict between impulsive and intentional desires as they independently drive behavior. Martingale, on the other hand, is more systematic and is intentional: players use it because they have been tricked by the seeming mathematical certainty of the system. Eventually they reject it, but sometimes even while still believing the system would work if they only had risk tolerance to continue with it.

That said, chasing, like the martingale system, provides a compelling example as to why a behavior that might look like it demonstrates a particular belief (say, the gambler’s fallacy) is often driven by something else altogether: a different set of beliefs, as with martingale, or the conflict between belief and impulse, as with chasing. And, again as with martingale, the defensible rationality of chasing, from the perspective of expected utility, provides a great example as to the complexity and difficulty of evaluating the rationality behind any of these beliefs.

While this is the last post using the “If it looks like a duck…” counter-analogy, the next few posts will continue to discuss the nature of the gambler’s fallacy in an attempt to convincingly argue that that fallacy is more myth than substance. The next essay will consider the inverse of this most resent set of essays: the common case among experienced gamblers when—despite often believing in the gambler’s fallacy—they behave as if they do not. That is because experienced gambler’s believe in a counter-vailing force affecting outcomes over the short-term that is stronger than the long-term “corrective force” associated with belief in the gambler’s fallacy: the belief that luck, like the tides, runs in waves that can be identified and used by skilled gamblers to anticipate short-term patterns. They often believe this while consciously “acknowledging” (incorrectly) that these likelihood-contradicting patterns in luck will eventually need to be corrected (the gambler’s fallacy). But the more important and diagnostic force, as far as most experienced gamblers are concerned, are the waves of luck that make unlikely patterns in previous outcomes more—rather than less—likely.

This is Part 2 of an essay about why it is problematic to infer belief in the gambler’s fallacy from gambling choices. If you haven’t already read Part 1, start there. It covers:

1. Pascal’s Casino Wager – The idea that once a gambler has committed to playing, even the smallest possibility that the gambler's fallacy isn't a fallacy can justify behaving as though it were true.

2. Hot Versus Cold Reasoning – The idea that we can both believe and disbelieve something depending on the interplay of rational thought and emotional impulses.

3. Beliefs Violating the Assumption of Event Independence – The idea that the gambler's fallacy assumes gamblers accept event independence, which is often not the case.

This post—Part 2 of the essay about the challenge of linking behaviors to beliefs—focuses on the Martingale betting system, a strategy often used by beginner roulette players. Superficially, it appears to reflect belief in the gambler’s fallacy. However, it has little to do with that fallacy.

Contents

1. The Gambler’s Fallacy: A Brief Recap

2. The Martingale Betting System

3. The Appeal of Martingale

4. Why Martingale Fails

   1. Casino Limits and Bankroll Constraints

   2. But Betting Limits Are Not the Only Problem

5. Given the Costs, Why Do Gamblers Use Martingale?

6. A Time When Martingale Makes Sense

7. Conclusion

The Gambler’s Fallacy: A Brief Recap

The gambler’s fallacy is the belief that after a streak of the same outcome—such as five reds in a row in roulette—the likelihood of that outcome occurring again decreases, even for independent events like a fair coin flip or a fair roulette wheel spin. If this belief doesn’t seem like a fallacy to you (if you think, for example, that heads is less likely after multiple heads in a row), you’re not alone. Many people do not believe it’s a fallacy—which, paradoxically, is part of why it's a well-documented fallacy.1

If you're skeptical, start with this essay on the law of large numbers, which explains why chance does not “self-correct.” Then read the first post in the gambler's fallacy series. If you'd prefer other sources, see this ChatGPT query, this Wikipedia page, or this peer-reviewed journal article on belief in the gambler’s fallacy.

The Martingale Betting System

Martingale systems involve increasing bet sizes after losses and decreasing them after wins, much as would be expected with belief in the gambler’s fallacy. The simplest form begins with the table minimum (one unit). After a loss, the player doubles their bet. After a win, they revert to the minimum. This ensures that each win results in a net profit of exactly one unit. To understand why, consider the following examples:

• Example 1: W1 – The player bets 1 unit and wins. Net profit: 1 unit.

• Example 2: L1, W2 – The player loses 1 unit, then doubles the bet and wins 2 units. Net profit: 1 unit.

• Example 3: L1, L2, W4 – The player loses 1, then 2, then wins 4. Net profit: 1 unit.

• … Example 11: L1, L2, L4, L8, L16, L32, L64, L128, L256, L512, W1024 – The player loses 10 times in a row before winning 1024 units, having lost a total of 1023 units. Net profit: 1 unit.

This system allows the player to win small amounts consistently—until they hit a table limit or run out of money.

The Appeal of Martingale

The appeal lies in the idea that a win is inevitable, provided there is no limit to the amount that can be bet. Consider, for example, if the gambler’s bankroll and casino limits were large enough to withstand 50 losses in row:

• Even in American roulette (which has both a "0" and "00" and an advantage to the casino of 5.26%), 50 consecutive losses will occur less than once for every 8.66 trillion wins.

• That’s rare enough that one could use this strategy to win 50 units every day for about as long as the earth has been a planet, and still only lose about once, on average.

• The minimum bet in roulette in today’s US casinos is usually $25 or more. With that minimum, players could expect to win $1250 every day, with essentially no chance of losing.

Most players who decide to give martingale a try do not think about it in such specific quantitative terms. Instead they might say something like this: “Come on! How often do you lose 10 times in a row? It never happens.” In either case, on first consideration, martingale can seem like a save method to beat the casinos. And it has nothing to do with belief in the gambler’s fallacy.

Why Martingale Fails

Casino Limits and Bankroll Constraints

Casinos impose table limits. In Las Vegas, the ratio between maximum and minimum bets is usually less than 1000:1. Suppose a casino allows a 1024-unit spread, letting a player withstand 10 losses before reaching the maximum, 11th, bet (as in Example 11 above):

• The chance of losing 11 times in a row is 0.086%.

• The player will win about 1164 sessions for every one losing session (where each losing session involves a streak of 11 lost bets in a row).

• But each win earns 1 unit, while each rare loss costs 2047 units (the sum of all 11 bets from 1 to 1024 units).

• Over time, the player would lose about 2047 units for every 1164 they win – a net loss of 883 units!

• With a $25 minimum bet, they'd need a $51,175 bankroll and they could expect to lose $22,075 ($25 * 883 units) after combining all the winning sessions with the one losing session.

• For a player who tries to win 50 units per playing session and who plays about 23 days per month, that would be a monthly loss.

But Betting Limits Are Not the Only Problem

People who accept the above shortcoming of the martingale system often still believe it would work, if only the casinos did not cap the maximum bet. That view is also flawed. Imagine, for example, if the player had a bankroll that could withstand 20 losses in a row instead of 11, and the casino was willing to allow it. Even with American roulette’s two zeroes, just adding those extra 9 losses would reduce the frequency of a bankroll-maximizing loss from once in every 1165 tries to once in every 375,900 tries, making the risk of losing trivial. The problem is that the bankroll required to cover that losing streak is 1,048,575 units (or $26.2 million given a $25 minimum bet). Players could earn significantly more by putting that money in a conservative interest-bearing government treasury bond while sipping daiquiris at the beach, rather than sitting for hours everyday at the roulette table.

Given the Costs, Why Do Gamblers Use Martingale?

The short answer is that experienced gamblers do not use martingale.

Beginners often do, however, for a few reasons:

• The face-value appeal of the system: The most obvious reason was already discussed: on first consideration, the martingale system can sound like a winning strategy. Few players sit down and work out the numbers and realize how high the risks are relative to the gains, even after just a month of play. They often have to learn that the hard way.

• Failure to commit: Many beginner players continue to use the martingale system for some time even after suffering extensive losses. The first reason for this is that they often get cold feet, failing to commit to the system till the very end, once their streak of losses gets close. Imagine, for example, that you’ve just lost the 256 unit bet (so perhaps $6400 given a $25 minimum bet). That’s already a total loss of $12,775. You have two more bets to go before you’re finished. That’s another $38,400 at risk, and now there’s more than a one in four chance it will happen (a 27.7% chance, to be exact). It is common at such a point for players to lose confidence in the system and stop betting, only to see that they would have won had they stuck with it to the end. That is enough for some to give the system another try.

• Using “bad luck” to rationalize a loss: Even players who do stick to the system to the end often rationalize their loss with reference to bad luck. The losing streak needed to lose with martingale is longer than most beginner roulette players have ever seen, much less experienced themselves. And—as all gamblers who have had a winning session are aware—luck matters. Even gamblers with authentically winning systems—such as skilled card counters in blackjack or professional poker players—sometimes have losing streaks. It is not unreasonable for a person convinced that the martingale system works to dismiss a loss as an unfortunate blip in an otherwise winning strategy.

That said, many beginners who try out the martingale system are fortunate enough to learn about its shortcomings before they every experience that first major loss (much less a second or third). That is because roulette, like other casino games, is played in a social setting, allowing the opportunity to learn from others. Players can learn vicariously by watching other gamblers try the system and fail. More often they learn from they advice of more experienced roulette players, nearly all of whom are familiar with the system and can tell compelling stories about its dangers.

In any case, whether players learn about the dangers of the martingale system from others or learn about it the hard way, with enough experience, roulette players eventually learn that martingale is to be avoided. Because of the harsh impact of this strategy, even though martingale can be confused for belief in the gambler’s fallacy, it tends to have far worse consequences. Most people who believe in the gambler’s fallacy just change their bet from one color to another (assuming the choice is between black and red in roulette). Perhaps they’ll increase their bets, too, but not so dramatically as with the martingale system. As such, belief in the gambler’s fallacy tends not to be so dramatically extinguished or adamantly opposed as belief in martingale betting systems.

A Time When Martingale Makes Sense

There is one situation where Martingale can be a rational choice: when a player desperately needs a small win with minimal risk. Suppose someone needs exactly $50 to afford a lifesaving medical procedure. Martingale will almost always succeed in this case. By contrast, switching between red and black based on recent outcomes (the gambler’s fallacy) provides no advantage.

Conclusion

While martingale can look like a strategy based on the gambler’s fallacy, its appeal is rooted in a different kind of misunderstanding of probability and risk. Unlike the gambler’s fallacy, it promises small, frequent wins with little chance of losing. The rare catastrophic loss, however, more than makes up for the small wins. Attempts to minimize the likelihood of such losses to make the risk seem trivial require bankrolls so large that the money could be invested elsewhere with less work and greater returns. The gambler’s fallacy, on the other hand, can coexist with betting strategies that do not have any impact on the frequency or size of wins relative to losses, as explained in a recent essay, “Belief in the Gambler’s Fallacy Is Trivial.”

The next essay will continue the discussion about why it is so difficult to infer belief in the gambler’s fallacy from behavior, starting with a discussion of chasing. Chasing behavior looks similar to martingale and has similar consequences (usually working, but occasionally having disastrous consequences). But its motivation is quite different from martingale, and is again unrelated to belief in the gambler’s fallacy.

This is essay three (Part 1) in a series of posts about the gambler’s fallacy: the belief that after an unlikely streak of the same outcome, such as five reds in a row in roulette, the likelihood that the outcome will occur again (red in the example just given) decreases, even when events are independent, such as with roulette or the flips of a fair coin.1 The conclusion of these posts will be that—for the most part—the gambler’s fallacy is a myth (not because people do not believe in it, but because experienced gamblers are particularly unlikely to believe in it, or at least to make choices that relfect that belief).

The first essay described three interrelated false conceptions that together make up the gambler’s fallacy: (1) Overestimating the extent to which small samples should be representative of the larger population from which they are drawn. In the case of roulette and other games of chance, “population” refers to outcome likelihoods, such that—for example—if there is a 50% likelihood of heads, we expect heads to occur closer to 50% of the time than it actually tends to with small samples. (2) Believing that chance itself can and will somehow self-correct when the small samples do not match the long-term likelihoods or the population distribution. And (3) giving more weight to recent outcomes than is warranted. That essay was essentially a description of the fallacy.

The second essay used text from Dostoevsky’s The Gambler to suggest that while belief in the gambler’s fallacy exists, it is trivial in two ways. First, it is trivial because it is just one of a wide range of beliefs gamblers hold that vary widely not just across gamblers but from moment to moment for individual gamblers. Second, it is trivial because the beliefs gamblers hold about the relationship between past and future outcomes are largely irrelevant: it doesn’t make a difference whether I incorrectly think red is more likely after a series of blacks, incorrectly think black is more likely, or correctly recognize that both black and red are equally likely: I still have the same chances of winning whether my next bet is on red or black. That irrelevance allows even gamblers who know better to engage in fantasy and superstition, in part because it adds fun and meaning to the game without any impact on expected return.2

This third essay points to the many ways in which it is problematic to use observation to assess belief in the gambler’s fallacy (such as when researchers assess belief in the fallacy from roulette players’ sequential choices). Gamblers often change their bet size or bet choice after a sequence of outcomes in ways that look like belief in the gambler’s fallacy, if that is what you are looking for, but for reasons unrelated to the gambler’s fallacy, or for reasons that are not well captured by the concept of belief. At other times, gamblers may believe in the fallacy but make choices that would seem to contradict that belief because they have other beliefs that are deemed more important. Below are three commonly occurring cases that support this contention. Several more are on the way soon, but I decided to just post the first three now, since the post is already quite long.

1. Pascal’s Casino Wager

2. Hot Versus Cold Reasoning

3. Beliefs Violating the Assumption of Event Independence

1. Pascal’s Casino Wager

Pascal’s wager refers to an argument made by the famous philosopher and mathematician, Blaise Pascal (Pensées, 1670), providing a pragmatic justification for belief in God (recall the previous discussion of Pascal’s role in the development of probability theory and the concept of expect value (EV) here). Pascal essentially made an argument from the perspective of maximizing EV: the cost of not believing in God if he exists is essentially infinite (eternal damnation), whereas the cost of believing in God if he does not exist is essentially nothing (just the decision to believe). With those values, as long as one believes the likelihood that God exists is any amount over 0, it makes pragmatic sense to believe in God. A similar argument can be made for believing in the gambler’s fallacy.

Near the end of the previous essay (five paragraphs from the bottom), I pointed out that in games of purse chance, what one believes about the implication of past events for future outcomes is in some ways irrelevant (as long as it does not impact how much one risks): if I start from the presupposition that the casino has an advantage and I am playing with the hope of getting lucky, whether or not I correctly believe that it does not matter whether I bet on red or black, or I incorrectly believe that a previous sequences of reds makes black more likely, or I incorrectly believe that previous sequence makes red more likely, it’s all the same in terms of expected value: the casino still has the same 5.26% advantage (or 2.7% in European roulette which has only one zero).

With that in mind, Pascal’s wager can be neatly applied to belief in the gambler’s fallacy. The gain from correctly believing in it is to potentially turn the casino’s expected value to the gambler’s favor. The cost of incorrectly believing in the gambler’s fallacy is essentially zero (again, with the caveat that the gambler has committed to risking the same amount either way). In that case, as long as one thinks there is any chance greater than zero that the gambler’s fallacy is true, it makes pragmatic sense to believe in the truth of the fallacy.

There are two glaring problems with this argument. The first is that, to the extent it is convincing, it could be applied to virtually an infinite number of beliefs. Yes, we should then believe in a Christian God, but we should also believe in any other God or any other thing we can imagine that might lead to eternal suffering if there’s essentially zero cost for believing it. And in the casino we should believe in any sort of magic or superstition or idea we might think up as long as the cost for not believing is essentially zero. We do not have such infinite resources, and even if we did, many such beliefs contradict one another.

The second problem is that it suggests people can choose to believe things they do not believe. There could be infinite expected value and zero cost for believing that 2+2 = 6, but I still think it’s false. With God, what one truly believes matters. And perhaps Pascal’s wager was enough motivation for him to find ways to truly believe (or, more likely, he already believed and the wager was a post-hoc attempt to convince others that they ought to believe as well).

In the casino, at least, neither of these two objections is a death sentence, however. With regard to the fact people cannot simply choose to believe things they in fact think are false just because there is some personal gain for the false belief: that is exactly the point here. What matters to success and failure in the casino, unlike in the eyes of God, is what one does, not what one believes. Players can act as though they believe in the gambler’s fallacy even if they do not. In the miniscule chance the gambler’s fallacy turned out not to be a fallacy at all, that behavior would still help them win. I can change my bet to black after a series of reds, even if I do not believe it will work, just as I can pray to God just before dying even if I do not believe in God, in case God values the act of desperate prayer more than authentic belief. As long as I think there is some chance greater than 0 that the gambler’s fallacy is true, why not act as though I believe in it? If I believe right, but choose wrong, I still lose. If I believe wrong, but choose right, I still win.

With regard to the point that this would justify belief in multiple, often conflicting, strategies (or in the modified version, acting as though one believes even if one does not), that’s a good point. It was the main observation from the discussion of Dostoevsky’s The Gambler in the previous post. The Grandmother’s changing strategies from moment to moment as she looks for any possible pattern, mirroring Skinner’s observation of superstition in Pigeons, suggests that gamblers might readily switch from belief to belief (or acting as if one believes in the Gambler’s fallacy to acting as if one believes the wheel favors zero). The fickle, half-committed beliefs are free to change depending on what the gambler thinks might possibly work.

2. Hot Versus Cold Reasoning

Reading the above might make you rightly uncomfortable with the simplistic treatment it gives to belief. Of course, there are many levels of belief and many senses in which one can believe. The beauty (or horror) of games of chance is that there is in some sense no cost at all for acting as though one believes nearly anything (as long as choices do not meaningfully impact expected return), while what one chooses can matter profoundly, depending on the potential wins or losses involved. This points to one of the key distinctions in ways of believing, in this case associated with hot (emotional, automatic, driven by less conscious, deliberative thinking; what Daniel Kahneman made famous as System 1) versus cold (non-emotional, slower, driven by more conscious, deliberative thinking; System 2) cognition. Consider a later dialogue between the grandmother and the hero / narrator Ivanovich in Doestovesky’s The Gambler:

“To think that that accursed zero should have turned up NOW!” [the Grandmother] sobbed. “The accursed, accursed thing! And, it is all YOUR fault,” she added, rounding upon me in a frenzy. “It was you who persuaded me to cease staking upon it.”

“But, Madame, I only explained the game to you. How am I to answer for every mischance which may occur in it?”

“You and your mischances!” she whispered threateningly. “Go! Away at once!”

“Farewell, then, Madame.” And I turned to depart.

“No — stay,” she put in hastily. “Where are you going to? Why should you leave me? You fool! No, no… stay here. It is I who was the fool. Tell me what I ought to do.”

What exactly does the grandmother believe here? Does she believe that Ivanovich is the cause of her losses or that it is she who is the fool? One can imagine that in that moment of great desperation immediately after losing, all she is aware of is the deeply felt belief that Ivanovich is in fact to blame. But yet, at some other level, a level that will be clear to her when she is far from the table and the emotional impact of the losses, she will likely recognize that she was to blame (or perhaps—motivated by a desire to justify her past behavior—she will become deeply convinced that Ivanovich was to blame for everything). The honest recognition that Ivanovich was not to blame is accessible to her even then, the moment her semi-bluff is called and Ivanovich decides to go, and she will be left without a guide. In the casino, with emotions running high and the cost of false beliefs often close to zero, one can deeply believe outrageous, conflicting things, while also not really believing them at all.

3. Beliefs Violating the Assumption of Event Independence

A required characteristic of the gambler’s fallacy is that events are independent—or at least that they are not negatively dependent—and that the gamblers have a reliable way to know about that non-negative dependence. Consider, for example, if red, black, and green outcomes in roulette were instead marbles in a jar: 18 red marbles, 18 black marbles, and one or two green marbles (for European or American roulette, respectively). If roulette worked that way, then it would not be a fallacy to switch one’s bet to black after a series of red outcomes: it would be the smart move. The problem with the independence assumption is that in real-world cases (as opposed to hypotheticals cases when events can be defined as independent), one can never be sure events are independent. Outside games of chance, events rarely are entirely independent. And while a person might be incorrect if they believe, for example, that a roulette wheel is rigged so that if red occurs too frequently the casino will use a magnet to draw the ball to black, such a false belief would not properly be called a fallacy: false beliefs do not necessarily imply faulty reasoning. Gamblers might be wrong in their beliefs, but we need more than that to call them irrational: they need maintain contradictory or otherwise incoherent beliefs.

This concern was discussed in some detail in an earlier essay about the law of large numbers. See that discussion for examples and for more extensive consideration of this issue. Here it is enough to point out that whether or not events are independent in games of chance is not a theoretical issue; it is an empirical one, and it depends on both beliefs about the environment and the actual state of the environment. Many casino games are negatively dependent, including card games where previous cards removed from the deck make those outcomes less likely, but also including games that might seem independent but that knowledgeable players know are not. For example, some slot machines turn out to have higher expected returns the more time has passed since a certain type of jackpot has been reached (progressive jackpots). UK slot machines are required to pay out across a payout table, so that in fact when the top jackpot hits, that jackpot becomes significantly less likely to occur for some time (and the longer it does not occur, the more likely it becomes). In the US, slot machines outcomes are, by law, independent.

You could argue that false beliefs have to be reasonable to avoid a justified “fallacy” label. If gamblers have direct access to information that would make that belief unreasonable, then it might not exactly be the gambler’s fallacy, but it should still be recognized as a fallacy. The problem is that there is no easy way to determine what is or isn’t reasonable for gamblers to know. Is it reasonable to believe that casinos rig the roulette wheels? Maybe not in today’s Las Vegas knowing what I know. But a lot of casino gamblers do not know nearly as much as I know about the changing regulations and norms of casinos in Las Vegas. And a lot of casino gamblers know much more than I do and have stories that would give you pause.

Here is an example from an interview with a slot machine player whose strategy might look like belief in the gambler’s fallacy, if the reasoning behind the strategy had not been provided. This player had worked for more than a decade as a slot machine technician and had relationships with many slot floor managers and casino managers. He assured me that the computer chips that operate the machines are now networked so that many features can be managed from a central computer. He claimed that casinos can change the payout percentage between a few choices on any single machine remotely, as long as they get permission for each of the payout percentages from the (state run) gambling commission. He also assured me that they can also change the payout percentages in other ways that are not legal if it’s done on the casino floor, but that are still possible and enabled for other reasons.

For example, casinos have slot machine tournaments where prizes are based on who gets the most credits. Those tournaments are limited to certain areas of the casino floor and the machines can be temporarily set to pay out with a positive return to the players. Since the payouts are just points that cannot be cashed out, it does not cost the casinos anything, and it makes the tournament more fun for the players, who are all having far greater success playing slot machines than they normally have. I know this not just from reports from gamblers, but also from repeated direct personal experience playing in casino slot machine tournaments, where essentially every player in the tournament finishes with more tournament credits than they started with.

As another example, the technician claimed—with convincing authority from my perspective—that the slot area manager can set any slot machine to deliver a jackpot remotely. The purpose is so that casinos can test the functionality of the machines. But though it is not legal to call up a jackpot for an actual player, he claimed that casinos nonetheless do it. For example, if a gambler has been losing consistently and is normally a big better, they may hand out a jackpot to keep that gambler coming back. I challenged him about the risk that casinos would face for breaking Nevada gambling laws and whether it would really be worth it. “Look at the [news]paper,” he said. “How often have you seen a Las Vegas casino lose a court case in Nevada?” The casinos were essentially more powerful than the government—or too entrenched within Nevada government—to be effectively regulated, he suggested.

One of the claims he made associated with the strategy he uses to reportedly win at casino slots was justified by an elaborate but compelling story about how casinos are like other businesses: they need to manage their cash flow; that is, they need to be sure they will have enough available cash each month to pay their monthly expenses. Since casinos rely on games with high variability, they can make dramatically different amounts from one month to another that cannot be predicted in advance. For that reason, he explained, they rightly worry about controlling that variability. He said that one way they do it is with the slot machines. They set them tighter—meaning with a lower payout percentage than the long-term percentage registered with the gambling commission—at the beginning of each accounting month to increase the certainty that they will make their monthly numbers.3 Because casinos do this, he explained, they must also loosen up the machines at the end of most months (once they know they have the cash flow to manage expenses), because otherwise it would be clear that the machines were set tighter than their payback percentages registered with the casino commission.

It is true that Las Vegas casinos must report their actual slot machine payback percentages on a monthly basis to the gambling commission and even register chip payout percentages for each machine (or at least this was independently reported to me by several interview participants). Monthly payout percentages by type of machine—quarter machines, nickel machines, dollar machines, etc.—are published in the local newspaper, not for each casino but by casino neighborhood (downtown, the Strip, etc.). Those regional payout percentages are news-worthy because local gamblers can use that information to find the best and worst regions for slot machines (hint: the casinos on the Strip tend to be the worst, the casinos in local’s neighborhoods tend to be the best).4 If casinos set their machines to be tighter during the first part of the month, and then return the machines to normal after making their numbers, then the gambling commission (or just a statistician looking at monthly payout rates published in the local paper over an extended period) could see that something was not right. “So what they do is, they make the machines looser than normal at the end of the month,” paying out a positive expected return to players so that the long-term payouts match what would be expected given the registered computer chip settings. He always plays slots at the end of the reporting month and, he claims, he is a long-term winner as a result. “Wouldn’t casinos lose money doing this, assuming you’re not the only person who knows about this?” I asked. “Oh, I’m not the only slot player who knows about this,” he assured me. “What do the casinos care? They get their numbers either way.” Better to give the better payouts back to the locals anyway, he suggested, they’re the return customers.

Do I believe in this strategy. No, absolutely not. I have heard dozens of stories that would be similarly compelling if I did not know they were false, some of which contradict this particular narrative. Could it possibly be true? I don’t see why not. I would guess that it could be spotted if the monthly numbers were combed over carefully enough by statisticians at the gambling commission. But he could also be right that the commission is not incentivized to make the casino’s legal breaches public. Even if they the commission is not in cahoots with the casinos, tourism revenue certainly depends on the belief that the casinos can be trusted, and publicizing breaches would be expensive for the industry. This technician certainly knew far more about how slot machines work and had greater insider knowledge about how the slot machines were managed than anyone else I interviewed.

If this gambler’s beliefs are true and his correspondent strategy in fact works, and if you were using gambling behavior to identify the gambler’s fallacy, this player might appear to believe in it. During most times (nearly the entire month, until the very end), gamblers like this would bet small amounts and less often, and they would tend to lose (since the machines are particular tight during those sessions). After these seemingly “unlikely” losses (near the end of the month), such players would increase their bets, seeming to believe that they were “due.”5 Then, after winning more than “expected” (by the observers standards), they would switch their strategy by reducing how much and how often they bet, again, seeming to confirm belief in the gambler’s fallacy.

This might seem like a rare, idiosyncratic exception but, while definitely idiosyncratic, it is not rare. Most of the frequent casino gamblers that I interviewed had their own “discovered” idiosyncratic strategies that they claimed improved their chances of winning and that were based on nuanced and plausible beliefs about unverifiable aspects of the casino environment that could be true but that I doubt are in fact true. Many would look like belief in the gambler’s fallacy if the observer was only looking for that particular explanation. But rarely do these strategies in fact entail event independence, a standard requirement for belief in the gambler’s fallacy.

The next post will continue this list of commonly seen behaviors that initially look like evidence for or against belief in the gambler’s fallacy, but that can be seen to mean something altogether different once the richness of gamblers’ beliefs and strategies are better understood.

[The Grandmother] “Just now I heard the flaxen-haired croupier call out ‘zero!’ And why does he keep raking in all the money that is on the table? To think that he should grab the whole pile for himself! What does zero mean?”

[Alexei Ivanovich, The Gambler and narrator] “Zero is what the bank takes for itself. If the wheel stops at that figure, everything lying on the table becomes the absolute property of the bank…

“Then I should receive nothing if I were staking?”

“No; unless by any chance you had PURPOSELY staked on zero; in which case you would receive thirty-five times the value of your stake.”

“Why thirty-five times, when zero so often turns up? And if so, why do not more of these fools stake upon it?”

“Because the number of chances against its occurrence is thirty-six.”

“Rubbish! Potapitch, Potapitch! Come here, and I will give you some money.” The old lady took out of her pocket a tightly-clasped purse, and extracted from its depths a ten-gulden piece. “Go at once, and stake that upon zero.”

“But, Madame, zero has only this moment turned up,” I remonstrated; “wherefore, it may not do so again for ever so long. Wait a little, and you may then have a better chance.”

“Rubbish! Stake, please.”

The above exchange in Dostoevsky’s The Gambler (1866) between Alexei Ivanovich (the protagonist and titular character) and “the grandmother” (the mother of the general whose children Ivanovich tutors) provides an interesting case. The grandmother has never played roulette, and Ivanovich serves as her guide, explaining how the game works and attempting to steer her away from poor choices. Despite his protests against betting on zero, she continues to do so. By the end of the night, she wins a small fortune in a run of luck that captures the entire casino’s attention. The next day, she returns and loses it all, eventually losing everything she has. Her episode foreshadows what will soon happen—more than once—to Ivanovich himself, ultimately leading to his ruin.

Recall from the previous post that the gambler’s fallacy involves the mistaken belief that chance will self-correct for an improbable series of past events, even when those events are independent.1 Can you spot the gambler’s fallacy in the passage above? Think about it for a moment, then scroll down for the answer.

.

.

.

.

.

.

.

.

.

.

Ivanovich expresses the fallacy when he says, “Wait a little, and you may then have a better chance.” The implication is that because zero has come up recently, the likelihood of it occurring again is lower than if one were to wait until zero had not appeared for some time. In fact, assuming the wheel is fair, the probability of the ball landing on zero is constant, regardless of how recently or frequently it has appeared.

Dostoevsky himself struggled with gambling addiction linked to roulette. The story of Ivanovich’s descent into compulsive gambling is partly autobiographical. Notably, it is this more experienced gambler, Ivanovich, who invokes the gambler’s fallacy to dissuade the inexperienced grandmother from what he perceives to be a poor strategy. The grandmother, however, appears wholly unconvinced and instead embrace a different false belief—that the zero is more likely than other numbers because it occurred more frequently than chance would predict.

This passage raises several questions about the gambler’s fallacy. Did Dostoevsky himself believe in it, as implied by Ivanovich’s endorsement of it while attempting to rein in the grandmother? Or do the two characters' subsequent experiences—fabulous winnings followed by devastating losses—suggest that gamblers’ beliefs, and reasoning more broadly, play a minor role in compulsive gambling? During the manic gambling episodes in the novella, the gambler’s fallacy is unmentioned. Instead, these episodes seem driven by compulsive behavior, with a desperate and irrational search for meaningful patterns overshadowing any consistent beliefs about the relationship between past and future outcomes.

The previous Substack essay recounted the famous historical event that gave the gambler’s fallacy its alternate name, the Monte Carlo fallacy. The ball landed on black 26 times in a row, and belief in the gambler’s fallacy reportedly earned the casino a fortune. That event, the enduring name of the fallacy, several studies in decision science (e.g., Ayton & Fischer, 2004; Caruso et al., 2010; Croson & Sundali, 2005; Keren & Lewis, 1994), and the apparent endorsement of the fallacy by a temporarily sensible Ivanovich all suggest that belief in the gambler’s fallacy is widespread in games of chance.

Dostoevsky’s rich gambling description, however, suggest that the issue is more complex. While some individuals, sometimes believe in the gambler’s fallacy, it is equally undeniably that others—or even the same individuals at different moments—form different false beliefs about the relationship between past and future outcomes in games of chance. Later in The Gambler (Chapter 14), Ivanovich (reflecting Dostoevsky’s own experience) presents a different perspective on the gambler’s fallacy, suggesting that naive, inexperienced gamblers are the ones most likely to believe in it, whereas seasoned gamblers understand its flaws. The passage, written almost 50 years before the Monte Carlo streak, serves as a fictional prelude to the fallacy’s famous historical namesake event2).

As though of set purpose, there came to my aid a circumstance which not infrequently repeats itself in gaming. The circumstance is that not infrequently luck attaches itself to, say, the red, and does not leave it for a space of say, ten, or even fifteen, rounds in succession. Three days ago I had heard that, during the previous week there had been a run of twenty-two coups on the red — an occurrence never before known at roulette — so that men spoke of it with astonishment. Naturally enough, many deserted the red after a dozen rounds, and practically no one could now be found to stake upon it. Yet upon the black also — the antithesis of the red — no experienced gambler would stake anything, for the reason that every practised player knows the meaning of “capricious fortune.” That is to say, after the sixteenth (or so) success of the red, one would think that the seventeenth coup would inevitably fall upon the black; wherefore, novices would be apt to back the latter in the seventeenth round, and even to double or treble their stakes upon it — only, in the end, to lose.

Here Ivanovich explicitly states that only a novice would bet as though they believed in the gambler’s fallacy. He does not necessarily claim the belief is false. Instead, he suggests an opposing force whereby luck attaches itself to things, such as colors in roulette.

B.F. Skinner’s classic 1948 paper, “‘Superstition’ in the pigeon” describes how even pigeons learn random associations between their actions and rewards. If a pigeon happens to be standing on one leg when it receives food, it will repeat that action in the expectation that it may increase the chances of getting more food. This aligns with the grandmother’s shifting superstitions in The Gambler, where her beliefs about winning evolve based on random past experiences.

The previous Substack essay described three interconnected false beliefs contributing to the gambler’s fallacy. Those beliefs do not, however, explain why people hold the fallacy, just as they do not explain why some people do not believe in it. If you are reading this Substack, you likely do not believe in the gambler’s fallacy, just as I do not.

So, who actually beliefs in the gambler’s fallacy? Is the belief widespread among gamblers, as suggested by the Monte Carlo event and academic studies? Do only experienced gamblers believe in it (as implied by Ivanovich in the first passage), while novices like the grandmother hold different misconceptions? Or is it the opposite—that experienced gamblers recognize the fallacy’s flaws, while only novices fall for it (as suggested by Ivanovich in the the second passage)? Alternatively, is the gambler’s fallacy just one of many superstitious false beliefs that arise when gambling but that shift depending on what—by chance—worked in the past (as seen in the grandmother’s fickle convictions and Skinner’s superstitious pigeons)?

One partial answer emerges from the discussion so far. The term, “the gambler’s fallacy” is misleading and oversimplifies the issue. It overgeneralizes the belief by suggesting that a tendency exhibited by some gamblers as certain times applies to all gamblers. Conversely, it underspecifies the belief, as many non-gamblers also exhibit the belief under conditions conducive to the fallacy. Dostoevsky’s depiction underscores a reality of gambling: there is a broad spectrum of beliefs about luck and probability, shaped by individual and cultural differences, education, and gambling experience. While belief in the gambler’s fallacy certainly exists among gamblers, this fact is somewhat trivial—little different from observing that many gamblers and many non-gamblers also believe in conspiracy theories or pseudoscience.

This diversity in gambling beliefs is worth emphasizing. People often attribute specific capacities—such as believing in the gambler’s fallacy—to entire groups, when in reality, these beliefs fluctuate among individuals and over time. While some gamblers fall for the gambler’s fallacy, others do not. Some even hold contradictory beliefs simultaneously.

There is another sense in which belief in the gambler’s fallacy—and similar ideas about the relationship between choices and outcomes in games of pure chance—is trivial. In such games, a gambler's beliefs ultimately make no difference. After an improbable streak of red outcomes, it does not matter whether I mistakenly believe black or red is more likely, or correctly understand that each outcome remains equally probable. The grandmother’s expected value was not impacted by whether or not she staked on zero (the strategy that she thought followed from reason) or on some other number that had not occurred for quite some time (the strategy that Ivanovich seemed to find more reasonable). In the end, it all boiled down to luck.3

When I play roulette, I like to bet on the number 29. Why? Because when I was 29 years old and in a casino, I placed bets on and around that number and walked away with one of my few big wins at the roulette table. Do I believe that betting on 29 improves my chances of winning? In a certain sense, yes. It makes the game more enjoyable if I can attach meaning to my bets, if I can pretend that my choice has significance. In moments of high emotion or when large sums are at stake, I might even feel as though I truly believe it.

Yet, in a deeper and more important sense—one in which I am not indulging in fantasy to make the game more meaningful—I do not believe it at all. Since my expected return remains unchanged regardless of my bets, embracing this illusion, even to the point of temporarily convincing myself it is real, enhances the experience. For someone like me, who understands that my beliefs do not affect the outcome, this kind of playful self-deception adds to the utility of gambling.4

The above discussion points appropriately to the triviality of the fact that gamblers believe in the gambler’s fallacy. Yes, sometimes they do, but sometimes they don’t. Sometimes non-gamblers do and do not as well, such that it is not in fact a gambler’s fallacy, except in the sense that the opportunity for the fallacy is rare outside the gambling context. And it mostly does not matter what gamblers believe in that context, because the success or failure of one’s choice is just luck anyway.

That, however, is an unsatisfying conclusion, begging for clarification. How widespread is belief in the gambler’s fallacy? Does it vary by game type? Are gamblers more prone to it than non-gamblers? Does it influence who chooses to gamble in the first place? Or who becomes addicted once that choice to gamble has been made? How does belief in the fallacy depend on the gamblers’ level of experience? Does belief in the fallacy impact the size and likelihood of wins and losses, as suggested by Footnote 2?5 The upcoming Substack essays will explore these issues in some depth, ultimately confirming Ivanovich’s observation in the second passage above: belief in the gambler’s fallacy is, for the most part, a mistake of the uninitiated, uncommon common among seasoned gamblers.

In 1913 in a Monte Carlo casino, a roulette ball landed on black 26 times in a row. With a fair roulette wheel, such a streak—of either color—should occur only once in 68.4 million attempts. Gamblers at that table reportedly lost fortunes betting on red during the course of that improbable streak, incorrectly believing that it must be “due.” The expression of this belief is most often seen among casino gamblers, as with this famous roulette example. As such, along with being called the Monte Carlo fallacy, it is most well-known by the moniker, “the gambler’s fallacy.” The gambler's fallacy is the mistaken belief that chance will self-correct for an improbable series of previous events, even when those events are independent. Gamblers who believe in the fallacy in turn believe that their chances of winning improve by betting against the continuation of the improbable sequence (in this case, by betting on red instead of black).

The gambler’s fallacy has been widely observed and discussed in popular culture and in research on the impact of previous independent events on decision making under risk. I introduced the concept briefly in an earlier post about the Law of Large Numbers noting, however, that most long-term gamblers do not make choices that correspond to belief in the gambler’s fallacy. Nonetheless, many novice gamblers (and non-gamblers) do believe in it. As such, it is worth considering why.

There are three interrelated cognitive “mistakes” associated with belief in the gambler’s fallacy, which I’ll refer to as (1) the representativeness heuristic and insensitivity to sample size; (2) “The Law of Small Numbers” and the belief that chance self-corrects; and (3) overweighting recency. Each will be considered in more detail below.

The Representativeness Heuristic and Insensitivity to Sample Size

Imagine the following scenario:

A certain town is served by two hospitals. In the larger hospital about 45 babies are born each day, and in the smaller hospital about 15 babies are born each day. As you know, about 50% of all babies are boys. However, the exact percentage varies from day to day. Sometimes it may be higher than 50%, sometimes lower.

For a period of 1 year, each hospital recorded the days on which more than 60% of the babies born were boys.

Which hospital do you think recorded more such days?

1. The larger hospital

2. The smaller hospital

3. About the same (that is, within 5% of each other)

Take your time and think about the answer. Scroll down when you’ve decided on the answer.

.

.

.

.

.

.

.

.

.

.

.

.

If you chose Option 3 (about the same), then you will be pleased to know that you answered the same as most participants in a study by the pioneering decision psychologists Amos Tversky and Daniel Kahneman (1974). 22% of their participants chose Option 1 and the same percentage chose Option 2. The other 56% chose Option 3.

Of course, as the discussion of the law of large numbers in the earlier essay suggests, smaller samples are, in fact, more likely to deviate from the population mean than larger samples, and so the correct answer is Option 2. On average, the smaller hospital will have more days with 60% boys than the larger hospital. Indeed, over the course of a year it would be extremely unlikely that the larger hospital would have as many days with more than 60% boys than the larger hospital.

To make this intuitive, imagine flipping a fair coin 10 times. How likely do you think it is that 100% of the outcomes will be heads (keeping in mind that by definition it is a fair coin)? Now imagine the coin has been flipped just three times. Now how likely do you think it is that all outcomes will be heads?

Here the intuition is obvious and correct. Of course, it is more likely that 100% of the outcomes will be heads with just three flips than it is with 10 flips, or, indeed, with any number of flips greater than 3, even though the likelihood of heads for each flip in both cases is 50:50. The math here is pretty simple (if you remember the rule from Probability 101). The chances of n outcomes in a row when the outcome has a probability of p is pⁿ. Thus, the odds of three heads in a row given that there is a 1/2 (50%) likelihood of heads is (1/2)³ (that is 1/8 or 12.5%), meaning that you’ll get all heads one out of each eight attempts, on average over the long run. The chances of ten heads in a row, however, is (1/2)¹⁰ (that is 1/1,024), meaning that you would get all heads just 0.10% of the time. You are 128 times more likely to get heads three times in a row than to get heads 10 times in a row. Small samples are more likely to be unrepresentative than large samples.

But in many cases, people are relatively immune to sample size when making likelihood predictions, as the hospital-baby example above demonstrates. In short, people often expect small samples to be more representative of the larger populations from which they are drawn (or—in the case of coin flips and roulette-wheel outcomes—of the probability distribution from which they are drawn). Tversky and Kahneman identify this as an example of what they call the representativeness heuristic, the tendency to assume samples will be representative of the populations from which they are drawn.

Part of the explanation for the gambler’s fallacy no doubt comes from the fact that people believe that an unlikely series of outcomes (say 3 reds in a row) is less normal than it actually is. They expect small samples to be more representative of the populations from which they are drawn than is justified, and so even short streaks are more surprising than they should be. For a live example of this effect with students asked to identify and predict random sequences in coin flips, listen to this section of a Radio Lab podcast, “Stochasticity.”

This is, however, at best a partial explanation for the gambler’s fallacy. It is true that people tend to underestimate the degree to which random, equiprobable outcomes will tend to cluster with small samples (the clustering illusion), as in the roulette and coin flip examples above. As a result, it is also true that people tend to be more surprised by streaks than they ought to be. This does not, however, explain why people would expect subsequent outcomes to be biased in the opposite direction. Indeed, with roulette and other games of chance, a more reasonable application of this incorrect understanding of small samples might be to update one’s beliefs about the game’s outcome likelihoods; that is, to assume that the coin must favor heads or the roulette wheel must be biased to favor black (or whatever color just occurred an unlikely number of times). In that case, people would not believe in the gambler’s fallacy but, on the contrary, they would falsely believe the opposite: that the streak is more likely to continue than chance would predict. After all, if I incorrectly think a coin or a roulette wheel is not behaving randomly, then I might be reasonable to conclude that that device is not random (as discussed here, in the earlier post on law of large number and the importance of experience in evaluating outcome likelihoods). More is needed to explain the gambler’s fallacy.

“The Law of Small Numbers” and the Belief that Chance Self-Corrects

Consider another scenario from Tversky and Kahneman (1971).

The mean IQ of the population of eighth graders in a city is known to be 100. You have selected a random sample of 50 children for a study of educational achievements. The first child tested has an IQ of 150. What do you expect the mean IQ to be for the whole sample?

Take your time and think about the answer before you scroll down to the answer below.

.

.

.

.

.

.

.

.

.

.

.

.

Kahneman and Tversky report that “a surprisingly large number of people believe that the expected IQ of the sample is still 100” (the population mean). The correct answer, they state, is 101.1 The first respondent had an IQ of 150. The other 49 respondents should be expected to have the population average of 100. The mean across all 50 participants would then be 101 [(150+100*49)/50 = 101]. In other words—not only do people expect small samples to be more representative of the larger populations from which they are drawn, as discussed earlier—they also expect chance to somehow self-correct with these small samples, so that the small sample will be representative of the population (or of long-term expectations). They note, as was also noted in the earlier essay about the Law of Large Numbers, that this is not how chance works: “deviations are not canceled as sampling proceeds, they are merely diluted” (p. 106). Yet people often believe that chance will self-correct so that even small samples will maintain the expected outcome ratios that large samples have. They ironically call this false conception of chance, “The Law of Small Numbers.”

It is this combination of both (1) the representativeness heuristic (and the corresponding belief that small samples should look more like the larger samples from which they are drawn), as described in the previous subsection, and (2) the incorrect belief that chance itself will somehow self-correct so that small samples will become representative, that together make up the law of small numbers. And Tversky and Kahneman rightly suggest this helps explain belief in the gambler’s fallacy. As with the representativeness heuristic by itself, however, the belief that chance self-corrects short-term unrepresentative outcomes is still not sufficient.

Overweighting Recency

People could—and often do—falsely believe that chance will self-correct over the long run, while not believing it will do so over the short run. Indeed, this turns out to be a common false belief among experienced gamblers, who instead often believe that chance does not self-correct immediately (that is, over the “short-run”) even while (incorrectly) acknowledging that it does over the long run, in line with their misunderstanding of the law of large numbers. This was referenced earlier and succinctly explained by Kahneman and Tversky with “deviations are not canceled as sampling proceeds, they are merely diluted”; again, see this discussion from the earlier essay on the law of large numbers for an explanation as to why the law of large numbers does not imply that chance must eventually self-correct for an unrepresentative sequence of outcomes). The law of small numbers described above is not just dependent on the false belief that chance will eventually self-correct, but rather that chance will self-correct over the short-term.

This belief relies on an overweighting of recent events. If people who believe in the gambler’s fallacy see a streak of red outcomes in roulette, they do not tend to consider the ratio of reds to blacks over the previous hundred (or thousand, or even just 10) outcomes. Instead, they tend to focus on the most recent streak, even if it is just a few reds in a row. If they were paying attention to a longer series of previous outcomes, they might correctly conclude that even three reds in a row would not be enough to balance out the ratio of reds to blacks, and conclude that more reds are still “due,” while continuing to believe in the gambler’s fallacy. Of course, gamblers might do that, and some gamblers would do that. In my opinion, it would be fair to call that an example of the gambler’s fallacy, since the fallacy itself is not consistently defined except to point to a false and unjustified belief that chance will self-correct. That said, most people who demonstrate belief in the gambler’s fallacy overweight the importance of recent events, as would be suggested by Tversky and Kahneman’s tongue-in-cheek “Law of Small Numbers.”

In summary, belief in the gambler’s fallacy usually corresponds to holding this trifecta of unjustified beliefs: (1) the expectation that small samples will be more representative of long-term likelihood distributions they tend to be; (2) the belief that chance must self-correct so that short-term unrepresentative distributions will not violate long-term expectations; and (3) the overweighting of recent outcomes, even when longer sequences of outcomes are available to memory and could be taken into account.

If you feel uneasy about the explanation so far, you should. The current explanation points to three false beliefs that together describe the gambler’s fallacy, but it does not explain why people have those false beliefs. Kahneman and Tversky’s representativness heuristic partly explains it, since that heuristic is justifiably posited to be generally adaptive even if it misfires in certain contexts, such as at the roulette table. But, again, the representativeness heuristic is not enough to explain why people would think chance self-corrects or why they pay more attention to immediately preceding outcomes than to longer sequences of events. The upcoming essays will address some of these concerns.

They consider how widespread belief in the gambler’s fallacy is, discussing examples from casino blackjack, roulette, and slot machines. Ultimately, these essays will argue that despite a variety of reasons to believe that the gambler’s fallacy is a widespread phenomenon in games of chance, the attrribution of the fallacy to gambler’s is largely itself a fallacy, resulting from a wide variety of mistaken assumptions or attributions.

If you would never willingly set foot in a casino or gamblers strike you as a particularly irrational lot, then you may be wondering what this Substack—with most examples taken from casino gambling—has to teach you about judgment and decision making more generally. This essay is an attempt to address that concern.

Contents

• The Gambler’s Fallacy and the Illusion of Cyclical Luck: Examples of Domain Specificity and Generality

  • Domain Specificity and Generality of the Gambler’s Fallacy

  • Domain Specificity and Generality of Belief in Cyclical Luck

• Casino Gambling as a Case Study

  • How the Case of Casino Gambling Generalizes

  • How the Case of Casino Gambling Does Not Generalize (and Why That May Be the Most Important Lesson)

• Too Long; Didn’t Read…

The Gambler’s Fallacy and the Illusion of Cyclical Luck: Examples of Domain Specificity and Generality

I want to start with a quick nod to the previous Substack essay which touched on a common misconception among gamblers about the law of large numbers, and on the relationship between that misconception and belief in both “the gambler’s fallacy” and “cyclical luck” (a.k.a., “the hot hand”) among casino gamblers. If you didn’t read that essay, you’ll want to take a look at it first, or this section won’t make as much sense as it might. Those two so-called fallacies are good examples that seem domain specific (to gambling), and so it is reasonable to wonder how does that discussion generalize to other people and to other kinds of decisions.

Domain Specificity and Generality of the Gambler’s Fallacy

While many experienced gamblers have a misconception of the law of large numbers and have false beliefs about luck, most non-gamblers misunderstand the law of large numbers too. When non-gamblers watch a roulette ball land on the same color repeatedly, the idea that black is due is common (I can attest from personal experience discussing the idea with non-gambling, non-decision-scientist friends and students). The initial understanding among non-math-nerds—who have not been regularly exposed to the distinction between frequencies (total numbers) and relative frequencies (ratios)—is that the rules of probability imply that unlikely sequences of events must somehow even out over time. It turns out to be non-gamblers or early-stage gamblers, when first exposed to roulette or other games of chance, who immediately make choices that suggest they believe in the gamblers fallacy. Experienced gamblers have usually learned that such a strategy has dangerous consequences.

There is something about games of chance such as roulette that is unlike most other daily experiences and in that sense is domain specific. That distinctiveness includes repeated exposure to the same kind of random, near 50:50 outcome, over and over again over a relatively brief period of time, with financial stakes that make those outcomes emotionally salient, and where it ultimately does not matter which choice is made (since continuing to be on the same color, say black, has the same likelihood of winning as changing to red). It provides the rare necessary environment to experience conditions that might elicit belief in the gambler’s fallacy. But the belief itself, once exposed to that environment, is not something unique to gamblers. On the contrary, it is beginner or early-stage casino gamblers who are most prone to it. In short, the gambler’s fallacy could be considered domain specific in the sense that it is only when playing certain games of chance that the fallacy has an opportunity to arise, but domain general in the sense that people across cultures are apt to use strategies that look like the gamblers fallacy when first exposed to such environments.

Domain Specificity and Generality of Belief in Cyclical Luck

Belief in the cyclical nature of luck itself (a.k.a., belief in the hot hand), however, is a good example of something that differentiates gamblers from many other people. This is true in the sense that the kind of people more prone to gambling are more likely to believe in it in the first place, whether due to cultural or individual differences in conceptions of the nature of luck. But it is also true in the sense that people with long-term exposure to games of chance tend to “learn” from experience that luck is cyclical in a way that and can be used to predict short-term future outcomes. Learn is in scare quotes because of course this lesson is false. But the perception that it is true, from both direct experience and vicariously, from repeated testimonials from other gamblers, is compelling—even to people who may have begun gambling convinced that there is no such thing as luck.

These two examples point to distinct aspects of gambling and gamblers that are both domain specific and distinct aspects that generalize to people and to risky decision making outside the casino.

Casino Gambling as a Case Study

A central theme of decision making in the wild, at least in this Substack, concerns the tension between how cases do and do not generalize, and the importance of both—of generalization and of uniqueness—to understanding and evaluating decision processes. Casino gambling was chosen because it’s a great example, a case, that generalizes in important ways to life in the contemporary industrialized world, and as a topic of scientific inquiry that generalizes to how we study understand and evaluate risky and uncertain decision making in other domains. It was also chosen because it is a wonderful demonstration of what does not generalize, about the hazards of domain-general and species-general models of rational choice or of human cognitive heuristics and biases. Much of what goes into understanding how and evaluating how well people make risky and uncertain decisions is either domain-specific (for example to blackjack or to gambling) or person-specific (for example, to blackjack players, or to a particular kind of blackjack player). Casino gambling helps highlight these contrasting goals of empiricism.

How the Case of Casino Gambling Generalizes

Casinos may seem like the extreme on the side of artificiality, deception, and manipulation, an unrepresentative outlier relative to the other environments in which members of industrialized societies spend their time. But casino environments are more like supermarkets, office buildings, and even our own homes than they may originally appear. More of what they offer is there because the users choose it and are willing to pay for it than might first meet the eye, and more of what we buy outside the casino is presented to us in deceptive and manipulative ways than we might initially suspect. Casinos may make these qualities more obvious than most environments, but it is in that transparency that they help illuminate similar characteristics of other built environments. Our decision contexts are a complex, co-evolving dance between buyers, sellers, creators, and intermediaries. Casinos are a great case because they lay bare this relationship more explicitly and transparently than most built environments do: partly built to deceive us and keep us coming back, and partly built to provide us what we are asking—and willing to pay—for.

Similarly, casino gamblers may seem like the extreme side of decision irrationality, unrepresentative outliers along the continuum from more to less rational, making risky choices that seem transparently designed to give the player a negative expected return. But studying gamblers’ beliefs and strategies up close helps teach us how often decisions that look nonsensical from a distance may have a rationality that the onlooker has simply failed to understand. However, it also demonstrates that all of us, not just gamblers, have persistent commitments to false or incoherent beliefs that can only be recognized as such from a more distant perspective. Gamblers are a great case because they illuminate general features of both how readily we attribute irrationality to that which we do not understand, and also how readily we ourselves may misapprehend a situation when we view it from a particular perspective.

How the Case of Casino Gambling Does Not Generalize (and Why That May Be the Most Important Lesson)

While there can be no doubt that many lessons from casino gambling and casino gamblers generalize to other domains and to other people, many of the lessons throughout this Substack are about the importance of domain specificity and the value of understanding what makes situations and people different as opposed to what makes us all the same. Gambling decisions have many characteristics that are unlike other risky or uncertain decisions. That is not a reason to conclude gambling is a poorly chosen case. Instead, it points to the fact that all cases are special. Important aspects of decision strategies are—in part—unique to the specialized situation. Decision processes develop with experience in a particular domain, and with the accumulated individual and cultural experience interacting with that domain. Understanding how those processes work, and evaluating how well they work, is enhanced by carefully considering the particular case, in the wild. Decision processes when playing blackjack or roulette are entirely unlike decision processes when deciding “whether or not to go to war or whether or not to carry an umbrella” (to use one of my favorite quotations from Tversky & Kahneman in justifying the use of simple gambles to study more general processes of decision making under risk and uncertainty).

Scientific psychology tends to look for domain- and species-general psychological processes, part of the inbuilt nature of the mind that we humans share. By searching specifically for those things, cognitive psychology also tends to miss what is unique and different: what varies across individuals, across cultures, and across decision domains. As the example at the top of this Substack essay emphasized, roulette and other games of chance provide one of the rare environments where people get to interact with near random, equiprobable, binary (red or black, odd or even, 1-18 or 19-36) repeated choices, over and over again. It is in that strange, artificial, intentionally designed domain that one can directly experience patterns in randomness and a disconnect between choice and consequence that is unlike almost any decisions under risk and uncertainty in the natural world. I cannot think of a single decision outside of intentionally, culturally designed contexts where repeated decisions have a near random and near equiprobable impact on outcomes; yet they are omnipresent in casinos… and in many sports.

But domain- and individual-specificity describes most decisions, in other “wilds.” Laboratory decisions making is unique because it is designed by scientists trying to disprove a particular hypothesis and lend support, instead to their favored hypothesis; because the scenarios tend to be hypothetical and are often unrealistic, because the participants tend to be university students, because learning and expertise tends to be absent or short lived, and because they are made outside a cultural and physical context that non-laboratory decisions tend to entail. Alternatively, supermarket shoppers make their decisions in part because of how the supermarket is designed, because of culture-specific practices, values, and beliefs that were encountered as part of growing up in a community of super-market shoppers, in part because of individual differences, including genetic differences, in part because of unique experiences outside the market, and in large part because of “expertise” and corresponding strategies developed after hundreds or thousands of hours spent in various supermarkets and in conversations with other shoppers. With the development of expertise in a domain, all of those unique features feed back on our neural development and change our brain structure and function so that even the cognitive processes are interdependent with the domain specificity. Both the decision processes and how well they work (their rationality) is interdependent with the domain, with experience, and with the individual and cultural differences that co-evolve with that domain over time. Focusing only on what is universal, shared across domains and across decision makers, misses much—arguably most—that is important to understanding and evaluating decision processes.

TL; DR

In short, I am convinced that casinos, gamblers, and gambling strategies and beliefs—together—have much to teach us about how we make decisions more generally, even if you have no interest in casinos or gambling, and even if you see yourself as nothing like a casino gamblers. But most of what it has to teach is about domain-specificity and non-generalization; about how different we all are and about how those differences develop with experience in distinct contexts; about the value of studying what makes us different and not just how we are all the same. Much of psychological science has been focused on what makes decisions all the same (e.g., the gambling metaphor and domain-general models of rational choice) or on what us, as humans, all the same (e.g., inbuilt cognitive processes associated with heuristics and biases), often with a naive assumptions that the psychologists’ point of view, or the undergraduate-psychology-student participants’ points of view, are more representative of people, as a species, than they actually are. Looking at the psychology of gamblers in the casino helps make clear both the truth and the absurdity in the idea that we are all alike.

Table of Contents

• The Law of Large Numbers

• Event Independence and the (False) Assumption that Devices Are Fair

  • 1. The Imperfection of Nature Means That Mechanical Gaming Devices are Never Fair

  • 2. Gamblers Are Often Mistaken About Seemingly Transparent Outcome Likelihoods: the Example of Slot Machines

  • 3. The Games Could Always Be Rigged

  • The Ambiguity of Event Independence in the Wild Complicates the Law of Large Numbers

• Gamblers’ (Mis)Conception about the Law of Large Numbers and Its Relationship to Two Fallacies Widely Associated With Gambling

  • The Fallacy of “the Gambler’s Fallacy”

  • “Belief in the Hot Hand”; or, More Accurately, Belief in Cyclical Luck

The Law of Large Numbers

Last week’s newsletter considered the concept of expected value (EV), the long-term average return from a risky investment or gamble, assuming the same choice could be repeated over and over again an infinite number of times. This is another way of describing the “law of large numbers,” a theorem proven by the mathematician Jacob Bernoulli and published posthumously in his seminal work, Ars Conjectandi, in 1713 (noted here in part because two of his nephews will be mentioned in an upcoming essay). Bernoulli proved that as the size of a sample of independent events increases, the sample mean will tend to approach and converge to the population mean. The law of large numbers says, for example, that as you increase the number of flips of a fair coin, the proportion of heads will converge to 50% over the long run. This principle helps explain why larger datasets provide more reliable estimates in statistics and probability.

Event Independence and the (False) Assumption that Devices Are Fair

Bernoulli’s proof starts with the assumption that events are mathematically independent. “Event independence,” by definition, means previous outcomes do not impact future outcome likelihoods. That assumption works for hypothetical scenarios, including laboratory experiments where participants can be told outcome likelihoods and told to assume that events are independent. But when making decisions in the wild—including among skeptical students participating in lab experiments that might involve deception—assumptions about outcome likelihoods and event independence are more problematic.

Card games like blackjack or poker are good examples where events are transparently dependent rather than independent. Card outcomes are negatively dependent: previous outcomes of one type of card make future outcomes of that type less likely. If I get a blackjack (an ace and a 10-value card) in the game of blackjack, an outcome that pays a bonus of 3:2 (meaning, for example, that for every 2 units bet the player will win 3 units instead of the usual even money win of just 2 units), then that immediately lessens the likelihood of subsequent blackjacks, reducing my expected value in the game. If I instead get cards other than aces and 10s, that increases the relative frequency of remaining aces and 10s, and a future blackjack becomes more probable.

On first consideration, it might seem equally obvious that events are independent for many other decisions, and in particular games of chance common in casinos. The flip of a fair coin is the standard example. Coins are used in many professional sports to randomly choose between two sides in the competition, since there is essentially a 50:50 chance that the coin will land on one side or the other (but note the slight bias to land on the same side that was facing up at the point when the flip was initiated). In the casino, the repeated rolls of a fair pair of dice in the popular American casino game craps, the repeated outcome of the spin of a fair roulette wheel, or the repeated outcome of the spin of a fair slot machine might all seem like non-controversial real-world examples of repeated independent events.

But note the qualifier “fair” in all of the above statements. If, by definition, the coin, the roulette wheel, and the dice, are fair, and if “fair” means that each outcome has the implied odds—50:50 for the coin; 18:20 for American roulette and 18:19 for European roulette1; and, for example, 1:35 for “snake eyes” (a pair of ones) or “box cars” (a pair of sixes) in craps2—then, yes, we can assume events are independent and match the promised outcome likelihoods. But that points back to the hypothetical but questionable assumption that the devices are fair, as defined above. Outside of human-designed environments or devices—casinos or coins—event independence is rare. If you find a fruit on a tree or catch a fish in a certain spot in the sea, it suggests that more fruits and more fish may be available. But once the availability of fruits or fish starts to noticeably diminish, the likelihood of finding more of them decreases for every additional fruit you pick or fish you catch. Indeed, in the wild, even with games of chance designed to appear fair—such that outcome likelihoods are transparent and outcomes are independent—they rarely are. Three reasons why this tends to be the case follow.

1. The Imperfection of Nature Means That Mechanical Gaming Devices are Never Fair

First, even if events are independent (from one roll of the dice or spin of the roulette wheel to the next), the outcome likelihoods will almost never perfectly match the likelihoods suggested by the game’s design. Coins have approximately a 50:50 chance of landing on heads or tails, and the empirical difference from 50:50 tends to be so small that it is irrelevant and imperceptible to the decision makers themselves. But no coin is perfectly balanced. Casinos go through a great deal of trouble to ensure that their dice are well balanced so that they have an equal likelihood of landing on any number. Roulette wheels are carefully designed so that each number has an equal likelihood, and casinos follow strict protocols to ensure the wheels remain balanced and that the croupiers do not develop a detectable “signature” that might allow gamblers to predict where the ball will land based on when it was released. This includes using different sized balls and requiring croupiers to switch the direction and speed of the ball. In practice, these methods are enough to ensure that the devices are “fair enough,” such that the vagaries of chance have a far larger impact on differences in outcomes across trials than any inherent bias in the devices themselves. But the biases still exist, no matter how small or imperceptible.

2. Gamblers Are Often Mistaken About Seemingly Transparent Outcome Likelihoods: the Example of Slot Machines

Second, players might not correctly assess outcome likelihoods, even when those likelihoods seem transparent. Previous outcomes can be a good guide for updating likelihood estimates given that potential for error. You might tell me you are flipping a fair coin, but if the coin lands on heads 100 times in a row, I would perhaps be foolish to take your promise at face value and to assume I had encountered a particularly improbable streak of heads. In that sense, outcomes can be considered dependent to the extent that previous outcomes can rightly inform and change our assessment of subsequent outcome likelihoods.3 Since many casino games are misleading about outcome likelihoods, this process of updating likelihood estimates based on observed outcomes becomes important. Slot machines provide a great example of just how nontransparent outcome likelihoods can be (for a more detailed discussion, see the sections on slot machines in this chapter).

Left: The “Liberty Bell,” the father of the contemporary slot machine (image courtesy of Marshall Fey), released to the public in 1899 (Legato, 2004). Right: A contemporary 25¢ banking slot machine with a siren light on top (image courtesy of Paul and Sarah Gorman).
From, Bennis et al., 2012, “Designed to fit minds: Institutions and ecological rationality.”

The surface structure of a slot machine is partly a historical artifact based on the original mechanical design. The first slots had three reels and each reel had a set number of stops (say 20) with a distinct symbol on each stop. Some symbols could be more common on the reel than others (so there might be seven cherries but only one red 7 among the 20 stops on each reel). Three red sevens would have a likelihood of just 1 in 8,000 (1/20 * 1/20 * 1/20 = 1/8,000) along with a jackpot payout, whereas three cherries—given the hypothetical example—would be 343 times more likely (7/20 * 7/20 * 7/20 = 343/8,000 ), but it would also have a much smaller payout. Before the computer age, players could calculate the EV of a slot machine. They could count the number and type of outcomes on each reel to determine the likelihoods of each winning combination. Then they could combine that information with the payouts for each type of win to calculate the precise expected loss for playing a particular slot machine (assuming, of course, that the machine was “fair”).

Today’s slot machines are designed to preserve that perception: a series of spinning reels, each with a set number of symbols equally spaced (often with “blank” between each symbol that the reel could stop on as well), with each reel seemingly stopping randomly on any of the equiprobable symbols or blanks. Today, however, that is just window dressing. The stops are still random, but they are programmed onto a computer chip, and the virtual reel can have thousands or millions of stops mapped onto each physical reel. There may be just one of a million virtual stops mapped onto the red seven jackpot symbol, but half of all stops mapped onto the blanks just above and below the red sevens. As long as the outcomes on the invisible virtual reel are random and possible—allowing the jackpot and various other pay lines to occur, however rarely—such machines are legal by Nevada standards. There are other requirements. Nevada slots must pay out a certain minimum percentage, and they usually pay out far more than the minimum because they are competing for business, the overall casino slot payout percentages are published, and higher payouts encourage customer loyalty. The only way to assess outcome likelihoods of specific machines is through inside information or experience playing. The surface structure of the outcomes (the proximity to a red 7, for example) provides little indication of how close players actually were to a jackpot. Importantly, frequent slot machine players know this very well. It is non-gamblers and beginner slot machine players who need to learn about the disconnect between the surface structure and the actual design of the machines.

3. The Games Could Always Be Rigged

Events may be dependent even on games that present outcomes as independent if the games themselves are rigged. The above discussion has assumed that casinos offer fair games. That is probably a good assumption in American casinos today: First, they are usually owned by publicly traded companies rather than private individuals and so both the motivation and opportunity to fix the games, and to do so undetected, would seem… more complex. Second, they already have a good system in place for making a profit by offering fair games that favor the casino without the need to risk cheating. Third, they have a large enough turnover to almost guarantee they reliably get into the long-run each month so that their actual returns correspond, more or less, to their expected returns.4 Finally, they have a lot at stake if they get caught cheating. Historically, however, casinos and other gambling halls did cheat. Loaded dice, magnetic roulette wheels, and stacked decks could all be used when bets are highest and casino losses are imminent. In smaller casinos markets, where minimizing the risk of big losses is paramount, fair games may be the exception rather than the rule, especially historically, when external measures to ensure fair games were less prevalent. This may also be true of today’s less regulated (and less regulatable) online casinos.

One might expect Czech casinos to be more prone to cheating for these reasons. They are less regulated, are more overtly associated with organized crime, and have fewer customers. As a result, getting into “the long run”—or simply earning enough profit each month to cover expenses, given expected returns—may be far less assured. Similarly, the competition for loyal customers the drives much of casino behavior in large gambling markets like Las Vegas is essentially non-existent. My own experience in Czech casinos supports such a conjecture. Dealers sometimes intentionally cheated by peaking at upcoming cards and helping us win to improve their tips (tips are uncommon in Czech, unlike in US casinos). There are other examples I will not go into here, but I have little doubt that cheating was widespread in some Prague casinos, at least in the early 2000s when I conducted my research.

The Ambiguity of Event Independence in the Wild Complicates the Law of Large Numbers

These three examples illustrate how (a) events might not be independent, or (b) assessing outcome likelihoods might otherwise depend on information from previous outcomes. The key point is that the theoretical assumption of event independence, required by Bernoulli’s proof of the law of large numbers, is rarely true in non-hypothetical situations. Consider a gambling-related example. If there were such a thing as luck (where luck refers to a condition that can be used to predict future outcomes), then even seemingly independent events—such as whether a roulette ball lands on red or black—could actually be dependent on recent previous outcomes. They would depend on the state of luck. In such a scenario, the law of large numbers would not predict average expected returns, at least in the short term.

For example, if luck were negatively dependent—meaning that a lucky previous outcome (or series of outcomes) predicted a shift in luck to the opposite direction—then even the results of fair coin flips would not be independent. That would not disprove Bernoulli and the law of large numbers, however. His proof is mathematical and assumes event independence. Bernoulli’s proof is not an empirical claim about whether or not particular repeated events in nature are in fact independent. The discovery of a negative dependence associated with uncanny luck would just demonstrate that event independence does not exist in nature (at least when luck gets involved). On the other hand, if luck were cyclical, like the seasons, with extended periods of “hot” and “cold,” then unlikely streaks associated with wins and losses would suggest that such streaks are apt to continue, at least in the short term, until the “season” changes. Luck presumably does not work in either of those two contradictory ways. But the point is that this is an empirical question rather than a purely mathematical or theoretical question. Whether luck is interdependent or independent is an empirical question, not something that can be deduced a priori. Of course, it is difficult to imagine how in practice that might work (whose luck is being tracked and for how long?) or why scientists would not have been able to measure its impact up to now.

Gamblers’ (Mis)Conception about the Law of Large Numbers, and Its Relationship to Two Fallacies Widely Associated With Gambling

The two empirical possibilities just discussed—concerning the nature of luck and its potential to negate event independence—are presumably false. However, they correspond to a common misconception about the law of large numbers among casino gamblers. That misconception has implications for two well-known fallacies associated with casino gamblers: The gambler’s fallacy and belief in the hot hand. The gambler’s fallacy is the belief that chance must correct for an improbable sequences of events. Belief in the hot hand is the opposite, the false belief that unrepresentative recent outcomes will continue into the future.

From my interviews and conversations with casino gamblers, it was evident that many long-time gamblers are familiar with the term “the law of large numbers.” They often referred to it—incorrectly—to explain their beliefs about how past unrepresentative patterns (such as an unlikely streak of wins or losses, or of red or black outcomes in roulette) influence future outcome likelihoods. Their misconception is that the law of large numbers indicates that these unrepresentative patterns must self-correct over time so that the long-term average will match the outcome likelihoods that are designed into the structure of the game. For example, if a roulette ball lands on red ten times in a row, many gamblers (and even non-gamblers) will invoke the law of large numbers. They use it to suggest that the streak must eventually be corrected, so that black outcomes catch up with reds, “since the law of large numbers says that with a large enough sample, red and black should each occur 50% of the time.”

To be clear, that understanding of the law of large numbers is mistaken. The difference between red and black outcomes could remain 10 forever, or even increase indefinitely. Even so—assuming the coin or roulette wheel are fair—subsequent outcome likelihoods remain unchanged: 1/2 in the case of a fair coin and 18/38 for red or black outcomes in American roulette. Furthermore, the relative frequency of reds to blacks will nonetheless approach the average expected outcome as the sample size increases, even after unlikely previous patterns in outcomes, despite common intuitions to the contrary.

To illustrate this, consider a small sample of just ten roulette wheel outcomes (ignoring the green zeros in roulette for simplicity): nine red and one black (a difference of eight). That’s 90% reds and only 10% black, not the 50:50 that would be expected over the long run. Now consider a sample with 1 billion outcomes, but now with 1000 more reds than blacks: 500,000,500 red outcomes and 499,999,500 black outcomes. That’s 50.00005% red and 49.99995% black. You’d be right to say I’m being ridiculous by not rounding to 50% each, even though red now has a 1000-outcome advantage over black, compared to the eight-outcome advantage after the first ten spins. In other words, the fact that large samples approach average expectation for independent events does not imply that previous outcomes impact future outcome likelihoods. On the contrary: by definition independent events do not impact future event likelihoods. With large samples, the relative frequency of each outcome approaches 50:50 even if the difference in the frequency of each outcome continues to increase over time. Due to the slight imperfections in nature, even with the most carefully designed roulette wheels, we can expect the difference in red and black outcomes to increase in the direction of any natural bias in the wheel. This can happen even as the relative frequency of red and black approaches something closer to 50:50.

As the earlier discussion suggests, however, this points to the problem of generalizing the law of large numbers to the real world. If events are independent, then previous outcomes do not impact future outcome likelihoods, and casino gamblers would be wrong to suggest otherwise. That said, casino gamblers could theoretically be correct that something in the nature of chance itself is self correcting.

1. The Fallacy of “the Gambler’s Fallacy”

On first consideration, this misconception about the law of large numbers—the idea that chance corrects itself to adjust for unrepresentative past outcomes—might seem like a clear endorsement of the gambler’s fallacy. However, experienced gamblers rarely make choices consistent with the gambler’s fallacy, nor do they invoke the law of large numbers to justify such a belief. This claim that experienced gamblers do not believe in the gambler’s fallacy (or at least do not make choices that reflect such a belief) contradicts several other sources of evidence: popular myth about the gambler’s fallacy5, historical examples suggesting gamblers believe in the hot hand6, and both laboratory and field research on how roulette players react to streaks7. See this later Substack newsletter for a partial discussion as to why non-gamblers and novice gamblers often do believe in the Gambler’s fallacy. An more complete explanation, that also considers why experienced gamblers tend not to believe in it, is still upcoming.

2. “Belief in the Hot Hand”; or, More Accurately, Belief in Cyclical Luck

Rather than using the law of large numbers to justify a belief in the gambler’s fallacy, experienced gamblers often refer to it as part of a nuanced explanation of why they do not believe in the gambler’s fallacy—or at least do not act on that belief. Instead their behavior corresponds to belief in “the hot hand.” The term “belief in the hot hand” originates from an influential and controversial paper by Gilovich, Vallone, and Tversky. That research provided compelling evidence that the widespread belief that basketball players get hot and cold in a way that can be used to predict their near-term future performance is false.

While I suspect the cause of this false perception in basketball is similar to the cause of the false perception in gambling, I believe that applying the term “hot hand” to gamblers is misleading. “Belief in cyclical luck” would be more accurate. The belief in basketball is about cyclical changes in human performance that can be used to predict near-term future performance. Many plausible psycho-physiological models could explain such performance changes. In fact, denying their existence seems to contradict direct subjective experience of our own changing ability to perform at our best depending on the circumstances. That is no doubt part of what makes their findings—that basketball player performance does not appear to be streaky beyond what would be predicted by chance alone—so surprising. The belief in gambling, however, is not about cyclical changes in human psycho-physiological performance at all, or even necessarily about human performance. Instead, in the casino domain, the belief is about cyclical patterns in luck itself. This luck can be associated with a “hot hand”—for example, a particular person rolling the dice in craps, or a particular roulette or blackjack dealer or player. But it is often associated with other, non-human things: black or red on a roulette wheel, the cards running hot or cold, a blackjack table, a lucky charm.

How does the aforementioned misconception about the law of large numbers—the mistaken belief that chance corrects short-term unrepresentative patterns—lead to a belief in cyclical luck? This too will wait for a soon to come future Substack essay. There is plenty to chew on in the current essay already.

Expected value (EV) refers to the long-term average return of a risky investment or gamble. It can be calculated with basic math when given both (1) outcome likelihoods (probabilities or other representations of outcome likelihood) and (2) outcome values (numerical quantities in a certain currency, such as dollars, pizzas, casino chips, or just points or units). If those numbers (outcome likelihoods and values) are not known, EV can be approximated by estimated likelihoods and values. EV can also be inferred from experience by calculating the mean return from a repeated risky decision, as lone as one has the opportunity to repeat the same decision over and over. See the next essay in this Substack for thoughts on a couple of the many challenges of estimating EV from experience. EV is one of the simplest measures to evaluate whether a decision is good or bad, with the choice that results in the highest EV being the best choice one can make, assuming maximizing average returns is the goal and one’s assessments of likelihoods and outcome values is accurate (two assumptions that are often false).

The dominant classical model of rational choice when making decisions under risk and uncertainty, Subjective Expected Utility (SEU) Theory, is a psychologically nuanced version of EV that takes into account the subjective nature of both probability assessment and of what one values. As such, understanding EV is central to understanding SEU, and in understanding its strengths and weaknesses as a normative and descriptive model of decision making under risk and uncertainty. Even if EV were not foundational to SEU Theory, it is unquestionably foundational to evaluating whether or not a casino gamble provides a positive or negative expected return to the gambler. One of the themes of the Casino Cognition Substack is that there is far more to decision making—including gambling decision making—than EV. But as a starting point, EV is arguably the best place to start.

The purpose of this essay is to provide a basic introduction to EV, enough about it to understand its meaning and relevance for both gambling and for describing how, and how well, people make decisions under risk and uncertainty. EV is a domain-general mathematical model, “experience far” relative to the kind of thinking that generally goes in to making decisions, so apologies in advance for that. That said, please bear with me (and feel free to skip the math if it doesn’t interest you): the concept is foundational to evaluating investments, gambles, and risky and uncertain decisions more generally, and so it’s a useful concept to understand.

Alternatively, if the math and background below is more than you’re looking for, but you’d like a quick description of EV and a simple calculation example, see the glossary entry from the Casino Cognition website, here.

The 1654 letters between Blaise Pascal and Pierre de Fermat: the birth of expected value and probability theory

Imagine you are playing a game of one-on-one basketball to 21 points for $100. You are leading 18 to 15 when it gets too dark to continue and so you and your opponent agree to split the pot based on each player’s likelihood of winning. For the sake of argument, imagine that the previous points should not be taken as an indicator of difference in skill. In other words, you assume that you each have a 50:50 chance of winning each subsequent point regardless of previous performance. How can you calculate your likelihood of winning so as to fairly divide the winnings?

A similar question was posed in 1654 by the French nobleman and noted gambler, the Chevalier de Méré, to the mathematician Blaise Pascal (of “Pascal’s Wager”), though not about basketball, of course. De Méré had been playing a game of points that had to end early, and wanted to know how to fairly divide the wager based on each player’s likelihood of winning (rather than canceling the bet altogether or giving the entire prize to the player with the early lead). The question led to a famous exchange of letters between Pascal and another noted mathematician, Pierre de Fermat (of, for example, “Fermat’s Last Theorem”). Their solution was to look at all possible ways the game could have finished if played to completion from the point when they had to stop, effectively giving birth to both probability theory, including “Pascal’s Triangle”, and the first mathematical framework for calculating expected value (EV).

EV for Roll Outcomes of a Pair of Dice

In its generalized form, expected value equals the sum of all products of (a) the likelihood of each outcome and (b) the value of that outcome. An easy example is the roll of a fair pair of dice, where fair implies that each of the six sides of the dice has an equal likelihood of occurring on each roll.

Photo by William Warby

A roll total of 12, which requires two sixes, will occur, on average over the long run, one in 36 times (1/6 * 1/6). If “roll totals of 12” has a value of $1 and all other rolls have a value of 0, then a roll of 12 has an expected value of 2.7 cents (1/36th of a dollar), since that’s the average amount you would win for a single roll of the dice, if the gamble could be repeated indefinitely. More generally, the expected value would be 2.7% of whatever the payout might be for rolling a pair of sixes. Of course, most gambles include the possibility of losing money, and not just the promise of an unlikely win. If you were to win $1 each time you rolled a dice total of 12 and lose $1 each time you rolled a dice total of anything other than 12, the expected value of that gamble would be -94.4 cents (- $1 * 35/36 + $1 * 1/36 = -$34/36 or -$0.944). In other words, you could expect to lose 94 cents for every dollar gambled in such a game of dice, on average, if you repeated that gamble over and over again, ad infinitum.

What if instead of “all other outcomes,” you were competing with just one other dice total? For example, say you win $1 every time you get a total of 12, and lose $1 every time you get a total of 7, and all other outcomes are a push (you don’t win or lose anything). Is the expected value even, in your favor, or against you? While it might not be immediately obvious, 7s are far more common than 12 because there are many more ways to roll a 7 than a 12. Only one of the 36 possible dice permutations can be a 12 (two sixes). A seven with a 2 and a 5 or a 5 and a 2 (already twice as likely as a 12 with 2 of 36 of the possible outcomes instead of just 1). But it can also be rolled with a 3 and 4, a 4 and 3, a 1 and 6, or a 6 and 1. Together that is 6 of the 36 possible roles, six times more likely than getting double sixes. If you accepted a gamble from me where I paid you a dollar each time you rolled a 12 and you paid me a dollar each time I rolled a 7 and neither of us won or lost for other rolls, you would be gambling with an expected loss of -13.9 cents per round ($1 * 1/36 [your chance of winning times the amount you would win] - $1 * 6/36 [my chance of winning times the amount you would lose] - 0 * 29/36 [the chance for all other outcomes times the amount that would cost you, nothing] = -5/36 or -$0.139). My expected value, on the other hand, would be the inverse of that, an average win of +13.9 cents per round. In more general terms, your expected value would be a cost of -13.9% of the amount risked, whereas my expected value would be a gain of 13.9% of the amount risked: a sucker’s bet.

Back to the Basketball Example

So what about that initial basketball example? There are a few ways to solve the problem but the easiest way to approach it may be to start by imagining how many total points at most could be scored before a winner must be determined. In the imagined game, the opponent has 15 points and you have 18 points and the game ends at 21. The highest possible score is 21 to 20, which could happen either if your opponent got 6 more points and you only 2 (with the opponent winning 21 to your 20), or if you got 3 more points and your opponent only 5 (with you winning). That is 8 total points more. Since each round is equally likely, it is a relatively straight forward—if time consuming—process to write out each possible set of (256!) permutations of 8 rounds of play and then simply count what percentage of those have 3 or more wins for you (in which case you’d be the one winning) and what percentage of those sets of 8 points your opponent got 6 or more points (in which case they’d be the winner). Of course, you’d often win before 8 rounds are played and your opponent could win in fewer than 8 rounds, too, but by imagining all 8 rounds are played to completion even if the game is won earlier, it allows for considering the entire problem space of equally probable outcomes, making it easier to conceptualize EV. In practice, however, it is easier to limit the analysis to the fewer scenarios when your opponent would win, which could only be accomplished with 6, 7, or 8 wins across the 8 total rounds, since your opponents likelihood of getting the 6 points needed to win before you get the 3 points needed to win is exactly the inverse of your likelihood of getting 3 points first, and it reguires fewer steps.

Luckily, there are nice mathematical formulas for calculating permutations and corresponding likelihoods thanks to that correspondence between Pascal and Fermat. The most general version for repeated events with exactly two possible outcomes (in this case either a point for you or a point for your opponent) is the binomial probability formula:

The formula would need to be applied repeatedly to count the probability for each value of k from 0 to 8 (where k is the number of points you won after 8 points have been scored by either you or your opponent), and then those likelihoods would need to be added up for each of the values of k when you received 2 or fewer points (and therefore your opponent received 6 or more points and won; that is, cases when k = 0, 1, or 2). Luckily with 50:50 likelihoods for each of the two possible outcomes (winning a point and losing a point both have a likelihood of 50% in this scenario), the formula can be simplified as the binomial coefficient divided by the total number of possible permutations across the 8 points (2⁸ or 256 permutations).

• When k = 0, there is 1 permutation (LLLLLLLL), and the likelihood is therefore 1/256.

• When k = 1, there are 8 permutations (LLLLLLLW, LLLLLLWL, etc.), so the likelihood is 8/256.

• When k = 2, there are 28 permutations, so the likelihood is 28/258.

• The total likelihood of your opponent winning, then, is the sum of those three probabilities (1/256 + 8/256 + 28/256 = 37/256 = 0.1445 = 14.45%). Your opponent could be expected to win 14.45% of the time and you could be expected to win 85.55% (100%-14.45%).

• In dollars, a fair division of the pot would be for you to finish with a gain of ($100 each or $200 total) or $171.09, ending with a gain of $71.09 on the hundred dollars bet. Your opponent, on the other hand, should get back 14.45% of that $200 or just $28.91, a loss of $71.09.

• Note that your EV is $71.09 (or 71.09% of the amount wagered), not the 85.55% representing your likelihood of winning. Likelihood of winning times the amount to be won ($100 in this example) is not the same thing as EV, because it does not take into account the amount you can lose. If you have a 50% chance of winning $100 but also a 50% chance of losing that amount, then your EV is not 50%, it is 0. On average, over the long run, you will end up with the same amount you started with. In this case, with an 85.55% chance of winning $100 and a 14.55% chance of losing $100, the formula for calculating EV follows: $100*85.55% - $100*14.55% = 71.09%.

Estimating EV has come to be a central first step in assessing the value of any potential investment or gamble. It answers the question of what percentage of one’s “investment” one can expect to gain or lose, on average. As suggested, it also turns out to serve as the basis for the dominant domain-general mathematical model of rational choice, foundational to game theory, to utility theory, and to much of the field of behavioral decision theory.

Table of Contents

• Decisions Under Certainty

• Decisions Under Risk

  • A narrow definition

  • Example 1: Overweighting small likelihoods and underweighting large likelihoods

  • Example 2: Risk aversion in the face of gains, but risk seeking in the face of losses

  • Risky decision making is uncommon in the wild

• Decisions Under Uncertainty

  • A narrow definition

  • Example of uncertainty even in a domain with given outcome probabilities

  • Radical uncertainty

  • My usage of risk and uncertainty throughout the Substack

• Game-Theoretical Decisions

Decisions Under Certainty

Decisions under certainty are choices between various items or outcomes. Choosing between jams, yogurts, or wines in a market, between vacation packages at a travel agency, or between available car models at a dealership are all examples of decisions under certainty. Although such decisions are referred to as “certain,” that is an idealization. They are not really certain. We cannot be sure that a presented item will really be provided as promised. There is also uncertainty about how much utility a choice will have even if it is provided as promised I might choose a company trip to Hawaii over a $1000 Christmas bonus, but in making that choice, I am making a guess that the trip to Hawaii will have more subjective value to me than the $1000. With that in mind, it should be noted that decisions under certainty is more a label about what the researchers themselves are trying to study than it is about something inherent to the decisions themselves. Such research often considers how different kinds of attributes are traded off (speed versus gas mileage versus price when buying a car, for example), with the intention that study participants take for granted that the given attributes are certain.

Decisions Under Risk

A narrow definition

In behavioral decision research, risky decisions usually refer to choices between options where at least one of the options has uncertain outcomes (likelihoods greater than 0% and less than 100%). For example: “With Option A, you get $100 for sure. With Option B (the risky option), you have a 50% chance of getting $0 and a 50% chance of getting $200. Which option would you prefer?” With risky decisions, likelihoods are known, or at least available.

That specialized usage of risk differs from every-day usage of the term in at least three ways. First, folk conceptions of risk rarely include known probabilities. Living near a live volcano might be considered risky even if those choosing to do so do not know the likelihood of a life- or home-threatening eruption. Second, risk tends not to be about trivial outcomes (magnitude matters). Third, risk implies potential costs and not just potential gains (valence matters). If someone were to say, “You could either win $1000 or lose a penny; are you willing to take that risk?” a reasonable response might be, “What’s the risk?” since the potential cost is trivial, despite the fact that the potential gain is significant. Scholars who study risk often adopt a broader definition that entails significant potential costs, aligning more closely with everyday usage. That said, in behavioral decision theory, the narrower conception of risk described above tends to prevail. The extensive amount of research focused on probabilistic outcomes has made it useful to differentiate between cases when probabilities are available (risky decisions) and cases when probabilities are unknown and the best one can do is to estimate them (uncertain decisions). Below are two examples of findings given this narrow conception of risk.

Example 1: Overweighting small likelihoods and underweighting large likelihoods

Description

Daniel Kahneman was awarded the Nobel Prize in Economics in 2002, in large part for his work with Amos Tversky developing prospect theory, a psychological model that integrates given outcome probabilities with the magnitude (size) and valence (gains or losses) of outcomes.1 One risk-relevant finding from prospect theory is that study participants have tended to overweight very small likelihoods (approaching but greater than 0%) and underweight large probabilities (approaching but less than 100%). Shark attacks provide a good real-world example, where the mere possibility of an attack, however remote, is enough to keep many people out of the ocean.

Criticism

See Hertwig et al.’s wonderful 2002 paper in Psychological Science, “Decisions from experience and the effects of rare events in risky choice,” for an important qualifier to the described finding. When people learn about outcome likelihoods from experience, rather than having those likelihoods stated as was the case in the development of prospect theory, research participants tend to do the opposite: overweighting large probabilities (less than 100%) and underweighting low probabilities (greater than 0%). On the one hand, that may be obvious when imagining the fat turkey trying to guess how good the day will be on Thanksgiving morning, after having woken up ever day being overfed—paraphrasing Taleb’s famous example from The Black Swan. Turkeys of course have no way to know the likelihood that the next day will be different from all the other days other than what they have experienced thus far. But inverse finding for decisions from experience is relevant even for outcomes that we decision makers may know are theoretically possible or theoretically not certain. Imagine a young doctor who may have learned about extremely rare diseases from text books while training to become a doctor. Those books will have mentioned the rarity of those exotic diseases; but the doctor, seeing patients with associated symptoms, may give such diseases nearly equal plausibility as more common ones, simply because both may have been given similar emphasis and study time in the textbook (indeed, the rare diseases may get more emphasis since the common ones will already be more familiar to doctors in training). That looks like a decision from description and matches the findings from prospect theory. A doctor nearing retirement, however, who has seen thousands of patients, and has long forgotten textbook training may not give rare disease that fit the symptoms even fleeting consideration, even after eliminating other more common diseases that fit the symptoms. That finding is directly contrary to prospect theory and suggests an important distinction in how we learn about likelihoods.

Example 2: Risk aversion in the face of gains, but risk seeking in the face of losses

Description

Another example from prospect theory is the now-qualified finding that people are risk seeking in the face of losses (preferring risky decisions over certain ones) but risk averse in the face of gains (preferring certain options over risky ones). The most famous example demonstrates this phenomenon as part of evidence for framing effects in what has become known as The Asian Disease Problem. Framing effects occur when two versions are considered indistinguishable in terms of meaningful content, but differ in terms of how that content is described or presented. In the Asian disease problem, two groups of participants are both told to “imagine that the U.S. is preparing for the outbreak of an unusual Asian disease, which is expected to kill 600 people.”

One group of participants was given a gain frame, where outcomes are described as lives being saved (rather than as deaths). That group is given the choice between two treatment options and asked which one they would prefer. Option 1: “200 people will be saved.” Option 2: “There is a 1/3 probability that 600 people will be saved and a 2/3 probability that no people will be saved.” If you do the math for Option 2, it turns out that the expected value (the long-term average) is equivalent to the the sure thing of Option A: 200 lives will be saved on average. Among those participants, 72% chose Option 1, the certain choice to save 200 lives; only 28% chose to gamble with the possibility of either saving everyone or saving no one.

The second group of participants was given a loss frame. Their Option 1 was, “400 people will die” (instead of 200 people being saved); and their Option 2 was, “there is a 1/3 probability that nobody will die, and a 2/3 probability that 600 people will die” (instead of there being “a 1/3 probability that 600 people will be saved and a 2/3 probability that no people will be saved”). Again, the expected value for the gamble (Option B) was the same as the certain outcome (Option A): 400 people die. Among those participants, however, 78% preferred the risky Option 2 and only 22% preferred the certain outcome, Option 1.

These findings might not initially seem compelling. After all, 200 lives saved is not directly comparable to 400 people dying; arguably it’s not even comparable to 200 people dying. But what is important about the study and makes it a compelling example of framing is that both Option 1 and Option 2 turn out to be identical across the two groups in terms of lives saved and people dying and the likelihoods or certainties associated with each. Given that both groups are told that the Asian disease is expected to kill 600 people, “Saving 200 people” (Group 1’s Option 1) is mathematically identical to “400 people dying” (Group 2s Option 1). In both cases, 200 people will be saved and 400 people will die given the 600 people whose lives are at stake.

Similarly, “a 1/3 probability that 600 people will be saved and a 2/3 probability that no people will be saved” (Group 1’s Option 2) is mathematically identical to “a 1/3 probability that nobody will die and a 2/3 probability that everyone will die” (Group 2’s Option 2). If there’s a 1/3 probability that 600 people will be saved (Group 1’s scenario), that means that there is a 1/3 probability that no people will die (Group 2’s scenario), given that the stated total population was 600 people. And if there is “a 2/3 probability that no people will be saved” (the second half of Group 1’s Option 2), that means that there is “a 2/3 probability that 600 people will die” (the second half of Group 2’s Option 2). Other than framing—using language that describes the outcomes in terms of how many people live or in terms of how many people die)—Option 1 and Option 2 are identical between groups, and the two options themselves are identical in terms of expected outcome value, except one (Option 1) is certain, and the other (Option 2) is risky. The participants, however, show vastly different preferences between groups, as evidenced by the 72% of participants in the gain frame who preferred the risk-free Option 1 and the 78% of participants in the loss frame who preferred the risky Option 2.

Criticism

The particular example has been expanded on in a variety of ways pointing to the fact that preference for risk in the face of losses and sure things in the face of gains depends on the subjective importance given to the items under consideration. If consequences are pennies on the dollar lost or gained, participants (for whom pennies are presumably deemed insignificant) tend to be risk seeking even in the face of gains (called the “peanuts effect”). At the other extreme (from peanuts), if the description of lives lost is highlighted to emphasize the sacredness of the lives lost, for example by showing images of individual lost lives, participants become risk seeking even in the face of gains (presumably with the idea that losing even a single life is deemed unacceptable, and a gamble at least makes that a possibility; see Tanner & Medin, 2004, “Protected values: No omission bias and no framing effects”). It seems there may be a sweet spot between mattering too little and mattering too much during which framing matters most, at least with respect to reversing preferences between certain and risky choices.

Alternatively, there have been several criticisms of the particular framing of the two scenarios in the Asian Disease Problem which seem to introduce several potential mediating variables that might largely explain the findings beyond people being risk averse in the face of gains and risk seeking in the face of losses. In particular, (1) the 600 total lives comes early in the set up and it seems reasonable to imagine that a certain percentage of participants forgot—or simply never attended to—that value when evaluating the hypothetical options; and (2) without integrating that total population size, 200 people being saved and 400 people dying are not identical at all, nor are the two versions of the gamble (Option 2) which potentially sounds far worse with Group 1 than Group 2 assuming one does not do the math integrating the total population. An alternative version where both scenarios involve a matched certainties and matched probabilities—lose or gain 300 lives for Option 1 versus a 50% chance of losing or gaining either 600 lives or 0 lives for Option 2—would presumably have a far less compelling result given that Tversky and Kahneman’s framing has an unusually large effect size (see, for example, Kühberger, 1998, “The Influence of Framing on Risky Decisions: A Meta-analysis”).

Risky decision making is uncommon in the wild

As suggested already—despite how often risky decision making has been studied experimentally—most decisions “in the wild” do not entail stated or known probabilities. One notable exception, however, is casino gambling; perhaps not surprising since gambling is the metaphor on which models of decision making under risk and uncertainty have been based. Although even with casino gambling the probabilities are rarely as transparent as the gambling metaphor would suggest, in many casino games outcome probabilities can be exactly calculated.

In the three casino games that make up the primary focus of this Substack—blackjack, roulette, and slot machines—this is true in distinct ways. In roulette, calculating outcome likelihoods is straightforward; a high-school knowledge of probability theory would be enough to assess outcome likelihoods of all roulette choices. In blackjack, the outcome likelihoods change as a function of the distribution of cards remaining to be dealt. Although they can theoretically be calculated, doing so requires computer analysis unavailable in the casino. Such calculations in real time in the casino are beyond the cognitive capacity of any known human. Analyses based on those likelihoods can nonetheless be used by researchers to evaluate how well casino blackjack players choose, assuming their goal is to maximize expected return.

In slot machines, the likelihoods are programmed into the software and—within legal limits that vary by jurisdiction—can be chosen by the casino management. But those likelihoods are entirely inaccessible to the slot-machine players themselves and to scholars and can only be guessed at based on previous outcomes, legal bounds, and published win-loss percentages, and individual reports which vary widely. Because the variability in slot machine outcomes is extremely high (with some progressive slot machines in Las Vegas offering jackpots in the millions of dollars), it is not possible to reliably estimate outcome likelihoods. The challenge is made harder by the fact that slots are designed such that the surface structure of the game does not map on to the actual outcome likelihoods (see, for example, Turner & Horbay, 2004, “How do slot machines and other electronic gambling machines actually work?”). Near misses are programmed into the software so that stops above and below jackpot symbols are more likely than the physical structure of the reels would suggest, as are stops on the jackpot symbol for the first and second reel.

Despite the challenges of integrating probabilities, particularly in blackjack and slot machines, consideration of outcome likelihoods is central to all three games. At the same time, despite the obvious importance of outcome probability, it is also less relevant than might be expected given the major role of probability in rational choice theory. Even in roulette, a game where probabilities can be perfectly calculated, strategies and beliefs that are only loosely connected to outcome likelihoods play an important role.

Decision Making Under Uncertainty

A narrow definition

“Risky decision making” is just one half of the duo of concepts that has been defined in an overly narrow way so as to differentiate between (1) decisions when likelihoods are known or knowable (risky decisions) and (2) decisions when likelihoods are unknown and can only be subjectively estimated. “Uncertainty” has come to refer spectifically to that second option. Of course, in everyday language and for many scholars, any non-definite outcome, whether the probabilities are known or unknown could be called “uncertain,” as could other aspects of uncertainty not specifically about outcome likelihoods (such as uncertainty about which choices are available or which outcomes might arise from each choice). In behavioral decision theory, however, this narrower meaning of uncertain has become useful to emphasize the dichotomy between known and unknown outcome probabilities.

Example of uncertainty even in a domain with given outcome probabilities

As has been suggested above, even with putative decisions under certainty, not just given likelihoods matter; uncertain likelihoods also matter (recall the idea that even with decision making under putative certainty, decision makers do not really know they are getting what has been promised, nor do they know how much they will like what has been promised if they get it. Recall that trip to Hawaii that might end in heartbreak. As such, as with decisions under certainty and decisions under risk, the emphasis on uncertainty here may says more about what the researchers are trying to study than about whether or not outcome likelihoods are probabilistic and whether those probabilities are known: at some level, all decisions involve some degree of uncertainty. Consider the following personal example which demonstrates the role of uncertainty about outcome likelihoods even when theoretical likelihoods have been exactly calculated.

A professional gambling friend had devised an optimal strategy for a blackjack-type game only offered in Czech casinos. I double-checked the calculation then shared it with a mathematician and blackjack computer-modeling colleague and we all agreed that the math was sound. It turned out that the game was beatable even without keeping track of cards removed from play (that is, even without counting cards). Several casinos in Prague offered the game, so one of the challenges was to choose where to play. The game differs from blackjack in some key ways, but most importantly in that the variability in outcomes is far higher, so that even with a positive expected return, any single gambling session tended to end with much higher wins, or much higher losses, than the mode gambling sessions in blackjack. From a theoretical calculation of expected return, taking into account outcome variability and the bankroll required, this game had the potential to pay handsomely and be well worth the risk given my own income at the time. On the other hand, there are several “uncertainties” that make those calculated likelihoods less relevant and likelihoods that can only be subjectively estimated far more relevant. Below is a consideration of a handful of those issues.

First, despite our best efforts, the possibility remained that my friend had made a mistake in his calculations that neither of the two of us who carefully checked it uncovered. I knew from my own research on gambler’s strategies and beliefs that many gamblers developed sophisticated mathematical models that were compelling on first consideration but that ended up entailing mistaken assumption with disastrous implications for expected return. Casinos excel at designing games that promote mistaken assumptions. Recall the example of slot machines where the surface structure showing reel outcomes does not match the virtual structure programmed into a computer chip but invisible to players.

Second, there is human playing error. How many playing or betting mistakes might we make each hour, and how would that impact the expected return? In blackjack, it is important to hide the fact that you are counting cards, since casinos that recognize a player as a card counter can easily make it impossible to count cards. One of the strategies for doing so is to occasionally make poor plays that no card counter would ever make, such as betting high of the top of a freshly shuffled stack of cards, or varying one’s bet before the odds could possibly have changed enough to justify it. Blackjack has been analyzed enough to make justified inferences about the impact of such deviations from the ideal strategy. This blackjack relative, however, has much higher variability in expected return than blackjack, and so individual errors could have a far bigger impact on expected return. On the other hand, we did not need to count cards at all to get an advantage, so that human error was likely to be less common. We could only guess at the impact of such errors on expected return.

Third, there seemed to be a much higher likelihood of being cheated or robbed during or after gambling in Czech casinos than in Las Vegas. I had read about the evolution of casino gambling in the US enough to believe that the large corporate-owned casinos today had more to gain by keeping the games fair than by cheating players, but also that that was not always the case, especially when there were fewer customers and less regulation. Czech casinos are more like casinos from a time long passed in the US: they have far fewer gamblers and so they are at higher risk of losing money if a couple of big players have big wins. They are also less regulated, partly as a result, so that cheating is easier. In Las Vegas casinos, for example, dealers are required to periodically change the blackjack cards and to display every single card to the players before a fresh shuffle. If a player is concerned about a marked or damaged card, they can bring the attention to the casino and get a fresh deck or have the cards checked. In some casinos in Prague, they did not every display the cards to players and refused to show the cards when I had serious concerns about cheating. Similarly, Czech casinos remain more closely associated with organized crime. During the time of my research, one casino owner was shot dead in a tunnel; in another case, a bomb was set of at a casino venue associated with organized crime and the casino industry. Employees themselves sometimes discussed the role of organized crime and cheating openly, and dealer’s intentionally cheated to help us win when we started tipping them, something that would almost surely never happen in Las Vegas where dealers are watched as closely as players to ensure fair play, and criminal prosecution is a real possibility. How should we evaluate the possibility that the casino is cheating in the game (such as by introducing extra cards that are unfavorable to the players)? Or the likelihood that someone may make a phone call when we leave the casino after a particularly successful night to get back that money in a less above-board way?

We did not have a clear way to assess those outcome likelihoods (or the potential negative values associated with them) into the decision. Our best guesses were, however, central to our choices about the casinos in which to play this particular game, about how much money we could safely arrive or leave with, and about whether to play the game at all. But in every case they were subjective estimates, and my best guess today—now that I have some distance—would be different than it had been then. Uncertainty prevails in the wild, even in the mathematically most well-defined games.

Radical uncertainty

The above narrow definition of uncertainty focuses on whether outcome likelihoods are known or unknown, but there are other aspects of decisions that tend to be uncertain in the wild other than just outcome likelihoods. In casino gambles, as in laboratory experiments, the range of possible choices tends to be well defined. But in most decisions in the wild, the range of choices is unknown, and arguably limited only by the imagination and by available resources and technology. What are my choices in deciding how to write an email? Whatever the answer, it is different now with the advent of LLMs than it was a couple of years ago. Just as choices are unknown, the possible outcomes of most choices are themselves unknown, even unknowable and—in some sense—unlimited, as are the values or utility associated with those outcomes. What will the utility of a trip to Hawaii be? That depends on whether I meet the love of my life or lose a limb to a shark, and on the combination of any other of the practically infinite number of events that might possibly occur during a trip to Hawaii. Going back to the blackjack-like casino game in the Czech Republic: where does the utility from the fun of trying to beat the casino at its own game come in? Or the danger of getting on the wrong side of a criminal gang? Or the utility (or disutility) associated with (not) having enough money to complete my dissertation research? Or the value of a good story associated with that game for future writing about that research? What role should the possibility that I will develop gambling problems play?

Gerd Gigerenzer has recently attempted to resolve this important ambiguity in the meaning of uncertainty by restricting the previous narrower definition of uncertainty to the term “ambiguity” and reserving the term “uncertainty” for this more radical sense of ambiguity common to decision making in the wild where choices and outcomes are uncertain, as is the utility that might be associated with envisioned outcomes (see “The rationality wars: A personal reflection,” 2024). It is not obvious to me that either term (ambiguity or uncertainty) is more well suited to either definition, and so rather than argue about correct terminology in a domain that is already infused with ambiguous meaning, I would tend toward simply recognizing the ambiguity of the terminology itself, and to trying to be clear in context. That said, Gigerenzer’s distinction between “small worlds” (well-constrained domains where choices, outcomes, and values are all well-constrained and transparent) and “large worlds” (largely corresponding to decisions in the wild, without such artificial constraints) is seems extremely important when considering decisions in the wild. Gigerenzer borrowed the small-world versus large-world distinction from Leonard Savage who used it to point out that his own simplifying model of rational choice, Subjective Expected Utility Theory, only holds up in a small subset of narrowly defined decision contexts (small worlds). Gigerenzer points out that such an observation calls into question the appropriateness of such normative models of rational choice for understanding and evaluating many—perhaps most—decisions.

My usage of risk and uncertainty throughout the Substack

I will tend to stick to the narrower senses of risk an uncertainty, where risk refers to cases where outcome likelihoods are available and uncertainty refers to outcome likelihoods that must be subjectively estimated. There are two main reasons, despite having just argued that a broader conception of uncertainty is important: (1) that is how the terms are commonly differentiated among behavioral decision scholars, and (2) unlike most decisions in the wild, gambling decisions tend to better fit the “small world” assumptions common to experimental studies. Choices, outcomes, and outcome values are well defined in most casino gambles. That said, as the example of the Czech blackjack-like game above suggests, it is more the rule than the exception that casino gambling choices end up involving the kind of radical uncertainty common to other large-world decisions, once the casino context is given careful consideration. That point will come up often throughout future essays in this Substack, since it turns out to be important for both understanding and evaluating gamblers’ decision processes.

Game-Theoretical Decisions

Many important decisions are interactive in the sense the best decision for one person depends on choices made by others, which are themselves uncertain. Casino poker is a good example, since the quality of each player’s choice depends on uncertain assessments of other players at the table. Game theory is not particularly relevant to blackjack, slot machines, or roulette, although there are a couple of exceptions. First, while other player’s choices do not impact outcomes in predictable ways, they do impact outcomes in unpredictable ways. If a player before me in blackjack decides to hit (take another card) or stand (stop taking additional cards), that will change the card I end up getting. And all of the blackjack players’ choices, including my own, will impact what cards the dealer eventually gets, which matters to the success of everyone at the table (see this summary of casino blackjack for context). Unless players are cheating, that impact will be random and unpredictable, but it is an impact nonetheless. Second, players themselves believe that random impact matters in a way that are not random and that can be used to one’s advantage, such that how other people play is seen by many to be an important factor.

Part of the value of sticking to games like blackjack, roulette, and slot machines is that they demonstrate how incredibly complicated even simpler decisions are, and do so with fewer degrees of freedom or potential confounding variables than game theoretical games such as poker. Without disrespect for scholars who focus on game-theoretical problems, this narrower domain turns out to be wonderful for moving the discussion from decisions in the lab to decisions in the wild while still allowing for meaningful commentary on the role of factors like environmental design, cultural and systems of meaning, and the wider conceptions of utility that are associated with such systems of meaning. As a result, game-theoretical decision making will rarely be considered in this Substack.

1. Cultural Psychology (what is it and why do I care about it)

2. The Psychology of Judgment and Decision Making (what is it and why do I care about it)

3. The Gambling Metaphor and Its Critics (the importance of the gambling metaphor to how scholars understand decision making under risk and uncertainty, and a consideration of important criticisms)

4. Playing Blackjack for a PhD (describes my specific dissertation research project, including a brief summary of methods)

Subscribe now

Cultural Psychology

My experience as a card counter in casino blackjack described in the previous post became relevant in graduate school. Several years after I had stopped counting cards. I was accepted into the interdisciplinary PhD program the Committee on Human Development at the University of Chicago. The Committee’s flagship areas of specialization were the closely related fields of psychological anthropology and cultural psychology (the main difference being their respective affinity to the fields of anthropology and psychology). I tended toward the more psychological side. Both disciplines emphasize the interdependence of culture (the unique distribution of practices, values, beliefs, and designed environments shared by a group) and psychology (mental processes impacting how people think, feel, judge, perceive, decide, and behave). I fell in love with cultural psychology and the many lessons it had to share about human behavior and cognition.

For some thought-provoking and accessible popular scientific books on the topic see Richard Nisbett’s (2003) The Geography of Thought (though there are good reasons to be skeptical of his origin story and the related East-West geographical divide), Nisbett and Dov Cohen’s (1996) Culture of Honor (a great example of compelling social psychology research and theory designed to establish the long-lasting impact of historical culture and geography on cognition), Joseph Henrich’s (2020) The Weirdest People in the World (though I am skeptical of his central thesis that Westerners are distinctly unique relative to other cultural groups for reasons described here), or Malcolm Gladwell’s Outliers (my favorite of the bunch, despite the book being written by a journalist rather than a scientist and not explicitly framed as a cultural psychology book).

The Psychology of Judgment and Decision Making

For my PhD dissertation research, I focused on the psychology of judgment and decision making (J/DM), which has broad overlap with the economics subdiscipline of behavioral economics. Much of the research in this field begins with normative (optimizing) models of rationality and explains deviations from those normative models with reference to mostly inbuilt, species-general cognitive heuristics (such as the availability and representativeness heuristics) and resulting biases (any systematic deviation from the dictates of rational choice). This is thanks, in no small part, to the pioneering work of the late Amos Tversky and Daniel Kahneman, although there is a much longer list of eminent scientists who have been foundational to the field, including a handful of Nobel laureates in economics. Kahneman (a Nobel laureate himself) sadly passed away on March 27th of this year. The emphasis on heuristics and biases has provided an important rejoinder to assumptions in classical economics that the accepted models of rational choice were not just normative but were also descriptive: they were taken not just as good models for how rational people ought to choose, but also as the best models we have to capture how people actually choose. Kahneman and Tversky, along with many other scholars, have convincingly demonstrated that these models do not capture how people make choices. Instead, they argue, inbuilt cognitive processes—most often rules of thumb called heuristics that usually work but that also lead to systematic and predictable errors—explain systematic deviations from these normative models.

Some of the fun (but also controversial) examples of this research concern what Kahneman and Tversky call the Representativeness Heuristic which is associated with a range of systematic errors or biases including, the base rate fallacy, the law of small numbers, the conjunction fallacy, and the hot-hand cognitive illusion.

The Gambling Metaphor and Its Critics

Models of heuristics and biases, and of decision processes more generally, have been strongly influenced by gambling as a metaphor (a) for how all decisions under risk and uncertainty ought to be made (normative models), such as utility theory (b) for assessing irrationality or bias with respect to how judgments and decisions are in fact made (descriptive models), and (c) for the design of research stimuli used to develop those descriptive models (for a rich and critical discussion of this approach, see Goldstein & Weber, 1995). The gambling metaphor has been used to describe risky or uncertain decisions far beyond the gambling domain to cases where choice options, probabilities, outcomes, and the associated costs and benefits of those outcomes are often poorly understood and play an ambiguous role in how people make decisions. Here’s a nice justification for the use of the gambling metaphor to create experimental stimuli from none other than Kahneman and Tversky, from their 1984 paper, “Choices, values, and frames” (p. 341):

Risky choices, such as whether or not to take an umbrella and whether or not to go to war, are made without advance knowledge of their consequences. Because the consequences of such actions depend on uncertain events such as the weather or the opponent’s resolve, the choice of an act may be construed as the acceptance of a gamble that can yield various outcomes with different probabilities. It is therefore natural that the study of decision making under risk has focused on choices between simple gambles with monetary outcomes and specified probabilities, in the hope that these simple problems will reveal basic attitudes toward risk and value.

To those long-trained in decision science, or anyone reading this essay who has inevitably been exposed to the analogy between life and gambling, it may seem to follow as a matter of course—to be “natural” as Kahneman and Tversky wrote—that risky or uncertain decisions can be understood as choices among gambles. But for those who have not been repeatedly exposed to the gambling metaphor, I suspect risky and uncertain decisions could just as convincingly have been framed in very different ways. After all, outside the domain of many popular games of chance, we usually have unreliable and ill-formed opinions about (a) the range of possible choices, (b) the possible outcomes that might arise from those choices, (c) the probabilities of those various possible outcomes (which themselves may vary widely across time depending on contextual changes), or (d) the utility that each of those potential outcomes might have for us. Given the radical uncertainty that obtains for even banal choices (“whether or not to take an umbrella”), not to mention for the really important decisions (“whether or not to go to war”), the idea that rational decision makers should frame risky or uncertain choices like gambles might seem just plain WEIRD (see the below section on comparative methods to make sense of this intentional pun).

One convincing criticism of the gambling metaphor comes from Goldstein and Weber’s (1995) paper, “Content and discontent: Indications and implications of domain specificity in preferential decision making.” They make a compelling case that the use of the metaphor, among other content-impoverished norms in cognitive psychology, may systematically misrepresent cognitive processes. As the write (p. 92):

The point is not merely that the experimental practice of using content-impoverished stimuli would have failed to discover a number of interesting phenomena, but that the particular phenomena that would have been overlooked are those that conflict with the overarching theoretical framework.

They point to a range of research suggesting that when researchers consider real-world decisions that are embued with meaning, it turns out that the meaning (the content) plays an important role in the cognitive processes. For a compelling experimental example in decision science, see Medin et al.’s, “The semantic side of decision making” (1999).

A second critical approach challenges the idea that the received normative models of rational choice are in fact normative. This criticism originates with the pioneering work of Herbert Simon (another Nobel Laureate) on bounded rationality. That work—without drawing a sharp line between normative and descriptive models of choice—argues that classic models of rationality make unrealistic assumptions about available time, information, and information processing capacity. Instead, Simon argued that models of rationality should be based on more realistic models of human cognition that take into account such limitations. Simon proposed heuristics as an example. For examples of Simon’s classic work on bounded rationality and heuristics see his classic essays, “A behavioral model of rational choice” (1955; introducing the idea of bounded rationality) or “Rational choice and the structure of the environment” (1956; introducing heuristics, and in particular satisficing, as a boundedly rational alternative to unrealistic classical models of rational choice).

Simon’s work was not critical of the heuristics and biases tradition—Simon has in fact praised that work—rather, it preceded and helped justify Kahneman, Tversky, and their colleagues’ separation of normative and descriptive processes; their emphasis on heuristics as an alternative to optimizing normative models of choice; and their findings that people’s choices systematically deviate from the predictions of normative models. But many contemporary scholars have referred back to Simon’s work to point out that the normative status of those models is also problematic and, therefore, to question the readiness to attribute deviations from those normative standards to bias or irrationality. The most persistent and often compelling criticisms come from Gerd Gigerenzer and his colleagues with work on smart heuristics (see for example, the books Simple Heuristics That Make Us Smart, Bounded Rationality: The Adaptive Toolbox, Ecological Rationality: Intelligence in the World)

My training and interest in cultural psychology fit well with both criticisms, convincing me that the gambling metaphor was ill-suited for either understanding or evaluating decision making under risk and uncertainty. I believed—and still do—that culture; culture-specific built environments within which humans spend an increasingly signification portion of their time (including casinos); and individual and social learning within those environments together play a far more proximal and illuminating role in both how—and how well—people make decisions than the current body of research—with its emphasis on domain- and species-general cognitive processes—would suggest. Examples that support this contention, taken mostly from the world of casino gambling, are essentially what this newsletter is about.

Playing Blackjack for a PhD

My experience counting cards suggested a creative way to address this topic. If culture and experience is central to how people make risky and uncertain decisions even in a well-constrained gambling domain such as casino blackjack, then it would make a compelling case for their centrality in non-gambling domains where probabilities, the range of choices and outcomes, and the costs and benefits of those outcomes are largely uncertain. In other words, if the gambling metaphor is inappropriate even for understanding and evaluating gambling decisions—once real-world content and context is considered—then surely it does not apply to more removed contexts of decision-making under risk and uncertainty. I applied for grants to support field work in casinos in Las Vegas and in Prague, in large part justifying spending extensive time as a participant-observer gambling by explaining the math behind card counting that would allow me to sit at the table without losing that grant funding to the casinos. I was lucky enough—luck, of course, will play a recurring theme in future newsletters—to get two distinguished fellowships that fully funded my research for the next year and a half.1 If I had known then what I know now, I would have advised against funding anyone to spend extensive time gambling in casinos, but that’s a story for another newsletter.

The research I conducted over the next two years had three distinct methodological components. First, it was intentionally comparative; second, it involved ethnographic participant-observation (dealer school, time as a blackjack dealer, and extensive time gambling in casinos); third it used both qualitative and closed-ended interviews. Those methods are described in more detail below.

Comparative Methods

Comparisons between groups is a central feature of experimental psychology. Cognitive psychology tends to compare groups of people sampled from a single population to measure the effects of different conditions on hypothesized outcomes as a means to generalize about built-in, species general cognitive processes. Scholars studying individual differences instead sample from distinct populations based on differences in personality, demographics (e.g., gender, socio-economic status, political preference, or education), or some other measures of individual difference. Psychologists focused on expertise compare samples based on their level of experience or expertise in the domain of study. Comparison is central to cultural psychology, as well, except that the comparisons are between populations with distinct cultural histories and environments. If we want to know what makes gamblers different from non-gamblers, or blackjack players different from roulette players, or Czechs different from Americans, or experienced gamblers different from inexperienced ones, comparative methods are essential.

Comparative methods are essential for recognizing what makes people different, but they’re also necessary for making generalizations about human cross-cultural universals. One of the noted shortcomings in cognitive psychology research seeking to make generalizations about universal, inbuilt, human cognitive processes is that the research has relied on a narrow sample of participants (Arnett, 2008; Henrich et al., 2010), often first-year undergraduate psychology students fitting the same demographic as the researchers themselves, a demographic to which Henrich and colleagues creatively gave the acronym WEIRD (Western, Educated, Industrialized, Rich, Democratic), pointing to the overwhelming bias in published psychology research in that both the researchers and the participants are from countries or populations that are unrepresentatively WEIRD. Importantly, research that has tested the extent to which WEIRD findings generalize across populations has found that they rarely do, with a compelling argument to be made that the WEIRD samples are among the least representative, and in that sense not just WEIRD, but also particularly weird (Henrich et al., 2010; Henrich, 2020; though, as noted above, I suspect this result may be a methodological artifact rather than a true indicator that WEIRD people are more psychologically unique than other cultural groups).

If the premise of cultural psychology is correct that culture matters to how we think, then generalizing to gamblers as a whole or to countries as a whole rather than just to blackjack players or to the Las Vegas Strip gamblers is not solved by getting a large random sample across games or across the two countries. That would just obscure the distinctive and sometimes contrasting influence of specific games, locations, or people. Even if I just want to know about blackjack, the more comparisons, the more I can learn about what is distinctly blackjack player or gambler or American, or human, rather than due to some other characteristic that happened to be common to my blackjack sample or population. Of course, resources are always limited and comparisons are costly, but the ideal of comparison is inherent to the insights of cultural psychology.

With the above in mind, my own research was intentionally comparative in several ways:

• it compared participants across different levels of experience;

• it compared blackjack players with other casino gamblers (roulette and slot machines);

• it compared players in distinct locations (Indiana, the Las Vegas Strip, downtown Las Vegas, and Prague, Czech Republic);

• and it compared the players’ strategies with mathematical normative models that seek to maximize expected value (basic strategy and card counting).

Participant-observation

Table 1 summarizes the various participant-observation activities and the amount of time devoted to them in each of the three main locations. The activities included time in dealer school (necessary so that I would be able to work as a dealer), work as a dealer (which was not possible in the Czech Republic due to visa restrictions), and time as a casino patron in all three locations. More detailed methods can be found in the methods section of my dissertation.

Table 1. Participant-Observation Activities & Locations

Interviews

The interviews were of three types: semi-structured interviews (with a mixture of closed- and open-ended questions that could be completed in under 20 minutes), blackjack strategy surveys (that assessed players’ conceptions of basic strategy, the best way to play each hand, all other things being equal, given the player’s hand total and the dealer’s exposed card), and ethnographic interviews (recorded, open-ended interviews that usually lasted a couple of hours and allowed me to go into depth about gambling-related questions in whatever direction the interviews went). Table 2 summarizes each type of survey by game and location (again, the dissertation methods section has more details about each type of interview).

Table 2. Interview Type by Games

*-Columns do not always sum to total because in many cases the same person participated in both the blackjack strategy survey and the ethnographic interview. In these cases the total has been counted as one interview.

†-Similarly, rows do always sum to total because the same person may have participated in the same interviews for more than one game. In these cases the total has been counted as one interview.

Newsletter Contents

1. Vegas Baby, Vegas!

2. Card Counting

3. Blackjack Rules and Basic Strategy

4. Card Counting is Simpler Than You Think

5. Card Counting is a Dumb Way to Make a Living

6. But I Didn’t Know That at the Time

7. Prague, Czech Republic

8. Culture and Experience Promote Systematic Deviations From Basic Strategy

9. Back to School

Vegas Baby, Vegas!

When I turned 21 (no pun intended!), I went on a road trip from Oregon to Colorado via Utah’s canyon country with my then girlfriend. As far back as I can remember, I had liked games and gambling, and so for me one of the highlights of the trip was an inconvenient detour through Las Vegas now that I was finally old enough to gamble. Though I don’t  trust my 30+ year old memory of events, I doubt the idea of gambling with the odds against me in the hope of getting lucky appealed to me much, and I doubt I thought I could get an advantage over the casino, but the general idea of checking out the over-the-top Las Vegas Strip and experiencing casino gambling somehow did, despite some part of me also thinking Las Vegas was a gaudy and immoral blemish on a beautiful natural landscape.

On the way we stopped at a used bookstore and I looked for books on how to gamble in casinos. I don’t remember if I already knew about card counting and was looking for books on the topic but, in any event, they had a used copy of the 2^nd edition of Beat the Dealer by Edward O. Thorp, which—lucky me—is the first (reliable) book on “Basic Strategy” and “card counting”. It is foundational to all subsequent math-based resources on blackjack strategy.

Card Counting

Card counting is a system for keeping track of the ratio of good to bad cards remaining to be played in casino blackjack, adjusting one’s playing strategy depending on that ratio, and—when that ratio is sufficiently good for the player, increasing one’s bets. Even though the casino usually has an advantage in blackjack, occasionally the ratio of good to bad cards changes enough before the dealer—a casino employee—shuffles the cards, and the player can have an advantage. By betting the minimum during those usual times when the casino has an advantage and betting as much as possible during those uncommon times when the player has an advantage, sufficiently skilled card counters, with the right table conditions, can get a long term positive expected return.

Blackjack Rules and Basic Strategy

Knowing the best way to play each hand when not counting cards—known as Basic Strategy—is essential, not just to successful card counting but also to minimizing the casino’s advantage for non-card counters. Players are competing against the dealer, and the dealer must play by set rules. Those rules often vary depending on the country and region, the particular casino within that region, and even specific tables within a casino, but the gist is that the goal of blackjack is to get more points than the dealer without getting more than 21 points total (getting more than 21 points is called busting and it results in an automatic loss for the player). Points correspond to the card values on a normal 52-card deck, except that face cards are all worth 10 and aces can be worth 1 or 11, depending on what makes a better hand. The dealer must take a card (hit) if their hand total is less than 17 and stop taking cards (stand) once their total reaches 17 or more. If the dealer busts, they must pay all remaining players. An ace and a ten-value card as the first two cards is called a blackjack. It is special in that it beats other types of 21 and it pays out 3 to 2 on the players’ bet (as long as the dealer doesn’t also have a blackjack. Unlike common home versions of blackjack, the dealer does not win ties: if the dealer and player have the same total, including if they both have blackjacks, it is called a push and the original bet is returned to the player.

While the dealer has to play every hand the same way, players do not. They also have several additional options that would seem to give them an advantage over the dealer. Players can hit or stand as they wish and can make other playing choices that the dealer cannot make, such as splitting, which involves separating any two identical-vlaue first cards into two separate hands and matching the original bet for the new second hand, or doubling down, which involves doubling one’s original bet after seeing one’s original two cards and the dealer’s first card, but then being forced to take one and only one additional card. Players make all of their choices after seeing the dealer’s first card, which is dealt face up. They do not see any other dealer cards until after all players have finished their turns, at which point the dealer plays his or her hand. For a more detailed description of blackjack rules, see the blackjack rules page on my companion website.

Given this wide variety of playing choices relative to the limited options for the dealer, the bonus payout for blackjacks, and the fact that the dealer does not win ties, why does the casino have an advantage at all, even against non-card counters? Even if the player just copied the dealer’s strategy (hitting values less than 17 and standing with 17 or more), shouldn’t the 3:2 payout for blackjacks give the player an advantage? The source of the casino’s advantage is one tie that is not a push: when players bust, the casino takes that bet and removes the player’s cards immediately, before the dealer plays. Given the blackjack rules where I played, dealer’s will bus more than 28% of the time, but they’ll win all of those times if the player busts first. That advantage for mutual busts is high enough that players who mimic the dealer have a disadvantage of about 6%. That’s far better than the approximate 28% advantage the casino would have if blackjack players chose randomly when making each type of possible decision1, but it’s far worse than the advantage the casino would have if players used perfect Basic Strategy, optimally varying when they hit, stand, split, or double down, depending on their cards and the dealer’s exposed card. Perfect basic strategy in casinos where I played generally reduced the casino’s advantage to 0.4-0.5% without any card counting at all. That is, for each $10 bet, the casino would only win 4 or 5 cents on average, over the long run.

Card Counting is Simpler Than You Think

Contrary to common depictions of card counting (such as Dustan Hoffman’s character in Rain Man), successful card counting does not rely on high-level math or exceptional memory. Instead, card counters do simple arithmetic, adding and subtracting small numbers, often just ones, to get a “running count” that is either positive or negative, and sometimes adjusting that count by multiplying by some fraction to adjust for the number of desk remaining (arriving at the “true count”). When that true count gets sufficiently high, card counters have a positive expected return and increase their bets. The main demands on memory are limited to memorization of the basic strategy given the particular casino rules, and of how to deviate from that strategy, depending on the count. But that memory comes from hours of training and practice that moves those rules into long-term memory before stepping foot in the casino, something that anyone able to get a college degree could do with enough practice. Indeed, the most brilliant mathematician on the planet could not calculate the basic strategy on the fly, much less adjust that basic strategy given which cards have been removed from the deck and evaluate the impact of those changes on expected return. The first statistically accurate Basic Strategy was published in 1956 and was limited to a specific set of casino rules, and it still had minor errors that were only identified after Monte Carlo simulations—repeatedly sampling particular card combinations millions of times to assess outcome probabilities—made it possible to check the math. This is where Thorpe’s 2^nd Edition of Beat the Dealer, and his collaboration with IBM employee Julian Braun, were so important. Card counters do not calculate probabilities at all. They just use simple heuristics (rules of thumb, such as adding or subtracting ones based on whether the card is good or bad for the player). Perfect assessment of the impact of all cards on outcome probabilities and optimal playing strategy still requires a computer.

Card counting strategies vary widely in terms of how sophisticated or complicated they are, balancing the limitations of human cognitive ability with the potential gain from using more complicated systems, a theme which has turned out to be central to the history of behavioral decision theory as well. The early trend was to focus on maximizing ideal expected return, and therefore the complexity of the card counting system, but later methods have focused on maximizing simplicity so as to limit mistakes, trying to limit any required real-time calculations and choices so players can spend more time faking it (looking like they’re not counting cards).

Card Counting is a Dumb Way to Make a Living

Part of the reason simple strategies are so important is that it is important not to look like a card counter, because once a casino suspects a player of counting cards, it’s easy to put a stop to it. In Nevada and the Czech Republic, casinos can simply kick players they suspect of counting cards out of the casino. In Atlantic City, they can’t kick players out, but they can change how they manage the table. For example, they can shuffle more often or limit the betting spread between maximum and minimum best or require all players to enter the table only immediately after the decks have been shuffled. All of these changes minimize the potential benefits of counting cards.

As such, card counters put a lot of effort into looking like they’re not counting cards, including doing things like occasionally drinking alcohol, sometimes betting large off the top of the deck, sometimes violating the Basic Strategy before any cards have been removed from play, and trying to make keeping track of the count sufficiently automatic that they can carry on conversations and look away from the table even while the cards are being dealt. Along with the potential for human error, all of those efforts masking card counting reduces the theoretical expected return from mathematical models, calling into question how readily practiced card counters—if not using other methods—actually play with a long-term advantage.

On top of that, as with casino gambling more generally, there is large degree in variability in outcomes just due to the vagaries of chance. Just as many gamblers can spend a week in Las Vegas and leave with a big win, many good card counters can spend a week counting cards and still lose a lot of money. A simulated example taken from Stanford Wong’s wonderful book on card counting, Professional Blackjack (1994), wonderfully quantifies the impact of variance. A card counter using one widely used system (“hi-lo”), without making any errors and spreading bets between $10 and $100, can expect to win $16 per hour on average but with a standard deviation of $415 per hour (meaning that about one third of the time, players will lose more than $399 per hour or win more than $431 in an hour). As such, many skilled but naïve card counters would lose an entire betting bankroll of thousands of dollars despite using a system with a positive long-term expected return.

Given the bankroll requirements, the unreliability of income, the amount of practice and time it takes to become a sufficient card counter, the unknown cost to expected return because of both human error and the need to not look like a card counter, and the somewhat limited expected return ($16/hour) even with perfect play and an impressive betting spread of 10-1 (which would not be allowed in most casinos remotely suspicious that someone is a skilled card counter), card counting is a lousy way to make a living. The vast majority of card counters do it because they’re naïve or because they’re motivated by other things besides earning a good living. I had a lot of fun counting cards, imagining myself up against the powerful casino with their mob roots, having to put on a show, and facing both financial risk and the fear of being found out and perhaps roughed up. The adrenaline, dopamine, and serotonin mixing around in my brain alongside large wins and losses no doubt added to that fun (and to the pain). But it’s clear to me now that making a good living was never in the cards.

Casinos make it possible to count cards by offering single-deck games and by not shuffling after every hand even though institutionalizing those changes and making card counting impossible would be easy. They do that because they make a heck of a lot of money from the prospect of profitable card counting which brings so many people to the table, along with the reality that very few of the people attracted by that prospect actually know Basic Strategy, much less how to count cards.

This is putting aside the non-trivial fact of problem and pathological gambling. Casinos are unhealthy environments, crafted in large part for their ability keep as much of the money from their patrons wallets as possible, while also keeping those patrons coming back repeatedly. The long-term impact on impulse control and well-being that comes with repeated wins and losses with games designed over generations of experimentation and refinement to condition players to stay as long as possible and come back repeatedly is not something to take lightly, and it has ruined many lives.

But I Didn’t Know That at the Time

I bought Beat the Dealer within a day or so of reaching Las Vegas. With limited time to learn the Basic Strategy and no time to practice, I learned the simplest and weakest card counting strategy in the book, which essentially amounted to (a) learning Basic Strategy for the tables I would play at by heart (though most Las Vegas casinos are happy to let players use Basic Strategy cards at the table, and many sell credit-card sized Basic Strategy cards in their casino bookstores or even give them away at the table), (b) finding a casino with a low minimum bet and single-deck blackjack (at the time I was able to find a single-deck game with $1 minimum bets on the Strip, but such low minimums with single-deck games may not be available anywhere but in downtown Las Vegas today, and they may not be available anywhere), (c) betting the minimum amount until all four 5s were dealt out of the deck before the next shuffle (a frustratingly uncommon event), and (d), betting as much as I could bear to risk when that happened (which was probably about $10 per hand). I ended up getting lucky and winning $50 during the couple of hours we spent in Las Vegas. I didn’t know it at the time, but I just as well might have lost that much or a lot more using the strategy I was using, and I might have easily won as much or more than that without counting cards at all. But winning $50 was exciting for me, I attributed it to my card-counting prowess, and I was hooked.

Over the next few years I studied increasingly advanced card counting strategies and went back to Vegas many times, culminating in the Uston Advanced Point Count system from Ken Uston’s influential, practical, and well-written book, Million Dollar Blackjack. That strategy assigns different values depending on the cards impact on the player’s expected return (each 10 removed from play is counted as minus 3, each 5 is counted as plus 3, etc.), it keeps a separate count of aces using the feet for making decisions about blackjack likelihoods, it adjusts the running count based on the number of half-decks remaining (to get the true count), and it requires memorization of 159 different deviations from the Basic Strategy depending on the player’s hand, the dealer’s upcard, and the count. Uston had a wonderful section on training, and I spent hundreds of hours practicing and memorizing so I could count through multiple decks in his prescribed times and make playing decisions based on counts without conscious cognitive effort.

I found “investors” (friends of my parents who were intrigued and convinced by my understanding of the math of the game and my ability to convey the ideas behind card counting, or maybe who just wanted to show support to their friend’s naive kid). They invested $1000, what I thought at the time was a meaningful bankroll along with a few hundred of my own dollars, and I went to back to Vegas and made what I remember to be a 50% (or 150%?) return on their investment. I didn’t know it then, but this was again primarily a result of luck. Given my bet size and range, I had been more likely to lose my entire bankroll over the course of the few days I was there than not. I also got banned from my first casino, which felt like a victory. I continued to read books and practice and visit Las Vegas, but as I learned more, I also realized I would need a bigger bankroll than I was comfortable risking for a lower expected return than the work could warrant, unless I wanted to join a card-counting team, which came with its own risks (see Uston’s book for more on teams and why they can be so effective; or for a romanticized Hollywood version about a famous team from MIT, see the (IMO unwatchable) film 21.

Prague, Czech Republic

I no longer remember how much I had won or lost playing blackjack by the time I decided it was not worth the time and effort, but my opportunities to play were few and far between and the lack of regular practice meant that I lost the ability to count cards well enough to hope for an advantage. Until I moved to the Czech Republic after college, that is. I moved there in December 1993, not long after graduating from college and about four years after the fall of the Berlin Wall and the Velvet Revolution that marked the peaceful end of communism in Czechoslovakia, and about a year and a half after the Velvet Divorce (the peaceful separation of the Czech and Slovak Republics into two separate countries). I knew there were opportunities to teach English in Prague and I was excited by the prospect of spending an extended period in a different culture and especially by this region where the changes were so new and so clearly transient and in a city that was beautiful, vibrant, friendly, and cheap.

Casinos offering blackjack were plentiful in Prague and the rules were similar to the best rules in Las Vegas (for 6-deck shoes). I decided to get back in to card counting, recognizing by then that it wouldn’t be a reliable or good source of income, but also enjoying the fun of it and believing the small advantage I had at least meant I was more likely to win than lose.

By then, I had learned enough to know that less is sometimes more in blackjack. There were newer systems that emphasized simplicity and minimizing errors over systems that flaunted their complexity and emphasized maximizing expected returns under idealized conditions. During the first several months in Prague, I played a lot of casino blackjack. But I also lost more than I won, more than I could afford, and enough to make me lose interest in the game (and doubt whether the casinos in Prague, with few tables, few patrons, and known to be run by organized crime) had the same incentives to run honest games as the casinos in Las Vegas.

Culture and Experience Promote Systematic Deviations From Basic Strategy

Several interesting observations (to me!) about casino blackjack players follow. First, despite the widely available and well-established mathematical credibility of the basic strategy, nearly none of the experienced casino blackjack players I met actually played according to Basic Strategy. This was true even on the Las Vegas Strip where Basic Strategy is available in casino bookstores, and despite the fact that it would probably take fewer than 10 minutes for most experienced blackjack players to learn how it differed from their existing strategy.

Second, deviations from Basic Strategy were not random: experienced blackjack players on the Las Vegas Strip and in Prague violated basic strategy in similar ways to one another and with shared conviction that those violations were correct, suggesting that through some unclear process of learning from the game or from one another they had developed similar false beliefs about how best to play.

Third, this wasn’t due to lack of awareness that there was a Basic Strategy. Experienced blackjack players know about card counting and the basic strategy. Many of them claim to play according to Basic Strategy and claim to count cards, but they do not play according to the actual Basic Strategy and they do not count cards in a way that could potentially give them a positive expected return (or even know enough about the math or theory behind it to understand that). Others knew that their playing strategy was different from Basic Strategy but believed Basic Strategy was simply wrong.

Fourth, experienced blackjack players in Prague and on the Las Vegas Strip deviated from basic strategy in systematically different ways. Players in Prague were far more concerned about how other players at the table played, and they were noticeably less likely to risk busting than players on the Las Vegas strip. In both locations, players would get sufficiently annoyed at me for how I was playing—even when I playing according to Basic Strategy without counting cards—that they would voice their annoyance and sometimes leave the table in frustration (despite my knowledge that the impact of other players’ choices on my own success was random), but they did so far more in the Czech Republic than in Las Vegas, and they did so for different reasons in Las Vegas than in Prague.

For what to me seemed to be a purely mathematical game with a statistically determinable best way to play each hand (assuming the goal is to maximize expected value, which eventually became obvious is not the goal for most blackjack players), it seemed extremely weird that experience playing would lead to shared false convictions about how to play that would differ from the widely available theory and compelling math that contradicted those shared beliefs. The fact that these shared false convictions differed across cultures seemed a clear indication that the learned suboptimal play was more than just a result of built-in characteristics of the human cognitive system (heuristics and biases) and that they reflected aspects of culture and social learning.

Back to School

Those observations, along with a variety of other experiences with casino gamblers and their strategies and beliefs about winning, contributed to a broader interest in the topic. It eventually led me to spend two years in casinos in northwestern Indiana (riverboats), in the Las Vegas area (as both a blackjack dealer and a player), and in Prague, researching the topic for a PhD in psychology. And it will provide most of the backbone for this Substack. The next newsletter will describe that dissertation research, along with its relevance to psychological science.

Subscribe now

Casino gambling has the characteristic of seeming eminently idiotic to people who would not consider it, but perfectly reasonable to many (though not all) of the people who do it. It points to ways in which the gamblers themselves are misguided because of their physical and social environment along with their faulty reasoning and their ignorance of the facts. But it also points to how outgroups looking in at gamblers from the outside are misguided because of their simplistic understanding of gamblers’ environments, values, and beliefs and their readiness to attribute judgments of truth and rationality that are a better reflection of their own limited experience and knowledge than they are of the limited experience and knowledge of the gamblers themselves. Because casino games are much simpler and more mathematical than most of the complex, real world environments in which social groups find themselves, focusing on those games allows for compelling analyses as to when and how both sides in this ingroup (casino gambler)–outgroup (non-casino gambler) comparison get it right and wrong. I hope and expect that a close look at this particular dynamic will provide insight and humility when assessing the rationality of judgments and beliefs, whether those assessments are of ourselves, of members of ingroups with whom we affiliate, or of members of outgroups against whom we differentiate ourselves, outgroups who often seem troublingly and unjustifiably off base.

Subscribe now

Despite the value that the science of judgment and decision making brings to our understanding of casino gamblers’ strategies and beliefs about winning—discussed in the previous post—there remains a long-running concern as to the extent to which that science generalizes beyond the lab to non-experimental contexts, including to casino gamblers. In more experimental, scientific language, these are questions about external and ecological validity.

Part of the criticism focuses on unrealistic simplifying assumptions in J/DM science and theory. Specifically:

• The use of abstract mathematical models to develop normative models of rational choice, such as subjective expected utility theory;

• And the use of carefully controlled experimental settings with hypothetical, intentionally content-free gambles, and novice decision makers to explain why people deviate from those normative models (such as the long list of putatively inbuilt, species-general, cognitive “heuristics and biases”).

Simplifying models and carefully controlled experiments are central to what most of us think of as good science and theory, but they risk defining the most important factors out of the equation.

Another part of the criticism emphasizes the complexity and richness of real-world decision contexts and how important that is to understanding processes in judgments and decision making. It points the importance of:

• decision makers’ experience and learning within specific decision domains;

• environments that are intentionally designed and that co-evolve with the decision makers, in part as a consequence of how those environments impact judgments and decisions;

• the rich social–cultural context and content, including multi-generational evolving practices, values, and beliefs, that are domain specific and that impact processes and outcomes in J/DM.

Casino gambling—with its close association to the simplifying assumptions in normative and descriptive models of decisions making under risk and uncertainty—is a wonderful domain by which to explore whether experimental findings do or do not generalize. Most importantly, the study of casino gambling in the wild provides opportunities to uncover important insights into processes in judgment and decision making that have been missed when studying J/DM in more controlled, experimental settings.

Subscribe now

Gambling has a long and rich relationship with the science and theory of judgment and decision making:

• Probability theory. It was a question from a gambler about how to divide the winnings in a prematurely-ended game of dice that is widely credited with Pascal and Fermat’s development of the early mathematics behind probability theory in the 17th century.

• Expected Utility Theory. In the 18th century, a simple gamble with a coin flip was central to the development of expected utility theory and the idea of marginal utility.

• Rational Choice Theory. In the 1940s and 50s that utility theory and the gambling metaphor it depends on was extended to contemporary rational choice theory (refined expected utility theory, game theory, and subjective expected utility theory) which forms the basis of economic models of decision making.

• Descriptive Decision Theory: Even among behavioral economists and psychologists who recognize that people systematically violate economic models of rational choice, those gambling-metaphor-informed normative models continue to serve as standards against which actual decisions are evaluated and alternative models are established.

• The experimental study of behavioral decision making. Finally, simple gambles have been a centerpiece in controlled experimental studies by cognitive psychologists and behavioral economists seeking to test the extent to which decisions correspond to rational choice theory and to the development of theories that explain why people deviate from those normative models, including, but not limited to prospect theory and some of the work in the “heuristics and biases” tradition.

Given this deep connection between gambling and the science and theory of judgment and decision making, it should not be surprising that this research and theory has a lot to say about gamblers strategies, whether or not they work, and why gamblers often use strategies that do not work and endorse beliefs that are not true. The essays in this Substack will regularly refer to that research and theory.

Subscribe now

This Substack is about four inter-related themes each of which will be described in a separate brief essay.

The first of these themes—the one described here—is the sometimes bizarre, often compelling beliefs about winning that propel so many people to risk their money gambling in casinos, despite the undeniable fact that most casino gamblers lose.

• What are those strategies and beliefs? Spoiler: there are a lot of them and they change with experience, differ across games, and vary widely across individuals and locations.

• Do the strategies work and are the beliefs true? Spoiler: sometimes yes and sometimes no but, in both cases, for reasons that non-gamblers would usually not expect.

• Why do casino gamblers endorse seemingly false beliefs and substandard strategies? Spoiler: the reason is usually not primarily because gamblers are poor reasoners or rely on readily dismissible evidence, and even though gamblers’ strategies are often sub-optimal and their beliefs are often false, the strategies work and the beliefs are true more often than most non-gamblers would expect, including decision scientists and clinical psychologists who think a lot about gambling decision making.

Every essay in this Substack will in some way or other be about at least one of the above topics.