If It Looks Like a Duck... It Might Not Be a Duck (Part 1)
What looks like the gambler's fallacy is often something else
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
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.
If the description of the gambler’s fallacy does not sound like a fallacy, then you are not alone. Many people do not believe it’s a fallacy. That’s part of the reason it is a well-known fallacy: it is widespread. In that case, please start with this earlier essay about the law of large numbers. It serves as background on the series of essays on the gambler’s fallacy and explains why the law of large numbers does not imply that chance must self-correct after an unlikely streak. Then continue from that essay on to the first essay in the-gambler’s-fallacy series. That essay defines and describes just what is required for belief in the gambler’s fallacy.
It is important to note that while these incorrect beliefs do not tend to matter with respect to which outcome is gambled on, they still do matter with respect to expected return and the variability of that return, because they influence how much gamblers risk on each bet and how many bets they make. But that’s for a later essay in this series.
“How do you know they do this?” I asked.
“Because I’ve had many conversations with different casino managers about it.”
Of course, it would be even better to know which individual casinos have the best payouts, but casino-specific reports are not made public.
The loses would be unlikely from the perspective of the researcher looking for behavior that fits the gambler’s fallacy, but not unlikely given the slot player’s inside information. Similarly, the belief that the gambler is “due” would only be from the perspective of the observer looking for the gambler’s fallacy. In fact, the slot player with this insider knowledge played more often and bet more money because they believed the payout percentages had been changed on the machines, nothing to do with belief in the gambler’s fallacy.