The Gambler’s Fallacy

Have you ever had a series of perfect days where everything seems to be going well for you, so you start growing suspicious of yourself? One might think, “Hm, the next day has to go terribly, right?”. Did you know there’s a name to this feeling? Let me introduce you to “The Gambler’s Fallacy” — the belief that if a particular event occurs more frequently than usual during the past, it is less likely to happen in the future.

It is also called the Monte Carlo fallacy after the belief was popularized during a game of roulette at the Monte Carlo casino on August 18th, 1913, when the ball fell into black TWENTY-SIX times in a row! If you’re unaware, roulette is a gambling game in which a ball is dropped onto a revolving wheel with numbered compartments, and the players bet on the number at which the ball comes to rest. The compartments are also colored red and black alternatively. Players can bet on numbers or color. Gamblers who bet against black that day lost millions of francs because of this highly uncommon occurrence.

Of course, it is a “fallacy” — a mistaken belief, but when has that stopped people from blindly believing in it? Amos Tversky, a cognitive and mathematical psychologist, and Daniel Kahneman, a psychologist, and economist, have explained the psychology behind this belief as a “cognitive bias produced by a psychological heuristic called the representative heuristic.” In layman’s terms, people judge probability by the events they have experienced before. We all know that the sum of all possibilities of an event is unity. Hence when one outcome of the event occurs consecutively, the human brain automatically expects the other outcome(s) to appear next to balance the probability out and add up to one!

Let us delve deeper into The Gambler’s Fallacy by looking at it from the perspective of statistics. The probability of getting a head while flipping an unbiased coin once is 0.5. If the same coin is flipped five times in a row, the probability of getting a head in each turn is still 0.5 for that particular flip. Alright, let’s take a real-life situation of this exact scenario. You flip a coin five times. Four times it results in a head. The human brain automatically expects the fifth turn to be a tail to “balance out the total probability,” but the probability of a head or a tail appearing is the same, and it is 0.5. The occurrence of one coin flip does not affect the probabilities involved in the second flip because they’re both independent events. Thus if two events are independent, the occurrence of one event does not affect the occurrence of the second! Thus the gambler’s fallacy has been debunked!

The gambler’s fallacy is no exception to the rule that every (or almost every) theorem in mathematics has an inverse — presenting the inverse gambler’s fallacy! Let’s say you walk into a casino and see the ball in a game of roulette fall into red twice consecutively. Your brain naturally concludes that the ball must have initially rolled into the black before falling into red sequentially. This is also called the “Retrospective Gambler’s Fallacy” — the existence of a bias for inference of unknown past events based upon known subsequent events!

One might wonder how all of these fallacies even came into being! Have you heard of “The law of small numbers”? This law states that if the sample size is large, every small change has an apparent shift in analyzing the events that occurred. Let’s bring our faithful old friend, the unbiased coin, back on the stage. The law of small numbers dictates that even if the coin is tossed a few times, it will equally land on heads and tails — which is incorrect! This law led to the erroneous belief we now know as “The Gambler’s Fallacy.”

Do you remember Amos Tversky and Daniel Kahneman? Yes, they’re the people who explained the origins of the Gambler’s fallacy. They’re also the same people who coined “The law of small numbers”! Kahneman gives a particular example in his book “Thinking, Fast and Slow.” He says that certain rural counties in the South USA had the lowest kidney cancer rate. What was unique about these counties — something about the rigorous hard work of farming or the free open air? He then studied counties with the highest kidney cancer rate, and guess what — they were also rural counties! Here we can infer that a rural lifestyle’s poverty and high-fat diet explain the high rates of kidney cancer. But we can’t have it both ways! It’s not sensible to attribute low and high cancer rates to rural lifestyles. The outliers in the high cancer areas appeared merely because the populations were so small. Small numbers mislead us and skew the results. This is the law of small numbers.

The gambler’s fallacy happens to have a competitor….which is also a fallacy. The Hot Hand Fallacy! It is the tendency to believe that if a particular event occurs more frequently than usual during the past, it is MORE likely to happen in the future.

This fallacy originates from basketball and is most commonly used for sports. If a player scores consecutively and has a “hot hand,” they are more likely to continue achieving and maintaining the streak in the future. Amos Tversky (yes, him, once again) claimed in a 1985 paper that basketball players have “hot hands” and that a player was more likely to make a successful shot if their previous attempt was successful. This fallacy is a cognitive, social bias, and one of its root causes is the inability to judge sequences appropriately. A belief in the hot hand fallacy affects a player’s perception of success. Everything I have mentioned here is a fallacy. Yet, almost every gambler believes in them, which keeps the casinos in Vegas running!

Enough of theory; let’s talk about the industry that exploits the gambler’s fallacy. Let me guess what you thought of — the casinos? Yes, you’re right, but let’s not be too predictable; it’s the video game industry. Have you heard of a specific genre of video games called “gacha games”?

The word “gacha” or “gachapon” originates from Japan and refers to the vending machines which dispense those tiny toys in capsules. It’s an onomatopoeic word where “gacha” is the sound that the crank on the device makes while “pon” is the sound of the capsule landing on the tray at the bottom.

A gacha game implements this mechanism. Players must “roll” several times before obtaining the character or item they want using in-game currency. In-game currency can be accepted by simply playing the game. However, when the player fails to get the object they wish for through the gacha, real-world money can be used to purchase in-game currency. And this is where the gambling part of it comes in.

Gacha games are incredibly addictive and can induce the player to spend tons of money on them just to obtain a particular character or item. Since they are technically games, many children are exposed to them and end up playing them and spending their parent’s money for a set of pixels on the screen. Gacha games are one of the most addictive kinds of microtransactions. Popular gacha games like Epic Seven and Arknights have millions of players around the globe and gain more and more revenue each day!

Most gacha games have something called the “pity system.” As the name suggests, the game takes “pity” on you after a certain number of rolls and gives you the character or item you desire. On the surface, this seems like a solution to a gambling addiction, but that is just the tip of the iceberg. In reality, the existence of a pity system further fuels the gambler’s fallacy. Let’s say the pity for a particular game is 90 rolls. A player has already rolled 60 times and is out of in-game currency. They think, “Just 30 more rolls!” swipe that credit card and spend 300 dollars. And the cycle continues. “Just ten more rolls!” another credit card swipe. These players fall prey to the gambler’s fallacy and delude themselves into thinking that they are the ones controlling the way the game works! Eventually, this turns out to be the first step to gambling addiction. Wherever gambling comes into play, the gambler’s fallacy will appear in one form or another!

Another industry that thrives off the gambler’s fallacy is the investing and finance industry. We often hear things like, “The market has been going up for the last four days, so it will definitely fall today!” when there’s actually equal probability for the market to fall or rise. The Inverse Gambler’s Fallacy has also made its way into investing.

Often, investors only tend to invest in the best-performing mutual funds of the previous year, assuming their good performance will continue in the current year. This is termed the practice of “chasing performance” and is generally frowned upon in the field.

Finally, the last fallacy — the retrospective gambler’s fallacy can also be seen in the investing arena. Let’s say we see a headline that says, “The GDP is down this quarter.” Most of us will interpret this as — the GDP was higher previously and is only down during this quarter. In reality, the GDP could have been higher previously and down in the current quarter, or it could’ve been down during both quarters. Thus, investors are advised to let their data and reasoning be the anchors for their decision-making, not their predictions!

The gambler’s fallacy is a natural psychological tendency — it’s tough to control! But if you still want to stop yourself from falling prey to this fallacy, the first step is to be aware of its existence. And if you’re reading this sentence, you’re already past that step!