Latticework of Mental Models: Gambler’s Fallacy

Let’s start with a small riddle.

A man, probably a statistician, believes that his next child will be a girl since his wife has already borne him three sons.

Do you find his argument convincing?

The argument intuitively doesn’t feel right. Isn’t it? But why?

We’ll circle back to this puzzle but before that let me indulge you in another interesting thought experiment.

Imagine yourself as a spectator in a coin flipping tournament. You notice that in one of the plays, the coin has landed on heads for 5 consecutive flips. If you were given an opportunity to bet on the next flip, would you bet on heads or tails?

I know you’re a value investor and don’t believe in speculating or gambling away your hard earned money on frivolous coin flipping tournaments, but this being a thought experiment I would request you to play along.

So what’s your answer?

The basic concepts of probability tells us that for random events like coin flips, both the head or tail are equal likely. In other words the probability of a head and a tail are both 1/2 (0.5).

So using my elementary knowledge of probability, I would reason that the universe will try to balance out the too many heads.

When I use this argument to put my bet on tails, I am falling for a bias called Gambler’s Fallacy.

The Gambler’s Fallacy is the mistaken belief that, if something happens more frequently than normal during some period, it will happen less frequently in the future, or that, if something happens less frequently than normal during some period, it will happen more frequently in the future (presumably as a means of balancing nature).
(source: Wikipedia)

That explains why our statistician friend’s argument is flawed. The gender of the fourth child is causally unrelated to any preceding chance events or series of such events. His chances of having a daughter are no better than 1 in 2 i.e., 50-50.

With independent events (the gender of kids, result of toss using a fair coin, etc.) there is no harmonising force at work. The coin doesn’t know that it had landed heads in the last 5 tosses.

The most famous example of the gambler’s fallacy occurred in a game of roulette at the Monte Carlo Casino in 1913 when the ball fell in black 26 times in a row. This was an extremely uncommon occurrence, although no more or less common than any of the other 67,108,863 sequences of 26 red or black. Gamblers lost millions of francs betting against black, reasoning incorrectly that the streak was causing an “imbalance” in the randomness of the wheel, and that it had to be followed by a long streak of red. (source: Wikipedia)

So why is it called a gambler’s fallacy? Because it’s rampant among gamblers and speculators.

There was a guy who claimed that he had a scientific way of playing the lottery. He diligently maintained a spreadsheet of winning numbers and would bet on those numbers which had appeared the least. Alas! another victim of Gambler’s fallacy.

Now if a coin (a fair one) has equal probability (50:50) of turning heads or tails then why is it fallacious to expect a tail after 5 consecutive heads? That’s a fair question to ask.

To answer that, let me take help from Daniel Kahneman. In his book, Thinking Fast and Slow, Danny writes …

People expect that a sequence of events generated by a random process [coin toss] will represent the essential characteristics [equal probability of head and tail] of that process even when the sequence is short [few tosses]. In considering tosses of a coin for heads or tails, for example, people regard the sequence H-T-H- T-T-H to be more likely than the sequence H-H-H-T-T-T, which does not appear random, and also more likely than the sequence H-H-H-H-T-H, which does not represent the fairness of the coin. Thus, people expect that the essential characteristics of the process will be represented, not only globally in the entire sequence, but also locally in each of its parts. A locally representative sequence [the sequence of 5 heads which you observed], however, deviates systematically from chance expectation: it contains too many alternations and too few runs.

Another consequence of the belief in local representativeness is the well-known gambler’s fallacy. After observing a long run of red on the roulette wheel, for example, most people erroneously believe that black is now due, presumably because the occurrence of black will result in a more representative sequence than the occurrence of an additional red. Chance is commonly viewed as a self-correcting process in which a deviation in one direction induces a deviation in the opposite direction to restore the equilibrium. In fact, deviations are not “corrected” as a chance process unfolds, they are merely diluted.

Kahneman’s insights are remarkable. So if you didn’t understand the above two paragraphs, please read them slowly and then re-read them.

Now if you were given an opportunity to bet on many such tosses, say 100 tosses, what would be your strategy for betting? Again the assumption being that it’s a fair coin (with no specific bias for either head or tail) and with the knowledge that probabilities are still 50:50.

Kahneman explains this using an anecdote about famous economist Paul Samuelson. He writes …

The great Paul Samuelson—a giant among the economists of the twentieth century—famously asked a friend whether he would accept a gamble on the toss of a coin in which he could lose $100 or win $200. His friend responded, “I won’t bet because I would feel the $100 loss more than the $200 gain. But I’ll take you on if you promise to let me make 100 such bets.” Unless you are a decision theorist, you probably share the intuition of Samuelson’s friend, that playing a very favorable but risky gamble multiple times reduces the subjective risk.

Samuelson’s friend was pretty smart. He understood that Gambler’s fallacy arises out of a belief in a law of small numbers, or the erroneous belief that small samples must be representative of the larger population. Hence he was willing to bet on the aggregate outcome of bigger sample size than a single outcome.

You probably noticed that I have been mentioning the use of a fair coin for our tosses. A fair coin ensures the pre-condition for a gambler’s fallacy to hold true, i.e. independent events.

Gambler’s Fallacy – Reversed!

If I told you that in the same coin tossing tournament, there is play where the coin has turned heads for 50 consecutive tosses. What would you bet on for the 51st coin toss?

The probability of that happening, using a fair coin, is 1 in a 1100 trillion but there’s also a chance that the coin is biased. A biased coin or an imperfect roulette wheel can also lead to outcomes like 5 consecutive heads but for totally different reasons. In these cases the gambler’s fallacy might superficially seem to apply, when it actually does not.

If you suspect that you’re dealing with a biased coin or a loaded dice, it’s reasonable to expect that long streak of skewed outcomes are not due to randomness. But sometimes human brain takes it too far in the other direction when a belief in the idea of hot hand takes root.

This is the reverse of gambler’s fallacy, also known as hot hand fallacy. It originates from basketball where players who scored several times in a row are believed to have a hot hand, i.e. are more likely to score at their next attempt.

In cricket, players are often declared to be “in-form” and are expected to consistently perform above average until the form lasts. Perhaps a hot-bat fallacy! 🙂

At first the idea of gambler’s fallacy might look paradoxical to the mental model of mean reversion but it’s not. In his book, The Art of Thinking Clearly, Rolf Dobelli explains the subtle difference between the two.

…if you are experiencing record cold where you live, it is likely that the temperature will return to normal values over the next few days, If the weather functioned like a casino, there would be 50% chance that the temperature would rise and a 50% chance that it would drop. But the weather is not like a casino. Complex feedback mechanisms in the atmosphere ensures that extremes balance themselves out.

So, take a closer look at the independent and interdependent events around you. Purely independent events really only exist at the casino, in the lottery and in theory. In real life, in the financial markets and in business, with the weather and your health, events are often interrelated.

Which means a domain prone to truly random and independent events, like a casino, is more prone to gambler’s fallacy. However, in systems which behave like complex adaptive systems i.e. prone to self correction, both mean reversion (negative feedback loops) and extreme outcomes (positive feedback loops) are a possibility.

In Investing

In stock markets how many times have we heard, ‘whatever goes up must come down’ or ‘how much more can it drop?’.

These beliefs are a manifestation of gambler’s fallacy in investor’s behaviour. Vishal is fond of quoting the example –

A stock that falls 95%, first fell 90% and from that level fell another 50%.

When Warren Buffett said, “turnarounds seldom turn” he probably had gambler’s fallacy in mind.

Expecting that the future events will smoothen out the stock prices, just because it has been falling for ‘n’ continuous trading days, is an irrational way of thinking.

Sometimes investors abruptly change their belief from gambler’s fallacy to hot-hand fallacy. When their patience runs out, having lost money by betting on outcomes which are supposedly “long overdue”, they tend to go overboard in assuming that a stock has come to its “form” and will continue it’s trend. They bet heavily on the new “trend” and right then the tide turns and the roaring market makes a mincemeat out of those unsuspecting hot-hand believers.

Conclusion

Typically humans intuitively find it hard to deal with randomness as a result we tend to put a tremendous amount of weight on previous events, believing that they’ll somehow influence future outcomes. Gambler’s fallacy dictates that the idea of “balancing force of the universe” is a baloney.

Today’s idea requires you to understand the basics of probability. If you aren’t acquainted with the basic probability concepts, you need to learn it else you might end up like the proverbial (as described by Charlie Munger) single legged man in an ass-kicking contest.

You must have noticed that I have referred to quite a few mental models which have appeared in this series before, and you might be feeling overwhelmed finding so many links.

Please understand that as you learn more ideas, you are bound to find more and more connections between those ideas. And that’s how a useful worldview is created – by constructing a framework of interconnected ideas.

That’s what a real latticework looks like. Building such a latticework serves the purpose of saving us from biggest errors in thinking and decision making – in our private lives, at work or in business.

Take care and keep learning.