Behaviouronomics: How to Win Russian Roulette

If you’ve been a reader of Safalniveshak for sometime you’d already be familiar with the game of Russian Roulette. For those who aren’t, here’s a quick explanation.

In fact I’ll let Nassim Taleb, author of Fooled By Randomness, do the explaining –

Imagine you are offered $10 million to play Russian roulette, i.e., to put a revolver containing one bullet in the six available chambers to your head and pull the trigger. Each realisation would count as one history, for a total of six possible histories of equal probabilities. Five out of these six histories would lead to enrichment; one would lead to a statistic, that is, an obituary with an embarrassing (but certainly original) cause of death.

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If you were to do the standard cost-benefit analysis, it tells you that your expected payoff is $8.3 million. Pretty nice, isn’t it?

But no one, even in the moments of insanity, will accept this kind of play because you know the downside is unacceptable. Unfortunately, in real world, the downside is not as clearly visible and our vision is often blurred by the average payoff numbers. In asymmetric games like Russian Roulette, averages conceal far more than they reveal.

Now there are two ways to imagine the outcome of Russian Roulette kind of games.

Let’s imagine a casino with normal roulette tables, i.e. no guns involved. The roulette table in our casino offers to triple your money if you win and the probability of winning is 90%.

The casino offers two ways to bet and you can choose only one of them. Plus, each day you can make a maximum of only 10 bets.

In type-1 bet, you have to choose one table and make all your bets on the same table for that day. Another condition is that you have to bet all your money in each bet. Which means, if you choose to start with $100 and type-1 bet, assuming you win all your 10 bets, you walk out with whopping $5.9 million.

In type-2 bet, you can split your money equally and bet on 10 tables simultaneously. In this case, starting with $100, assuming you win all your 10 bets, you end up with $300.

It’s clear that type-1 is far more lucrative, only until we’re looking at the happy scenario, i.e. you winning all 10 bets. What if you win only 9 bets and lose just one of them. Now, you see the catch, don’t you?

In type-1 bet, losing just one bet turns you into a pouper. Since you lose all, you can’t bet anymore and finish the day with zero. In contrast, losing just one 1 bet in type-2 means you still end the day with $270.

Same payoff, same instrument, same number of bets, but widely different risk and returns. What’s going on?

The type-1 bet is called the time scenario and type-2 is called the ensemble scenario. Russian roulette is the example of time scenario. And when you say that probability of winning in the Russian roulette is 83.33%, you’re thinking about the ensemble scenario.

In simple words, the probabilities of success from the collection of people do not apply to one person.

Taylor Pearson explains —

Any actor taking part in a system can be defined as either ergodic or non-ergodic.

In an ergodic scenario, the average outcome of the group is the same as the average outcome of the individual over time. An example of an ergodic systems would be the outcomes of a coin toss (heads/tails). If 100 people flip a coin once or 1 person flips a coin 100 times, you get the same outcome. (Though the consequences of those outcomes (e.g. win/lose money) are typically not ergodic)!

In a non-ergodic system, the individual, over time, does not get the average outcome of the group. This is what we saw in our gambling thought experiment.

A way to identify an ergodic situation is to ask do I get the same result if I:

look at one individual’s trajectory across time2
look at a bunch of individual’s trajectories at a single point in time

If yes: ergodic.

If not: non-ergodic.

Why is this idea of ergodicity important?

Because while thinking about probabilistic outcomes, we often forget that two options with equal expected payoff can have widely different risks.

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