Gambler’s Ruin

Aaron Brown demonstrates that ruination can lead to enlightenment; tough but so true

The term “gambler’s ruin” is used for a number of statistical ideas whose common denominator is predicting the eventual outcome of a series of repeated bets. There are some beautiful and counterintuitive results in which bets that seem unappealing individually lead inevitably to good outcomes, and bets that seem too attractive to refuse, lead just as inexorably to disaster.

The main gambler’s ruin results, from most to least obvious, are:

• If a bet has an absorbing state and you make the bet often enough, you will reach the absorbing state. Technically we need a condition that the
probability of the absorbing state is non-zero and does not decrease too quickly, but I will ignore such technicalities for this column.
• If a bet has a negative expected return and you bet enough, you will be ruined.

The next two results apply even to positive expected return bets:

• If you raise your bet proportionate to your winnings, but do not cut it when you lose, you will eventually be ruined.
• If you bet more than the Kelly amount, you will eventually be ruined.

Each of these results can be applied in either forward or inverse form. Sometimes we have good information about the statistical properties of individual bets, and can use it to predict the eventual outcomes for people who make the bets repeatedly.

In other cases, we have good data on eventual outcomes, and can use it to deduce properties of the individual bets. It’s a good statistical principle to
study what you want to know about – individual bets if you want to know individual bet properties, or eventual outcomes if you want to predict eventual outcomes. But sometimes we have better information on the thing we are less interested in, and that is when gambler’s ruin can be helpful.

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