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Gambler's Ruin

© Copyright 1999, Jim Loy

A phenomenon of probability called "gambler's ruin" is one of the things that keeps me from gambling. Essentially, gambler's ruin says that, if you play long enough, you will go bankrupt, and have to quit (as the house will not extend you credit). "Long enough" may be a very long time. It mainly depends on how much money you start with, how much you bet, and the odds of the game. Even with better than even odds, you will eventually go bankrupt, and have to quit. But, this may take very long, indeed.

Let's flip quarters, and see how this works. You flip one of your quarters, I will call "heads." If you win (tails) I pay you a quarter; if I win (heads) you pay me a quarter. We quit when one of us runs out of quarters. I start with a lot of quarters. You start with one quarter. The top row shows the number of flips. The left column shows your current amount of money. The other numbers are probabilities.

After seven flips, your probability of going broke (at some time during those seven flips) is .7265625. Your expected return, surprisingly (perhaps) is $.25, your original stake. You probably already knew that starting with only one quarter was dangerous. You would rather start with 2 quarters, or more. But, the same thing happens, just more slowly. You still go broke, eventually. Here is a similar table, for a starting stake of 50 cents:

With a stake of 50 cents, your chances of going broke within 7 flips is still less than 50% (and your expected return is still 50 cents, your initial stake). In the case of an initial stake of 25 cents, I never actually proved that you would eventually go broke. I just showed you that there is that tendency. It is somewhat less obvious that you go broke in this table. Let me assure you that you do. But, you go broke much more slowly. You probably are not going to stop at 7 flips. A computer simulation becomes very convincing. After many many flips, your chances of going broke get near 100%.

Much bigger initial stakes slow this process, quite a bit. Again, you eventually go broke (especially with a game with worse odds than flipping a coin). But, the process may take a very long time, maybe longer than you will ever play, maybe longer than you will live.

With a very very large initial stake, such as the casino has, you will also go broke. But, it may take thousands or millions of years. And they are playing a different game than you are, one with better odds.

Addendum:

Let's say you are offered a one time deal. We will roll one die (singular of dice), and if it comes up one, you lose the bet. If it comes up any other number, two through six, you win even money. How much would you like to bet? Let's say you have $10000. Should you bet it all? To maximize your expected return, yes you should bet it all. You should probably go borrow some more money, if you can. Some people would even steal money to invest into this one time bet, and hope to pay it back with the winnings. The money may go away, one chance in six. But, it is more likely to be doubled. This is a good bet, unlike almost all bets in Vegas.

Now let's say that you have the same deal, but you can keep playing until you break the bank or go broke. Again, you have $10000. You bet it all. If you win, you let it ride, double or nothing. If you keep doing this, you will go broke, very soon. Gambler's ruin. Each time, you are making the recommended bet for a one time deal, all your money. But, within only a few rolls of the die, you will very likely go broke. Your chances are more than 1/2 (.5177) that you will lose your all your money in four rolls of the die. How long do you think you would last if you wagered all your money at roulette?

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