Friday, January 23, 2009

Gambler's Ruin

Here is a tip for this Chinese New Year gambling.

The concept of Gambler's Ruin was mentioned in a very insightful book called Fortune's Formula which I thought deserve more scruntiny, both from gambling and investing point of view.

The story goes as follows. Imagine you have some money and you decide to bet a fixed amount in a game where your probability of winning is 50%. Say you have $100, you want to bet $1 on "Big" and "Small" (and in this case, there is no "House Win" like double sixes. Bcos if there is "House Win", your winning probability would be less than 50% which we do not want in this story yet).

So what is the probability that you will lose all your money after some time?

Well it's 100%!

This is intuitively illogical bcos the odds are 50% right? Why should one lose everything? Well the caveat here is "after some time" which is as good as saying "playing forever". If you play forever, you are bound to lose everything, which can be shown mathematically and that's what we are gonna do in this post. If you decide to stop after winning a certain amt of money, then good for you, it's possible that you achieve your goal and leave the casino with some money.

The mathematical proof of why you can expect to lose everything if you play for a long time (or rather forever) goes as follows:

The probability of either losing your money or doubling your money is 0.5. Consider these mutually exclusive cases:

Case 1: Probability of losing all your money after X1 no. of bets = 0.5
Case 2: After X1 no. of bets, you have doubled your money (0.5), but you continue to play for another X2 bets, so probability of later losing everything again = 0.5*0.5 = 0.25
Case 3: After X1 + X2 no. of bets, you double your money yet again, lucky you! (0.5*0.5), but you continue to play another X3 bets, and the probability of later losing everything = 0.5*0.5*0.5 = 0.125

The no. of cases go on and you add up all the probability that you will go broke = 0.5+0.25+0.125+... = 1

So, as long as you bet the same absolute amt, even in an even odds game, you WILL lose everything in the end.

This link allows you to simulate exactly what will happen and I tried it and recorded down that it takes about 20,000 games to go broke if you have $100 and the odds are 50% (which is quite a lot, but still the math is against you). If the house odds just goes up by 5% (ie your probability of winning is 45%), you lose everything in less than 1,000 games.

So what to do? Well the book says that you should follow this formula called Kelly's Formula to decide how much to bet (which is a % of your money rather than a fixed absolute amount). In this case, the formula actually says don't bet though...

So don't keep betting $1 during the usual CNY Big/Small game. Vary your bet size according to Kelly's Formula!
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