Gambler’s Ruin

I hope that everyone knows the first rule of gambling: “The longer you play, the more you lose.”

The odds are (only) the first obstacle for a gambler. The games are designed so that the house takes in more money than it pays out. Even in games that are known for more generous odds, a small edge favoring the house makes plenty of difference.

The second problem for the gambler is related to his finite resources. No matter how successful a gambler has been, he is always a certain number of losses away from going completely bankrupt. The casino has comparatively limitless resources and is not similarly vulnerable. If the gambler plays long enough, then he will eventually have a string of losses that are large enough to eradicate all previous gains. This is true even if the odds of the game are in the gambler’s favor. Another way to analyze this problem is to imagine a simple game. In this game, a bettor starts with $400. The bettor calls two coin flips. If he is correct on either one, he doubles his money. If he is wrong on both, he loses half of whatever amount he last won, or else $100 if he hasn’t won yet. For example, if he calls the first or the second flip correctly, then he wins $400, so that he has a new stake of $800. If he calls both the third and the fourth flips incorrectly, then he loses half of $400, which is $200, so his stake drops to $600. The game continues until the bettor has no money. What is the probability that the game will eventually have to end? If the bettor is ever wrong on 9 consecutive coin flips, it will always break him. Even though it may take awhile, if the bettor does not leave the game voluntarily he is guaranteed to eventually hit a rocky streak that costs him everything.

The third problem for the gambler is based on utility theory. One dollar is not always equal to another dollar. Suppose that a gambler has a net worth of $10K. And suppose that the gambler is willing to stake half of it on a “fair” game, meaning that the odds of winning are 50%. If the gambler wins, he stands to gain enough money to live in a more comfortable apartment or to drive a more expensive car. If the gambler loses, he stands to lose the car and some of the furniture that he already owns. The discomfort associated with losing would be greater than the reward for winning. Because of diminishing returns, the money in the gambler’s pocket is more valuable than the same amount of money in the pot. A rational gambler would not be willing to play this game even if the odds gave him an edge.

A gambler going against the house is mathematically predisposed to lose. A gambler who plans on winning is mathematically handicapped. Luck favors the ones who walk away earliest.

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