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# Gambler’s ruin

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The basic meaning of gambler’s ruin is a gambler’s loss of the last of his bank of gambling money and consequent inability to continue gambling. “Gambler’s ruin” is also sometimes used to refer to a final large losing bet placed in the hopes of winning back all the gambler has lost during a gambling session.More generally however the phrase refers to the ever decreasing expected value of a gambler’s bank as he continues to gamble with his winnings.

## Examples

### Coin flipping

Consider a flipping coin game with two players where each player has a 50% chance of winning each flip. After a flip the loser transfers one penny to the winner. The game ends when one player has all the pennies. If there is no other limit on the number of flips, the probability that the game will eventually end this way is 100%. If player one has n1 pennies and player two n2 pennies, the chances P1 and P2 that players one and two, respectively, will end penniless are:

P1 =  n2/(n1+n2)

P2 = n1/(n1+n2)

It follows that the player that starts with fewest pennies is most likely to fail. Even with equal odds, the longer one gambles, the greater the chance that the player starting out with the most pennies wins. However, this does not imply positive expected value for richer player since for each complete game (many flips) that the richer player loses, he will forfeit more pennies than his poorer playmate.

Consider players with 90 and 10 pennies respectively, repeating the game 100 times. The player with 90 pennies is expected to win 90 out of 100 complete games, winning 10 pennies each game. However, he is also expected to lose 10 games, each time forfeiting all 90 of his pennies. So after the series of 100 games, the richer player is expected to win 90*10=900 pennies, and lose 10*90=900 pennies. Despite the fact that after any single game, one player ends up with all the pennies, the expected result over many games is for both players to break even.

A casino generally has:

• many more pennies than any player thus ensuring that the player is much more likely than the casino to experience gambler’s ruin;
• odds that favor the casino resulting in negative expected return for the player; and
• various risk management techniques that limits their maximum loss.

The combination of above ensures that the casino will in the vast majority of cases come out ahead in the long run. For an illustration, see this Gambler’s Ruin simulation: [1]

### Casino games

A typical casino game has a slight house advantage. The advantage is the long-run expectation, most often expressed as a percentage of the amount wagered. It remains constant from one play to the next. If the long-run expectation is expressed as a percentage of the amount that the player starts with, however, then the house advantage increases the longer the player continues.

For example, the official house advantage for a casino game might be 1%, and thus the expected value of return for the gambler is 99%. However, this math would only be true if the gambler never used the results of a winning bet again. Thus after gambling 100 dollars the idealized average gambler would be left with 99 dollars, but, if he continued to bet using his 99 dollars in winnings, he would again lose 1% on average and his expected value would go down to 98.01 dollars. This downward spiral continues until the gambler’s expected value approaches zero: gambler’s ruin.

The long-run expectation will not necessarily be the result experienced by any particular gambler. The gambler who plays for a finite period of time may finish with a net win, despite the house advantage, or may go broke much more quickly than the mathematical prediction.

### Speculation

It might be pointed out that where economic activity is concentrated in transfers of wealth rather than its creation, gambler’s ruin results in most of the wealth being held by a very small number of participants. We see this in the stock market when speculative activity is the norm rather than long-term dividend producing investment.