Gambler's ruin
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The basic meaning of gambler's ruin is when a gambler loses the last of his bank of gambling money and is unable 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 the ever decreasing expected value of return on a gambler's bank as he continues to gamble with his winnings.
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.