Gambler's ruin

The player's ( eng. gambler 's ruin ) Ruin means the loss of the last game capital and the associated opportunity to continue playing in gambling. Moreover, the term sometimes refers to the last, very high loss bet placed by a player in the hope of regaining all his previous gambling losses.

In game theory is the steady decline expected value of the game capital in the course of the game, " the player's Ruin", if the profits are reinvested.

Examples

Münzwurfspiel

Alice and Bob possess cents cents. It is repeatedly tossed a fair coin. Depending on the outcome, the loser pays the winner a dime. The game ends when a player has no more money. If the number of litters is unlimited, the probability for this game end is 100%. For the odds applies:

The winning probabilities relate to one another as the stakes.

Is Alice the richer player so this means that they must not make a positive profit expectation, because in every game they lost lost more money than their poorer opponents Bob.

For the expectation value and variance of the playing time is valid:

The Münzwürfspiele obtained from the stronger player will take on average less long.

This means in particular again the expected value of playing time

Own example, Alice and Bob only 1000 cents 1 cent starting capital, so the game takes 1000 coin flips on average. If Bob is ruined with 50 % probability after the first toss, the game when it first developed in favor of Bob, also take a long time. In this example, the last won by Alice games on average 667 throws, and won by Bob Games 334000 throws.

Should the chance per litter be other than 50 %, then the probability of ruin can be represented schematically by the following table:

The cases in which or is infinite, or where, and thus, should be considered as the limit. See also Markov chain.

Casino Games

A typical casino game ( "Big Game") includes a low house edge. This advantage lies in the long-term expected value and can be expressed as a percentage of the amount deposited. It remains unchanged from game to game, but increases arithmetically with increasing playing time, if he is related to the start-up capital of the player.

For example, be 1% of the official house advantage for the casino game; the expected value of the payout for the player accordingly 99%. This equation works when the player would never use a betting profit to continue playing. An idealized weather, which employs 100 euros would retain 99 euros after the game. But if he continues this 99 euro again, he would again lose an average of 1% and decrease its expected value at 98.01 euros. The downward spiral continues until the expected value of the approaches zero: the player's ruin.

The long-term expected value does not necessarily correspond to the result which is experienced by a particular player. Players who play a finite period of time, may, notwithstanding the house edge, achieve a net profit, or they may perish much faster than in the mathematical prediction.

A casino usually has

• ... Much more capital than any player, so a player is much more likely to suffer " the player Ruin" as the casino;
• ... Odds that favor the casino and create a negative expected value for the player;
• Different risk ... strategies that limit its maximum loss.

This ensures that the casino will win in almost all cases in the long run.

Speculation

It can be shown that where economic activities are focused on the transfer of assets, instead of acting on the structure of assets, the player's Ruin, with the result that most of the wealth is held by very few market participants. This is visible in the stock market, if speculative strategies towards long-term dividend-oriented investment predominate.