Ruin the player

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The ruin of the player ( English gambler's ruin ) means in gambling the loss of the last game capital and thus the possibility to continue playing. In addition, the term sometimes refers to the last, very large losing bet that a player places in the hope of regaining all of their previous gambling losses.

In game theory , "ruin of the player" stands for the steadily decreasing expected value of the game capital in the course of the game when the winnings are reinvested.

Examples

Coin toss game

Alice owns cents and Bob owns cents. A fair coin is tossed repeatedly. Depending on the outcome, the loser pays the winner a cent. The game ends when a player runs out of money. If the number of throws is unlimited, the probability that the game will end is 100%. The following applies to the chances of winning: ${\displaystyle a}$ ${\displaystyle b}$

${\displaystyle {\text{P(Alice gewinnt)}}={\text{P(Bob verliert)}}={\frac {a}{a+b}}}$

${\displaystyle {\text{P(Bob gewinnt)}}={\text{P(Alice verliert)}}={\frac {b}{a+b}}}$

The odds of winning are related to each other like the stakes.

${\displaystyle {\frac {P({\text{Alice gewinnt}})}{P({\text{Bob gewinnt}})}}={\frac {a}{b}}}$

If Alice is the richer player, however, this does not mean a positive profit expected value for her, because with every lost game she loses more money than her poorer opponent Bob. ${\displaystyle (a>b)}$

${\displaystyle {\text{E(Gewinn von Alice)}}=b\cdot P({\text{Alice gewinnt}})-a\cdot P({\text{Alice verliert}})={\frac {b\cdot a}{a+b}}-{\frac {a\cdot b}{a+b}}=0}$.

The following applies for the expected value and variance of the game duration:

${\displaystyle {\text{E(Spieldauer)}}=ab\,}$

${\displaystyle {\text{Var}}({\text{Spieldauer}})={\frac {ab\,(a^{2}+b^{2}-2)}{3}}}$

The coin tossing games won by the richer player take less time on average.

${\displaystyle {\text{E(Spieldauer}}|{\text{Alice gewinnt)}}={\frac {b\,(b+2a)}{3}}}$

${\displaystyle {\text{E(Spieldauer}}|{\text{Bob gewinnt)}}={\frac {a\,(a+2b)}{3}}}$

This results in particular again in the expected value of the playing time

${\displaystyle {\text{E(Spieldauer}}|{\text{Alice gewinnt)}}\cdot {\text{P(Alice gewinnt)}}+{\text{E(Spieldauer}}|{\text{Bob gewinnt)}}\cdot {\text{P(Bob gewinnt)}}}$

${\displaystyle ={\frac {b\,(b+2a)}{3}}\cdot {\frac {a}{a+b}}+{\frac {a\,(a+2b)}{3}}\cdot {\frac {b}{a+b}}=ab\,{\frac {(b+2a)+(a+2b)}{3(a+b)}}=ab}$

For example, if Alice has 1000 cents and Bob only has 1 cent starting capital, the game takes an average of 1000 coin flips. Although there is a 50% chance that Bob will be ruined after the first coin toss, the game can take a long time if it initially works in Bob's favor. In this example, the games Alice won take an average of 667 throws and the games Bob won 334,000 throws.

If the chance per throw is not equal to 50%, the ruin probability can be shown schematically using the following table:

	Alice	bob
Seed capital	a	b
Chance per coin toss	p	q
Probability of ruin	${\displaystyle {\frac {\left({\frac {p}{q}}\right)^{b}-1}{\left({\frac {p}{q}}\right)^{a+b}-1}}}$	${\displaystyle {\frac {\left({\frac {q}{p}}\right)^{a}-1}{\left({\frac {q}{p}}\right)^{a+b}-1}}}$

The cases in which or is infinite, or in which , and thus is, are to be regarded as Limes. See also the Markov chain . ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle p=q=50\,\%}$ ${\displaystyle {\frac {q}{p}}=1}$

Casino games

A typical casino game ("Big Game") contains a small house edge . This advantage lies in the long-term expected value and can be expressed as a proportion of the sum used. It remains unchanged from game to game, but increases mathematically with increasing game duration if it is related to the player's starting capital.

For example, the official house edge on a casino game can be 1%; the expected value for the payout for the player is 99%. This calculation works if the player would never use a bet to continue playing. An idealized bettor who stakes 100 euros would keep 99 euros after the game. If he bets this 99 euros again, however, he would lose another 1% on average and his expected value would drop to 98.01 euros. The downward spiral continues until the expected value approaches zero: the ruin of the player.

The long-term expected value does not necessarily correspond to the outcome that a particular player experiences. Players who play for a finite amount of time can win net regardless of the house edge, or they can perish much faster than mathematical prediction.

A casino usually has

• ... much more capital than any player, so a player is far more likely to "ruin the player" than the casino;
• ... odds that favor the casino and create a negative expectation for players;
• ... various risk strategies that limit your maximum loss.

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

speculation

It can be shown that where economic activity is focused on transferring wealth rather than building wealth, the ruin of the player works with the result that most of the wealth is held by very few market participants. This becomes evident in the stock market when speculative strategies outweigh long-term dividend - oriented investments.

See also

• Player fallacy
• Damage actuarial

Web links

A simulation of the "ruin of the player