Gambler's ruin

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. In probability theory, the term sometimes refers to the fact that a gambler will almost certainly go broke in the long run against an opponent with much more money, even if the opponent's advantage on each turn is small or zero.

Examples

Coin flipping

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

How do you calculate this certainty?

If player one has "n"₁ pennies and player two "n"₂ pennies, the chances "P"₁ and "P"₂ that players one and two, respectively, will end penniless are:

:$P\_1=\; frac\{n\_2\}\{n\_1+n\_2\}$:$P\_2=\; frac\{n\_1\}\{n\_1+n\_2\}$

Two examples of this are if one player has more pennies than the other; and if both players have the same number of pennies.In the first case say player one $(P\_1)$ has 8 pennies and player two ($P\_2$) were to have 5 pennies then the probability of each losing is:

:$P\_1=frac\{5\}\{8+5\}$ $=frac\{5\}\{13\}$ = 0.3846 or 38.46%

:$P\_2=frac\{8\}\{8+5\}$ $=frac\{8\}\{13\}$ = 0.6154 or 61.54%

It follows that even with equal odds of winning the player that starts with fewer pennies is more likely to fail.

In the second case where both players have the same number of pennies (in this case 6) the likelihood of each losing is:

:$P\_1=frac\{6\}\{6+6\}$ = $frac\{6\}\{12\}$ = $frac\{1\}\{2\}$ = 0.5

:$P\_2=frac\{6\}\{6+6\}$ = $frac\{6\}\{12\}$ = $frac\{1\}\{2\}$ =0.5

In the event of an unfair coin, where player one wins each toss with probability p, and player two wins with probability q = 1-p < p, then the probability of each winning is:

:$P\_1=\; frac\{1-(frac\{q\}\{p\})^\{n\_1\{1-(frac\{q\}\{p\})^\{n\_1+n\_2$:$P\_2=\; frac\{(frac\{q\}\{p\})^\{n\_1\}-(frac\{q\}\{p\})^\{n\_1+n\_2\{1-(frac\{q\}\{p\})^\{n\_1+n\_2$

This can be shown as follows: Consider the probability of player 1 experiencing gamblers ruin having started with $n\; >\; 1$ amount of money, $P(R\_n)$. Then, using the Law of Total Probability, we have

:,

where W denotes the event that player 1 wins the first bet. Then clearly $P(W)\; =\; p$ and . Also $P(R\_n\; |\; W)$ is the probability that player 1 experiences gambler's ruin having started with $n+1$ amount of money: $P(R\_\{n+1\})$; and is the probability that player 1 experiences gambler's ruin having started with $n-1$ amount of money: $P(R\_\{n-1\})$.

Denoting $q\_n\; =\; P(R\_n)$, we get the linear homogenous recurrence relation

:$q\_n\; =\; q\_\{n+1\}\; p\; +\; q\_\{n-1\}\; q$,

which we can solve using the fact that $q\_0\; =\; 1$ (i.e. the probability of gambler's ruin given that player 1 starts with no money is 1), and $q\_\{n\_1\; +\; n\_2\}\; =\; 0$ (i.e. the probability of gambler's ruin given that player 1 starts with all the money is 0.) For a more detailed description of the method see e.g. Feller (1957).

N-player ruin problem

The above described problem (2 players) is a special case of the so-called N-Player ruin problem.Here $N\; geq\; 2\; ,\; ,$ players with initial capital $x\_1,\; x\_2,\; cdots,\; x\_N,,$ dollars, respectively, play a sequence of (arbitrary) independent games and win and lose certain amounts of dollars from/to each other according to fixed rules.The sequence of games ends as soon as at least one player is ruined. Standard Markov chain methods can be applied to solve in principle this more general problem, but the computations become quickly prohibitive as soon as the number of playersor their initial capital increase. For $N\; =\; 3\; ,$ and large initial capitals $x\_1,\; x\_2,\; cdots,\; x\_N\; ,$the solution can be well approximated by using two-dimensional Brownian motion. (For $N\; >\; 3$ this is not possible.)In practice the true problem is to find the solution for the typical cases of $N\; geq\; 3$ and limited initial capital.Swan (2006) proposed an algorithm based on Matrix-analytic methods (Folding algorithm for ruin problems) whichreduces, in such cases, the order of the computational task significantly.

Casino games

A typical casino game has a slight house advantage, which is the long-run expectation, most often expressed as a percentage of the amount wagered. In most games, this edge remains constant from one play to the next (blackjack being one notable exception).

Calculating House Edge

If the game played has an actual pay off of "w" and true odds of "t" then the house edge can be calculated from the formula::$Edge=\; frac\{t-w\}\{t+1\}$

For example, if you have true odds of 15 to 1 and a payoff of 14 to 1 then the house advantage is:

:$frac\{15-14\}\{15+1\}$ = $frac\{1\}\{16\}$ = 0.0625 = 6.25%

For example, the official house advantage for a casino game is 6.25%, and thus the expected value of return for the gambler is 93.75% of the total capital wagered. However, this calculation would be exactly true only if the gambler never re-wagered the proceeds of a winning bet. Thus after gambling 100 dollars (called "action") the idealized average gambler would be left with 93.75 dollars in his bankroll. If he continued to bet (using his 93.75 dollars in proceeds as his new bankroll), he would again lose 6.25% of his action on average and the expected value of his bankroll would go down to 87.89 dollars. If the proceeds are continually re-wagered, this downward spiral continues until the gambler's expected value approaches zero. Gambler's ruin would occur the first time the bankroll reaches exactly 0, which could occur earlier or later but must occur eventually.

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 average.

Gambler's ruin can also occur if the player has an advantage over the house, as in Blackjack card counting, because the player has a limited bankroll. [ [http://www.blackjackincolor.com/blackjackrisk1.htm BlackjackinColor.com] ]

Notes

See also

* Gambler's fallacy
* Martingale (betting system)
* Gambler's conceit
* Fixed-odds betting

References

Epstein, R. The Theory of Gambling and Statistical Logic. Academic Press; Revised edition (March 10, 1995)

Ferguson T. S. Gamblers Ruin in Three Dimensions. Unpublished manuscript: http://www.math.ucla.edu/~tom/

Swan, Y. and F. T. Bruss . A Matrix-Analytic Approach to the N-Player Ruin Problem. Journal of Applied Probability43, 755-766 (2006)

External links

* [http://math.ucsd.edu/~anistat/gamblers_ruin.html Illustration of Gambler's Ruin]
* [http://www.mathpages.com/home/kmath084/kmath084.htm The Gambler's Ruin] at MathPages