Gambler's ruin
Encyclopedia
The term gambler's ruin is used for a number of related statistical ideas:
While the first three meanings have some relevance for gamblers
, they are also general theorem
s with wide application and many related results in probability
and statistics
. Huygens' result in particular led to important advances in the mathematical theory of probability
.
when he wins, but does not reduce his stake when he loses. This general pattern is not uncommon among real gamblers, and casinos encourage it by "chipping up" winners (giving them higher denomination chips). Under this betting scheme, it will take at most N losing bets in a row to bankrupt him. If his probability of winning each bet is less than 1 (if it is 1, then he is no gambler), he will eventually lose N bets in a row, however big N is. It is not necessary that he follow the precise rule, just that he increase his bet fast enough as he wins. This is true even if the expected value of each bet is positive.
The gambler playing a fair game (with 0.5 probability of winning) will eventually either go broke or double his wealth. These events are equally likely, or the game would not be fair (ignoring the fact that his bankroll might jump over one event or the other, this is a minor complication to the argument). So he has a 0.5 chance of going broke before doubling his money. Once he doubles his money, he again has a 0.5 chance of doubling his money before going broke. Overall, there is a 0.25 chance that he will go broke after doubling his money once, but before doubling it twice. Continuing this way, his chance of going broke is 0.5 + 0.25 + 0.125 + . . . which approaches 1.
Huygens' result is illustrated in the next section.
The eventual fate of a player at a negative expected value
game cannot be better than the player at a fair game, so he will go broke as well.
If there are no other limitations on the number of flips, the probability that the game will eventually end this way is 100%. (One way to see this is as follows. Any given finite string of heads and tails will eventually be flipped with certainty: the probability of not seeing this string, while high at first, decays exponentially. In particular, the players would eventually flip a string of heads as long as the total number of pennies in play, by which time the game must have already ended.)
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:
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
has 8 pennies and player two (
) were to have 5 pennies then the probability of each losing is:
= 0.3846 or 38.46%
= 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:
= 
= 
= 0.5
= 
= 
=0.5
This can be shown as follows: Consider the probability of player 1 experiencing gamblers ruin having started with
amount of money, 
. Then, using the Law of Total Probability, we have
,
where W denotes the event that player 1 wins the first bet. Then clearly
and 
. Also 
is the probability that player 1 experiences gambler's ruin having started with 
amount of money: 
; and 
is the probability that player 1 experiences gambler's ruin having started with 
amount of money: 
.
Denoting
, we get the linear homogenous recurrence relation
,
which we can solve using the fact that
(i.e. the probability of gambler's ruin given that player 1 starts with no money is 1), and 
(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).
Here
players with initial capital 
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 quickly become prohibitive as soon as the number of players
or their initial capital increase. For
and large initial capitals 
the solution can be well approximated by using two-dimensional Brownian motion
. (For
this is not possible.)
In practice the true problem is to find the solution for the typical cases of
and limited initial capital.
Swan (2006) proposed an algorithm based on Matrix-analytic methods (Folding algorithm for ruin problems) which
reduces, in such cases, the order of the computational task significantly.
- The original meaning is that a gamblerGamblingGambling is the wagering of money or something of material value on an event with an uncertain outcome with the primary intent of winning additional money and/or material goods...
who raises his bet to a fixed fraction of bankroll when he wins, but does not reduce it when he loses, will eventually go broke, even if he has a positive expected valueExpected valueIn probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...
on each bet. - Another common meaning is that a gamblerGamblingGambling is the wagering of money or something of material value on an event with an uncertain outcome with the primary intent of winning additional money and/or material goods...
with finite wealth, playing a fair game (that is, each bet has expected valueExpected valueIn probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...
zero to both sides) will eventually go broke against an opponent with infinite wealth. - The result above is a corollaryCorollaryA corollary is a statement that follows readily from a previous statement.In mathematics a corollary typically follows a theorem. The use of the term corollary, rather than proposition or theorem, is intrinsically subjective...
of a general theorem by Christiaan Huygens which is also known as gambler's ruin. That theorem shows how to compute the probability of each player winning a series of bets that continues until one's entire initial stake is lost, given the initial stakes of the two players and the constant probability of winning. This is the oldest mathematical idea that goes by the name gambler's ruin, but not the first idea to which the name was applied. - The most common use of the term today is for the unsurprising idea that a gambler playing a negative expected valueExpected valueIn probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...
game will eventually go broke, regardless of betting system. This is another corollaryCorollaryA corollary is a statement that follows readily from a previous statement.In mathematics a corollary typically follows a theorem. The use of the term corollary, rather than proposition or theorem, is intrinsically subjective...
to Huygens' result.
While the first three meanings have some relevance for gamblers
Gambling
Gambling is the wagering of money or something of material value on an event with an uncertain outcome with the primary intent of winning additional money and/or material goods...
, they are also general theorem
Theorem
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...
s with wide application and many related results in probability
Probability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...
and statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
. Huygens' result in particular led to important advances in the mathematical theory of probability
Probability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...
.
Reasons for the four results
Let "Bankroll" be the amount of money a gambler has at his disposal at any moment, and let "N" be any positive integer. Suppose that he raises his stake to when he wins, but does not reduce his stake when he loses. This general pattern is not uncommon among real gamblers, and casinos encourage it by "chipping up" winners (giving them higher denomination chips). Under this betting scheme, it will take at most N losing bets in a row to bankrupt him. If his probability of winning each bet is less than 1 (if it is 1, then he is no gambler), he will eventually lose N bets in a row, however big N is. It is not necessary that he follow the precise rule, just that he increase his bet fast enough as he wins. This is true even if the expected value of each bet is positive.The gambler playing a fair game (with 0.5 probability of winning) will eventually either go broke or double his wealth. These events are equally likely, or the game would not be fair (ignoring the fact that his bankroll might jump over one event or the other, this is a minor complication to the argument). So he has a 0.5 chance of going broke before doubling his money. Once he doubles his money, he again has a 0.5 chance of doubling his money before going broke. Overall, there is a 0.25 chance that he will go broke after doubling his money once, but before doubling it twice. Continuing this way, his chance of going broke is 0.5 + 0.25 + 0.125 + . . . which approaches 1.
Huygens' result is illustrated in the next section.
The eventual fate of a player at a negative expected value
Expected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...
game cannot be better than the player at a fair game, so he will go broke as well.
Fair 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%. (One way to see this is as follows. Any given finite string of heads and tails will eventually be flipped with certainty: the probability of not seeing this string, while high at first, decays exponentially. In particular, the players would eventually flip a string of heads as long as the total number of pennies in play, by which time the game must have already ended.)
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:
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
has 8 pennies and player two () were to have 5 pennies then the probability of each losing is: = 0.3846 or 38.46% = 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:
= = = 0.5 = = =0.5Unfair coin flipping
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, then the probability of each ending penniless is:This can be shown as follows: Consider the probability of player 1 experiencing gamblers ruin having started with
amount of money, . Then, using the Law of Total Probability, we have,where W denotes the event that player 1 wins the first bet. Then clearly
and . Also is the probability that player 1 experiences gambler's ruin having started with amount of money: ; and is the probability that player 1 experiences gambler's ruin having started with amount of money: .Denoting
, we get the linear homogenous recurrence relation,which we can solve using the fact that
(i.e. the probability of gambler's ruin given that player 1 starts with no money is 1), and (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
players with initial capital 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
Markov chain
A Markov chain, named after Andrey Markov, is a mathematical system that undergoes transitions from one state to another, between a finite or countable number of possible states. It is a random process characterized as memoryless: the next state depends only on the current state and not on the...
methods can be applied to
solve in principle this more general problem, but the computations quickly become prohibitive as soon as the number of players
or their initial capital increase. For
and large initial capitals the solution can be well approximated by using two-dimensional Brownian motion
Wiener process
In mathematics, the Wiener process is a continuous-time stochastic process named in honor of Norbert Wiener. It is often called standard Brownian motion, after Robert Brown...
. (For
this is not possible.)In practice the true problem is to find the solution for the typical cases of
and limited initial capital.Swan (2006) proposed an algorithm based on Matrix-analytic methods (Folding algorithm for ruin problems) which
reduces, in such cases, the order of the computational task significantly.
See also
- Gambler's fallacyGambler's fallacyThe Gambler's fallacy, also known as the Monte Carlo fallacy , and also referred to as the fallacy of the maturity of chances, is the belief that if deviations from expected behaviour are observed in repeated independent trials of some random process, future deviations in the opposite direction are...
- Martingale (betting system)Martingale (betting system)Originally, martingale referred to a class of betting strategies popular in 18th century France. The simplest of these strategies was designed for a game in which the gambler wins his stake if a coin comes up heads and loses it if the coin comes up tails...
- Gambler's conceitGambler's conceitGambler’s conceit is the fallacy described by behavioral economist David J. Ewing, where a gambler believes they will be able to stop a risky behavior while still engaging in it. This belief frequently operates during games of chance, such as casino games...
- Fixed-odds betting
External links
- Illustration of Gambler's Ruin
- The Gambler's Ruin at MathPages