Gambler's Ruin
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In
statistics Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...
, gambler's ruin is the fact that a
gambler Gambling (also known as betting or gaming) is the wagering of something of value ("the stakes") on a random event with the intent of winning something else of value, where instances of strategy are discounted. Gambling thus requires three ele ...
playing a game with negative
expected value In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first Moment (mathematics), moment) is a generalization of the weighted average. Informa ...
will eventually go
bankrupt Bankruptcy is a legal process through which people or other entities who cannot repay debts to creditors may seek relief from some or all of their debts. In most jurisdictions, bankruptcy is imposed by a court order, often initiated by the de ...
, regardless of their betting system. The concept was initially stated: A persistent gambler who raises his bet to a fixed fraction of the gambler's bankroll after a win, but does not reduce it after a loss, will eventually and inevitably go broke, even if each bet has a positive expected value. Another statement of the concept is that a persistent gambler with finite wealth, playing a fair game (that is, each bet has expected value of zero to both sides) will eventually and inevitably go broke against an opponent with infinite wealth. Such a situation can be modeled by a
random walk In mathematics, a random walk, sometimes known as a drunkard's walk, is a stochastic process that describes a path that consists of a succession of random steps on some Space (mathematics), mathematical space. An elementary example of a rand ...
on the real number line. In that context, it is probable that the gambler will, with virtual certainty, return to their point of origin, which means going broke, and is ruined an infinite number of times if the random walk continues forever. This is a
corollary In mathematics and logic, a corollary ( , ) is a theorem of less importance which can be readily deduced from a previous, more notable statement. A corollary could, for instance, be a proposition which is incidentally proved while proving another ...
of a general theorem by
Christiaan Huygens Christiaan Huygens, Halen, Lord of Zeelhem, ( , ; ; also spelled Huyghens; ; 14 April 1629 – 8 July 1695) was a Dutch mathematician, physicist, engineer, astronomer, and inventor who is regarded as a key figure in the Scientific Revolution ...
, 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 term's common usage today is another corollary to Huygens's result. The concept has specific relevance for gamblers. However it also leads to mathematical
theorem In mathematics and formal logic, a theorem is a statement (logic), statement that has been Mathematical proof, proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to esta ...
s with wide application and many related results in
probability Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an e ...
and
statistics Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...
. Huygens's result in particular led to important advances in the mathematical theory of probability.


History

The earliest known mention of the gambler's ruin problem is a letter from
Blaise Pascal Blaise Pascal (19June 162319August 1662) was a French mathematician, physicist, inventor, philosopher, and Catholic Church, Catholic writer. Pascal was a child prodigy who was educated by his father, a tax collector in Rouen. His earliest ...
to
Pierre Fermat Pierre de Fermat (; ; 17 August 1601 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he is recognized for his d ...
in 1656 (two years after the more famous correspondence on the
problem of points The problem of points, also called the problem of division of the stakes, is a classical problem in probability theory. One of the famous problems that motivated the beginnings of modern probability theory in the 17th century, it led Blaise Pascal ...
). Pascal's version was summarized in a 1656 letter from Pierre de Carcavi to Huygens:
Let two men play with three dice, the first player scoring a point whenever 11 is thrown, and the second whenever 14 is thrown. But instead of the points accumulating in the ordinary way, let a point be added to a player's score only if his opponent's score is nil, but otherwise let it be subtracted from his opponent's score. It is as if opposing points form pairs, and annihilate each other, so that the trailing player always has zero points. The winner is the first to reach twelve points; what are the relative chances of each player winning?
Huygens reformulated the problem and published it in ''De ratiociniis in ludo aleae'' ("On Reasoning in Games of Chance", 1657):
Problem (2-1) Each player starts with 12 points, and a successful roll of the three dice for a player (getting an 11 for the first player or a 14 for the second) adds one to that player's score and subtracts one from the other player's score; the loser of the game is the first to reach zero points. What is the probability of victory for each player?
This is the classic gambler's ruin formulation: two players begin with fixed stakes, transferring points until one or the other is "ruined" by getting to zero points. However, the term "gambler's ruin" was not applied until many years later. The gambler's ruin problem is often applied to gamblers with finite capital playing against a bookie or
casino A casino is a facility for gambling. Casinos are often built near or combined with hotels, resorts, restaurants, retail shops, cruise ships, and other tourist attractions. Some casinos also host live entertainment, such as stand-up comedy, conce ...
assumed to have an “infinite” or much larger amount of capital available. It can then be proven that the probability of the gambler's eventual ruin tends to 1 even in the scenario where the game is fair or what mathematically is defined as a martingale.


Reasons for the four results

Let d be the amount of money a gambler has at their disposal at any moment, and let N be any positive integer. Suppose that they raise their stake to \frac when they win, but do not reduce their stake when they lose (a not uncommon pattern among real gamblers). Under this betting scheme, it will take at most ''N'' losing bets in a row to bankrupt them. If their probability of winning each bet is less than 1 (if it is 1, then they are no gambler), they are virtually certain to eventually lose ''N'' bets in a row, however big ''N'' is. It is not necessary that they follow the precise rule, just that they increase their bet fast enough as they win. This is true even if the expected value of each bet is positive. The gambler playing a fair game (with probability \frac of winning) will eventually either go broke or double their wealth. By symmetry, they have a \frac chance of going broke before doubling their money. If they double their money, they repeat this process and they again have a \frac chance of doubling their money before going broke. After the second process, they have a \left(\frac\right)^2 = \frac chance that they have not gone broke yet. Continuing this way, their chance of not going broke after n processes is \left(\frac\right)^n, which approaches 0, and their chance of going broke after n successive processes is \sum_^\left(\frac\right)^i which approaches 1. Huygens's result is illustrated in the next section. The eventual fate of a player at a game with negative
expected value In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first Moment (mathematics), moment) is a generalization of the weighted average. Informa ...
cannot be better than the player at a fair game, so they will go broke as well.


Example of Huygens's result


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 1. (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 ''n''1 pennies and player two ''n''2 pennies, the probabilities ''P''1 and ''P''2 that players one and two, respectively, will end penniless are: \begin P_1 & = \frac \\ ptP_2 & = \frac \end 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: \begin P_1 & =\frac =\frac = 0.3846 \text 38.46\% \\ ptP_2 & =\frac =\frac = 0.6154 \text 61.54\% \end 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: \begin P_1 & =\frac = \frac = \frac = 0.5 \\ ptP_2 & =\frac = \frac = \frac = 0.5 \end


Unfair 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: \begin P_1 & = \frac \\ ptP_2 & = \frac \end An argument is that the expected hitting time is finite and so with a martingale, associating the value \left(\frac\right)^l with each state so that the expected value of the state is constant, this is the solution to the system of equations: \begin P_1 + P_2 & = 1 \\ pt\left(\frac\right)^ & = P_1 + P_2 \left(\frac\right)^ \end Alternately, 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 P(R_n) = P(R_\mid W)P(W) + P(R_\mid\bar)P(\bar), where W denotes the event that player 1 wins the first bet. Then clearly P(W) = p and P(\bar) = 1 - p = q. Also P(R_, W) =P(R_) is the probability that player 1 experiences gambler's ruin having started with n+1 amount of money: and P(R_ \mid \bar) = P(R_) is the probability that player 1 experiences gambler's ruin having started with n-1 amount of money. Denoting q_n = P(R_n), we get the linear homogeneous recurrence relation q_n = q_ p + q_ 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_ = 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 (1970), ''An introduction to probability theory and its applications'', 3rd ed.


''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, \ldots, x_N dollars, respectively, play a sequence of (arbitrary) independent games and win and lose certain amounts of dollars from and to each other according to fixed rules. The sequence of games ends as soon as at least one player is ruined. Standard
Markov chain In probability theory and statistics, a Markov chain or Markov process is a stochastic process describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally ...
methods can be applied to solve this more general problem in principle, but the computations quickly become prohibitive as soon as the number of players or their initial capitals increase. For N = 2 and large initial capitals x_1, x_2 the solution can be well approximated by using two-dimensional
Brownian motion Brownian motion is the random motion of particles suspended in a medium (a liquid or a gas). The traditional mathematical formulation of Brownian motion is that of the Wiener process, which is often called Brownian motion, even in mathematical ...
. (For N \geq 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) which significantly reduces the order of the computational task in such cases.


See also

* *
Fixed-odds betting Fixed-odds betting is a form of gambling where individuals place bets on the outcome of an event, such as sports matches or horse races, at predetermined odds. In fixed-odds betting, the odds are fixed and determined at the time of placing the b ...
*
Gambler's conceit Gambler'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. The gambler's conceit frequently works in conjunction with the gamb ...
*
Gambling Gambling (also known as betting or gaming) is the wagering of something of Value (economics), value ("the stakes") on a Event (probability theory), random event with the intent of winning something else of value, where instances of strategy (ga ...
*
Gambler's fallacy The gambler's fallacy, also known as the Monte Carlo fallacy or the fallacy of the maturity of chances, is the belief that, if an event (whose occurrences are Independent and identically distributed random variables, independent and identically dis ...
* Impossibility of a gambling system *
Kelly criterion In probability theory, the Kelly criterion (or Kelly strategy or Kelly bet) is a formula for sizing a sequence of bets by maximizing the long-term expected value of the logarithm of wealth, which is equivalent to maximizing the long-term expected ...
*
Martingale (betting system) A martingale is a class of betting strategies that originated from and were popular in 18th-century France. The simplest of these strategies was designed for a game in which the gambler wins the stake if a coin comes up heads and loses if it co ...
*
Online gambling Online gambling (also known as iGaming or iGambling) is any kind of gambling conducted on the internet. This includes virtual poker, casinos, and sports betting. The first online gambling venue opened to the general public was ticketing for th ...
*
Risk of ruin Risk of ruin is a concept in gambling, insurance, and finance relating to the likelihood of losing all one's investment capital or extinguishing one's bankroll below the minimum for further play. For instance, if someone bets all their money on a s ...
*
Volatility tax The volatility tax is a mathematical finance term first published by Rick Ashburn, CFA in a 2003 column, and formalized by hedge fund manager Mark Spitznagel, describing the effect of large investment losses (or volatility) on compound returns.


Notes


References

* * Ferguson T. S. ''Gamblers Ruin in Three Dimensions''. Unpublished manuscript: https://www.math.ucla.edu/~tom/ * * * *{{cite journal, first1=Yves C. , last1=Swan , authorlink2=F. Thomas Bruss , first2=F. Thomas, last2=Bruss , title= A Matrix-Analytic Approach to the N-Player Ruin Problem , journal= Journal of Applied Probability , volume=4 , issue=3 , pages=755–766 , year=2006 , doi=10.1017/S0021900200002084 , doi-access=free


External links


Illustration of Gambler's Ruin
at MathPages
The Gambler’s Ruin Simulation
at Wolfram Demonstration Project Gambling terminology Probability problems Causal fallacies Variants of random walks