|
The term ''gambler's ruin'' is used for a number of related statistical ideas: * The original meaning is that a gambler 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 value on each bet. * Another common meaning is that a gambler with finite wealth, playing a fair game (that is, each bet has expected value zero to both sides) will eventually go broke against an opponent with infinite wealth. Such a situation can be modeled by a random walk on the real number line. In that context it is provable that the agent will return to his point of origin or go broke and is ruined an infinite number of times if the random walk continues forever. * The result above is a corollary 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 that a gambler playing a negative expected value game will eventually go broke, regardless of betting system. This is another corollary to Huygens' result. The ideas have specific relevance for gamblers; however they are also general theorems 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. ==History== The earliest known mention of the gambler's ruin problem is a letter from Blaise Pascal to Pierre Fermat in 1656 (two years after the more famous correspondence on the problem of points). 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?〔Jan Gullberg, Mathematics from the birth of numbers, W. W. Norton & Company; ISBN 978-0-393-04002-9〕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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Gambler's ruin」の詳細全文を読む スポンサード リンク
|