Gambler’s Ruin: A Random Walk on the Simplex
Gambler’s Ruin: A Random Walk on the Simplex
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赌徒的毁灭：单纯形上的随机游走
DOI：
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发表时间：
1987
期刊：
影响因子：
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通讯作者：
B. Hajek
中科院分区：
文献类型：
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作者：
B. Hajek
本文的目的是解决托马斯M.封面（见第五章第5.4节）。假设有三个赌徒，他们的资本分别为p a、p B和p c，其中p a + p B + p c = 1。参与者在二维单纯形Pi ≥ 0，pa + p B + pc = 1中进行对称的三方博弈。当其中一个玩家破产时，剩下的两个玩家之间继续游戏，其中游戏现在由一维布朗运动建模，直到第二个玩家输了，剩下的玩家被宣布为赢家。Doob的可选抽样定理意味着参与人i将以概率pi成为赢家Cover的问题是找出玩家在特定顺序中输的概率。例如，我们想知道参与人3先输然后参与人2输的概率。我们提供了一个“混乱”的解决方案。
The purpose of this note is to give a solution to a problem of Thomas M. Cover (see Chapter V, Section 5.4). Suppose there are three gamblers with respective capital p a , p b , and p c , where p a + p b + p c = 1. The players engage in a symmetric three-way game modeled by Brownian motion in the two-dimensional simplex P i ≥ 0, p a + p b + p c = 1. When one of the players goes broke, play continues between the remaining two players, where the play is now modeled by a Brownian motion in one dimension, until a second player loses, and the remaining player is declared a winner. Doob’s optional sampling theorem implies that player i will be a winner with probability p i . Cover’s problem is to find the probability that the players lose in a specific order. For example, we would like to find the probability that player 3 loses first and then player 2 loses. We provide a “messy” solution.