提前终止的博弈：公平分配赌注

Divide up Gains

专题: Strategy / 策略
难度: L4
来源: QuantQuestion

题目详情

Peter and Paula play a game of chance that consists of several rounds. Each individual round is won, with equal probabilities of, by either Peter ord Paula; the winner then receives one point. Successive rounds are independent. Each has staked 50 for a total of 100, and they agree that the game ends as soon as one of them has won a total of 5 points; this player then receives the 100. After they have completed four rounds, of which Peter has won three and Paula only one, a fire breaks out so that they cannot continue their game.

a. How should the 100 be divided between Peter and Paula?

b. How should the 100 be divided in the general case, when Peter needs to win a more rounds and Paula needs to win b more rounds?

解析

(a) Peter 还需 2 胜、Paula 还需 4 胜。等价于最多再打 5 局：Peter 在 5 局内赢至少 2 局则先到达。\n\nPeter 获胜概率\n\n $\n1-\\mathbb{P}(\\text{5 局内 Peter \\le 1 胜})=1-\\left[\\binom{5}{0}+\\binom{5}{1}\\right]\\left(\\frac12\\right)^5=\\boxed{0.8125}.\n$ \n\n因此 100 应按 81.25 与 18.75 分。\n\n(b) 一般：Peter 还需 $a$ 胜、Paula 还需 $b$ 胜，则\n\n $\n\\boxed{\\mathbb{P}(\\text{Peter})=\\sum_{k=a}^{a+b-1}\\binom{a+b-1}{k}\\left(\\frac12\\right)^{a+b-1}}.\n$