[ustctraining][13.05.09]F - 分蛋糕
2013-05-09 23:04
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记蛋糕总量为1。设在第i轮给对手分Xi的蛋糕。Xi小于某个临界值,对手就会拒绝,而对自己来说,希望Xi越小,但同时希望对手不要拒绝(如果说希望对手拒绝,由第i轮与第i+1轮的相似性,推知游戏将无限轮,自己收益为0)。记Xi为这个临界值,即如果从第i 轮开始的话的最优策略。假设X(i+1)已知,那么X(i) = 1/(i+1) – X(i+1),因为1/(i+1) – X(i+1)是如果对手拒绝的话的最大获得。于是,X1=1/2–X2 , X2=1/3-X3, ... 又因为Xi一定存在,且0<Xi<1/i,故Xi收敛于0。故X1=1/2–1/3+1/4- … = 1–ln(2)。故小明分了m*ln2的蛋糕,小方分了m*(1-ln2)的蛋糕。代码略。
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