Bachet's game is a variant of the game of Nim. There are $n$ objects in one pile. Two players take turns to remove any positive number of objects not exceeding some fixed number $m$. The player who takes the last object loses. We consider a variant of Bachet's game in which each move is a lottery over set $\{1,2,\ldots, m\}$. The outcome of a lottery is the number of objects that player takes from the pile. We show that under some nondegenericity assumptions on the set of available lotteries the probability that the first player wins in subgame perfect Nash equilibrium converges to $1/2$ as $n$ tends to infinity.

中文翻译：

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巴切特的抽奖动作游戏

Bachet 的游戏是 Nim 游戏的一个变种。一堆中有 $n$ 个对象。两名玩家轮流移除不超过某个固定数量 $m$ 的任何正数量的物体。拿走最后一个物体的玩家输了。我们考虑 Bachet 游戏的一个变体，其中每一步都是对 $\{1,2,\ldots, m\}$ 集合的抽奖。彩票的结果是玩家从堆中拿走的物品数量。我们表明，在对可用彩票集的一些非退化假设下，第一个玩家在子博弈完美纳什均衡中获胜的概率收敛到 $1/2$，因为 $n$ 趋于无穷大。