Abstract We prove a counterintuitive result concerning the expected hitting/absorption time for a class of Markov chains. The “paradox” already shows itself in the following elementary example that is suitable for undergraduate teaching: Batman and the Joker perform independent discrete-time random walks on the vertices of a square until they meet, starting from opposite vertices. Batman always moves (and clockwise and anticlockwise steps are equally likely), while the Joker remains still on any given step with probability On average the Joker survives for twice as long by staying still with arbitrarily small but positive probability (in the limit as ) than by always moving (when q = 0).

中文翻译：

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预期命中时间的悖论

摘要 我们证明了关于一类马尔可夫链的预期命中/吸收时间的违反直觉的结果。“悖论”已经在以下适合本科教学的基本示例中体现出来：蝙蝠侠和小丑在正方形的顶点上执行独立的离散时间随机游走，直到它们相遇，从相反的顶点开始。蝙蝠侠总是移动（顺时针和逆时针步骤的可能性相同），而小丑以概率保持在任何给定的步骤上平均而言，小丑以任意小但为正的概率（在极限为通过始终移动（当 q = 0 时）。