一种贪婪的芯片发射博弈
A Greedy Chip-firing Game
AI总结：
论文提出确定性的 hunger game，以转子路由式机制模拟有限常返和吸收马尔可夫链，并证明其分布、击中测度与期望击中时间的偏差以 N^{-1} 收敛。
摘要：
我们引入了马尔可夫链的一种确定性类似物，并将其称为 hunger game（饥饿博弈）。与 rotor-routing（转子路由）一样，hunger game 以确定性方式模拟常返马尔可夫链和吸收马尔可夫链的行为。对于具有有限多个状态的常返马尔可夫链，hunger game 模拟集中在平稳分布附近，其偏差以类似 N^{-1} 的速度衰减，其中 N 是模拟步数；对于具有有限多个状态的吸收马尔可夫链，hunger game 模拟在击中测度和期望击中时间方面也表现出集中性，其偏差以 N^{-1} 而非 N^{-1/2} 的速度衰减。当有限马尔可夫链中的转移概率为有理数时，该博弈最终是周期的；其周期似乎对所有初始构型都相同，而且吸引盆似乎通过平移铺满构型空间（即 hunger 向量集合），但我们尚未证明这一点。
英文摘要：
We introduce a deterministic analogue of Markov chains that we call the hunger game. Like rotor-routing, the hunger game deterministically mimics the behavior of both recurrent Markov chains and absorbing Markov chains. In the case of recurrent Markov chains with finitely many states, hunger game simulation concentrates around the stationary distribution with discrepancy falling off like $N^{-1}$, where $N$ is the number of simulation steps; in the case of absorbing Markov chains with finitely many states, hunger game simulation also exhibits concentration for hitting measures and expected hitting times with discrepancy falling off like $N^{-1}$ rather than $N^{-1/2}$. When transition probabilities in a finite Markov chain are rational, the game is eventually periodic; the period seems to be the same for all initial configurations and the basin of attraction appears to tile the configuration space (the set of hunger vectors) by translation, but we have not proved this.
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