发表机构
Shanghai Jiao Tong University; Moscow Institute of Physics and Technology（上海交通大学; 莫斯科物理技术学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究贪心芯片博弈的周期性质，证明饥饿博弈及单调得分推广中所有周期轨道的最小周期均为 T，并解决了 Li 和 Propp 的两个猜想，同时揭示了与主席分配问题的联系。
AI 中文摘要
对于定义在 $n$ 个状态上的有限不可约有理转移矩阵 $P=(P_{ij})$，由 Li 和 Propp 引入的饥饿博弈是一种贪心芯片博弈过程，其中每个状态 $i$ 携带一个实值饥饿度 $h_i$。在每一步中，选择一个具有最大饥饿度的状态 $i$，若出现平局则按状态上的固定顺序解决。发射 $i$ 会从其饥饿度中减去 1，然后向每个状态 $j$ 的饥饿度加上 $P_{ij}$。设 $\pi$ 为 $P$ 的平稳分布，$T$ 为使得 $T\pi$ 为整数向量的最小正整数。我们证明每个周期轨道的周期均为 $T$，其中状态 $i$ 发射 $T\pi_i$ 次，并且总饥饿度为零的周期状态集合通过格平移铺满相应的超平面。这些结果解决了 Li 和 Propp 的两个猜想。我们还证明了有限强连通有向图上固定优先级芯片博弈的非平凡周期轨道具有类似的周期定理，这是另一种贪心芯片博弈模型。随后，我们发现了饥饿博弈与主席分配问题之间的联系：总是选择最低于其比例目标的状态的贪心规则恰好是一个秩一饥饿博弈。受此应用及相关分配规则的启发，我们将饥饿博弈推广为：为每个状态分配一个得分，该得分是其饥饿度的非递减函数，每次发射具有最大得分的状态。对于这些单调得分饥饿博弈，我们证明每个周期轨道仍然具有周期 $T$，并给出了从任意初始状态最终进入周期性的充分条件。
英文摘要
For a finite irreducible rational transition matrix $P=(P_{ij})$ on $n$ states, the hunger game introduced by Li and Propp is a greedy chip-firing process in which each state $i$ carries a real-valued hunger $h_i$. At each step, a state $i$ of maximal hunger is selected, with ties resolved by a fixed order on the states. Firing $i$ subtracts one from its hunger and then adds $P_{ij}$ to the hunger of each state $j$. Let $π$ be the stationary distribution of $P$, and let $T$ be the least positive integer such that $Tπ$ is integral. We prove that every periodic orbit has least period $T$, with state $i$ firing $Tπ_i$ times, and that the set of periodic states of total hunger zero tiles the corresponding hyperplane by lattice translations. These results settle two conjectures of Li and Propp. We also prove an analogous least-period theorem for nontrivial periodic orbits of fixed-priority chip-firing on finite strongly connected digraphs, another greedy chip-firing model. We then find a connection between the hunger game and the chairman assignment problem: the greedy rule that always chooses the state furthest below its proportional target is precisely a rank-one hunger game. Motivated by this application and related allocation rules, we generalize the hunger game by assigning each state a score that is a nondecreasing function of its hunger and firing a state of maximal score each time. For these monotone-score hunger games, we prove that every periodic orbit still has least period $T$ and give a sufficient condition for eventual periodicity from every initial state.
Comments15 pages
