追寻丢失的硬币：玩家资助的整数 tug-of-war 游戏
On the Trail of Lost Pennies: player-funded tug-of-war on the integers
AI总结：
本文研究了玩家资助的整数 tug-of-war 游戏，揭示了纳什均衡的存在条件及特征，推进了资源分配博弈的相关理论。
摘要：
我们研究随机轮换资源分配博弈。在追寻丢失的硬币游戏中，一个计数器在$\mathbb{Z}$上移动。每一轮，Maxine 会下注$a \in [0,\infty)$，Mina 会下注$b \in [0,\infty)$。计数器$X$则会相邻移动，向右移动的概率为$\tfrac{a}{a+b}$。如果在无限轮游戏中$X_i$趋向于$-\infty$，Mina将获得一个单位，Maxine获得零；如果$X_i$趋向于$\infty$，则这些收益为零和$x$。因此，给定玩家的净收益为$-A+B$，其中$A$是她的下注总和，$B$是她的终端收益。该游戏受到2009年[PSSW]中无偏 tug-of-war 的启发，但实际上与1987年在经济学文献中引入的原始 tug-of-war 非常相似。我们展示了该游戏具有令人惊讶的特征。对于一个自然的策略类，纳什均衡存在当且仅当$x$位于$[λ,λ^{-1}]$内，其中$λ\in (0,1)$是一个特定的值。我们指出$λ$非常接近于1，证明了$λ\leq 0.999904$，并提供了明确的数值证据表明$λ\geq 1 - 10^{-4}$。对于每个$x \in [λ,λ^{-1}]$，我们找到了可数多个纳什均衡。每个大致由一个积分战场指数特征化：当计数器靠近时，双方都激烈下注，随着计数器远离，下注迅速但不对称地衰减。我们的结果推进了[HarrisVickers87,Konrad12]关于基金管理和激励-结果关系的前提，这些前提可能适用于许多玩家资助的下注游戏。在与[HP22]中分配预算的游戏配套研究的同时，我们因此为一种示例性的 tug-of-war 游戏提供了详细的数学分析。我们还回顾了 tug-of-war 在经济学和数学中的独立发展，希望数学家们能进一步关注 tug-of-war 在其原始资源分配形式中的应用。
英文摘要：
We study random-turn resource-allocation games. In the Trail of Lost Pennies, a counter moves on $\mathbb{Z}$. At each turn, Maxine stakes $a \in [0,\infty)$ and Mina $b \in [0,\infty)$. The counter $X$ then moves adjacently, to the right with probability $\tfrac{a}{a+b}$. If $X_i \to -\infty$ in this infinte-turn game, Mina receives one unit, and Maxine zero; if $X_i \to \infty$, then these receipts are zero and $x$. Thus the net receipt to a given player is $-A+B$, where $A$ is the sum of her stakes, and $B$ is her terminal receipt. The game was inspired by unbiased tug-of-war in~[PSSW] from 2009 but in fact closely resembles the original version of tug-of-war, introduced [HarrisVickers87] in the economics literature in 1987. We show that the game has surprising features. For a natural class of strategies, Nash equilibria exist precisely when $x$ lies in $[λ,λ^{-1}]$, for a certain $λ\in (0,1)$. We indicate that $λ$ is remarkably close to one, proving that $λ\leq 0.999904$ and presenting clear numerical evidence that $λ\geq 1 - 10^{-4}$. For each $x \in [λ,λ^{-1}]$, we find countably many Nash equilibria. Each is roughly characterized by an integral {\em battlefield} index: when the counter is nearby, both players stake intensely, with rapid but asymmetric decay in stakes as it moves away. Our results advance premises [HarrisVickers87,Konrad12] for fund management and the incentive-outcome relation that plausibly hold for many player-funded stake-governed games. Alongside a companion treatment [HP22] of games with allocated budgets, we thus offer a detailed mathematical treatment of an illustrative class of tug-of-war games. We also review the separate developments of tug-of-war in economics and mathematics in the hope that mathematicians direct further attention to tug-of-war in its original resource-allocation guise.
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