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 $[\lambda,\lambda^{-1}]$, for a certain $\lambda \in (0,1)$. We indicate that $\lambda$ is remarkably close to one, proving that $\lambda \leq 0.999904$ and presenting clear numerical evidence that $\lambda \geq 1 - 10^{-4}$. For each $x \in [\lambda,\lambda^{-1}]$, 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.

翻译：我们研究随机回合的资源分配博弈。在“丢失便士的踪迹”中，一个计数子在$\mathbb{Z}$上移动。每回合，Maxine押注$a \in [0,\infty)$，Mina押注$b \in [0,\infty)$。计数子$X$随后相邻移动，以概率$\tfrac{a}{a+b}$向右。在此无限回合博弈中，若$X_i \to -\infty$，则Mina获得1单位，Maxine获得0；若$X_i \to \infty$，则收益分别为0和$x$。因此，给定玩家的净收益为$-A+B$，其中$A$是她押注的总和，$B$是她的终端收益。该博弈受2009年[PSSW]中无偏拉锯战启发，但实际更接近1987年经济学文献[HarrisVickers87]中引入的原始拉锯战版本。我们证明该博弈具有惊人特征。对于一类自然策略，纳什均衡恰好存在于$x$位于$[\lambda,\lambda^{-1}]$时，其中$\lambda \in (0,1)$为特定常数。我们指出$\lambda$极其接近1，证明$\lambda \leq 0.999904$，并给出清晰数值证据表明$\lambda \geq 1 - 10^{-4}$。对于每个$x \in [\lambda,\lambda^{-1}]$，我们找到可数无限个纳什均衡。每个均衡大致由积分“战场”指标刻画：当计数子靠近时，双方激烈押注，随着其远离，押注迅速但不对称衰减。我们的结果推进了[HarrisVickers87,Konrad12]关于资金管理和激励-结果关系的假定，这些假定合理适用于许多玩家出资的押注主导博弈。结合对预算分配博弈的配套研究[HP22]，我们为说明性的拉锯战博弈类提供了详细的数学处理。我们还回顾了经济学和数学中拉锯战的独立发展，希望数学家们进一步关注其原始资源分配形式下的拉锯战。