We study a class of 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~\cite{PSSW09} from 2009 but in fact closely resembles the original version of tug-of-war, introduced~\cite{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 [HV87,K12] 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：我们研究一类随机回合的资源分配博弈。在"遗失便士追踪"中，计数器在$\mathbb{Z}$上移动。每回合，Maxine下注$a \in [0,\infty)$，Mina下注$b \in [0,\infty)$。随后计数器$X$相邻移动，以概率$\tfrac{a}{a+b}$向右移动。若在此无限回合博弈中$X_i \to -\infty$，则Mina获得一个单位收益，Maxine获得零；若$X_i \to \infty$，则二者收益分别为零和$x$。因此，给定玩家的净收益为$-A+B$，其中$A$为其下注总额，$B$为其终端收益。该博弈受2009年文献\cite{PSSW09}中无偏拔河博弈的启发，但实际上与1987年经济学文献\cite{HarrisVickers87}引入的原始版本拔河博弈高度相似。我们证明该博弈具有惊人特性。对于某类自然策略，当且仅当$x$位于区间$[\lambda,\lambda^{-1}]$（其中$\lambda \in (0,1)$为特定常数）时，纳什均衡存在。我们指出$\lambda$极其接近1，证明$\lambda \leq 0.999904$，并通过清晰的数值证据表明$\lambda \geq 1 - 10^{-4}$。对于每个$x \in [\lambda,\lambda^{-1}]$，我们找到可数无穷多个纳什均衡。每个均衡大致由整数型"战场"指数刻画：当计数器接近该指数时，双方激烈下注，且随着计数器远离，下注额快速但非对称衰减。我们的研究推进了[HV87,K12]关于资金管理与激励-结果关系的设想，这些设想对许多玩家注资型下注博弈普遍成立。结合关于预算分配博弈的配套研究[HP22]，我们为这类具有代表性的拔河博弈提供了详细的数学分析。我们还回顾了经济学与数学领域中拔河博弈的独立发展历程，期望数学家们更多关注拔河博弈在资源分配领域的原始形态。