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.

翻译：我们研究随机轮次资源分配博弈。在"失落的便士"博弈中，一个计数器在$\mathbb{Z}$上移动。每轮中，Maxine下注$a \in [0,\infty)$，Mina下注$b \in [0,\infty)$。随后计数器$X$以概率$\tfrac{a}{a+b}$向右移动一步。若在此无限轮次博弈中$X_i \to -\infty$，则Mina获得1单位收益而Maxine获得零收益；若$X_i \to \infty$，则收益分配为0和$x$。因此给定玩家的净收益为$-A+B$，其中$A$是其总下注额，$B$是其最终收益。该博弈受启发于2009年[PSSW]中的无偏拔河博弈，但实际上与1987年经济学文献[HarrisVickers87]提出的原始拔河博弈版本高度相似。我们证明该博弈具有令人惊异的特性：对于一类自然策略，纳什均衡当且仅当$x$属于$[λ,λ^{-1}]$时存在，其中$λ\in (0,1)$为特定常数。我们指出$λ$惊人地接近1，证明$λ\leq 0.999904$并提供明确数值证据表明$λ\geq 1 - 10^{-4}$。对每个$x \in [λ,λ^{-1}]$，我们找到可数多个纳什均衡。每个均衡大致由整数型{\em战场}指标刻画：当计数器临近时，双方玩家激烈下注，随着计数器远离，下注额呈快速但非对称衰减。我们的结果推进了[HarrisVickers87,Konrad12]中关于资金管理与激励-结果关系的前提假设，这些假设可能适用于众多玩家资助型注额调控博弈。结合对预算分配博弈的配套研究[HP22]，我们为此类阐释性拔河博弈提供了详尽的数学处理。我们还回顾了经济学与数学中拔河博弈的独立发展历程，期望数学家能进一步关注原始资源分配形式的拔河博弈。