On the Trail of Lost Pennies: player-funded tug-of-war on the integers

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.