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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 theright with probability $\tfrac{a}{a+b}$. If $X_i \to -\infty$ in thisinfinte-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 givenplayer is $-A+B$, where $A$ is the sum of her stakes, and $B$ is her terminalreceipt. The game was inspired by unbiased tug-of-war in~[PSSW] from 2009 butin fact closely resembles the original version of tug-of-war, introduced[HarrisVickers87] in the economics literature in 1987. We show that the gamehas surprising features. For a natural class of strategies, Nash equilibriaexist 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 evidencethat $\lambda \geq 1 - 10^{-4}$. For each $x \in [\lambda,\lambda^{-1}]$, wefind countably many Nash equilibria. Each is roughly characterized by anintegral {\em battlefield} index: when the counter is nearby, both playersstake intensely, with rapid but asymmetric decay in stakes as it moves away.Our results advance premises [HarrisVickers87,Konrad12] for fund management andthe incentive-outcome relation that plausibly hold for many player-fundedstake-governed games. Alongside a companion treatment [HP22] of games withallocated budgets, we thus offer a detailed mathematical treatment of anillustrative class of tug-of-war games. We also review the separatedevelopments of tug-of-war in economics and mathematics in the hope thatmathematicians direct further attention to tug-of-war in its originalresource-allocation guise.

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