On the Trail of Lost Pennies: player-funded tug-of-war on the integers
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arXiv
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| Natura: | Preprint |
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2022
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| _version_ | 1866914312340111360 |
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| author | Hammond, Alan |
| author_facet | Hammond, Alan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_07451 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the Trail of Lost Pennies: player-funded tug-of-war on the integers Hammond, Alan Probability Computer Science and Game Theory Theoretical Economics 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. |
| title | On the Trail of Lost Pennies: player-funded tug-of-war on the integers |
| topic | Probability Computer Science and Game Theory Theoretical Economics |
| url | https://arxiv.org/abs/2209.07451 |