论文标题
在失落的便士的踪迹上：球员资助的拔河战在整数上
On the Trail of Lost Pennies: player-funded tug-of-war on the integers
论文作者
论文摘要
我们研究随机转移资源分配游戏。在失落的便士的踪迹中，一个计数器在$ \ mathbb {z} $上移动。在[0，\ infty）$中的Maxine Stakes $ a \ in [0，\ infty）$中的Maxine Stakes $ a \。然后，计数器$ x $将相邻移动，以概率$ \ tfrac {a} {a+b} $向右移动。如果$ x_i \ to- to- \ infty $在此Infinte -Turn游戏中，Mina将获得一个单位，Maxine Zero;如果$ x_i \ to \ infty $，则这些收据为零，$ x $。因此，给定播放器的净收入为$ -a+b $，其中$ a $是她的股权，而$ b $是她的终端收据。该游戏的灵感来自于2009年的〜[PSSW]中的无偏拔比赛，但实际上与1987年的经济学文献中引入了[harrisvickers87]的原始版本。我们显示游戏具有令人惊讶的功能。对于天然策略，当$ x $置于$ [λ，λ^{ - 1}] $中，对于某个$λ\ in（0,1）$，nash equilibria恰好存在。我们表明$λ$非常接近一个，证明$λ\ leq 0.99904 $，并提供清晰的数值证据，表明$λ\ geq 1-10^{ - 4} $。对于[λ，λ^{ - 1}] $中的每个$ x \，我们发现许多nash equilibria。每个玩家都在附近，每个玩家都大致刻有一个整体{\ em战场}索引的整体特征，两个玩家都持有迅速但不对称的衰变，因为它移开了。我们的业绩为基金管理和激励结果关系，我们的结果是[Harrisvickers87，Konrad12]，这对于许多球员资助的股份进行了保险，这是合理的。除了具有分配预算的游戏的伴侣处理[HP22]外，我们还提供了详细的数学处理，对拔河游戏的说明性类别。我们还回顾了拔河比赛中经济学和数学的单独发展，希望数学家以其原始的资源分配式的幌子将进一步的关注引起拔河的关注。
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.
