In tug-of-war, two players compete by moving a counter along edges of a graph, each winning the right to move at a given turn according to the flip of a possibly biased coin. The game ends when the counter reaches the boundary, a fixed subset of the vertices, at which point one player pays the other an amount determined by the boundary vertex. Economists and mathematicians have independently studied tug-of-war for many years, focussing respectively on resource-allocation forms of the game, in which players iteratively spend precious budgets in an effort to influence the bias of the coins that determine the turn victors; and on PDE arising in fine mesh limits of the constant-bias game in a Euclidean setting. In this article, we offer a mathematical treatment of a class of tug-of-war games with allocated budgets: each player is initially given a fixed budget which she draws on throughout the game to offer a stake at the start of each turn, and her probability of winning the turn is the ratio of her stake and the sum of the two stakes. We consider the game played on a tree, with boundary being the set of leaves, and the payment function being the indicator of a single distinguished leaf. We find the game value and the essentially unique Nash equilibrium of a leisurely version of the game, in which the move at any given turn is cancelled with constant probability after stakes have been placed. We show that the ratio of the players' remaining budgets is maintained at its initial value $λ$; game value is a biased infinity harmonic function; and the proportion of remaining budget that players stake at a given turn is given in terms of the spatial gradient and the $λ$-derivative of game value. We also indicate examples in which the solution takes a different form in the non-leisurely game.

翻译：在拔河博弈中，两位参与者通过沿图边移动一枚棋子的方式展开竞争，每回合的移动权由一枚可能带有偏置的硬币投掷结果决定。当棋子到达边界——即顶点的一个固定子集时，博弈终止，此时一方需根据边界顶点向另一方支付指定金额。经济学家与数学家已各自研究拔河博弈多年：前者主要关注资源分配形式的博弈，即参与者通过迭代消耗有限预算来影响决定回合胜负的硬币偏置；后者则致力于研究欧几里得场景下常偏置博弈在精细网格极限中产生的偏微分方程。本文对一类具有预算分配的拔河博弈进行数学化处理：每位参与者初始获得固定预算，并在每回合开始时从中提取筹码进行下注，其赢得该回合的概率等于其筹码与双方筹码总和之比。我们研究该博弈在树结构上的实现，其中边界为叶节点集合，支付函数为单个特定叶节点的指示函数。通过分析该博弈的休闲版本（即在筹码下注后每个回合的移动动作以恒定概率被取消），我们得到了博弈值及其本质唯一的纳什均衡。研究证明：参与者剩余预算之比始终维持初始值$λ$；博弈值为偏置无穷调和函数；且参与者在每回合下注的剩余预算比例可由博弈值的空间梯度与$λ$导数表示。本文亦通过示例说明非休闲博弈情形下解可能呈现的不同形式。