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 $\lambda$; 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 $\lambda$-derivative of game value. We also indicate examples in which the solution takes a different form in the non-leisurely game.

翻译：在拉锯战中，两名玩家通过沿图边移动一个计数器进行竞争，每轮根据可能偏斜的硬币抛掷结果获得移动权。当计数器到达边界（顶点的固定子集）时，游戏结束，此时一名玩家根据边界顶点向另一名玩家支付特定金额。经济学家和数学家多年来独立研究了拉锯战，分别关注游戏的资源分配形式（玩家反复消耗宝贵预算以影响决定回合胜者的硬币偏斜）以及欧几里得环境下恒定偏斜游戏细网格极限中出现的偏微分方程。本文提出了一类具有分配预算的拉锯战游戏的数学处理：每位玩家初始获得固定预算，并在整个游戏中每轮开始时据此提供赌注，其赢得该轮的概率等于其赌注与双方赌注之和的比值。我们考虑在树上进行的游戏，其中边界为叶节点集，支付函数为单个特定叶节点的指示函数。我们找到了该游戏休闲版本的游戏价值和本质唯一的纳什均衡——在该版本中，每轮在赌注已下后，移动以恒定概率被取消。我们证明：玩家剩余预算之比保持初始值λ；游戏价值为有偏无穷调和函数；玩家在一轮中下注的剩余预算比例由游戏价值的空间梯度及其对λ的导数给出。我们还指出非休闲游戏中解可能采取不同形式的示例。