Cooperative games form an important class of problems in game theory, where a key goal is to distribute a value among a set of players who are allowed to cooperate by forming coalitions. An outcome of the game is given by an allocation vector that assigns a value share to each player. A crucial aspect of such games is submodularity (or convexity ). Indeed, convex instances of cooperative games exhibit several nice properties, e.g. regarding the existence and computation of allocations realizing some of the most important solution concepts proposed in the literature. For this reason, a relevant question is whether one can give a polynomial-time characterization of submodular instances, for prominent cooperative games that are in general non-convex. In this paper, we focus on a fundamental and widely studied cooperative game, namely the spanning tree game . An efficient recognition of submodular instances of this game was not known so far, and explicitly mentioned as an open question in the literature. We here settle this open problem by giving a polynomial-time characterization of submodular spanning tree games.

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子模块生成树游戏的有效表征

合作博弈形成了博弈论中的一类重要问题，其中一个关键目标是在允许通过形成联盟进行合作的一组参与者之间分配价值。游戏的结果由分配向量给出，该向量为每个玩家分配价值份额。这种游戏的一个关键方面是子模块性（或凸性）。事实上，合作博弈的凸实例展示了几个很好的特性，例如关于分配的存在和计算，实现了文献中提出的一些最重要的解决方案概念。出于这个原因，一个相关的问题是，对于通常是非凸的突出合作游戏，是否可以给出子模实例的多项式时间特征。在本文中，我们专注于一个基础的和广泛研究的合作博弈，即生成树游戏。迄今为止，对该游戏的子模块实例的有效识别尚不清楚，并且在文献中明确提到是一个悬而未决的问题。我们在这里通过给出子模生成树游戏的多项式时间特征来解决这个开放问题。