The problem of fair division of payoff is one of the key issues when considering cooperation of strategic individuals. It arises naturally in a number of applications related to operational research, including sharing the cost of transportation or dividing the profit among supply chain agents. In this paper, we consider the problem of extending the Shapley Value—a fundamental payoff division scheme—to cooperative games with externalities. While this problem has raised a lot of attention in the literature, most works focused on developing alternative axiomatizations for an extension. Instead, in this paper we focus on the coalition formation process that naturally leads to an extended payoff division scheme. Specifically, building upon recent literature, we view coalition formation as a discrete-time stochastic process, characterized by the underlying family of probability distributions on the set of partitions of players. Given this, we analyse how various properties of the probability distributions that underlie the stochastic processes relate to the game-theoretic properties of the corresponding payoff division scheme. Finally, we prove that the Stochastic Shapley value—a known payoff division scheme from the literature—is the only one that satisfies all aforementioned axioms.

中文翻译：

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存在外部性时的公平划分

收益的公平分配问题是考虑战略个体合作的关键问题之一。它自然出现在与运筹学相关的许多应用中，包括分担运输成本或在供应链代理之间分配利润。在本文中，我们考虑将 Shapley 值（一种基本的收益分配方案）扩展到具有外部性的合作博弈的问题。虽然这个问题在文献中引起了很多关注，但大多数工作都集中在为扩展开发替代公理化。相反，在本文中，我们专注于自然导致扩展收益分配方案的联盟形成过程。具体来说，基于最近的文献，我们将联盟的形成视为一个离散时间的随机过程，以玩家分区集上的潜在概率分布族为特征。鉴于此，我们分析了作为随机过程基础的概率分布的各种特性如何与相应收益分配方案的博弈论特性相关。最后，我们证明了 Stochastic Shapley 值（文献中已知的收益分配方案）是唯一满足所有上述公理的方法。