The matching game is a cooperative game where the value of every coalition is the maximum revenue of players in the coalition can make by forming pairwise disjoint partners. The multiple partners matching game generalizes the matching game by allowing each player to have more than one possibly repeated partner. In this paper, we study profit-sharing in multiple partners matching games. A central concept for profit-sharing is the core which consists of all possible ways of distributing the profit among individual players such that the grand coalition remains intact. The core of multiple partners matching games may be empty [Deng et al., Algorithmic aspects of the core of combinatorial optimization games, Math. Oper. Res., 1999.]; even when the core is non-empty, the core membership problem is intractable in general [Biro et al., The stable fixtures problem with payments, Games Econ. Behav., 2018]. Thus we study approximate core allocations upon which a coalition may be paid less than the profit it makes by seceding from the grand coalition. We provide an LP-based mechanism guaranteeing that no coalition is paid less than $2/3$ times the profit it makes on its own. We also show that $2/3$ is the best possible factor relative to the underlying LP-relaxation. Our result generalizes the work of Vazirani [Vazirani, The general graph matching game: approximate core, arXiv, 2021] from matching games to multiple partners matching games.
翻译:匹配游戏是一个合作游戏, 每一个联盟的价值都是联盟中玩家的最大收入。 匹配游戏的多个伙伴通过建立对称脱节伙伴可以形成联盟中玩家的最大收入。 匹配游戏的多个伙伴通过让每个玩家拥有不止一个可能的重复伙伴来概括匹配游戏的匹配游戏; 在本文中, 我们研究的是多个伙伴的利润分享。 利润分享的核心概念是各个玩家之间分配利润的所有可能方式的核心, 使大联盟保持完好。 匹配游戏的多个伙伴的核心可能是空的 [Deng et al., 组合优化游戏核心的算法方面, Math. Oper. Res. 1999] ; 即使核心玩家的核心游戏是非空的, 匹配游戏中的核心成员问题在总体上是棘手的 [Biro et al., 支付稳定固定的固定问题, Econ. Behav., 2018] 。 因此我们研究的是核心分配的大约核心资金, 其支付额可能低于从大联盟获得的利润。 我们提供了一个基于LP- 机制, 保证联盟不会得到超过2/3美元, 美元, 我们的匹配游戏的比值 的比值的比值的比值的比值。 我们的比值 3 显示的计算结果可能显示的比值的结果。