Approximate Core Allocations for Multiple Partners Matching Games

The multiple partners matching game is a cooperative profit-sharing game, which generalizes the classic matching game by allowing each player to have more than one partner. The core is one of the most important concepts in cooperative game theory, which consists all possible ways of allocating the total profit of the game among individual players such that the grand coalition remains intact. For the multiple partners matching game, the core may be empty in general [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 for the multiple partners matching game, and provide an LP-based mechanism guaranteeing that no coalition is paid less than $2/3$ times the profit it makes on its own. Moreover, we show that the factor $2/3$ is best possible in general, but can be improved depending on how severely constrained the players are. Our result generalizes the recent work of Vazirani [Vazirani, The general graph matching game: approximate core, arXiv, 2021] from matching games to multiple partners matching games.