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The matching game is a cooperative game where the value of every coalition isthe maximum revenue of players in the coalition can make by forming pairwisedisjoint partners. The multiple partners matching game generalizes the matchinggame by allowing each player to have more than one possibly repeated partner.In this paper, we study profit-sharing in multiple partners matching games. Acentral concept for profit-sharing is the core which consists of all possibleways of distributing the profit among individual players such that the grandcoalition remains intact. The core of multiple partners matching games may beempty [Deng et al., Algorithmic aspects of the core of combinatorialoptimization games, Math. Oper. Res., 1999.]; even when the core is non-empty,the core membership problem is intractable in general [Biro et al., The stablefixtures problem with payments, Games Econ. Behav., 2018]. Thus we studyapproximate core allocations upon which a coalition may be paid less than theprofit it makes by seceding from the grand coalition. We provide an LP-basedmechanism guaranteeing that no coalition is paid less than $2/3$ times theprofit it makes on its own. We also show that $2/3$ is the best possible factorrelative to the underlying LP-relaxation. Our result generalizes the work ofVazirani [Vazirani, The general graph matching game: approximate core, arXiv,2021] from matching games to multiple partners matching games.

Approximate Core Allocations for Multiple Partners Matching Games