  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.
