The General Graph Matching Game: Approximate Core

The classic paper of Shapley and Shubik \cite{Shapley1971assignment} characterized the core of the assignment game using powerful ideas from matching theory and LP-duality theory and their highly non-trivial interplay. Whereas the core of the assignment game is always non-empty, that of the general graph matching game the core can be empty. This paper salvages the situation by giving an imputation in the $2/3$-approximate core for the latter. We show that this bound is best possible -- it is the integrality gap of the natural underlying LP. Our profit allocation method goes further: the multiplier on the profit of an agent lies in the interval $[{2 \over 3}, 1]$, depending on how severely constrained the agent is. The core is a quintessential solution concept in cooperative game theory. It contains ways of distributing the total worth of a game among agents in such a way that no sub-coalition has incentive to secede from the grand coalition. Our $2/3$-approximate imputation implies that a sub-coalition will be able to gain at most a $3/2$ factor by seceding, and less in typical cases.