The General Graph Matching Game: Approximate Core

The classic paper of Shapley and Shubik [SS71] characterized the core of the assignment game using 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 can be empty. This paper salvages the situation by giving an imputation in the 2/3-approximate core for the latter. This bound is best possible, since 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 3 , 1], depending on how severely constrained the agent is. The core is a key solution concept in cooperative game theory. It contains all 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 imputation, in the 2/3-approximate core, implies that a sub-coalition will gain at most a 3/2 factor by seceding, and less in typical cases. *Supported in part by NSF grant CCF-1815901.