The classic paper of Shapley and Shubik 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.
The classic paper of Shapley and Shubik Shapley1971assignment 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 \over 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.
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