Fall 2019

The General Graph Matching Game: Approximate Core

Monday, Mar. 1, 2021 9:40 am10:10 am

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Vijay Vazirani (UC Irvine)



The matching game, which can also be viewed as a transferable utility matching market, forms a cornerstone of cooperative game theory. For the bipartite case, the classic paper of Shapley and Shubik \cite{Shapley1971assignment} characterized the core using ideas from matching theory and LP-duality theory and their highly non-trivial interplay. Whereas the core of that 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.  Our imputation implies that a sub-coalition will gain at most a $3/2$ factor by seceding, and less in typical cases.