So the dual problem of max-flow is min-cut. What's the dual problem of stable matching?

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    $\begingroup$ Have you tried writing stable matching as a linear program? If you succeed, you can just dualize it and see what you get. $\endgroup$ – Yuval Filmus Feb 5 '14 at 4:37

From http://kuznets.fas.harvard.edu/~aroth/papers/1993_MOR_StableMatchingsOptimal.pdf :

Shapley and Shubik (1972) observed that [the dual] formulates the problem of finding payoff vectors in the core of the above assignment game.

The assignment game described is that two agents who choose to match get a total "value" equal to the edge weight between them, and they can split this value between them in some way; these are the "payoffs".

It might be nice for future readers if you put the LP and dual in your question....

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