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