In economics and social choice theory, an envy-free matching (EFM) is a matching between people to "things", which is envy-free in the sense that no person would like to switch their "thing" with that of another person.
Moreover, if |NG(X)| ≥ |X| ≥ 1 (where NG(X) is the set of neighbors of X in Y), then G admits a nonempty EFM.
[1] This is a relaxation of Hall's marriage condition, which says that, if |NG(X')| ≥ |X'| for every subset X' of X, then an X-saturating matching exists.
[2] An example of this setting is the rental harmony problem - matching tenants (the agents) to rooms (the items) while setting a price to each room.
It is a relaxation of a Walrasian equilibrium: a Walrasian equilibrium consists of an EF price and EF matching, and in addition, every item must either be matched or have zero price.