I just discovered an application of Galois connection to economics (or, more precisely, mechanism design) in a

newly revised paper by Georg Noldeke and Larry Samuelson on ["The Implementation Duality"](https://cowles.yale.edu/sites/default/files/files/pub/d20/d2091.pdf).

They use the "antitone" definition of Galois connection, though (i.e., \\( f(p) \leq q \Leftrightarrow p \geq g(q) \\) ).

Here is a quote from p. 8 of the paper (a "profile" u gives utility u(x) to an agent of type x, (Φv)(x) is the highest utility that an agent of type x can get when trading/being matched to a counterpart; similarly for v(y) and Ψu):

> Suppose we have a pair of profiles u and v such that each buyer x ∈ X is content to obtain u(x) rather than matching with any seller y ∈ Y and providing that seller with utility v(y), that is, the inequality u ≥ Φv holds. It is then intuitive that every seller y ∈ Y would similarly weakly prefer to obtain utility v(y) to matching with any buyer x ∈ X who insists on receiving utility u(x), that is, the inequality v ≥ Ψu holds. Reversing the roles of buyers and sellers in this explanation motivates the other direction of the equivalence.

newly revised paper by Georg Noldeke and Larry Samuelson on ["The Implementation Duality"](https://cowles.yale.edu/sites/default/files/files/pub/d20/d2091.pdf).

They use the "antitone" definition of Galois connection, though (i.e., \\( f(p) \leq q \Leftrightarrow p \geq g(q) \\) ).

Here is a quote from p. 8 of the paper (a "profile" u gives utility u(x) to an agent of type x, (Φv)(x) is the highest utility that an agent of type x can get when trading/being matched to a counterpart; similarly for v(y) and Ψu):

> Suppose we have a pair of profiles u and v such that each buyer x ∈ X is content to obtain u(x) rather than matching with any seller y ∈ Y and providing that seller with utility v(y), that is, the inequality u ≥ Φv holds. It is then intuitive that every seller y ∈ Y would similarly weakly prefer to obtain utility v(y) to matching with any buyer x ∈ X who insists on receiving utility u(x), that is, the inequality v ≥ Ψu holds. Reversing the roles of buyers and sellers in this explanation motivates the other direction of the equivalence.