Stable matchings with correlated preferences

Suppose I use the Gale-Shapley (GS) algorithm to find a stable matching between two sets P and R of size n. Proposer p∈P gets utility urp=αwr+(1−α)xrp from being matched with reviewer r∈R, where wr is common to all proposers, xrp is specific to proposer p, and α∈[0,1] controls the correlation in utilities across proposers.¹ Likewise, reviewer r gets utility vpr=βyp+(1−β)zpr from being matched with proposer p, where yp is common to all reviewers, zpr is specific to reviewer r, and β∈[0,1] controls the correlation in utilities across reviewers. The wr, xrp, yp, and zpr are iid standard normal. I run the GS algorithm 200 times, each time (i) simulating new utility realizations and (ii) computing the means U≡1n∑p∈Purp and V≡1n∑r∈Rvpr of utilities under the resulting matching. I then compute the grand means of U and V across all 200 simulations. The chart below shows how these grand means vary with α and β when n=50.

Proposers and reviewers tend to be better off when (i) utilities on their side of the market are less correlated and (ii) utilities on the other side of the market are more correlated. Intuitively, same-side correlations induce competition that makes the most desirable people on that side better off but the rest much worse off. This competition benefits the other side of the market because it gives people on that side more power to choose “winners” according to their preferences.