Unbalanced random matching markets

We analyze large random matching markets with unequal numbers of men and women. We find that with high probability, a vanishing fraction of agents have multiple stable partners, i.e., the ‘core ’ is small. Further, we find that being on the short side of the market confers a large advantage. For each agent, assign a rank of 1 to the agent’s most preferred partner, a rank of 2 to the next most preferred partner and so forth. If there are n men and n + 1 women then, we show that with high probability, in any stable matching, the men’s average rank of their wives is no more than 3 log n, whereas the women’s average rank of their husbands is at least n/(3 log n). If there are n men and (1 + λ)n women for λ> 0 then, with high probability, in any stable matching the men’s average rank of wives is O(1), whereas the women’s average rank of husbands is Ω(n). Simulations show that our results hold even for small markets. These findings for unbalanced markets contrast sharply with known results for balanced markets: in particular, our results show that the large core in the balanced case is a knife edge phenomenon that breaks with the slightest imbalance.