Unbalanced Random Matching Markets: The Stark Effect of Competition

Stability is used often as a criterion in organizing clearinghouses for two-sided matching markets, where agents on both sides of the market have preferences over potential matches. We study competition in matching markets with random heterogeneous preferences by considering markets with an unequal number of agents on either side. First, we show that even the slightest imbalance yields an essentially unique stable matching. Second, we give a tight description of stable outcomes, showing that matching markets are extremely competitive. Each agent on the short side of the market is matched to one of his top preferences and each agent on the long side does almost no better than being matched to a random partner. Our results suggest that any matching market is likely to have a small core, explaining why empirically small cores are ubiquitous and solving a longstanding puzzle.