Which Random Matching Markets Exhibit a Stark Effect of Competition

We revisit the popular random matching market model introduced by Knuth (1976) and Pittel (1989), and shown by Ashlagi, Kanoria and Leshno (2013) to exhibit a "stark effect of competition", i.e., with any difference in the number of agents on the two sides, the short side agents obtain substantially better outcomes. We generalize the model to allow "partially connected" markets with each agent having an average degree $d$ in a random (undirected) graph. Each agent has a (uniformly random) preference ranking over only their neighbors in the graph. We characterize stable matchings in large markets and find that the short side enjoys a significant advantage only for $d$ exceeding $\log^2 n$ where $n$ is the number of agents on one side: For moderately connected markets with $d=o(\log^2 n)$, we find that there is no stark effect of competition, with agents on both sides getting a $\sqrt{d}$-ranked partner on average. Notably, this regime extends far beyond the connectivity threshold of $d= \Theta(\log n)$. In contrast, for densely connected markets with $d = \omega(\log^2 n)$, we find that the short side agents get $\log n$-ranked partner on average, while the long side agents get a partner of (much larger) rank $d/\log n$ on average. Numerical simulations of our model confirm and sharpen our theoretical predictions. Since preference list lengths in most real-world matching markets are much below $\log^2 n$, our findings may help explain why available datasets do not exhibit a strong effect of competition.