Rank-optimal assignments in uniform markets

We prove that in a market where agents rank objects independently and uniformly at random, there exists an assignment of objects to agents with a constant average rank (i.e., an average rank independent of the market size). The proof builds on techniques from random graph theory and the FKG inequality (Fortuin et al., 1971). When the agents’ rankings are their private information, no Dominant Strategy Incentive Compatible mechanism can implement the assignment with the smallest average rank; however, we show that there exists a Bayesian Incentive Compatible mechanism that does so. Together with the fact that the average rank under the Random Serial Dictatorship (RSD) mechanism grows infinitely large with the market size, our findings indicate that the average rank under RSD can take a heavy toll compared to the first-best, and highlight the possibility of using other assignment methods in scenarios where average rank is a relevant objective.