The size of the core in assignment markets

TL;DR

This paper analyzes the size of the core in assignment markets with stochastic agent productivity, showing that the variation in utility diminishes as the market grows, which supports the assumption of a unique core outcome in empirical research.

Contribution

It characterizes how the size of the core shrinks in large assignment markets with random productivity, providing bounds and tightness results.

Findings

The variation in utility per agent is bounded by O*(1/n^{1/K})

The bound on core size is tight in the worst case

Results justify the common assumption of a unique core outcome in empirical studies.

Abstract

Assignment markets involve matching with transfers, as in labor markets and housing markets. We consider a two-sided assignment market with agent types and stochastic structure similar to models used in empirical studies, and characterize the size of the core in such markets. Each agent has a randomly drawn productivity with respect to each type of agent on the other side. The value generated from a match between a pair of agents is the sum of the two productivity terms, each of which depends only on the type but not the identity of one of the agents, and a third deterministic term driven by the pair of types. We allow the number of agents to grow, keeping the number of agent types fixed. Let $n$ be the number of agents and $K$ be the number of types on the side of the market with more types. We find, under reasonable assumptions, that the relative variation in utility per agent over core outcomes is bounded as $O^*(1/n^{1/K})$, where polylogarithmic factors have been suppressed. Further, we show that this bound is tight in worst case. We also provide a tighter bound under more restrictive assumptions. Our results provide partial justification for the typical assumption of a unique core outcome in empirical studies.