Greedy Allocations and Equitable Matchings

TL;DR

This paper introduces a new method for characterizing and approximating the set of feasible allocations in matching problems, especially with capacity constraints, and explores implications for mechanism design and equity.

Contribution

It develops a novel approach to interim realizability, provides approximate characterizations in complex settings, and links these to greedy algorithms and mechanism design strategies.

Findings

Necessary conditions for interim realizability identified.

Scaled allocations by 1/2 are sufficient for certain characterizations.

Deferred acceptance can guarantee at least half of the optimal payoff.

Abstract

I provide a novel approach to characterizing the set of interim realizable allocations, in the spirit of Matthews (1984) and Border (1991). The approach allows me to identify precisely why exact characterizations are difficult to obtain in some settings. The main results of the paper then show how to adapt the approach in order to obtain approximate characterizations of the interim realizable set in such settings. As an application, I study multi-item allocation problems when agents have capacity constraints. I identify necessary conditions for interim realizability, and show that these conditions are sufficient for realizability when the interim allocation in question is scaled by 1/2. I then characterize a subset of the realizable polytope which contains all such scaled allocations. This polytope is generated by a majorization relationship between the scaled interim allocations and allocations induced by a certain ``greedy algorithm''. I use these results to study mechanism design with equity concerns and model ambiguity. I also relate optimal mechanisms to the commonly used deferred acceptance and serial dictatorship matching algorithms. For example, I provide conditions on the principal's objective such that by carefully choosing school priorities and running deferred acceptance, the principal can guarantee at least half of the optimal (full information) payoff.