The combinatorial assignment problem: approximate competitive equilibrium from equal incomes

TL;DR: A solution to the combinatorial assignment problem by proposing two new criteria of outcome fairness, the maximin share guarantee and envy bounded by a single good, which weaken well-known criteria to accommodate indivisibilities and formalize why dictatorships are unfair.

Abstract: Impossibility theorems suggest that the only efficient and strategyproof mechanisms for the problem of combinatorial assignment - e.g., assigning schedules of courses to students - are dictatorships. Dictatorships are mostly rejected as unfair: for any two agents, one chooses all their objects before the other chooses any. Any solution will involve compromise amongst efficiency, incentive and fairness considerations.This paper proposes a solution to the combinatorial assignment problem. It is developed in four steps. First, I propose two new criteria of outcome fairness, the maximin share guarantee and envy bounded by a single good, which weaken well-known criteria to accommodate indivisibilities; the criteria formalize why dictatorships are unfair. Second, I prove existence of an approximation to Competitive Equilibrium from Equal Incomes in which (i) incomes are unequal but arbitrarily close together; (ii) the market clears with error, which approaches zero in the limit and is small for realistic problems. Third, I show that this Approximate CEEI satisfies the fairness criteria. Last, I define a mechanism based on Approximate CEEI that is strategyproof for the zero-measure agents economists traditionally regard as price takers. The proposed mechanism is calibrated on real data and is compared to alternatives from theory and practice: all other known mechanisms are either manipulable by zero-measure agents or unfair ex-post, and most are both manipulable and unfair.