The complexity of fairness through equilibrium

Competitive equilibrium with equal incomes (CEEI) is a well-known fair allocation mechanism [Foley67:Resource, Varian74: Equity, Thomson85:Theories]; however, for indivisible resources a CEEI may not exist. It was shown in Budish [2011] that in the case of indivisible resources there is always an allocation, called A-CEEI, that is approximately fair, approximately truthful, and approximately efficient, for some favorable approximation parameters. This approximation is used in practice to assign business school students to classes. In this paper we show that finding the A-CEEI allocation guaranteed to exist by Budish's theorem is PPAD-complete. We further show that finding an approximate equilibrium with better approximation guarantees is even harder: NP-complete.