Competitive Equilibria with Indivisible Goods and Generic Budgets: Settling the Open Cases

We study competitive equilibria in the basic Fisher market model, but with indivisible goods. Such equilibria fail to exist in the simplest possible market of two players with equal budgets and a single good, yet this is a knife’s edge instance as equilibria exist once budgets are not precisely equal. Is nonexistence of equilibria also a knife-edge phenomenon in complex markets with multiple goods? Our computerized search has indicated that equilibria often exist when budgets are “generic”. We prove several existence results both for the case of general preferences and for the special case of additive preferences, and relate competitive equilibria to notions of fair allocation of indivisible items. ∗Part of this work was done at Microsoft Research, Herzliya. The authors wish to thank Hervé Moulin and Ariel Procaccia for very helpful conversations on the fairness questions that arise in this research. The authors also thank Ariel Procaccia and Nisarg Shah for their kind permission to use Spliddit data and for providing such data. We thank Erel Segal-Halevi for his comments and his simplified proof of Proposition 3.2. We also thank Eric Budish for raising the issue of identical items for additive preferences. †Microsoft Research, [email protected]. ‡The Hebrew University of Jerusalem and Microsoft Research, [email protected]. §The Hebrew University of Jerusalem, [email protected]. ar X iv :1 70 3. 08 15 0v 1 [ cs .G T ] 2 3 M ar 2 01 7