Competitive equilibrium for almost all incomes

Competitive equilibrium from equal incomes (CEEI) is a well-known rule for fair allocation of resources among agents with different preferences. It has many advantages, among them is the fact that a CEEI allocation is both Pareto efficient and envy-free. However, when the resources are indivisible, a CEEI allocation might not exist even when there are two agents and a single item. In contrast to this discouraging non-existence result, Babaioff, Nisan and Talgam-Cohen (2017) recently suggested a new and more encouraging approach to allocation of indivisible items: instead of insisting that the incomes be equal, they suggest to look at the entire space of possible incomes, and check whether there exists a competitive equilibrium for almost all income-vectors (CEFAI) - All income-space except a subset of measure zero. They show that a CEFAI exists when there are at most 3 items, or when there are 4 items and two agents. They also show that when there are 5 items and two agents there might not exist a CEFAI. They leave open the cases of 4 items with three or four agents. This paper presents a new way to implement a CEFAI, as a subgame-perfect equilibrium of a sequential game. This new implementation allows us both to offer much simpler solutions to the known cases (at most 3 items, and 4 items with two agents), and to prove that a CEFAI exists even in the much more difficult case of 4 items and three agents. Moreover, we prove that a CEFAI might not exist with 4 items and four agents. Thus, this paper completes the characterization of CEFAI for monotone preferences.