Fair Allocation through Competitive Equilibrium from Generic Incomes

Two food banks catering to populations of different sizes with different needs must divide among themselves a donation of food items. What constitutes a "fair" allocation of the items among them? Competitive equilibrium from equal incomes (CEEI) is a classic solution to the problem of fair and efficient allocation of goods among equally entitled agents [Foley 1967, Varian 1974]. Every agent (foodbank) receives an equal endowment of artificial currency with which to "purchase" bundles of goods (food items). Prices for the goods are set high enough such that the agents can simultaneously get their favorite within-budget bundle, and low enough such that all goods are allocated (no waste). A CEEI satisfies mathematical notions of fairness like fair-share, and also has built-in transparency -- prices can be published so the agents can verify they're being treated equally. However, a CEEI is not guaranteed to exist when the items are indivisible. We study competitive equilibrium from generic incomes (CEGI), which is based on the idea of slightly perturbed endowments, and enjoys similar fairness, efficiency and transparency properties as CEEI. We show that when the two agents have almost equal endowments and additive preferences for the items, a CEGI always exists. We then consider agents who are a priori non-equal (like different-sized foodbanks); we formulate a new notion of fair allocation among non-equals satisfied by CEGI, and show existence in cases of interest (like when the agents have identical preferences). Experiments on simulated and Spliddit data (a popular fair division website) indicate more general existence. Our results open opportunities for future research on fairness through generic endowments, and on fair treatment of non-equals.