Fair Shares: Feasibility, Domination, and Incentives

We consider fair allocation of indivisible goods to n equally entitled agents. Every agent i has a valuation function v_i from some given class of valuation functions. A share s is a function that maps (v_i, n) to a nonnegative value. A share is feasible if for every allocation instance, there is an allocation that gives every agent i a bundle that is acceptable with respect to v_i, one of value at least her share value s(v_i, n). We introduce the following concepts. A share is self-maximizing if reporting the true valuation maximizes the minimum true value of a bundle that is acceptable with respect to the report. A share s ρ-dominates another share s^′ if s(v_i, n) ≥ρ · s^′ (v_i, n) for every valuation function. We initiate a systematic study of feasible and self-maximizing shares and a systematic study of ρ-domination relation between shares, presenting both positive and negative results.