Nondiscrimination and monotonicity in fair division

Abstract

Conditions one might impose on fair allocation procedures are introduced. Nondiscrimination requires that agents share an item in proportion to their entitlements if they receive nothing else. The “price” procedures of Pratt (J. Risk Uncertain. 35:203–236, 2007), including the Nash bargaining procedure, satisfy this. Other prominent efficient procedures do not.

In two-agent problems, reducing the feasible set between the solution and one agent’s maximum point increases the utility cost to that agent of providing any given utility gain to the other and is equivalent to decreasing the dispersion of the latter’s values for the items he does not receive without changing their total. One-agent monotonicity requires that such a change should not hurt the first agent, limited monotonicity that the solution should not change. For prices, the former implies convexity in the smaller of the two valuations, the latter linearity. In either case, the price is at least their average and hence spiteful.