Some Neglected Axioms in Fair Division

by John W. Pratt

Overview — This paper considers allocation and bargaining problems, and introduces conditions that one might expect fair procedures to satisfy. However, not all conditions one might hope for can be satisfied simultaneously. Furthermore, some apparently plausible and widely proposed axioms and procedures have consequences whose undesirability clearly goes far beyond what can be excused in this way. Thus pitfalls lurk in the field of fair division. Key concepts include:

The first condition, "nondiscrimination," asserts that, in an allocation problem, if two agents receive probability shares of the same item and no chance of any other, then their shares should be proportional to their entitlements.
The remaining, "monotonicity" conditions apply to two agents and assert that a change in the feasible set that increases the utility cost to one agent of providing any given utility gain to the other should not hurt the first agent, or at least the solution should not change.

Author 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 (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.