Fair (and Not So Fair) Division

by John W. Pratt

Overview — "Fair" could be defined as what people of good will would want to be. This does not constitute an operational definition, however. This paper provides a specific procedure to calculate what could be considered fair and reasonable for various situations that require a fair division. A simple example would be a family that has inherited objects of artistic and/or sentimental value and wants to divide them up fairly while taking into account differences in taste. Laymen, mathematicians, and economists all have their own proposals for creating a fair division. Pratt suggests a procedure that, when put to the test of a range of examples, produces outcomes that accord with our intuitive sense of what is fair and desirable while previously proposed procedures do not. Key concepts include:

The procedure measures the value of each object in terms of its desirability to the various participants. It allocates the objects so that the participants receive the same total value (or value proportional to their entitlements if they are unequal), without envy or waste ("money left on the table"). Randomization is used if needed to accomplish this.
Many procedures work well on average problems. Indeed, all reasonable procedures are much alike in near-symmetric problems. It is the lopsided examples that test the procedures, especially with more than two participants.
Participants are not penalized for receiving objects of no value to anyone else or for being honest about their values for such objects.

Author Abstract

Drawbacks of existing procedures are illustrated and a method of efficient fair division is proposed that avoids them. Given additive participants' utilities, each item is priced at the geometric mean (or some other function) of its two highest valuations. The utilities are scaled so that the market clears with the participants' purchases proportional to their entitlements. The method is generalized to arbitrary bargaining sets and existence is proved. For two or three participants, the expected utilities are unique. For more, under additivity, the geometric mean separates the prices where uniqueness holds and where it fails; it holds for the geometric mean except in one case where refinement is needed.