Competitive Fair Division under linear preferences

The fair division of a bundle of goods (manna) among agents with heterogenous preferences is an important challenge for normative economic analysis. The difficulty is to combine effi ciency (Pareto optimality) with some convincing notion of fairness. When agents are responsible for their ordinal preferences, and cardinal measures of utility are not relevant, the division rule favored by economist is the Competitive equilibrium with equal incomes, thereafter CEEI, invented almost 50 years ago (Varian, Kolm). Here we discuss this rule in the important special case where individual preferences are linear. This means that the goods are perfect substitutes and each participant in the division needs only report p − 1 rates of substitution if there are p goods to share. This assumption is less drastic than may appear. On the one hand it is the only practical approach in several free web sites (Adjusted Winner, Spliddit) computing fair solutions for concrete division problems: siblings are sharing family heirlooms, partners divide the common assets upon dissolving their partnership, and so on. They are asked to distribute 100 points over the (divisible) goods, and the resulting distribution is interpreted as an additive utility function representing the underlying linear preferences. Anything more sophisticated involving comparisons of subsets of goods is impractical when we have more than a couple of goods. On the other hand perfect substitutability is realistic when we divide inputs into a production process such as land, machines with same function but different specifications, hours of work with different skills, computing resources, etc.. One critical property of the CEEI rule when preferences are linear (or more generally homothetic) is that it maximizes the product of the canonical linear (homothetic) utility functions over all feasible allocations (Gale). Therefore the