Absence of envy does not imply fairness.

• Author: Holcombe, Randall G.

1. Introduction

   Berliant, Thompson, and Dunz [4, 202] accurately state, "There now exists in economics a well-developed literature devoted to the formulation and the analysis of equity concepts. The concept that has played the central role is that of an envy-free allocation, that is, an allocation such that nobody prefers what someone else receives to what he receives." The idea that envy-free divisions are fair has been promoted by many authors [1; 3; 8; 14] in addition to Berliant, Thompson, and Dunz. This concept has a long history, but is still current.(1) Even if this definition of envy is accepted, the equation of lack of envy to fairness is fundamentally flawed because it judges the fairness of the outcome without considering the procedure that produced the outcome. Fair outcomes are outcomes that are produced by fair procedures, and envy-free outcomes may not be fair, if they are produced by unfair procedures.

   A number of factors can lead to envy-free outcomes that are not fair. Most obviously, by some measure of merit (such as working harder), some people may deserve more than others, which would mean that fair allocations are not envy-free, and envy-free allocations are not fair. Furthermore, the literature on the subject shows that when the economy includes production as well as distribution, the concept of an envy-free outcome is not so clearly defined [12; 15]. At first it might seem that these are complications that obscure the fairness of envy-free divisions, and that in the absence of overruling factors, envy-free distributions would be fair. In order to avoid these problems initially, the next section considers an example of an unfair envy-free outcome in a purely distributional setting without production, and where no participant is any more deserving than any other. The problem is simply one of fair division.

2. An Example

   Consider the simplest case of a more general problem of fair division discussed by Brams and Taylor [6]. Two people must divide a bundle of two goods between them. The Brams and Taylor solution is for one of them to divide the bundle in two, and then allow the other to choose which of the two bundles he would rather have.(2) The solution is envy-free because the chooser gets the bundle he most prefers, and the divider has an incentive to divide the bundles so that she is not left with a bundle she prefers less than the one the chooser takes. The conventional wisdom, typified by the quotation that opened the paper, is that because the result is envy-free, it is fair. A simple example can show that it is not.

   Consider two individuals, Caruso and Tuesday, dividing two goods, bananas and coconuts. Assume that the individuals know each other so well that they know exactly each other's utility functions. Playing this game of division, Tuesday divides the bananas and coconuts into two bundles and Caruso chooses the bundle he most prefers. The process is depicted in the Edgeworth box diagram in Figure 1. Tuesday could simply put half the bananas and half the coconuts in each pile, placing them at point M, but there is another strategy she can follow that can make her better off. Knowing Caruso's utility function, she knows that he loves coconuts, whereas Tuesday relatively prefers bananas. Thus, Tuesday divides the bundles so that one has more coconuts and the other more bananas, like point A, such that Caruso is on the same indifference curve as he would have been had the goods been divided equally. At point A, Tuesday is better off than an equal division.

   If Caruso were to choose the bundle Tuesday intended for herself, that would place them at the other side of the Edgeworth box, at point B. Point B is found by extending a straight line from point A through point M such that the line segments AM and MB are equal in length. Thus, given Tuesday's division, Caruso's choice is point A or point B, and getting more utility at point A, Caruso chooses A.

   If Caruso were the divider and Tuesday the chooser, he could follow a similar strategy, giving Tuesday the choice between C and D. By the same logic Tuesday will choose C...