Absence of envy does not imply fairness.

Holcombe, Randall G.

I. 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.

II. 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. These are not necessarily the best strategies for the divider, but they do illustrate that this division procedure systematically favors the divider over the chooser.(3) If a procedure systematically favors one participant over another, then it could not be considered fair, in the way that the term fair is normally used, even though it is envy-free.

III. Is This Procedure Fair?

If fairness is defined as the absence of envy, then by definition the procedure would be fair, even though one participant is systematically favored over the other. This reinforces the point that the absence of envy is not an appropriate definition for fairness. As Rawls [13] uses the term fair, it would be hard to imagine that from behind a veil of ignorance individuals would agree to a procedure that would systematically benefit some over others after the veil was lifted. Indeed, it was just this type of systematic advantage that the veil of ignorance construct was intended to eliminate.(4)

Luce and Raiffa [10] discuss this game of fair division and note what they call the divider's advantage. They advocate selecting the divider by some unbiased procedure, such as a flip of a coin, in order to analyze the properties of this method of division further. Similarly, Crawford [7, 238] shows that an individual is at least as well off being the divider as the chooser, yet concludes [7, 245] that this is a process of fair division, without further addressing the divider's advantage. Holcombe [9, 1155] also remarks on the divider's advantage, yet the correspondence between fair outcomes and envy-free outcomes has continued to be accepted in the literature, perhaps because in none of these cases was the issue directly addressed.

The divider-chooser method of "fair" division focuses directly on this issue. There is no production, and no participant is any more deserving than any other participant. The outcome is envy-free. Yet one participant is systematically favored over the other. Is this fair? The division of heterogeneous bundles of land is often used to illustrate this method of division. Assume that two brothers inherit a tract of heterogeneous land that is to be divided between them. The land contains some very developable land near a main road and a secluded area with a nice fishing pond somewhere else on the tract. One brother has expressed an interest in developing the property near the road, while the other more contemplative brother loves to fish at the secluded pond. If the contemplative brother were the divider, he would place more land in the parcel with the pond, confident that his brother would pick the land that could be readily developed, whereas if the other brother were the divider, he would place more land in the parcel near the road, confident that the contemplative brother would prefer to retain ownership of the secluded lake. Would this method of division really be considered fair when we know in advance that it is biased to favor one of the participants over the other?

The right to be the divider could be arbitrarily determined, perhaps by the flip of a coin as Luce and Raiffa [10] have suggested. But one would be hard-pressed to think of even this as fair because one brother would benefit relative to the other based on an arbitrary event. In order to analyze this more completely, one must separate the fairness of the coin toss from the fairness of the divider's advantage that is assigned by the coin toss. Then one can see that an unbiased event - the coin toss - determines which player is advantaged relative to the other. It is the coin toss that is unbiased, and presumably fair, but the divider-chooser algorithm still favors one participant over the other. If one simply wants to assign an advantage in an unbiased way, then the brothers could flip a coin and give all of the property to the winner, leaving the other with nothing.(5) Before the coin toss, neither brother has an advantage over the other, so the expected outcome is the same either way.

Nobody in the literature has suggested that a fair way to divide an estate between two individuals is to give the entire estate to one of them based on a random event such as a coin toss. Why would this not be fair? One reason might be that if the individuals are risk-averse, ex ante they would both prefer a method of division that would produce a lower variance. But if one would not want to use the flip of a coin to assign all of the property to one of the individuals, then the same objections should apply to using a coin toss to assign the divider's advantage to one of the individuals. The divider-chooser method produces an outcome that is envy-free, unlike giving all of the property to one individual, but it is still unfair because it is biased in favor of one of the participants.

Baumol [2, 1161] has referred to this as a philosophical and linguistic issue, and he is undoubtedly right. For most of history, people accepted their inherited stations in life as fair, even though they were not envy-free as the contemporary literature defines it. It is equally reasonable to accept envy-free outcomes as fair, even though some people are systematically favored over others. When confronted with an example like that illustrated in Figure 1, however, perhaps some who were willing to accept without question that envy-free allocations are fair will question whether they would be willing to defend that proposition in every one of its potential real-world applications. Certainly envy-free allocations can be fair, but the divider's advantage in the above example illustrates that envy-free allocations are not necessarily fair. If this is true, then freedom from envy cannot be the criterion for determining fairness.

IV. What Is the Problem with That Definition?

If one is willing to question whether freedom from envy implies fairness, then it is worth analyzing why that is not a good definition of fairness. The example above helps illustrate the problem. In most cases, envy-free allocations are likely to be fair, unless some individuals deserve more than others. But in some cases, like the one above, an envy-free allocation can be generated that treats some people (the choosers) unfairly. The reason is that the process that generated the outcome was unfair. In most real-world distributional issues, the concept of fairness will be obscured if one tries to judge fairness only by looking at the outcome, without examining the process that leads to the outcome.

Assume that one watches a football game in which the outcome is 48-7. Was the game fair? One would naturally be inclined to look beyond the score to see if the officiating was fair, if both teams wanted to play the game, and so forth before passing judgment. Is the distribution of income fair? Is the burden of taxes fairly distributed? Are the prices of goods and services fairly determined? Again, one cannot judge the fairness of the outcome without judging the fairness of the process. This principle is illustrated by showing that envy-free outcomes are not necessarily fair.

The idea that fair outcomes are the result of fair processes has been discussed before. Rawls [13] developed his well-known device of the veil of ignorance to represent a fair procedure. Rawls goes further, to suggest some fair outcomes, such as the maximin rule (which would maximize the well-being of the least well-off individual), that he believes would be the result of fair procedures. Nozick [11] is critical of Rawls on this count, and develops a pure procedural theory of justice, where the fairness of the outcome is strictly judged by the fairness of the procedure. Rawls and Nozick consider more complex environments where individuals differ in their productive abilities, and where merit can play a role in the fairness of outcomes. The divider-chooser example discussed above shows that even in a very simple purely distributional environment where there is no production and nobody deserves any more than anyone else, the fairness of an outcome still cannot be judged without looking at the procedure that produced the outcome. This is especially relevant to an ongoing literature that uses the characteristic of freedom from envy as a criterion for determining whether an outcome is fair.

The problem with using freedom from envy as a criterion for fairness is that the criterion examines only the outcome of the process rather than the process itself. Any equity criterion that proposes to look only at an outcome will suffer the same flaw. One cannot judge the equity of an income distribution by looking at a Gini coefficient or a Lorenz curve, for example, because those measures of income inequality look only at the outcome, without considering the process that produced the outcome.(6) Similarly, one cannot judge the fairness of prices without considering the process that created those prices.(7) Fair outcomes are the result of fair processes.(8)

V. Conclusion

The processes that produce prices or income distributions are complex and involve aspects of luck, hard work, and differing initial endowments, among other factors, and one might be tempted to conclude that these complicating factors obscure the merits of using freedom from envy as an equity criterion. For that reason, the study of the divider-chooser game examined above is especially illuminating. In the divider-chooser game nobody deserves more than anybody else, and nobody produces anything, so merit and effort - important factors in most real-world distributional issues - are irrelevant. Still, in a purely distributional context, the distributional process can be unfair, as in the divider-chooser game, leading to an envy-free but unfair outcome.

As the quotation at the beginning of this paper states, there is a well-developed literature on equity concepts that uses freedom from envy as its criterion for fairness. However, freedom from envy is neither necessary nor sufficient for fairness. It is not necessary because in many circumstances some people deserve more than others. The divider-chooser game shows that it is not sufficient for fairness even when merit is not an issue. The absence of envy does not imply fairness.

Randall G. Holcombe Florida State University Tallahassee, Florida

The author gratefully acknowledges helpful comments from Bruce Benson, Steven Brams, Steven Caudill, Barry Hirsch, David Macpherson, David Rasmussen, Kevin Reffett, Tim Sass, Russell Sobel, and an anonymous reviewer of this Journal.

1. Steinhaus [14] refers to an envy-free solution to a fair division problem that is centuries old, and Baumol [3] gives some additional history to the idea. The history of problems of fair division that build on the concept of freedom from envy is covered well by Brams and Taylor [6].

2. This solution was not devised by Brams and Taylor, but has a long history. This divider-chooser method is exactly the one analyzed by many others [7; 10; 14]. Brams and Taylor [6] place this problem in a more complete context, both by illustrating its shortcomings and by analyzing other methods of fair division.

3. By looking at the diagram, one can see that it would be possible for the divider to choose points A and B such that the chooser would be worse off at either point than if the two goods were equally divided. That is really beside the point, however. The important thing is that the procedure systematically favors one of the participants (the divider) over the other (the chooser).

4. This reference to Rawls only refers to his procedure of obtaining agreement from behind a veil of ignorance, and not to the outcomes, such as the maximin criterion, that Rawls conjectured would be agreed to from behind the veil of ignorance. The distinction between procedures and outcomes in Rawls is discussed further below.

5. Giving all of the property to one party easily solves another problem dealt with in this literature. Ideally, economists would like the division to be Pareto optimal in addition to fair, and giving all the property to one party is Pareto optimal. In many circumstances, such as dividing up property among heirs, a fair outcome is probably more important than a Pareto optimal outcome because after a fair division participants can trade to make the outcome Pareto optimal.

6. Certainly one would be justified in looking at Gini coefficients and Lorenz curves as suggestions that the process that produced the income distribution might not be fair, but looking at those outcome measures alone would not be sufficient to judge the fairness of the distribution of income.

7. Baumol [2] applies the criterion of freedom from envy to evaluate the fairness of prices and rationing schemes.

8. In order to concentrate on the problem with freedom from envy as a definition of fairness, this paper suggests no alternatives other than the general one of selecting a process that all participants agree is fair. Brams and Taylor [6] discuss a number of interesting alternatives in problems of fair division that are not biased toward one participant, and extend their discussion to problems involving division among more than two individuals.

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