On the Fairness Literature: Comment.

• Author: Berliant, Marcus
• Position: Fair allocation

Marcus Berliant [*]

Karl Dunz [+]

William Thomson [++]

We offer a guide to the recent literature on the problem of fair allocation, and we present various concepts that have been developed in this literature and the various economic environments in the context of which they have been analyzed.

1. Introduction

   In a recent communication to this journal, Holcombe (1997) raised a number of criticisms of the theory of fairness. Although we will certainly not claim that this theory is in a completely satisfactory state, a considerable literature has appeared over the last 20 years in which the concerns that Holcombe expressed have been addressed on multiple occasions. In a number of important cases, it is now well understood how they can be resolved or the extent to which they can be accommodated. We offer a short guide to the relevant papers that should help introduce the reader to this body of work.

2. Limitations of the Fairness Literature

   Holcombe opens with the following quotation from our earlier work (Berliant, Dunz, and Thomson 1992): "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 (Foley 1967), that is, an allocation such that nobody prefers what someone else receives to what he receives" (Holcombe 1997, p. 797).

   He then proceeds with the following criticisms:

   (a) An allocation may be envy-free but violates "fairness." He gives the example of the two-person divide-and-choose scheme (one agent divides and the other chooses). This game generates an envy-free allocation, but it does not seem to treat the two agents symmetrically enough. Indeed, it gives an advantage to the divider (if he knows the chooser's preferences).

   (b) Some agents may deserve more than others, and envy might well be justified then.

   (c) Random devices may appear fair, but once implemented, they are not.

   (d) The fairness of the process that leads to the final allocations should not be ignored.

3. No-Envy Is Not the Only Concept that Has Been Analyzed

   In reply to point a, that divide-and-choose suffers from a basic asymmetry, we will first note that this difficulty has been known for many years (see, for example, Kolm 1972 and Crawford 1977). It is in fact one of Crawford's motivations in his subsequent work (Crawford 1979).

   The comment does bring up an interesting conceptual issue, which has broader relevance than to the theory of fairness, namely, is an allocation rule as good as any of its selections? That very example (divide-and-choose) is used by Thomson (1996a) to make the point that the answer is, in general, no. [1] When the concept of an allocation rule is properly understood as a mapping from a domain of economies to some allocation space, and in particular when a rule has been chosen on the basis of its behavior in response to changes in the parameters of the economy under consideration, a rule is not equivalent to its selections. In the case of divideand-choose, this argument can be illustrated most simply by bringing up the familiar property of anonymity, that is, invariance of a rule under "renamings" of agents. Obviously, the equilibrium correspondence of divide-and-choose is not an anonymous solution even though the noenvy solution itself satisfies this property.

   Moreover, Holcombe's exclusive focus on the no-envy solution is not warranted. A number of other criteria have indeed been proposed and analyzed in detail. In Berliant, Dunz, and Thomson (1992), we also discuss the notion that is arguably the main alternative to no-envy, namely "egalitarian equivalence" (Pazner and Schmeidler 1978). An allocation is egalitarian equivalent if there is a "reference bundle" that each agent finds indifferent to the bundle he receives. Since the publication of our paper, the inventory of available concepts has been further enriched. Here is a brief survey of the ones we consider to be the most important.

   * For models in which equal division is meaningful, tests on allocations have been formulated based on comparisons to this natural reference point (the equal division lower bound, the equal division core; see Moulin 1995 or Thomson 1999c for references).

   * Various selections from egalitarian equivalence have recently played a prominent role in the literature. They are defined by placing certain restrictions on the reference bundle. Which restrictions are the most interesting depends on the class of economies under consideration, in particular whether public goods are present and on the properties of the resulting allocation rule (Moulin 1987b, c).

   * The central idea underlying egalitarian equivalence, namely that an allocation can be meaningfully evaluated by reference to hypothetical...