Implementability and Horizontal Equity Imply No-Envy

Abstract

THE REQUIREMENT OF NO-ENVY is at the heart of recent equity theory. An allocation is free from envy if no agent strictly prefers the bundle of goods which is assigned to another agent to the one she/he gets. An allocation rule satisfies No-Envy if it only selects envy-free allocations. In this paper, we examine the relationship between No-Envy and implementability in a general model. Our main result is that in monotonically closed domains the No-Envy property is satisfied by any allocation rule which is both horizontally equitable and Nash Implementable. The requirement of horizontal equity, called Equal Treatment of Equals, simply states that two agents having the same preferences should be treated equally, i.e., should be assigned the same welfare level. The monotonic closedness condition on the domain of admissible preferences is satisfied in many private and/or public good environments, as discussed below. Quasi-linear domains, however, are examples of nonmonotonically closed domains. Our result confirms the widespread intuition that the No-Envy requirement is justified not only from an equity point of view but also from an implementation standpoint (see Hammond (1979) and Champsaur and Laroque (1981)). Moreover, it throws some light on several previous results where specific allocation rules defined over monotonically closed domains are characterized by Nash Implementability among other axioms. As a consequence of our analysis, No-Envy can be weakened into Equal Treatment of Equals in these characterizations (see, e.g., Thomson (1990) and Nagahisa and Suh (1995)). Similar arguments apply to decentralization problems where informational efficiency is the primary concern. For instance, Calsamiglia and Kirman (1993) characterized the Equal Income Walrasian rule on the basis of informational efficiency, Pareto Optimality, and No-Envy. Again, No-Envy can be replaced by Equal Treatment of Equals in this result.2 On the other hand, our result also explains why Nash Implementable allocation rules violating No-Envy over monotonically closed domains all fail to satisfy Equal Treatment of Equals. Examples include the Lindahl solution, the ratio equilibrium solution (Kaneko (1977)) and the balanced linear cost share solution (Mas-Colell and Silvestre (1989)); see Corchon (1989) and Wilkie (1990). At the end of the paper, we show that if we restrict ourselves to allocation functions (that is, allocation rules selecting one and only one allocation per economy), then a similar result holds for Strategy-Proofness, provided the Satterthwaite-Sonnenschein (1981) property of Non-Bossiness is also imposed. That is, in monotonically closed