A Concept of Constrained Egalitarianism in Fair Division Problems

This paper is really a modification of a paper by Diamantaras and Thomson (1990). In that paper, the no-envy concept due to Foley (1967) was refined to accommodate some kind of a radial) no-envy comparison, inspired by Chaudhuri (19986). Simply put, each person compares his/her own consumption bundle with all possible radial expansions and contractions of every other person’s consumption bundle. A weakly Pareto Optimal allocation which is envy free against such a maximal expansion is the one selected by Diamantaras and Thomson (1990). Our framework differs from the Diamantaras and Thomson (1990) framework in that we consider only the pure exchange situation. With the possibility of quantity constraints on consumption. Thus, since such technical issues with regard to existence of envy free allocation in the sense of Foley (1967) are somewhat secondary (though present) in our framework, we view this no-envy concept as a new equity criterion. In this framework, we prove the Diamantaras and Thomson (1990) result regarding the existence of an envy free allocation on a somewhat larger domain of preferences. We also feel that our existence proof is much simpler than the one due to the two authors, although it is difficult to say whether our proof would extend to the economies with production as studied by them.