Welfare egalitarianism in surplus-sharing problems and convex games

We show that the constrained egalitarian surplus-sharing rule, which divides the surplus so that the poorer players’ resulting payoffs become equal but not larger than any remaining player’s status quo payoff, is characterized by Pareto optimality, path independence, both well-known, and less first (LF), requiring that a player does not gain if her status quo payoff exceeds that of another player by the surplus. This result is used to show that, on the domain of convex games, Dutta-Ray’s egalitarian solution is characterized by aggregate monotonicity (AM), bounded pairwise fairness, resembling LF, and the bilateral reduced game property (2-RGP) à la Davis and Maschler. We show that 2-RGP can be replaced by individual rationality and bilateral consistency à la Hart and Mas-Colell. We prove that the egalitarian solution is the unique core selection that satisfies AM and bounded richness, requiring that the poorest players cannot be made richer within the core. Replacing “poorest” by “poorer” allows to eliminate AM.