Tempered Utilitarianism, Tempered Egalitarianism, and the Nash Bargaining Solution

Abstract: A solution to Nash’s bargaining problem satisfies utilitarian monotonicity (egalitarian monotonicity) if it makes the utilities-sum (utilities-minimum) non-decreasing with respect to set inclusion. There does not exist a solution that satisfies utilitarian monotonicity, midpoint-domination, and conflict-freeness. The last axiom is the requirement that the solution selects the ideal point when doing so is feasible. An analogous impossibility result holds for egalitarian monotonicity. Distributive justice monotonicity is the requirement that when the feasible set expands, it is not the case that both the utilitarian and egalitarian objectives weakly decrease, with (at least) one of these decreases being strict. Based on distributive justice monotonicity, an impossibility result is derived for bargaining among n ≥ 3 players, and a characterization of the Nash bargaining solution is derived for n = 2 players.