Sensitivity of bargaining solutions to set curvature

This paper studies the distortion of the asymmetric Nash and Kalai-Smorodinsky bargaining solutions that arise from differences in the shape of the bargaining set. In a two-person bargaining framework, we compare outcomes across different feasible sets for fixed bargaining power and repeat this for all possible power constellations between the players. Two arbitrary bargaining sets are selected, and the distortion of the asymmetric Nash solution, the symmetric Kalai-Smorodinsky solution, and the Nash solution with a shifted disagreement point is measured relative to the asymmetric Kalai-Smorodinsky solution, which serves as a reference. We compare pairs of bargaining problems that differ only in curvature and analyze how outcomes vary across solutions. This yields a quantitative measure of relative distortion and shows how sensitive bargaining solutions are to changes in set structure. For all solutions under observation, a weaker (stronger) player always prefers the more (less) curved bargaining set, as the distortion increases (decreases) in their favor. Indifference between two sets occurs when the distortion is equal for a given power constellation. For the asymmetric Nash and Kalai-Smorodinsky solutions, indifference occurs exactly when the solution points exhibit the same slope on their respective Pareto frontiers. Finally, the number of indifference points is always odd if one bargaining set contains the other. The results highlight how the shape of the Pareto frontier can introduce additional distortion in already unequal situations, suggesting that the structure of the bargaining set plays a crucial role in determining the fairness of outcomes.