Efficiency and the Redistribution of Welfare

We examine how the relation between individual and social utility affect the efficiency of game-theoretic solution concepts. We first provide general results for monotone utility-maximization games, showing that if each player’s utility is at least his marginal contribution to the welfare, then the social welfare in any Strong Nash Equilibrium is at least half of the optimal. The efficiency degrades smoothly as the marginal contribution assumption is relaxed. For non-monotone utility maximization games, we manage to give efficiency results if the game is also a potential game. We also extend previous results on efficiency of Nash Equilibria for the case when social welfare is submodular. We then focus on Effort Market Games, a general cooperative setting where players exert effort in projects and then the produced value is redistributed to them using some simple local mechanism. Our results for general utility maximization games are instrumental in reasoning about efficiency in Effort Market Games. We show that in a concave environment, splitting locally according to the Shapley value achieves globally at least half of the optimal social welfare at any Nash Equilibrium of the game (our proof is of a smoothness type and hence generalizes also to no-regret learning outcomes). We generally characterize the properties that a distribution mechanism has to satisfy to achieve good efficiency. Moreover, we show that equal splitting of the value works well only when the projects have small number of participants. In a convex environment, Nash Equilibria can have unbounded inefficiency regardless of the mechanism. Hence, we focus on coalitionaly robust solution concepts that allow for group deviations of the players a natural assumption in such a cooperative setting. We show that every local mechanism is bound to have an inefficiency of at least the maximum number of participants per project at the worst Strong Nash Equilibrium. Further, both the Shapley value mechanism and the equal splitting mechanism achieve this lower bound. We also draw strong connections of our results with the axiomatic surplus sharing literature.