Nash Social Welfare Approximation for Strategic Agents

The fair division of resources among strategic agents is an important age-old problem that has led to a rich body of literature. At the center of this literature lies the question of whether there exist mechanisms that can implement fair outcomes, despite the agents' strategic behavior. A fundamental objective function used for measuring the fairness of an allocation is the geometric mean of the agents' values, known as the Nash social welfare (NSW). This objective function is maximized by widely known solution concepts such as Nash bargaining and the competitive equilibrium with equal incomes. In this work we focus on the question of (approximately) implementing this objective. The starting point of our analysis is the Fisher market, a fundamental model of an economy, whose benchmark is precisely the (weighted) Nash social welfare. We begin by studying two extreme classes of valuations functions, namely perfect substitutes and perfect complements, and find that for perfect substitutes, the Fisher market mechanism yields a constant approximation: at most 2 and at least e1/e (≈ 1.44). However, for perfect complements, the Fisher market mechanism does not work well, its bound degrading linearly with the number of players. Strikingly, the Trading Post mechanism---an indirect market mechanism also known as the Shapley-Shubik game---has significantly better performance than the Fisher market on its own benchmark. Not only does Trading Post achieve an approximation of 2 for perfect substitutes, but this bound holds for any concave utilities, and it becomes essentially optimal for perfect complements, where it reaches (1+ε) for any ε>0. Moreover, we show that all the Nash equilibria of the Trading Post mechanism are pure (hence the approximation factors extend to all Nash equilibria), and satisfy an important notion of individual fairness known as proportionality.