Adaptive maximization of social welfare

TL;DR

This paper develops algorithms for repeatedly selecting policies to maximize social welfare, balancing private utility and public revenue, with theoretical bounds on regret and extensions to taxation scenarios.

Contribution

It introduces a novel approach to welfare maximization with indirect utility inference, providing regret bounds and optimal algorithms for both stochastic and adversarial settings.

Findings

Regret grows as T^{2/3} in adversarial setting

Optimal rate of T^{1/2} achieved in stochastic setting with concavity

Algorithm outperforms traditional multi-armed bandit approaches

Abstract

We consider the problem of repeatedly choosing policies to maximize social welfare. Welfare is a weighted sum of private utility and public revenue. Earlier outcomes inform later policies. Utility is not observed, but indirectly inferred. Response functions are learned through experimentation. We derive a lower bound on regret, and a matching adversarial upper bound for a variant of the Exp3 algorithm. Cumulative regret grows at a rate of $T^{2/3}$. This implies that (i) welfare maximization is harder than the multi-armed bandit problem (with a rate of $T^{1/2}$ for finite policy sets), and (ii) our algorithm achieves the optimal rate. For the stochastic setting, if social welfare is concave, we can achieve a rate of $T^{1/2}$ (for continuous policy sets), using a dyadic search algorithm. We analyze an extension to nonlinear income taxation, and sketch an extension to commodity taxation. We compare our setting to monopoly pricing (which is easier), and price setting for bilateral trade (which is harder).