Adaptive maximization of social welfare
Nicolo Cesa-Bianchi Roberto Colomboni Maximilian Kasy
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).