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Adaptive maximization of social welfare

Nicolo Cesa-Bianchi, Roberto Colomboni, Maximilian Kasy

TL;DR

This work studies adaptive policymaking aimed at maximizing a welfare-based objective when welfare is unobserved and must be inferred from counterfactual demand. It develops a tempered Exp3 algorithm that discretizes policy space and deliberately explores counterfactual policies, achieving a worst-case regret of order $T^{2/3}$ (up to a $\log^{1/3}T$ factor), and proves a matching lower bound, establishing rate-optimality. In the stochastic setting with concave welfare, it shows that regret improves to $O(T^{1/2})$ via a Dyadic Search procedure that maintains a shrinking active interval around the optimum. The paper also extends the framework to nonlinear income taxation and sketches a commodity-taxation extension, connecting welfare learning to classical economic models like Mirrlees-Saez and Ramsey, and discusses potential Bayesian alternatives such as Thompson sampling. These results have practical implications for data-driven, welfare-aware policy design under uncertainty and nonstationarity.

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).

Adaptive maximization of social welfare

TL;DR

This work studies adaptive policymaking aimed at maximizing a welfare-based objective when welfare is unobserved and must be inferred from counterfactual demand. It develops a tempered Exp3 algorithm that discretizes policy space and deliberately explores counterfactual policies, achieving a worst-case regret of order (up to a factor), and proves a matching lower bound, establishing rate-optimality. In the stochastic setting with concave welfare, it shows that regret improves to via a Dyadic Search procedure that maintains a shrinking active interval around the optimum. The paper also extends the framework to nonlinear income taxation and sketches a commodity-taxation extension, connecting welfare learning to classical economic models like Mirrlees-Saez and Ramsey, and discusses potential Bayesian alternatives such as Thompson sampling. These results have practical implications for data-driven, welfare-aware policy design under uncertainty and nonstationarity.

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 . This implies that (i) welfare maximization is harder than the multi-armed bandit problem (with a rate of 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 (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).
Paper Structure (66 sections, 11 theorems, 137 equations, 3 figures, 1 table, 5 algorithms)

This paper contains 66 sections, 11 theorems, 137 equations, 3 figures, 1 table, 5 algorithms.

Table of Contents

  1. Introduction
  2. Why welfare, why adversarial guarantees?
  3. Lower and upper bounds on regret
  4. A lower bound on stochastic regret
  5. A matching upper bound on adversarial regret for modified Exp3
  6. Improved stochastic bounds for concave social welfare
  7. Extensions to non-linear income taxation and to commodity taxation
  8. Roadmap
  9. Background and literature
  10. Optimal taxation
  11. Multi-armed bandits
  12. Adversarial decision-making
  13. Bandit approaches for economic problems
  14. Setup
  15. Notation
  16. ...and 51 more sections

Key Result

Theorem 1

Consider the setup of Section sec:setup. There exists a constant $C>0$ such that, for any randomized algorithm for the choice of $x_1, x_2, \dots$ and any time horizon $T \in \mathbb{N}$, the following holds.

Figures (3)

  • Figure 1: Construction for proving the lower bound on regret
  • Figure 2: Tempered Exp3 for Social Welfare -- Numerical example
  • Figure 3: Construction for proving the lower bound on regret for the concave case

Theorems & Definitions (18)

  • Theorem 1: Lower bound on regret
  • Theorem 2: Adversarial upper bound on regret of Tempered Exp3 for Social Welfare
  • Corollary 1: Stochastic upper bound on regret of Tempered Exp3 for Social Welfare
  • Theorem 3: Lower bound on regret for the concave case
  • Theorem 4: Stochastic upper bound on regret of Dyadic Search for Social Welfare
  • Theorem 5: Adversarial upper bound on regret of Tempered Exp3 for Optimal Income Taxation
  • proof : Proof of Claim \ref{['eq:Pinsker']}
  • Lemma 1: Confidence intervals contain true welfare differences with high probability
  • Lemma 2: Confidence intervals shrink with epoch length
  • proof
  • ...and 8 more