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Policy Learning with Distributional Welfare

Yifan Cui, Sukjin Han

Abstract

In this paper, we explore optimal treatment allocation policies that target distributional welfare. Most literature on treatment choice has considered utilitarian welfare based on the conditional average treatment effect (ATE). While average welfare is intuitive, it may yield undesirable allocations especially when individuals are heterogeneous (e.g., with outliers) - the very reason individualized treatments were introduced in the first place. This observation motivates us to propose an optimal policy that allocates the treatment based on the conditional quantile of individual treatment effects (QoTE). Depending on the choice of the quantile probability, this criterion can accommodate a policymaker who is either prudent or negligent. The challenge of identifying the QoTE lies in its requirement for knowledge of the joint distribution of the counterfactual outcomes, which is not generally point-identified. We introduce minimax policies that are robust to this model uncertainty. A range of identifying assumptions can be used to yield more informative policies. For both stochastic and deterministic policies, we establish the asymptotic bound on the regret of implementing the proposed policies. The framework can be generalized to any setting where welfare is defined as a functional of the joint distribution of the potential outcomes.

Policy Learning with Distributional Welfare

Abstract

In this paper, we explore optimal treatment allocation policies that target distributional welfare. Most literature on treatment choice has considered utilitarian welfare based on the conditional average treatment effect (ATE). While average welfare is intuitive, it may yield undesirable allocations especially when individuals are heterogeneous (e.g., with outliers) - the very reason individualized treatments were introduced in the first place. This observation motivates us to propose an optimal policy that allocates the treatment based on the conditional quantile of individual treatment effects (QoTE). Depending on the choice of the quantile probability, this criterion can accommodate a policymaker who is either prudent or negligent. The challenge of identifying the QoTE lies in its requirement for knowledge of the joint distribution of the counterfactual outcomes, which is not generally point-identified. We introduce minimax policies that are robust to this model uncertainty. A range of identifying assumptions can be used to yield more informative policies. For both stochastic and deterministic policies, we establish the asymptotic bound on the regret of implementing the proposed policies. The framework can be generalized to any setting where welfare is defined as a functional of the joint distribution of the potential outcomes.
Paper Structure (29 sections, 8 theorems, 44 equations, 6 figures, 8 tables)

This paper contains 29 sections, 8 theorems, 44 equations, 6 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Related Literature
  3. Organization of the Paper
  4. Treatment Rules and Distributional Welfare
  5. Introducing Distributional Welfare
  6. Policies Robust to Model Ambiguity
  7. Possible Identifying Assumptions
  8. Theoretical Properties of Estimated Policy
  9. Regret Bounds with Unconstrained Policy Class
  10. Regret Bounds with Constrained Policy Class
  11. Calculating Bounds
  12. Numerical Illustrations
  13. Empirical Applications
  14. Generalization
  15. More on Possible Identifying Assumptions
  16. ...and 14 more sections

Key Result

Theorem 2.1

Suppose $Y_{d}$ is continuously distributed and $\mathcal{A}$ is either $[0,1]$ or $\{0,1\}$. Then, the first best rule $\delta_{\tau}^{\dagger}(x)\equiv1\{Q_{\tau}(x)\ge0\}$ for $\tau=0.5$ satisfies

Figures (6)

  • Figure 1: Bounds on the QoTE of Six Representative Patients
  • Figure 2: Treatment Decisions for Male Patients with Specific Health Conditions
  • Figure 3: Treatment Decisions Based on the QTEs and the ATE
  • Figure 4: Bounds on the QoTE of Six Representative Workers
  • Figure 5: Treatment Decisions for Female Workers Without High School Diploma
  • ...and 1 more figures

Theorems & Definitions (17)

  • Theorem 2.1
  • Lemma 3.1
  • Lemma 4.1
  • Theorem 4.1
  • Corollary 4.1
  • Theorem 4.2
  • Definition 5.1: Bernstein Copula
  • Lemma 5.1
  • Example 1
  • Example 2
  • ...and 7 more