Mind the Gap: Optimal and Equitable Encouragement Policies

1. Problem Statement

In many consequential domains (e.g., healthcare, social services, criminal justice), decision-makers cannot compel individuals to accept a treatment; they can only provide encouragements (recommendations, nudges, reminders). This leads to non-adherence (or non-compliance), where the actual treatment uptake ($T$) differs from the recommendation ($R$).

Existing policy learning methods often optimize based on the Intention-to-Treat (ITT) effect (the effect of the recommendation itself) or assume full control over treatment assignment. However, this creates two critical gaps:

1. Fairness Gaps: Naive targeting based on ITT can exacerbate disparities if certain groups are less responsive to nudges due to structural barriers (e.g., administrative burden, lack of trust) rather than lower treatment efficacy.
2. Operational Blindness: Optimizing for outcomes without separating responsiveness to encouragement from treatment efficacy fails to identify the root cause of inequities, leading to suboptimal interventions.

The paper addresses the problem of learning personalized encouragement policies ($\pi$) that maximize policy value (downstream outcomes) while satisfying constraints on treatment take-up parity (fairness in who actually receives the treatment) and budget constraints.

2. Methodology

A. Causal Framework and Identification

The author adopts a potential outcomes framework with a Covariate-Conditional No-Direct-Effect Assumption (Exclusion Restriction).

• Assumption: The encouragement $R$ affects the outcome $Y$ only through its effect on treatment take-up $T$. There is no direct path $R \to Y$.
• Decomposition: Under this assumption, the "Encouragement Score" ($s(x,a)$), which drives optimal targeting, decomposes into two distinct components:
  $$s(x, a) = \underbrace{\kappa(x, a)}_{\text{Responsiveness}} \times \underbrace{\tau(x, a)}_{\text{Treatment Efficacy}}$$
  • $\kappa(x, a) = P(T=1|R=1) - P(T=1|R=0)$: The causal effect of the encouragement on take-up (compliance).
  • $\tau(x, a) = E[Y(1) - Y(0) | X, A]$: The heterogeneous treatment effect (HTE) of the treatment itself.

This decomposition allows the planner to audit whether disparities arise from differences in efficacy (the treatment works differently) or access/compliance (the treatment is harder to take up).

B. Policy Characterization

The paper derives the structure of optimal policies under various constraints:

1. Budget-Constrained (Naive): The optimal policy is a threshold rule on the encouragement score $s(x,a)$.
2. Demographic Parity (Recommendation): To ensure equal recommendation rates across groups, the optimal policy uses group-specific thresholds on the score distribution.
3. Treatment Parity (Take-up): To ensure equal actual treatment rates across groups (the natural fairness target in encouragement settings), the optimal policy is a threshold on a Lagrangian score:
   $$\ell(\lambda, x, a) = \kappa(x, a) \left( \tau(x, a) - \lambda \frac{\mathbb{I}(A=a) - \mathbb{I}(A=b)}{P(A)} \right)$$
   This explicitly penalizes policies that induce disparate take-up rates.

C. Robust Optimization for Algorithmic Advice

In settings where recommendations are deterministic functions of covariates (e.g., risk scores), the overlap assumption (positivity) is violated. Standard robust optimization would require extrapolating the entire outcome model, which is highly conservative.

• Innovation: By decomposing the problem, the method only requires robust extrapolation of the responsiveness model ($\kappa$), not the outcome model ($\mu$).
• Technique: The author uses Lipschitz continuity assumptions on the responsiveness function to construct uncertainty sets for the no-overlap regions. This yields a tractable Linear Programming (LP) formulation that is significantly less conservative than previous approaches.

D. Variance-Sensitive Two-Stage Learning

To handle finite-sample variance and ensure fairness constraints hold out-of-sample, the paper proposes a Two-Stage Constrained Policy Optimization algorithm:

1. Stage 1: Estimate nuisance functions and solve a preliminary optimization to identify "nearly binding" constraints.
2. Stage 2: Solve a localized optimization on a held-out dataset, incorporating variance-based slacks derived from Stage 1.

• Benefit: This approach achieves tighter control over fairness constraints with lower regret compared to single-stage methods, as it adapts to the variance of near-optimal policies rather than worst-case variance.

3. Key Contributions

1. Theoretical Decomposition: Establishes that in encouragement designs, policy value is the product of responsiveness and efficacy. This distinction is crucial for diagnosing the source of inequity.
2. Fairness Targeting: Argues that induced treatment take-up (not recommendation rates) is the natural fairness target. It provides closed-form characterizations for optimal policies under take-up parity constraints.
3. Robustness without Conservatism: Demonstrates that decomposing the problem allows for robust optimization under overlap violations (common in algorithmic advice) with much milder penalties than standard methods.
4. Algorithmic Innovation: Introduces a two-stage, variance-regularized learning algorithm that improves out-of-sample fairness constraint satisfaction.
5. Empirical Validation: Provides rigorous case studies showing that naive targeting often fails to achieve equity due to compliance gaps, and that the proposed methods can rectify this.

4. Results and Case Studies

Case Study 1: SNAP Recertification Reminders

• Context: Text message reminders for Supplemental Nutrition Assistance Program (SNAP) recertification.
• Finding: Naive ITT targeting (optimizing for total benefits) creates racial disparities.
  • Diagnosis: The disparity is driven almost entirely by differential responsiveness (compliance), not treatment efficacy. Non-white beneficiaries benefit significantly from the treatment but are less likely to act on the reminder due to structural barriers (e.g., language, administrative burden).
  • Result: A fairness-constrained policy that equalizes take-up rates reduces the racial gap in treatment access significantly with only a marginal loss in total program value. The paper suggests that operational redesign (e.g., better navigation support) is needed rather than just better targeting.

Case Study 2: Pretrial Supervised Release (Electronic Monitoring)

• Context: Judicial decisions on supervised release based on algorithmic risk scores (PSA-DMF).
• Challenge: The algorithmic recommendation is deterministic (no overlap), and unconfoundedness is questionable.
• Method: Applied the robust Lipschitz extrapolation method to optimize interpretable scorecards.
• Result:
  • Local changes to simple, interpretable decision matrices (scorecards) can reduce worst-case racial disparities in supervision rates by more than half (from ~5.7% to ~2.2%) while maintaining similar failure-to-appear (FTA) rates.
  • The method successfully handles the lack of overlap by only extrapolating the response to the recommendation, not the outcome.

Simulation Results

• The Two-Stage Algorithm significantly outperforms naive and one-stage baselines in finite samples. It achieves better fairness constraint coverage (closer to the target $\epsilon$) and lower Lagrangian regret, particularly in moderate sample sizes ($n \approx 750-2000$).

5. Significance

This paper bridges the gap between causal inference (encouragement designs) and algorithmic fairness (constrained optimization). Its primary significance lies in shifting the focus from "who gets a recommendation" to "who actually receives the treatment."

• Managerial Impact: It provides a diagnostic framework for organizations. If disparities exist, they must first audit whether the issue is efficacy (the treatment doesn't work for a group) or access (the group can't take the treatment). The remedy differs fundamentally: efficacy issues require scientific intervention, while access issues require operational redesign (e.g., reducing administrative burdens).
• Policy Design: It offers a mathematically rigorous way to design "nudge" policies that are both efficient and equitable, even when data is observational or recommendations are deterministic.
• Robustness: By localizing the robustness requirement to the behavioral response model, it makes data-driven decision-making feasible in high-stakes, low-overlap environments like criminal justice and healthcare.