ACPW Estimator for Causal & Welfare Analysis

• ACPW is a doubly robust estimator that combines outcome regression with cumulative propensity weighting to assess causal effects and welfare metrics.
• It attains semiparametric efficiency and consistency by requiring correct specification of either the propensity function or the outcome model, making it reliable even under misspecification.
• The estimator is applied in longitudinal causal inference, consumer surplus evaluation, and fairness-aware auditing, offering stable inference where traditional methods often falter.

The Augmented Cumulative Propensity Weighting (ACPW) estimator is a doubly robust, semiparametrically efficient methodology for causal effect and welfare estimation under general off-policy and longitudinal settings, especially when positivity violations or policy adaptation complicate classical causal inference or welfare calculations. ACPW extends cumulative propensity weighting (CPW) by incorporating an augmentation term based on outcome regression, thereby achieving consistency if either the propensity function or the outcome model is correctly specified. Applications span longitudinal treatment effect estimation, consumer surplus evaluation under random pricing, and inequality-aware welfare auditing in algorithmic decision-making platforms (McClean et al., 14 Jul 2025, Bian et al., 3 Jan 2026).

1. Estimation Framework and Target Functionals

ACPW is designed to estimate causal contrasts or welfare quantities in settings characterized by stochastic policies and potentially time-varying structures. In longitudinal causal inference, the target is the cumulative cross-world weighted effect between two fixed treatment regimes $\overline a_T$ and $\overline a_T'$, formally: $$\psi(\overline a_T,\overline a_T') = \mathbb E\Biggl[\bigl\{Y(\overline a_T)-Y(\overline a_T')\bigr\} \prod_{t=1}^T w_t(p_t\{\overline X_t(\overline a_{t-1})\}) w_t'(p_t'\{\overline X_t(\overline a_{t-1}')\})\Biggr],$$ where $Y(\overline a_T)$ and $Y(\overline a_T')$ denote potential outcomes, $p_t$ and $p_t'$ are regime-specific propensity scores, and $w_t$, $w_t'$ are tuning weight functions ensuring proper support (McClean et al., 14 Jul 2025).

In consumer surplus evaluation for algorithmic pricing, ACPW targets the population mean surplus under a target pricing distribution $\pi$, given observed covariates $X$, randomized prices $P$ under policy $\pi_D$, and binary purchase indicator $Y$: $$\mathcal S(\pi) = E_X\left[\int_{p=0}^\infty \pi(p|X) E[(V-p)_+|X]\,dp\right] = E\left[\int_0^\infty \pi(p|X)\int_{z=p}^\infty \mu(X,z)\,dz\,dp\right],$$ with $\mu(x,z) = P(V > z|X = x) = E[Y|X = x, P = z]$ (Bian et al., 3 Jan 2026).

2. Identification and Efficient Influence Function

ACPW leverages the weighted g-formula for identification under nonparametric structural equation models and sequential randomization, expressing the target functional as an expectation weighted by cross-world propensity and covariate densities: $$\psi(\overline a_T) = \int m_{T+1}(\overline x_{T+1}) \prod_{t=1}^T w_t\{\pi_t(\overline x_t)\} w_t'\{\pi_t'(\overline x_t)\} dP(\overline x_{T+1}|\overline A_T=\overline a_T),$$ where $m_t$ are recursively defined sequential outcome regressions.

The efficient influence function (EIF) for the ACPW functional is: $$\psi^\pi(X,P,Y) = h(X) + \frac{F^\pi(P|X)}{\pi_D(P|X)} (Y-\mu(X,P)) - \mathcal S(\pi),$$ with augmentation term $h(X)$, CPW reweight factor $F^\pi$, and model-based outcome estimates (Bian et al., 3 Jan 2026).

In longitudinal ACPW, the EIF contains additional terms accounting for density ratio fluctuations and weight-score derivatives, preserving sensitivity to positivity and mechanistic regime differences (McClean et al., 14 Jul 2025).

3. Estimation Procedure and Nuisance Function Learning

The ACPW estimator is constructed as the empirical average of the plug-in EIF, incorporating cross-fitting to mitigate overfitting. For i.i.d. data $\{Z_i\}$ or $(X_i,P_i,Y_i)$, one computes

$$\widehat\psi_{\mathrm{ACPW}} = \frac{1}{n} \sum_{i=1}^n \widehat\varphi(Z_i),$$

with $\widehat\varphi$ indexed by estimated nuisance functions: propensity scores $\widehat\pi_t$, $\widehat\pi_t'$, covariate density ratios $\widehat\rho_t$, and outcome regressions $\widehat m_t$ (McClean et al., 14 Jul 2025).

In the consumer pricing context, ACPW employs: $$\widehat{\mathcal S}_{\mathrm{ACPW}}(\pi) = \frac1n\sum_{i=1}^n \left[\widehat h^{-i}(X_i) + \frac{F^\pi(P_i|X_i)}{\widehat\pi_D^{-i}(P_i|X_i)} (Y_i-\widehat\mu^{-i}(X_i,P_i))\right],$$ where fold-excluded nuisances are fit on hold-out subsets (cross-fitting), and $\widehat h^{-i}(X_i)$ is the model-based direct term for observation $i$ (Bian et al., 3 Jan 2026).

Flexible machine learning tools (random forests, boosting, ensemble regression) are recommended for nuisance function estimation, with mild complexity control to guarantee suitable error rates ($o_P(n^{-1/4})$ or $\alpha_1+\alpha_2>1/2$ convergence) (McClean et al., 14 Jul 2025, Bian et al., 3 Jan 2026).

4. Doubly Robustness, Efficiency, and Asymptotics

The ACPW estimator exhibits the doubly robust property: it is consistent and $\sqrt n$-asymptotically normal if either the outcome regression or propensity (density ratio) model is correctly specified. Explicitly,

$$\sqrt{n}(\widehat\psi_{\mathrm{ACPW}} - \psi) \xrightarrow{d} N(0, \sigma^2),$$

with $\sigma^2 = \mathrm{Var}\{\varphi(Z)\}$ under the conditions that either all propensity/density-ratio models or all outcome regression models are consistent. Further, ACPW achieves the semiparametric efficiency bound provided the product of nuisance convergence rates exceeds $n^{-1/2}$ (McClean et al., 14 Jul 2025, Bian et al., 3 Jan 2026).

This efficiency and robustness pertain both to longitudinal causal effects (where positivity violations can cause classical estimators to fail) and to welfare evaluation under algorithmic pricing, where model misspecification or inadequate demand representation induce substantial bias in non-augmented methods.

5. Comparison with Direct and Pure CPW Methods

Direct-Method (DM) plug-in estimators rely exclusively on model-based outcome estimates, thereby incurring bias if misspecified and lacking weighting-induced variance reduction. Pure CPW estimators weight observed outcomes using cumulative propensity factors, retaining unbiasedness under correct policy estimation but suffering increased variance and bias if the propensity model is inaccurate.

ACPW integrates both approaches, achieving consistency if either nuisance is correct (double robustness) and delivering more stable inference and lower mean squared error in finite samples. Empirical studies demonstrate that ACPW retains validity under misspecification of either demand or propensity models, whereas DM and CPW fail if their respective models are incorrect (Bian et al., 3 Jan 2026). Convergence rates for ACPW are $O(n^{-1})$ MSE when both nuisances are correctly specified, outperforming DM and CPW in stability and finite-sample inference.

6. Extensions: Fairness-Aware and Longitudinal Generalizations

ACPW extends naturally to inequality-aware (fairness-aware) surplus measures via Atkinson-type indices: $$\mathcal S^r(\pi) = \left(E_X[S(\pi|X)^r]\right)^{1/r}, \quad S(\pi|X) = \int \pi(p|X)\int_p^\infty \mu(X,z)\,dz\,dp,$$ with corresponding influence functions and estimators. The Inequality-Aware ACPW (IA-ACPW) estimator reduces first-order bias and enables valid confidence intervals, though full consistency in nonlinear functionals requires correct outcome regression model alone. Empirically, IA-ACPW achieves lower MSE than naive DM for $r<1$ (Bian et al., 3 Jan 2026).

In longitudinal causal inference, ACPW overcomes classical positivity limitations by weighting mechanistic regime differences across counterfactual worlds, enabling identification of effects under partial support conditions and exposing interpretability-implementability tradeoffs in effect definition (McClean et al., 14 Jul 2025).

7. Practical Implementation and Empirical Insights

Practical implementation of ACPW entails cross-fitting with fold-excluded nuisance function learning, assembly of plug-in influence function contributions, and empirical mean-based inference. For warranted confidence intervals, variance is estimated by: $$\widehat\sigma^2 = \frac1n \sum_i [\widehat\varphi(Z_i) - \widehat\psi]^2,$$ yielding intervals $\widehat\psi \pm 1.96\,\widehat\sigma/\sqrt n$.

Numerical studies in algorithmic pricing validate ACPW’s robustness against model misspecification, fast convergence rates, and superior MSE performance for both aggregate and fairness-aware surplus measures. In auto-loan policy analysis, ACPW quantifies the tradeoff between aggregate surplus reduction and fairness improvement across distinct consumer segments, demonstrating its utility in regulatory audits and profit-equity evaluations (Bian et al., 3 Jan 2026).

In sum, the Augmented Cumulative Propensity Weighting estimator constitutes a rigorous, adaptable, and statistically robust solution for causal effect identification and welfare auditing in complex, data-driven decision environments characterized by longitudinal structure, randomization, and fairness concerns.