Unified Inference Framework for Single and Multi-Player Performative Prediction: Method and Asymptotic Optimality

Performative prediction characterizes environments where predictive models alter the very data distributions they aim to forecast, triggering complex feedback loops. While prior research treats single-agent and multi-agent performativity as distinct phenomena, this paper introduces a unified statistical inference framework that bridges these contexts, treating the former as a special case of the latter. Our contribution is two-fold. First, we put forward the Repeated Risk Minimization (RRM) procedure for estimating the performative stability, and establish a rigorous inferential theory for admitting its asymptotic normality and confirming its asymptotic efficiency. Second, for the performative optimality, we introduce a novel two-step plug-in estimator that integrates the idea of Recalibrated Prediction Powered Inference (RePPI) with Importance Sampling, and further provide formal derivations for the Central Limit Theorems of both the underlying distributional parameters and the plug-in results. The theoretical analysis demonstrates that our estimator achieves the semiparametric efficiency bound and maintains robustness under mild distributional misspecification. This work provides a principled toolkit for reliable estimation and decision-making in dynamic, performative environments.

💡 Research Summary

This paper tackles the emerging problem of performative prediction, where a deployed predictive model actively shapes the data distribution it later observes, creating a feedback loop. While prior work has treated the single‑agent and multi‑agent (multiplayer) settings as separate problems, the authors propose a unified statistical inference framework that subsumes both, viewing the single‑player case as a special instance of the multiplayer scenario.

The core contributions are twofold. First, the authors introduce a repeated risk minimization (RRM) procedure for estimating the performative stable point and develop an empirical version called Empirical Repeated Risk (ERR). By iteratively minimizing the empirical risk on the distribution induced by the previous iterate, ERR produces a sequence of estimators (\hat\theta_t). Under a set of regularity conditions—Lipschitz continuity of the distribution map, smoothness of the loss, and strong monotonicity of the gradient—the authors prove a central limit theorem (CLT) for each iteration: (\sqrt{N}(\hat\theta_t-\theta_t)) converges in distribution to a normal law with a covariance matrix (\Sigma_t) that aggregates the covariances from all earlier steps. They further derive a lower bound on the asymptotic covariance for any estimator of the stable point and show that ERR attains this bound, establishing its semiparametric efficiency.

Second, the paper addresses performative optimality. Here the goal is to estimate the parameter (\theta^{PO}) that minimizes the performative risk over the true (unknown) distribution map. The authors first estimate the distribution‑mapping parameters (\beta_i) for each player using a three‑fold cross‑fitted Recalibrated Prediction‑Powered Inference (RePPI) procedure, which leverages surrogate outcomes and machine‑learning predictions to achieve efficient estimation. They prove a CLT for (\hat\beta_i) and demonstrate that it reaches the semiparametric efficiency bound. Using these (\hat\beta_i), they construct a plug‑in estimator (\hat\theta^{PO}{\hat\beta}) by solving the empirical risk minimization problem under the estimated map. Because direct sampling from the induced distribution is generally infeasible, they incorporate importance sampling to generate weighted samples, enabling practical computation. The authors establish a second CLT for (\hat\theta^{PO}{\hat\beta}), with asymptotic covariance (\Sigma_\theta) directly linked to the covariance of the (\hat\beta_i). Under differentiability of the solution map with respect to (\beta), they show that both (\hat\beta_i) and (\hat\theta^{PO}_{\hat\beta}) achieve the theoretical lower bounds, confirming semiparametric optimality for the optimality estimator as well.

The framework naturally extends to the multiplayer setting. Each player runs its own ERR and RePPI procedures, and the Nash equilibrium is defined as the collection of parameters where each player’s performative risk (conditioned on the others’ strategies) is minimized. The authors prove that the same CLTs and efficiency results hold for the joint estimator of the Nash equilibrium, and that the single‑player results are recovered when the number of players reduces to one.

From a practical standpoint, the work provides tools for uncertainty quantification (confidence intervals) in environments where model deployment influences future data—examples include loan‑interest‑rate setting, criminal‑justice risk scoring, and public‑policy design. The importance‑sampling‑augmented plug‑in estimator is particularly valuable when data collection is costly or when the distribution map is only partially known. Moreover, the robustness of the CLTs to mild misspecification of the distribution map ensures that the methodology remains reliable in realistic, noisy settings.

In summary, this paper delivers a comprehensive, theoretically rigorous, and practically applicable inference suite for performative prediction. By unifying single‑ and multi‑player contexts, establishing asymptotic normality, and achieving semiparametric efficiency for both stability and optimality estimators, it sets a new benchmark for statistical analysis in feedback‑driven predictive systems.