📊 statistics

Microfoundation Inference for Strategic Prediction

This paper proposes a methodology using optimal transport and cost-adjusted utility maximization to infer the distribution map of strategic agent responses, thereby enabling the prediction of long-term social impacts in performative prediction scenarios.

Original authors: Daniele Bracale, Subha Maity, Felipe Maia Polo, Seamus Somerstep, Moulinath Banerjee, Yuekai Sun

Published 2026-07-28

📖 3 min read☕ Coffee break read

CC BY 4.0

Original authors: Daniele Bracale, Subha Maity, Felipe Maia Polo, Seamus Somerstep, Moulinath Banerjee, Yuekai Sun

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a weather forecaster. You look at the clouds and predict rain. But here's the twist: as soon as you shout "Rain is coming!", everyone grabs an umbrella and stays inside. Because everyone stayed inside, it didn't rain as much as you predicted. Your prediction actually changed the weather! This is the strange world of "performative prediction." It happens whenever a computer model makes a guess about people, and those people change their behavior because of the guess. It's like a game of cat and mouse where the cat (the model) keeps trying to catch the mouse (the data), but the mouse keeps running away and changing shape every time the cat gets close. This isn't just a math problem; it's a social one. If a bank's computer says you are a risky borrower, you might try harder to fix your credit score, or you might give up entirely. The model's prediction shifts the reality it's supposed to measure. The big question for scientists is: how do we build models that understand these changes and still make good predictions, even when people are actively trying to game the system?

This paper tackles that puzzle by trying to figure out why people change their behavior. The authors suggest that when people see a prediction, they act like rational shoppers: they weigh the "benefit" of changing their actions against the "cost" or effort it takes to do so. Think of it like a student deciding whether to study harder for a test. If the teacher says, "I'll give you an A if you memorize this list," the student calculates: "Is getting that A worth the headache of memorizing?" The paper argues that while we can usually see what the "benefit" is (the A grade), we often have no idea what the "cost" is (how hard the student finds the work). Previous methods tried to guess this cost, but they often got it wrong, leading to bad predictions.

The authors propose a new way to "infer" or figure out this hidden cost. They use a clever mathematical tool called "optimal transport," which is like a logistics company trying to move piles of dirt from one shape to another with the least amount of work. In their case, they are moving the "before" picture of a population (how people looked before the model existed) to the "after" picture (how they look after the model is deployed). By matching these two pictures perfectly, they can reverse-engineer the "cost" that made people move. They tested this idea on a dataset about credit scores, simulating how people might try to game a loan system. Their method suggests that they can accurately estimate how much effort people are willing to put in, even if they don't know exactly what the people are trying to achieve.

The paper finds that this new method is quite robust. Even if the researchers guessed the "benefit" part of the equation wrong (like thinking the reward was a gold star when it was actually a cash prize), their method could still figure out the "cost" well enough to predict how people would move. In their simulations, this allowed them to find a better, more stable prediction model much faster than older methods that just kept guessing and retraining. The authors show that by understanding the hidden "price tag" people put on changing their behavior, we can build AI that doesn't just chase a moving target, but actually understands the game it's playing.

Technical Summary: Microfoundation Inference for Strategic Prediction

1. Problem Statement

The paper addresses the challenge of performative prediction, a scenario where a predictive model $\theta$ influences the distribution of the target variable because stakeholders (agents) take strategic actions to maximize their utility in response to the model. This phenomenon, often described by Campbell's law, creates a distribution shift where the data distribution $Q_\theta$ is a function of the predictor $\theta$ .

A critical bottleneck in applying performative prediction theory is the unknown mechanism of this distribution shift. Existing approaches generally fall into two categories, both with significant limitations:

Zeroth-order optimization: Treating the problem as derivative-free optimization (e.g., using finite differences). This suffers from slow convergence and the curse of dimensionality because it does not explicitly model the distribution map $\theta \mapsto Q_\theta$ .
Microfoundation modeling: Assuming agents follow a specific response model (e.g., cost-adjusted utility maximization). While this allows for faster optimization algorithms by providing "white-box" access to the distribution map, these models are often misspecified. If the assumed agent cost function is incorrect, the resulting distribution map is erroneous, leading to suboptimal or inconsistent predictors.

The authors propose a methodology to infer the microfoundation of agent responses directly from data. Specifically, they aim to learn the unknown cost function $c$ that governs how agents modify their attributes $Z$ in response to a model $\theta$ , assuming the benefit function $B_\theta$ is known.

2. Methodology

2.1 Agent Response Model

The authors adopt the standard performative prediction framework where agents are rational. Upon exposure to a model $\theta$ , an agent with attributes $Z$ updates to $Z'$ by solving:
$T_\theta(Z) \in \arg\max_{z' \in \mathcal{Z}} \{ B_\theta(z') - c(Z, z') \}$
Here, $B_\theta$ is the known benefit (e.g., the model's output), and $c$ is the unknown cost of changing attributes. The resulting distribution is the push-forward measure $Q_\theta = (T_\theta)_\# P$ , where $P$ is the pre-model (ex-ante) distribution.

2.2 Identifiability and Bregman Divergences

The authors note that estimating a general bivariate cost function $c(z, z')$ is impossible with finite samples from a finite number of distributions. To address this, they restrict the cost function to the class of Bregman divergences:
$c_\phi(z, z') = \phi(z') - \phi(z) - \nabla\phi(z)^\top(z' - z)$
where $\phi$ is a strictly convex potential function. This reduces the estimation problem from a complex bivariate function to estimating the scalar potential $\phi(z)$ , mitigating identifiability issues.

2.3 Optimal Transport Estimation

The core of the proposed method leverages Optimal Transport (OT) to align distributions. Based on the first-order optimality condition of the agent's maximization problem, the authors derive that for the true potential $\phi^*$ :
$(\nabla\phi^* - \nabla B_\theta)_\# Q_\theta = (\nabla\phi^*)_\# P$
This implies that the push-forward of the ex-post distribution $Q_\theta$ under the map $(\nabla\phi^* - \nabla B_\theta)$ is identical to the push-forward of the ex-ante distribution $P$ under $\nabla\phi^*$ .

The authors propose estimating $\phi$ by finding a "barycenter" distribution $\mu$ that minimizes the sum of squared 2-Wasserstein distances to these push-forward distributions. The optimization problem is:
$\arg\min_{\phi \in \Phi_{cvx}} \min_{\mu \in \Delta(\mathcal{Z})} \left[ W_2^2(\mu, (\nabla\phi)_\# P) + \sum_{k=1}^m W_2^2(\mu, (\nabla\phi - \nabla B_{\theta_k})_\# Q_{\theta_k}) \right]$
This is solved via a block coordinate descent algorithm (Algorithm 1), alternating between updating the barycenter $\mu$ and the potential $\phi$ .

2.4 Scenarios

The method is applicable in two settings:

Ex-ante and Ex-post: The learner has samples from both the pre-deployment distribution $P$ and multiple post-deployment distributions $Q_{\theta_k}$ .
Ex-post only: The learner only has access to post-deployment distributions $Q_{\theta_k}$ , estimating $\phi$ by aligning the push-forwards of these distributions under different $\theta_k$ .

3. Key Contributions

Estimation Method: A novel method for estimating the cost function (within the class of Bregman divergences) of a utility-maximizing microfoundation model. The method relies on matching pre-model and post-model distributions using optimal transport.
Theoretical Analysis:
- Identifiability: The authors provide conditions under which the cost function is identifiable from the data (e.g., Corollary 4.2 shows identifiability with one ex-ante and one ex-post distribution if the benefit function is strictly concave).
- Convergence Rates: They establish a convergence rate of $O(n^{-2/d})$ for the estimated parameters under strong convexity assumptions, derived from empirical Wasserstein distance convergence results.
Empirical Validation: Numerical experiments on a credit scoring dataset demonstrate the method's ability to estimate costs and improve downstream tasks (performative risk minimization).

4. Experimental Results

The authors evaluated their method using a semi-synthetic setup based on a real credit scoring dataset (Kaggle), where agents strategically manipulate features to avoid default classification.

Robustness to Misspecification: In experiments where the benefit function $B_\theta$ was misspecified (e.g., assuming a logarithmic benefit when the truth was square-root), the estimation of the potential derivative $\phi'$ became biased. However, the resulting response map $T_\theta$ remained robust, and the predictive performance was not significantly degraded. This suggests the method can tolerate errors in the benefit function specification.
Convergence: The estimation error for the response map $T_\theta$ decreased as the sample size increased, consistent with theoretical predictions.
Performative Risk Minimization: When applied to minimize performative risk (finding the optimal classifier $\theta$ $θ$ ), the proposed method converged significantly faster to the optimal solution (Oracle) compared to baselines:
- Repeated Gradient Descent (RGD) and Performative Gradient Descent (PerfGD): These zeroth-order or partially specified methods showed slower convergence and, in some cases, failed to reach the oracle due to assumptions (e.g., Gaussianity in PerfGD) that did not hold.
- Least Squares (LS): A two-step method assuming a linear response.
- Result: The proposed method achieved competitive performance, converging to the oracle as the number of published classifiers (sample size) increased.

5. Significance and Claims

The paper claims that accurate inference of agent microfoundations is a crucial step for deploying socially responsible and effective models in strategic settings. By learning the cost function from data rather than assuming a misspecified model, practitioners can:

Utilize fast-converging optimization algorithms that require explicit knowledge of the distribution map, avoiding the slow convergence of derivative-free methods.
Deploy models with peace of mind regarding misspecification, as the method is robust to errors in the benefit function.
Better understand the long-term impacts of predictive models on populations, facilitating the enforcement of fairness and other social constraints.

The authors conclude that while their work focuses on strategic agents, the broader problem of learning distribution shifts in non-strategic performative settings remains an open area for future research. The code for the implementation is made publicly available.