💻 computer science

Actionable Recourse in Competitive Environments: A Dynamic Game of Endogenous Selection

This paper proposes a dynamic game framework demonstrating that in competitive environments, the strategic interplay between candidates exerting effort to improve their features and the resulting endogenous evolution of selection thresholds amplifies initial disparities and creates persistent performance gaps.

Original authors: Ya-Ting Yang, Quanyan Zhu

Published 2026-03-19

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Ya-Ting Yang, Quanyan Zhu

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 a high-stakes game of musical chairs, but instead of chairs, there are only a few spots at the "Good Life" table (like getting into a top university or landing a dream job).

In this paper, the authors ask a tricky question: What happens if everyone is told exactly how to move to get a chair, and then they all try to move at the same time?

Here is the story of the paper, broken down into simple concepts and analogies.

1. The Setup: The "Smart" Gatekeeper

Imagine a university admissions office run by an AI. This AI looks at your resume (your "features") and gives you a score.

The Old Way: If you get rejected, the AI might say, "You need to improve your test scores." You study, your score goes up, and you get in. Simple.
The New Reality: The problem is that there are 1,000 applicants and only 100 spots. It's a competitive environment. If everyone studies and improves their test scores, the average score goes up. Suddenly, the "passing line" moves higher. You improved, but you're still rejected because everyone else improved too.

2. The Core Concept: "Endogenous Selection" (The Moving Goalpost)

The paper calls this Endogenous Selection. Think of it like a moving goalpost in a soccer game.

The referee (the AI) sets the goalpost based on where the players are standing right now.
If all the players run forward to get closer to the goal, the referee moves the goalpost further away to keep the same number of people scoring.
The Result: You are running faster and sweating more (exerting more effort), but you aren't actually getting any closer to the goal. The "bar" keeps rising to match the crowd's effort.

3. The Trap: "Involution" (Running in Place)

The authors use a sociological term called Involution. Imagine a hamster wheel.

At first, the hamster runs, and the wheel turns.
But if every hamster runs faster, the wheel spins so fast that the hamster is just running in place, exhausted, with no progress.
In the paper's model, as candidates try harder to fix their "actionable" features (like test scores or resume keywords), the collective benchmark rises. The system becomes a closed loop:
1. AI sets a rule.
2. People try to beat the rule.
3. The rule changes because the people changed.
4. People have to try even harder.
5. Repeat until everyone is exhausted.

4. The "Rich Get Richer" Effect (Social Stratification)

Here is the most surprising part of the paper. The people who already got the "Good Life" spots (the winners) actually control the game.

The AI looks at the winners to decide what a "good candidate" looks like.
If the winners happen to have high test scores (even if they got lucky or had better resources), the AI decides that high test scores are the only thing that matters.
The losers are then forced to run toward that specific metric.
The Analogy: Imagine a race where the winners happen to be tall. The referee decides that "height" is the only thing that matters. Now, short people are forced to wear stilts (spending huge effort) to compete, while tall people just stand there. The gap between tall and short people gets wider, not smaller. The initial advantage becomes a permanent wall.

5. The "Ceiling" Problem

The paper also introduces a limit. Imagine you are trying to improve your test score, but the maximum score is 100.

If you are at 90, it's easy to get to 95.
But if you are at 99, getting to 100 is incredibly hard and expensive (like trying to push a boulder up a mountain).
The math shows that as people get closer to this "ceiling," the effort required becomes infinite. Eventually, people stop trying because it's too costly, but they still can't get in because the "winners" are still slightly ahead.

6. The Conclusion: A Warning

The paper concludes that giving everyone "actionable recourse" (a clear path to improvement) in a competitive system doesn't necessarily create fairness. Instead, it can create a stratified society:

The Winners: They set the rules. They define what "success" looks like.
The Losers: They run themselves ragged trying to meet those rules, but the goalposts keep moving.
The Gap: The distance between the winners and losers becomes permanent, based on things that cannot be changed (like innate talent, background, or immutable traits), because the system eventually stops caring about the things people can change.

Summary in One Sentence

When an AI tells everyone how to improve to get a limited prize, and everyone tries to improve at once, the prize requirements just get higher, leaving the original winners ahead and the rest exhausted, creating a permanent divide that effort alone cannot fix.

1. Problem Statement

The paper addresses a critical gap in the literature on Actionable Recourse (the ability of individuals to modify features to overturn unfavorable AI decisions). While existing research focuses on static, individual-level recourse, real-world decision systems (e.g., university admissions, hiring, grants) are inherently competitive. In these settings:

Only a fixed fraction ( $\rho$ ) of candidates are selected.
Improvements by one candidate affect the relative standing and outcomes of others.
The "success benchmark" is not fixed but evolves endogenously as the population adapts.

Core Question: What happens when actionable recourse is available to everyone in a competitive environment? The authors hypothesize that this leads to endogenous selection, where the decision rule and selection threshold are determined by the population's current state, potentially amplifying initial disparities and creating persistent performance gaps (a phenomenon akin to "involution").

2. Methodology

The authors propose a formal framework modeling recourse as a closed-loop dynamical system involving two interacting agents: a System Designer and a population of Candidates.

A. The Designer's Problem (Risk-Based Selection)

Objective: The designer selects the top $\rho$ fraction of candidates based on a linear scoring rule $s_i = w^\top x_i$ .
Optimization Formulation: The selection of the top- $\rho$ tail is modeled using Conditional Value-at-Risk (CVaR). Specifically, the designer maximizes the average score of the top $\rho$ fraction.
Regularization: To ensure the optimization problem is well-posed (avoiding infinite weights), a quadratic regularization term ( $\lambda \|w\|^2$ ) is added.
Result: The optimal scoring vector $w^*$ is proportional to the centroid of the selected top- $\rho$ subset. The selection boundary is a hyperplane determined by this centroid.

B. The Candidates' Problem (Strategic Recourse)

Features: Features are partitioned into Actionable ( $J_A$ ) and Immutable ( $J_N$ ) coordinates.
Recourse Mechanism: Rejected candidates attempt to move across the decision boundary by modifying actionable features.
Cost Function:
- Initially, a quadratic cost is used to define minimal recourse.
- In the dynamic setting, a Logarithmic Barrier cost function is introduced: $C(\gamma) = \frac{k}{2}\gamma^2 - \theta \log(\Delta - \gamma)$ .
- This models a "ceiling" effect (e.g., a maximum test score), where the marginal cost of improvement approaches infinity as the candidate nears the physical limit of the feature.
Strategy: Candidates solve an optimization problem to maximize their score improvement net of effort costs, given the current direction $d_t$ implied by the classifier.

C. The Dynamic System

The system evolves in discrete time steps $t$ :

Designer Step: Observes population state $X_t$ , solves the CVaR optimization to find $w_t$ and threshold $\eta_t$ .
Signal: The system communicates the actionable direction $d_t$ (projection of $w_t$ onto $J_A$ ) to rejected candidates.
Candidate Step: Rejected candidates update their features $x_i^{t+1} = x_i^t + \gamma_i^* d_t$ based on the optimal effort $\gamma_i^*$ .
Feedback: The new population distribution $X_{t+1}$ shifts the CVaR centroid, altering $w_{t+1}$ and the decision boundary for the next round.

3. Key Contributions

Dynamic Framework for Competitive Recourse: The paper moves beyond static recourse analysis to model the interaction between a selection rule and a population of strategic agents as a dynamical system.
Endogenous Selection Mechanism: It formally demonstrates that the selection threshold and the direction of improvement are not exogenous but are determined by the population's current feature distribution.
Characterization of Equilibrium: The authors define Recourse Equilibrium (a fixed point where no candidate benefits from further effort) and analyze the conditions under which the system stabilizes.
Theory of Social Stratification: The paper introduces a formal definition of Social Stratification in this context: a state where the selection rule depends entirely on immutable features ( $d^* = 0$ ), yet a persistent gap remains between selected and rejected groups.

4. Key Results and Findings

A. Amplification of Initial Disparities

The study shows that the group initially selected defines the "standard of success." Because the scoring direction $w^*$ is derived from the centroid of the selected group, the system incentivizes all other candidates to compete in the specific direction defined by the winners. This creates an asymmetric feedback loop where the winners set the rules, and the losers must chase a moving target.

B. Emergence of Involution

The system exhibits "involution": as candidates invest increasing effort to improve, the collective benchmark rises.

Initially, rejected candidates improve rapidly.
As the population improves, the decision boundary rotates to maintain the top- $\rho$ selection.
Eventually, the marginal cost of improvement (due to the ceiling constraint) outweighs the benefit, leading to a slowdown in progress despite high effort.

C. Structural Accountability Failure

The analysis reveals two failure modes:

Infeasibility: If the optimal weight vector lies entirely in the immutable subspace ( $w_A^* = 0$ ), recourse is structurally impossible regardless of effort.
Stratification: In equilibrium, if $d^* = 0$ (the direction of improvement vanishes) but rejected candidates remain ( $R^* \neq \emptyset$ ), the system has locked into a state where success depends solely on immutable traits (e.g., GPA), creating a permanent, unbridgeable gap between groups.

D. Numerical Case Study

Simulations using GRE (actionable) and GPA (immutable) scores demonstrate:

Phase 1: Rapid improvement in actionable features as candidates chase the initial boundary.
Phase 2: The decision boundary rotates (becomes steeper) as the population shifts.
Phase 3: Convergence to a stable state where the actionable signal norm declines, and the gap between groups is sustained by immutable features.

5. Significance

Theoretical: It bridges Strategic Classification, Performative Prediction, and Game Theory, showing how competitive constraints transform recourse from a tool for individual empowerment into a mechanism for collective stratification.
Practical: It warns policymakers and system designers that providing "actionable recourse" in competitive environments without addressing resource constraints or selection ratios may inadvertently widen inequality. If the system rewards specific mutable traits that are correlated with immutable advantages, it can create a "rat race" where the relative gap persists even if absolute scores rise.
Governance: The findings suggest that fairness metrics in competitive AI must account for dynamic feedback loops. Static fairness checks are insufficient; the long-term equilibrium of the system must be analyzed to prevent the entrenchment of bias through endogenous selection.