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

Abstract: Actionable recourse studies whether individuals can modify feasible features to overturn unfavorable outcomes produced by AI-assisted decision-support systems. However, many such systems operate in competitive settings, such as admission or hiring, where only a fraction of candidates can succeed. A fundamental question arises: what happens when actionable recourse is available to everyone in a competitive environment? This study proposes a framework that models recourse as a strategic interaction among candidates under a risk-based selection rule. Rejected individuals exert effort to improve actionable features along directions implied by the decision rule, while the success benchmark evolves endogenously as many candidates adjust simultaneously. This creates endogenous selection, in which both the decision rule and the selection threshold are determined by the population's current feature state. This interaction generates a closed-loop dynamical system linking candidate selection and strategic recourse. We show that the initially selected candidates determine both the benchmark of success and the direction of improvement, thereby amplifying initial disparities and producing persistent performance gaps across the population.

• The paper develops a CVaR-based framework showing that top-ρ competitive selection chooses the subset with the largest feature-centroid norm, making the scoring direction endogenous to current winners.
• The paper derives closed-form quadratic-cost recourse and identifies actionable signal strength, ||w_A||², as a diagnostic for effort burdens and structural infeasibility when decisions rely on immutable features.
• The paper shows that endogenous rule updates can produce stratified equilibria in which immutable-feature gaps persist or widen, while simulations reveal classifier rotation, declining incentives, and convergence near actionable ceilings.

This paper develops a formal framework for actionable recourse in competitive selection environments, where only a fixed fraction $\rho$ of candidates can succeed. The central contribution is to model recourse as a closed-loop dynamical system in which the selection benchmark and the recommended improvement direction are both endogenous to the population's feature distribution. The authors show that this coupling can amplify initial disparities and produce persistent, structurally supported performance gaps—a formal counterpart to the sociological phenomenon of "involution."

Competitive selection as upper-tail CVaR maximization

The designer selects the top $\rho$ fraction of a population of $n$ candidates under a linear scoring rule $s_i = w^\top x_i$. The paper's first technical move is to recast top-$\rho$ selection as maximization of a customized upper-tail conditional value-at-risk, $\mathrm{CVaR}^{\mathrm{up}}_\rho(w)$, defined via a variational form analogous to the Rockafellar–Uryasev representation. A lemma establishes the equivalence between this risk measure and the average score of the top $\rho n$ order statistics. Because the measure is positively homogeneous of degree one, the raw maximization is ill-posed; quadratic regularization with parameter $\lambda$ restores well-posedness.

The regularized problem admits a dual formulation in which the inner minimization over a threshold and slacks yields dual weights $\alpha_i$ on a capped simplex ($0 \le \alpha_i \le 1/(\rho n)$, $\rho$0). Combining the inner and outer maximizations produces a max–max problem that reduces, after eliminating $\rho$1, to maximizing the squared norm of the weighted feature centroid over the capped simplex. By complementary slackness and the geometry of extreme points, the optimal dual solution places weight $\rho$2 on exactly $\rho$3 candidates, and the optimal scoring direction is proportional to the empirical centroid of this upper-tail subset:

$\rho$4

This is a clean structural result: competitive selection is equivalent to choosing the subset of size $\rho$5 whose centroid has maximal norm, with dual variables acting as endogenous selection weights. It also makes explicit that the classifier itself is a function of who is currently winning—already foreshadowing the endogeneity that drives the dynamic analysis.

Recourse under the induced rule

Given the linear rule and threshold $\rho$6, a rejected candidate with margin $\rho$7 seeks the minimal-cost action in the actionable subspace $\rho$8 (features in $\rho$9 are immutable). Under quadratic effort cost, the recourse problem has a closed-form solution: the optimal action is proportional to the actionable weight projection, $n$0, with cost $n$1.

Two consequences follow directly. First, recourse cost grows quadratically in the margin, so candidates far from the boundary face disproportionately large effort burdens. Second, the denominator $n$2 acts as a diagnostic of "actionability strength": when the optimal scoring direction lies entirely in the immutable subspace ($n$3), recourse is infeasible for every rejected candidate, and rejection becomes structurally final rather than performance-based. The paper frames this as a structural accountability failure—contestability can be denied not by an explicit rule but by the geometry of the learned classifier. A further normative observation is that because $n$4, the classifier explicitly directs effort; legitimacy would require that this direction align with socially productive dimensions, a condition the framework can measure but does not enforce.

Closed-loop dynamics and endogenous direction

The dynamic model proceeds in discrete time. At each step, the designer recomputes $n$5 from the current population $n$6; rejected candidates then move along the normalized actionable direction $n$7, the projection of $n$8 onto $n$9. Effort is modeled with a logarithmic barrier cost,

$s_i = w^\top x_i$0

where $s_i = w^\top x_i$1 is the remaining gap between candidate $s_i = w^\top x_i$2's actionable feature and a hard ceiling $s_i = w^\top x_i$3. The barrier term is strictly convex and its marginal cost diverges as $s_i = w^\top x_i$4, so improvement becomes infinitely expensive near the ceiling. The candidate's strictly concave problem yields a unique interior optimum given in closed form via the first-order condition, and the resulting closed-loop recursion expresses each candidate's update in terms of two endogenous objects: the actionable projection of the current top-$s_i = w^\top x_i$5 tail centroid, and the individually optimal effort.

The induced map $s_i = w^\top x_i$6 is deterministic but only piecewise smooth: small perturbations of $s_i = w^\top x_i$7 that swap the $s_i = w^\top x_i$8-th ranked candidate can discontinuously change the selected set, and hence the centroid and communicated direction. This nonsmoothness is a genuine obstacle to classical convergence analysis, and the paper does not resolve it.

Equilibrium and stratification

A recourse equilibrium is a fixed point of $s_i = w^\top x_i$9. The fixed-point characterization is sharp: $\rho$0 is an equilibrium if and only if $\rho$1 for all rejected candidates. This admits exactly two regimes. A structural equilibrium has $\rho$2—selection depends only on immutable features. An effort-suppressed equilibrium has $\rho$3 but zero optimal effort from every rejected candidate, driven by unfavorable marginal tradeoffs (e.g., a strong barrier).

The paper's central qualitative claim emerges here: at a stratified equilibrium, the inter-group gap is supported entirely on the immutable subspace, $\rho$4. Because both the success threshold and the improvement direction are determined by the selected tail $\rho$5, the initially favored group defines not only the standard of success but the direction in which others must compete. Whenever the tail centroid moves faster than the rejected group improves, the gap $\rho$6 widens monotonically. This is the mechanism by which competitive recourse amplifies initial disparities rather than correcting them—a direct contrast with the individual-level promise of recourse as a tool for upward mobility.

Numerical case study

The simulation uses a two-feature setting (immutable GPA, actionable GRE capped at $\rho$7) with the logarithmic barrier effort rule. The dynamics exhibit three phases. In the early phase ($\rho$8 to $\rho$9), the oblique boundary gives rejected candidates a clear incentive, and GRE scores rise along the communicated direction. In the intermediate phase, the improving candidates change the top-$\mathrm{CVaR}^{\mathrm{up}}_\rho(w)$0 tail, causing the classifier to rotate and the boundary to steepen—direct evidence of endogenous co-evolution of rule and population. In the final phase ($\mathrm{CVaR}^{\mathrm{up}}_\rho(w)$1 to $\mathrm{CVaR}^{\mathrm{up}}_\rho(w)$2 and beyond), mean GRE flattens, GRE variance contracts, and the norm of the actionable signal $\mathrm{CVaR}^{\mathrm{up}}_\rho(w)$3 declines. Convergence arises from two complementary forces the framework distinguishes explicitly: attenuation of the actionable signal itself, or suppression of effort via the barrier near the ceiling. The case study is illustrative rather than a systematic empirical validation; no sensitivity analysis over $\mathrm{CVaR}^{\mathrm{up}}_\rho(w)$4, $\mathrm{CVaR}^{\mathrm{up}}_\rho(w)$5, or cost parameters is reported.

Limitations and open questions

Several assumptions bear directly on the results. The scoring rule is restricted to be linear, and the designer is assumed to re-optimize exactly at every step; both the CVaR duality and the closed-form recourse rely on this. The candidate model is deterministic best-response with a single actionable coordinate in the dynamic section, and the barrier parameters $\mathrm{CVaR}^{\mathrm{up}}_\rho(w)$6 are exogenous. The equilibrium analysis characterizes fixed points but does not establish existence, uniqueness, or convergence of $\mathrm{CVaR}^{\mathrm{up}}_\rho(w)$7 to an equilibrium—nonsmoothness of $\mathrm{CVaR}^{\mathrm{up}}_\rho(w)$8 from set membership changes is acknowledged but not addressed. The stratification result identifies when gaps persist but not their magnitude as a function of $\mathrm{CVaR}^{\mathrm{up}}_\rho(w)$9 or the cost structure. The authors explicitly defer deeper dynamical-systems analysis and the connection to feedback-form dynamic games to future work.

Conclusion

The paper reframes actionable recourse from an individual-versus-fixed-classifier question into a dynamic game of endogenous selection. Its main analytical results—CVaR representation of competitive selection, closed-form recourse with an actionability diagnostic, the two-regime fixed-point characterization, and the immutable-subsupport of stratified equilibria—jointly show that universally available recourse does not guarantee mobility when selection is competitive. The framework offers concrete quantities ($\rho n$0, $\rho n$1) that could serve as audit metrics for competitive decision systems, while leaving convergence theory and welfare-optimal designer policy as open problems.