Conformal Prediction at Performative Equilibrium: The Price of Performativity

Ahmed Mahrous; Roberto Di Pietro

math.STNov 3, 2026

Conformal Prediction at Performative Equilibrium: The Price of Performativity

Authors: Ahmed Mahrous, Roberto Di Pietro

Organizations: King Abdullah University of Science and Technology (KAUST), Thuwal 23955, Saudi Arabia

Abstract

Conformal prediction gives finite-sample coverage when calibration and test points are exchangeable, but exchangeability can fail when deployment changes the data distribution. We study conformal prediction in performative settings, where the deployed predictor affects the data distribution, and seek prediction sets valid at performative equilibrium: coverage must hold under $D(f^\star)$ for a performatively stable predictor $f^\star$ . Under total-variation $L$ -sensitivity of the distribution map, we prove a phase transition at $L=1$ . If $L \ge 1$ , no finite-sample procedure can uniformly guarantee both equilibrium-valid coverage and nontrivial prediction sets. If $L<1$ , the minimax excess miscoverage over informative procedures is $\widetildeΩ(1/((1-L)\sqrt{n}))$ , implying a sample-complexity inflation of order $1/(1-L)^2$ for fixed excess miscoverage. Our main technical tool is a transfer theorem: any procedure returning a predictor within $\varepsilon$ of equilibrium yields equilibrium-valid coverage up to an additive $L\varepsilon$ term after split-conformal calibration. We instantiate the theorem with two retraining algorithms: a model-free method for discretizable classes and a parametric tail-averaged method that matches the $n$ - and $L$ -dependence of this rate up to logarithmic factors. Experiments, including semi-synthetic credit scoring, demonstrate the predicted threshold behavior and scaling laws.

Figures & tables

f_{t}\leftarrow\arg\min_{f\in\mathcal{F}}\widehat{R}(f;S_{t})

Algorithm 2 Parametric Performative Conformal Prediction (P-PCP)

	Model-free (Alg. 1 )	Parametric (Alg. 2 )
Contraction	$L<1$	$\rho=L\beta/\gamma<1$
Output	Last iterate	Average after stabilization
Rate	$\widetilde{O}\!\bigl(\frac{1}{(1-L)^{3/2}\sqrt{n}}\bigr)$	$\widetilde{O}\!\bigl(\frac{1}{(1-\rho)\sqrt{n}}\bigr)$
Loss assumptions	Bounded	Smooth, strongly convex
Relation to lower bound	Suboptimal by a factor $(1-L)^{-1/2}$ , up to logs	Matches lower bound up to logs when $\rho\asymp L$

Table 1: Comparison of the two algorithms

Panel of Figure 1 — Figure 1: Synthetic validation of the theory. (a) Excess miscoverage versus sample size $n$ (log–log) for $L\in\{0,\ldots,0.7\}$ ; the dashed line shows the $n^{-1/2}$ reference slope. (b) Threshold sample size $n^{\star}(L)$ , defined as the smallest $n$ for which excess miscoverage is at most $0.02$ , plotted against $1/(1-L)^{2}$ ; the dashed line is a linear fit. (c) Coverage versus $L\varepsilon$ for five values of $L$ . The dashed line is the first-order lower bound $1-\alpha-L\varepsilon$ .

Figure 2: Phase transition at

L=1

. Bounded Bernoulli sweep based on

g_{L,\pm}(\theta)=\operatorname{clip}(c+L(\theta-c)\pm\varepsilon,\,[\varepsilon,1-\varepsilon])

with

c=0.05

and

\varepsilon=0.01

. The green curve is conservative equilibrium coverage (minimum over the

+

and

-

branches); the blue curve is the unrestricted full-set rate (maximum over the two branches). The supercritical regime is therefore well-defined for all plotted

L

, with explicit equilibria throughout. The dashed vertical line marks the threshold

L=1

Panel of Figure 3 — Figure 3: Semi-synthetic strategic regression experiment. (a) Equilibrium coverage versus manipulation strength $\epsilon$ . The naive split conformal baseline undercovers as performativity increases, whereas P-PCP remains closer to the nominal $90\%$ target. (b) Median prediction interval width versus $\epsilon$ for the two procedures.

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

Simulation	Figures	Purpose / key parameters	Reps
1	Fig. 1 (a,b)	$L\in\{0,0.1,\ldots,0.7\}$ , $n\in\{50,100,200,500,1000,3000,10^{4},3\times 10^{4},10^{5}\}$ , Gaussian location model, $a=0.3$ ; tests rate and price of performativity	2000
2	Fig. 1 (c)	$L\in\{0.1,0.3,0.5,0.7,0.9\}$ , $\varepsilon\in\{0,0.05,0.1,0.2,0.5,1,2\}$ , $m=2000$ , Gaussian location model, $a=0.3$ ; tests $L\varepsilon$ collapse	2000
3	Fig. 2	$L\in\{0.7,0.8,0.9,0.95,0.98,0.99,$ $1.0,1.01,1.02,1.05,1.1,1.2\}$ , bounded Bernoulli sweep at $n=300{,}000$ with $c=0.05$ and $\varepsilon=0.01$ ; tests the phase boundary with explicit equilibria for all $L$	2000
4	Fig. S1	$L\in\{0,0.3,0.5,0.7,0.9\}$ , $n\in\{100,500,1000,5000,10^{4},5\times 10^{4}\}$ , $\delta=0.5/\sqrt{n}$ ; Le Cam two-hypothesis construction; worst-case excess miscoverage and scaled rate check	2000
5	Fig. S2	Finite-class Gaussian location model for MF-PCP; $L\in\{0,0.3,0.5,0.7\}$ , $N\in\{10,50,200\}$ , $n\in\{200,500,1000,5000,10^{4},5\times 10^{4}\}$	2000
6	Fig. S3	Impossibility construction at $L=1$ using $D_{+}$ and $D_{-}$ , $n\in\{100,500,1000,5000,10^{4},5\times 10^{4},10^{5}\}$ , $\varepsilon=0.5/n$	2000

Appendix

Table S1 : Synthetic experiments reported in the paper and appendix.

Panel of Figure 7 — Figure S1 : Simulation 4: Le Cam lower bound validation. (a) Worst-case excess miscoverage (maximum over $D_{0},D_{1}$ ) vs. $n$ on a log-log scale. The $L=0.9$ curve saturates at the trivial level for $n\leq 1000$ (algorithm has not converged). (b) Scaled quantity $\mathrm{excess}\times(1{-}L)\times\sqrt{n}$ ; approximate flatness in $[0.13,0.35]$ confirms the $\Omega(1/((1{-}L)\sqrt{n}))$ rate.

Panel of Figure 8 — Figure S2 : Simulation 5: Model-free algorithm validation. (a) Excess miscoverage vs. $n$ for $N=50$ , confirming $n^{-1/2}$ convergence across $L$ values. (b) Effect of class size $N$ at $L=0.5$ ; curves nearly overlap, indicating the $\log N$ factor is small.

Figure S3 : Simulation 6: Impossibility at

L=1

. Failure rate (trivial set or coverage failure) vs.

n

for maps

D_{+}

and

D_{-}

. The rate stays bounded away from zero at all sample sizes, consistent with Theorem 3.1 . Dashed line: Le Cam floor at

0.5

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Split conformal prediction provides finite-sample marginal coverage under exchangeability, but this guarantee averages over the random calibration sample. We study instead the law of the calibration-conditional coverage induced by a realized conformal threshold. In the continuous i.i.d. setting this law is exactly

Beta(k,n+1-k)

, so the usual marginal guarantee corresponds to its mean. We take this beta law as a finite-sample reference object and quantify departures from it using Wasserstein distances on

[0,1]

. The framework yields direct bounds on marginal coverage gaps and on bad-calibration probabilities, and separates different sources of non-i.i.d. behavior according to how they deform the beta reference: test-side shift acts through a transport map on the coverage scale, while calibration dependence changes the order-statistic law itself. We instantiate the framework in scale-shift, clustered, and stationary mixing settings, where the induced deformations can be characterized explicitly or through Berry-Esseen approximations. Simulations on dependent processes confirm that the first-order approximation tracks the empirical Wasserstein distance even at moderate sample sizes.

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