As platforms increasingly rely on learning algorithms, collectives may form and seek ways to influence these platforms to align with their own interests. This can be achieved by coordinated submission of altered data. To evaluate the potential impact of such behavior, it is essential to understand the computations that collectives must perform to impact platforms in this way. In particular, collectives need to make a priori assessments of the effect of the collective before taking action, as they may face potential risks when modifying their data. Moreover they need to develop implementable coordination algorithms based on quantities that can be inferred from observed data. We develop a framework that provides a theoretical and algorithmic treatment of these issues and present experimental results in a product evaluation domain.

We introduce *Backward Conformal Prediction*, a method that guarantees conformal coverage while providing flexible control over the size of prediction sets. Unlike standard conformal prediction, which fixes the coverage level and allows the conformal set size to vary, our approach defines a rule that constrains how prediction set sizes behave based on the observed data, and adapts the coverage level accordingly. Our method builds on two key foundations: (i) recent results by Gauthier et al. [2025] on post-hoc validity using e-values, which ensure marginal coverage of the form $\mathbb{P}(Y_{\rm test} \in \hat C_n^{\tilde{\alpha}}(X_{\rm test})) \ge 1 - \mathbb{E}[\tilde{\alpha}]$ for any data-dependent miscoverage $\tilde{\alpha}$, and (ii) a novel leave-one-out estimator $\hat{\alpha}^{\rm LOO}$ of the marginal miscoverage $\mathbb{E}[\tilde{\alpha}]$ based on the calibration set, ensuring that the theoretical guarantees remain computable in practice. This approach is particularly useful in applications where large prediction sets are impractical such as medical diagnosis. We provide theoretical results and empirical evidence supporting the validity of our method, demonstrating that it maintains computable coverage guarantees while ensuring interpretable, well-controlled prediction set sizes.