Conformal Risk Sharing: Certified Cost Allocation with Participation Guarantees

Abstract: Sharing the financial impact of rare adverse events across a group can soften extreme individual burdens, but any participant made worse off by the arrangement has reason to leave. A credible mechanism must therefore provide each agent with a trustworthy cap on their future obligation and should be deployed only if the aggregate harm across participants is bounded. We formalise this as the Certified Allocation Problem: from finite data and without distributional assumptions, find a redistribution rule, produce obligation caps for every participant, and verify that no participant is made materially worse off. We propose Conformal Risk Sharing, which solves this problem by pairing an interpretable sharing policy with split conformal calibration. The sharing intensity is tuned on training data, while held-out calibration data produces distribution-free per-agent guarantees (valid under exchangeability). Experiments on synthetic and real-world data, including precipitation and energy-cooperative data, confirm that the framework can substantially reduce extreme obligations for high-risk agents while controlling harm to others.

• The paper proposes a three-stage conformal calibration framework that achieves finite-sample, distribution-free per-agent coverage while enforcing explicit participation constraints.
• It employs an interpretable linear policy blending identity and pooling rules, optimized via split conformal prediction to manage tail risk and ensure regulated harm constraints.
• Empirical results reveal up to 50% tail cap reductions across applications like parametric insurance, spatial risk management, and energy pooling.

Conformal Risk Sharing: Formalizing Distribution-Free Certified Cost Allocation

Problem Formulation and Motivation

This paper addresses a core challenge in multi-agent systems: redistribution of realized stochastic costs while maintaining individual agent participation and providing robust, high-confidence guarantees on future obligations. The "Certified Allocation Problem" is rigorously defined at the intersection of cooperative cost sharing, distribution-free statistical inference, and mechanism design under uncertainty. Standard cooperative game-theoretic and actuarial risk-sharing frameworks require strong distributional assumptions and typically yield population-level (asymptotic or parametric) risk guarantees. Conformal prediction, by contrast, enables finite-sample marginal coverage but has rarely been operationalized in multi-agent redistribution with mutually coupled obligations and explicit participation constraints.

The motivating use cases are broad, spanning P2P parametric insurance, cooperative energy communities, and shared computing infrastructure. Each scenario exhibits heavy-tailed, temporally- or spatially-correlated risks and necessitates non-parametric per-agent guarantees to prevent adverse selection and unraveling of pooling arrangements. The Certified Allocation Problem requires redistributing a random vector of costs among $n$ agents: finding a feasible allocation rule $A$, producing per-agent finite-sample obligation caps, and ensuring that no participant's high-confidence obligation materially exceeds that of stand-alone exposure (the "identity" allocation).

Solution Framework: Conformal Risk Sharing

The solution, Conformal Risk Sharing, leverages a three-stage policy learning and certification pipeline involving training/validation/calibration splits. An interpretable linear allocation policy is parameterized as a convex combination of the identity and a base pooling rule (e.g., uniform, locality-based, or variance-optimal doubly-stochastic). The mutualisation intensity $\alpha$ determines the fraction of each agent's exposure pooled. This policy family supports direct operational and regulatory interpretation, crucial for real-world risk pooling mechanisms.

Calibration proceeds via split conformal prediction: high-confidence ($1-\delta$) per-agent obligation caps are computed as order statistics in held-out blocks, yielding marginal coverage for each agent under exchangeability. The selection of pooling intensity is completed on the validation set, optimizing aggregate tail risk subject to a certified harm constraint—limiting the aggregate cap increase relative to baseline, controlled by governance parameters ($\eta, \varepsilon$). The design enforces conservation (total obligations equal total losses) and requires that the empirical harm does not exceed the assigned budget for pooling to be deployed; otherwise, it transparently reverts to the identity allocation.

Key properties and theoretical guarantees include:

• Marginal finite-sample, distribution-free $1-\delta$ coverage for each agent’s cap at deployment (Theorem 1).
• System-level coverage for user-chosen functionals (Corollary 1), with no reliance on parametric or distributional assumptions.
• Separation of the policy-learning and certificate-calibration steps ensures the statistical validity of guarantees, even when adaptivity is present in the former.

Empirical Evaluation and Numerical Results

Experiments span (1) synthetic heavy-tailed, zero-inflated risk data; (2) real-world spatially correlated precipitation blocks in a parametric insurance context; and (3) deseasonalized cooperative electricity consumption data. Across these regimes, the framework exhibits robust marginal coverage, substantial certified relief for high-risk agents, and active enforcement of participation constraints.

Prominent numerical findings include:

• In synthetic settings, global pooling yields up to 9% reduction in top-decile agent tail caps and a 23% rejection rate at the certification audit, confirming the practical significance of participation constraints.
• On E-OBS precipitation triggers, global pooling achieves a 27% aggregate tail cap reduction and a striking 50% reduction for high-risk deciles, with empirical coverage near the nominal level. Under strong temporal nonstationarity, calibration blocks distant from the deployment regime result in coverage degradation for all policies, confirming the necessity for periodic re-certification.
• In energy cooperative data, sparse cross-agent dependence allows for over 50% aggregate cap reductions when participation budgets are permissive, with all policies passing the harm constraint due to effective diversification. Coverage remains at or near the nominal level for all agent classes.

These results systematically illustrate the trade-off between efficiency (cap relief) and conservatism (coverage, harm constraint), governed by the participation parameters and the risk dependence structure.

Theoretical and Practical Implications

The framework formalizes a new principled approach to multi-agent cost sharing under uncertainty and finite data. It demonstrates, both theoretically and empirically, that marginal finite-sample guarantees are achievable without distributional assumptions or model risk, at the cost of potential conservatism in small samples or under strong dependence. It also reveals intrinsic limitations: joint (system-wide) high-confidence guarantees are not implied by per-agent conformal caps, and aggressive pooling under strong correlation can render participation constraints infeasible.

From a practical standpoint, the method is directly applicable to risk sharing in decentralized insurance, energy pooling, and resource sharing, where regulatory or contractual requirements preclude model-based or parametric guarantees. Its modularity allows deployment with user-specified pooling rules, governance constraints, and recertification frequency; operational transparency is ensured by auditability of the policy and certificate computation.

Future Directions

Future developments may enhance the expressivity and adaptivity of both the allocation policy class and the certification layer:

• Bayesian and parametric alternatives offer tighter caps when the model is well-specified, but forfeit worst-case distribution-free validity.
• Learning richer, low-dimensional policy structures (beyond scalar mutualisation) or end-to-end differentiable policy optimization, with fixed calibration, may unlock additional efficiency while retaining finite-sample guarantees.
• Covariate-conditional certificates (e.g., via conformalized quantile regression) could adapt tail caps to side information, and sequential deployment could benefit from online conformal prediction frameworks under temporal or spatial drift.
• Nonlinear pooling mechanisms, such as deductibles, may offer further capital relief, motivating extensions of the certification procedure to nonlinear policies.

Conclusion

Conformal Risk Sharing introduces a precise framework for distribution-free, certified cost allocation in uncertain, heavy-tailed, and multi-agent environments. By embedding conformal calibration in cost-sharing policy selection under explicit participation constraints, it provides operationally auditable guarantees and quantifiable safety/efficiency trade-offs without model risk. While current guarantees are marginal and the policy family is simple, the approach sets a formal foundation for robust, data-driven risk pooling mechanisms and motivates a range of extensions in statistical and economic multi-agent design.