Linear Risk Sharing (LRS)

• Linear Risk Sharing is a method where each participant’s share is a linear function of total risk, ensuring fair and efficient allocation.
• It relies on convex analysis, duality, and infimal convolution techniques to derive optimal, tractable risk-sharing contracts.
• Applications span insurance, finance, and decentralized networks, demonstrating both practical policy design and theoretical robustness.

Linear Risk Sharing (LRS) refers to a class of methodologies, models, and allocation rules in which aggregate risk—typically arising from uncertain losses or financial exposures—is split among participants through linear, affine, or proportionally structured mechanisms. The linearity arises both in the allocation rules (each participant's share is a linear function of total risk or state variables) and, in many formulations, in the dual mathematical representations stemming from classical convex analysis, stochastic optimization, or network-theoretic perspectives. LRS serves as both a normative benchmark for fair and efficient risk transfer and a basis for designing practically implementable risk-sharing contracts and systems across insurance, finance, and decentralized economic networks.

1. Mathematical Foundations of Linear Risk Sharing

The mathematical infrastructure of LRS is grounded in convex analysis, duality, and the theory of risk measures. The archetypal setup involves $n$ agents and an aggregate risk $X$ (such as a random loss or claim), with the goal of finding an allocation $(X_1, \ldots, X_n)$ such that $\sum_{i=1}^n X_i = X$ and each $X_i$ is assigned via a linear or affine rule: $X_i = \alpha_i X + \beta_i.$ In classical complete markets, this structure emerges naturally from utility maximization, super-hedging duality, and convex optimization:

• Dual characterizations: In super-hedging and utility maximization, the optimal cost or allocation splits risk linearly with respect to state variables or Lagrange multipliers, as in

$p(G) = \sup_{Q\in\mathcal{Q}} E^Q[\beta_T G]$

or by first-order conditions on risk-tolerances $\gamma_i$,

$$X_i^* = \left(\frac{1/\gamma_i}{\sum_{j} 1/\gamma_j}\right) X + \kappa_i.$$

(Bouchard, 2013)

• Infimal convolution of risk measures: In the allocation of risk under different risk measures $\rho_i$, the infimal convolution $\bigotimes_{i=1}^n \rho_i$ preserves linearity under certain regularity. That is, for "cash-additive" or convex risk measures,

$$\left(\bigotimes_{i=1}^n \rho_i\right)(X) = \inf\left\{\sum_{i=1}^n \rho_i(X_i): \sum X_i = X\right\},$$

leading to explicit linear sharing rules for classes of risk measures including entropic and expected shortfall (Melnikov, 25 May 2025).

• Stochastic target problems and HJB PDEs: The value function for reaching a set of acceptable terminal states via portfolios or allocations often satisfies a Hamilton-Jacobi-BeLLMan equation where, after dual transformations, the linearity in state variables is preserved (Bouchard, 2013).
• Axiomatizations: Linear and affine rules (including uniform, mean-proportional, and covariance-based) are uniquely characterized by properties such as reshuffling invariance, source-anonymity, and strong aggregate dependence (Dhaene et al., 9 Nov 2024).

2. Game Theory, Equilibrium, and Strategic Considerations

LRS is not only a matter of efficient allocation but also of strategic behavior among participants:

• Nash equilibria in risk-sharing games: When agents possess market power, strategic reporting leads to risk-sharing allocations that may be linear but are subject to endogenous truncation (limited liability) and inefficiency (utility loss relative to the Arrow-Debreu optimum). The equilibrium contract in such models is of the form

$$C_i = \delta_i \log\left(\frac{R_i}{Q}\right) + \delta_i H(Q, R_i),$$

where $R_i$ is the reported belief and $Q$ is the equilibrium pricing kernel (Anthropelos et al., 2014).

• Fairness and Surplus Sharing: Auction-based mechanisms allocate first-mover advantage and ensure surplus is shared equally, providing decentralized, strategy-proof LRS implementations (Ogaku, 7 May 2025).
• LRS under ambiguity or model uncertainty: When agents face sets of priors or multiple models, the equilibrium combines utility maximization with penalizations for model divergence, producing (through dual representations) optimal allocations that respect both risk aversion and ambiguity attitudes (Kroell et al., 3 Apr 2025).

3. Design and Analysis of Linear Risk Sharing Rules

LRS rules can be formulated and evaluated according to structural, axiomatic, and performance-based criteria:

• Uniform and Proportional Rules: Fundamental LRS schemes include:
  • Uniform: Equal division, $X_i = X/n$.
  • Mean-proportional: $X_i = (\mathbb{E}X_i/\mathbb{E}X) X$.
• Covariance-based and scenario-based rules: More sophisticated affine rules adjust for agents' risk contributions based on covariance or scenario outcomes,

$$X_i = \mathbb{E}[X_i] + \frac{\mathrm{cov}(X_i, X)}{\mathrm{var}(X)} (X - \mathbb{E}[X]).$$

(Dhaene et al., 9 Nov 2024)

• Distortion risk metrics and comonotonicity: When agents use distortion riskmetrics (e.g., Gini deviation, mean-median deviation, or quantile-based measures), the Pareto-optimal allocation is determined by the aggregation (infimal convolution) of their metrics, which, for concave distortions, results in comonotonic (linear) allocations, and for non-concave cases, in mixtures involving countermonotonic structures (Lauzier et al., 2023).
• Acceptance sets and dual penalties: In continuum models, the aggregate acceptance set is the $L^\infty$-closure of the Aumann integral over individual agent sets, while the global dual penalty is the $\mu$-weighted average of the individual penalties (Melnikov, 25 May 2025).

4. LRS on Networks and Peer-to-Peer Systems

The network-theoretic generalization of LRS analyzes how topological and stochastic properties of agent connections affect risk redistribution:

• Matrix formalism: Allocations are effected via $\xi = M X$, where $M$ is a nonnegative (row-, column-, or doubly-stochastic) matrix. Key constraints include:
  • Row-stochasticity enforces convex pooling per agent.
  • Column-stochasticity ensures global budget balance.
  • Doubly-stochastic matrices guarantee full risk pooling and fairness (majorization, convex order dominance).
• Topological effects: Complete graphs maximize diversification; star and scale-free networks induce heterogeneity favoring high-degree nodes; ring and random graphs interpolate between these extremes.
• Random networks and two-layer randomness: Randomness in both $X$ and the network (random $M$) leads to final exposure variances reflecting degree distribution and network heterogeneity, e.g. $\mathrm{Var}(\xi_i) \approx \sigma^2 / (d_i+1)$ (Charpentier et al., 25 Sep 2025).
• Self-retention vs. diversification: Convex combinations of the identity and network matrix, $M(\lambda) = (1-\lambda) I + \lambda P$, encode the trade-off between retaining own risk and pooling.

5. Risk Measures, Convex Order, and Asymptotic Properties

LRS interacts with the statistical evaluation of risk via coherent and distortion-based measures:

• Value-at-Risk (VaR), Conditional Tail Expectation (CTE), and Lambda-VaR: In tail-heavy, multivariate setups, linear sharing preserves explicit bounds for individual and aggregate VaR/CTE, determined by transformation of the exponent measure under the sharing matrix $A$ (Kley et al., 2015, Liu et al., 6 Aug 2024).
• Inf-convolution for Law-Invariant Measures: For functionals such as expected shortfall or Choquet quantiles, the infimal convolution over agents' measures yields explicit aggregate measures and optimal linear allocations—robust even in ambiguous, non-additive or heterogeneous belief settings (Liu et al., 27 Dec 2024, Melnikov, 25 May 2025).
• Convex order and majorization: Linear sharing via doubly-stochastic matrices ensures ex-post allocations are less risky (in the convex order) than pre-sharing allocations, and variance or higher-moment reductions follow naturally from matrix theory (Birkhoff–von Neumann decomposition, etc.) (Charpentier et al., 25 Sep 2025).

6. Limitations, Constraints, and Extensions

The theory and application of LRS are subject to multiple important constraints:

• Limited liability and bounded contracts: Imposing lower/upper bounds on exposures (as in wage floors or maximum coverages) necessitates modifications—e.g., the optimal contract becomes an "option" on the linear contract, truncated at the boundaries (Martin, 2020).
• Market power, strategic misreporting: In OTC or thin markets, agents may distort beliefs or allocations strategically, leading to bounded, inefficient (non-Pareto optimal) but still essentially linear risk-sharing rules (Anthropelos et al., 2014).
• High dimensionality and lack of robust improvements: In large state spaces, the set of possible welfare-improving allocations after shocks or renegotiation shrinks exponentially with dimension, limiting the relevance or probability of nontrivial renegotiations or dynamic improvements (Echenique et al., 14 Jan 2024).
• Computational challenges: Infinite-dimensional allocation spaces, as in risk sharing among a continuum of agents or with infinite menus, require sophisticated compactness, duality, or measurable selection tools (Melnikov, 25 May 2025, Ogaku, 7 May 2025).

7. Practical Applications and Impact

LRS has been applied in:

• Insurance: Classic pooling, mutual and P2P insurance, and decentralised risk sharing on networks.
• Financial contracts: Pricing and hedging of derivatives under utility/risk minimization and in bilateral negotiation with funding differences and regulatory constraints (full margin requirements) (Lee et al., 2019).
• Regulatory and capital adequacy frameworks: Aggregating capital requirements across agents with heterogeneous risk measures or regulatory regimes (Liebrich et al., 2018).
• Decentralised allocation and consensus formation: Mechanism design for risk sharing without central enforcement—via sequential pricing, auctions, and decentralized choice (Ogaku, 7 May 2025).

LRS also underpins design principles for fair, transparent, and non-punitive risk redistribution in decentralized finance, community insurance, and network-driven economic collaborations. The mathematical structure not only yields tractable and robust allocation rules but also translates to practical, interpretable policies for exposure and premium setting in contemporary risk systems.