Cheapest Feasible Matching (CFM)

• Cheapest Feasible Matching (CFM) is a framework for minimal, risk-adjusted premium and reward allocation that fosters participation and fairness in risk-sharing pools.
• It integrates coherent, law-invariant risk measures with game-theoretic equilibrium analysis to set the lowest feasible margins, ensuring stability and Sybil-resilience.
• CFM is employed in Proof-of-Stake blockchains and insurance pooling to achieve efficient capital allocation, decentralized control, and scalable risk mitigation.

The Cheapest Feasible Matching (CFM) paradigm encompasses a family of incentive-compatible reward sharing schemes in collaborative risk-bearing, staking, and insurance settings. CFM mechanisms ensure that risk or rewards are split in a way that sustains participation of all needed members, using the minimum feasible premium or margin while maintaining strong stability properties, allocation fairness, and resistance to manipulative attacks such as Sybil splits. CFM-type rules are prominent in the design of Proof-of-Stake blockchains, pooling mechanisms in insurance, and surplus allocation in multi-participant risk pools. The key objectives of CFM include decentralization, fairness, efficient capital allocation, and robust scaling of per-member premia as the participation pool grows.

1. Mechanistic Foundations of CFM

The CFM principle assigns to each agent or participant a minimal, risk-adjusted share of premium or reward, subject to feasibility constraints dictated by risk-sharing, cost-recovery, or reward-pooling requirements. In insurance and risk-sharing pools, the CFM approach uses law-invariant or coherent risk measures—monotone, translation-invariant and (often) convex and positive homogeneous functionals—allocating capital and premium in an economically minimal yet feasible way. In proof-of-stake or collaborative blockchain protocols, CFM instances are obtained via mechanisms that assign rewards to pool leaders and delegates in a way that achieves Nash equilibrium at the lowest sustainable premium level, while maintaining Sybil-resilience and decentralization (Coculescu et al., 2018, Brünjes et al., 2018, Kiayias et al., 7 May 2025).

Key mathematical foundations include:

• Subadditive capital requirements (e.g., TVaR, variance-based premia).
• Law-invariant risk functionals (via Kusuoka or dual representations).
• Game-theoretic equilibrium analysis to ensure no beneficial deviation below the CFM allocation.

2. Risk Measures and Premium Computation

In classical pooling (insurance, mutuals), the CFM emerges from Pareto-optimal equal-sharing contracts under general law-invariant risk measures. Given i.i.d. risks $X_1,\dots,X_n$ with mean $\mu$ and variance $\sigma^2$, the total capital required (premium) for the pool is

$\rho(L_n) = \text{risk measure of } L_n = S_n/n,$

where $L_n = S_n/n$ is the per-member loss. The "cheapest" premium per agent, ensuring feasibility, is then $\pi_n = \rho(L_n) - \mathbb{E}[L_n]$.

Notably, under certain law-invariant risk measures (e.g., rank-dependent utility or concave distortions), the CFM satisficing premium decays as $O(n^{-1/2})$ in the number of risks pooled. In expected-utility or variance-premium principles, the decay is faster, $O(n^{-1})$. This differential scaling is a central feature of the CFM landscape: for rank-dependent or distortion risk measures, the minimal sustainable premium shrinks only as $n^{-1/2}$, indicating persistent first-order frictions in risk sharing, as opposed to the full efficiency attained in expected-utility or quadratic settings (Knispel et al., 2021).

3. Implementation in Pooling and Blockchain Systems

The modern realization of CFM in blockchains appears in premium/reward sharing schemes for staking pools. The cap-and-margin scheme, as specified in (Brünjes et al., 2018), features a pool reward $r(\sigma,s)$ (with $\sigma$ total stake, $s$ operator stake) capped at a target pool size $\beta$, a margin $m$, and explicit cost-recovery for the operator. The distribution rule allows operators to set the lowest margin sustaining equilibrium participation:

• Each player selects pools with maximal per-unit desirability, $D_j = (1-m_j)\max(P_j,0)$, where $P_j = r(\beta,S_j)-C_j$ is the saturated-pool profit net of operator costs.
• Non-myopic Nash equilibrium is achieved with exactly $k$ pools, each of size $\beta=1/k$, and each operator's margin set to

$m_i = 1 - \frac{D_{k+1}}{P_i}.$

• This guarantees that no further reduction in premium is feasible without destabilizing the pooling structure, i.e., allocations are CFM in that context.

The situation is directly analogous in insurance surplus-sharing: the minimal risk-adjusted premium allocation for each participant is set by the Euler (marginal) allocation corresponding to a coherent risk measure, e.g., $T_i = E_{Q^*}[L_i]$, where $Q^*$ is the optimally-risky measure. Premiums above this threshold constitute capital contributions, subsequently shared via realized surpluses (Coculescu et al., 2018).

4. Equilibrium, Decentralization, and Manipulation Resistance

CFM mechanisms are tightly bound to equilibrium analysis: allocations are set so that no agent can benefit by unilaterally reducing their premium or margin below the feasible threshold without destabilizing the equilibrium (pool dissolution or running at a loss). In the context of staking pools, this ensures exactly $k$ viable pools with operator margins at the "lowest feasible" level—hence, the CFM label (Brünjes et al., 2018).

Sybil-resilience critically depends on rule structure. Proportional allocation (splitting rewards by stake) achieves perfect decentralization (Price of Stability, PoS = 1) but fails resistance to Sybil splits in adversarial settings. The Shapley-value rule, however, achieves both low PoS (≤4/3 in the oceanic regime, ≤2 with atomics) and Sybil-proofness at CFM-level margins (Kiayias et al., 7 May 2025). The CFM equilibrium thus aligns economic efficiency, decentralization, and attack resistance.

5. Scaling Laws and Practical Implications

A central insight from CFM analyses is the scaling law for achievable premium reductions. Under distortion-type risk measures or for blockchains with inherent first-order risk-sharing friction, halving per-member premium requires quadrupling pool size ($n \propto 1/\varepsilon^2$ for target residual premium $\varepsilon$). By contrast, under expected-utility or variance-type allocation, only a doubling is required ($n \propto 1/\varepsilon$) (Knispel et al., 2021). This constraint fundamentally limits the cost-effectiveness of mutualization in models with persistent risk aversion or concave sharing distortions.

Practically, in insurance or blockchain pooling:

• The minimal feasible margin (premium) is often above zero and decreases only slowly with scale.
• Capital requirements, surplus sharing, and decentralization are best secured by CFM-compatible allocations.
• Hybrid schemes approximate ideal CFM with computational tractability by adding operator bonuses to proportional rules and setting minimal acceptable margins to mimic Shapley allocation (Kiayias et al., 7 May 2025).

6. Special Cases and Variants

CFM reduces to classical proportional sharing in settings with linearity or additivity, and to more complex iterative allocations (e.g., Shapley value, marginal contributions) when superadditive/subadditive effects, complementarities, or pivotality are present. Variants of basic proportional sharing—including superadditive (proportional-to-square or higher) and subadditive (proportional-to-root) rules—typically yield degraded decentralization and worse manipulation resistance, and are generally avoided for robust CFM design. Hybridization using operator bonuses retains the bulk of CFM benefits while improving implementation efficiency (Kiayias et al., 7 May 2025, Brünjes et al., 2018).

7. Summary and Outlook

Cheapest Feasible Matching constitutes a fundamental paradigm for robust, efficient, and manipulation-resistant allocation of risk, capital, and rewards in stochastic, adversarial, or game-theoretic pool settings. CFM allocations reflect the minimal incentive-compatible margin structure supporting stable participation and capital sufficiency, across diverse applications in insurance, finance, and blockchain consensus. The literature indicates that practical adoption of CFM requires careful selection of risk measures, equilibrium computational methods, and design parameters (e.g., operator bonuses, cap settings) to optimize the trade-off between decentralization, fairness, and resilience to strategic exploitation (Knispel et al., 2021, Coculescu et al., 2018, Kiayias et al., 7 May 2025, Brünjes et al., 2018).