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arxiv: 2606.27150 · v1 · pith:DH2JCN6Snew · submitted 2026-06-25 · 💱 q-fin.RM · math.OC

Endogenous Reinsurance Pricing in Large Competitive Insurance Markets: Finite-Player and Mean Field Analysis

Ruimeng Hu , Byungdoo Kong This is my paper

Pith reviewed 2026-06-26 01:29 UTC · model grok-4.3

classification 💱 q-fin.RM math.OC
keywords reinsurance pricingStackelberg equilibriummean field gamesrelative performancethreshold structurecompetitive insuranceendogenous premiumspillover effects
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The pith

Relative performance concerns among insurers generate a spillover mechanism that produces a threshold retention structure, reducing the reinsurer's Stackelberg pricing problem to one-dimensional optimization over a compact interval.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies a market in which one reinsurer sets a common premium and insurers choose retention levels while tracking both their absolute performance and their performance relative to the rest of the population. For any fixed premium the insurers' equilibrium retention is characterized by the solution of a scalar fixed-point equation whose dependence on the premium is monotone and exhibits three distinct regimes. This structure converts the reinsurer's problem into a one-dimensional search whose solution yields Stackelberg equilibria in both the finite-player setting and its mean-field limit. The analysis supplies an efficient continuation algorithm for the finite case and proves convergence of equilibria without requiring uniqueness of the mean-field premium.

Core claim

For any fixed premium the insurers' Nash equilibrium retention levels solve a scalar fixed-point equation; the solution is monotone in the premium and therefore partitions the premium axis into three intervals in which all insurers fully cede, retain partially, or retain fully. The reinsurer, acting as Stackelberg leader, optimizes its premium over the compact interval delimited by the two critical thresholds, producing equilibria for both finite populations and the corresponding mean-field model; finite equilibria converge to mean-field equilibria without any uniqueness assumption on the limit premium.

What carries the argument

scalar fixed-point equation whose monotone solution induces a three-regime threshold structure in retention as a function of premium

If this is right

  • The reinsurer's optimization reduces to a search over a single compact interval rather than over all possible retention configurations.
  • A threshold-continuation algorithm computes finite-player equilibria without enumerating retention patterns.
  • Finite-player equilibria converge to mean-field equilibria without requiring uniqueness of the limiting premium.
  • Relative-performance concerns can produce strictly positive retention even when reinsurance is offered at an actuarially favorable rate.
  • Heterogeneity in insurer risk profiles amplifies the spillover effects that shift the location of the retention thresholds.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same fixed-point reduction may allow similar Stackelberg analyses in other markets where agents care about relative performance, such as asset-management tournaments.
  • Regulatory caps on reinsurance premiums could be mapped directly onto the threshold intervals to predict which retention regime prevails.
  • The continuation procedure offers a template for computing equilibria in other mean-field games whose best responses admit scalar fixed-point representations.
  • If claim risks are correlated across insurers, the scalar fixed point would become a vector fixed point whose dimension equals the number of distinct risk classes.

Load-bearing premise

For every fixed premium the insurers' game possesses an equilibrium retention vector that reduces to a scalar fixed point whose solution increases with premium through the claimed full-cession, partial-retention, and full-retention regimes.

What would settle it

Fix two insurers with different risk exposures, compute their best-response retention functions explicitly, and verify whether their intersection crosses from full cession to partial retention to full retention at two distinct premium thresholds.

Figures

Figures reproduced from arXiv: 2606.27150 by Byungdoo Kong, Ruimeng Hu.

Figure 1
Figure 1. Figure 1: considers three insurers with (a 1 , a2 , a3 ) = (1, 1.30, 1.35) and (θ 1 , θ2 , θ3 ) = (0.5, 0, 1). Consistent with Proposition 2.6 and Corollary 2.8, each qb i (p) is continuous and monotonically increasing, and is affine on each subinterval of [p min, pmax] = [1, 1.880], where the follower retention configuration is fixed. Here, p min = 1 and p max = 1.880 follow from (2.16), while the insurer-specific … view at source ↗
Figure 2
Figure 2. Figure 2: Two-insurer examples with θ 1 = 0 and varying θ 2 . Each panel plots the follower equilibrium retentions qb i (p), i = 1, 2, and the reinsurer’s reduced objective J L (p). The graphical conventions are the same as in [PITH_FULL_IMAGE:figures/full_fig_p027_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Two-insurer Stackelberg–Nash equilibrium maps o [PITH_FULL_IMAGE:figures/full_fig_p028_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Mean field equilibrium and finite-player convergen [PITH_FULL_IMAGE:figures/full_fig_p029_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Mean field equilibria and finite-player convergenc [PITH_FULL_IMAGE:figures/full_fig_p030_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Finite-player premium diagnostics for the nonuni [PITH_FULL_IMAGE:figures/full_fig_p031_6.png] view at source ↗
read the original abstract

We study endogenous reinsurance pricing in a competitive insurance market with one strategic reinsurer and many heterogeneous insurers. The reinsurer acts as a Stackelberg leader by choosing a common premium rate and an investment strategy, while insurers decide how much risk to retain and how to invest, taking into account their own performance, their performance relative to the insurer population, and common insurance-claim and financial-market noise. This creates a feedback loop absent from standard reinsurance models with exogenous premiums: a premium change affects insurers directly through the cost of reinsurance, and indirectly through the population's aggregate exposure to common insurance-claim risk. For a fixed premium, we characterize the insurers' equilibrium retention through a scalar fixed point and establish its monotone premium response. This characterization reveals a spillover mechanism generated by relative performance concerns and leads to a threshold structure in which insurers move from full cession to partial retention and then to full retention as the premium increases. Using this structure, we reduce the reinsurer's premium problem to a one-dimensional optimization over a compact premium interval and characterize Stackelberg equilibria in both finite-player and mean field models. In the finite-player case, we develop an efficient threshold continuation procedure that determines equilibrium premiums without enumerating all retention configurations. We also prove convergence from finite-player equilibria to mean field equilibria without requiring the mean field equilibrium premium to be unique. Numerical illustrations show how relative performance concerns amplify spillover effects and can induce retention even when reinsurance remains actuarially favorable. They also demonstrate that Stackelberg equilibria need not be unique in either setting.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper models endogenous reinsurance pricing as a Stackelberg game between one strategic reinsurer (choosing premium and investment) and many heterogeneous insurers (choosing retention and investment) who care about both absolute and relative performance under common claim and market noise. For fixed premium the insurers' game is reduced to a scalar fixed-point equation whose solution is shown to be monotone in premium and to exhibit a three-regime threshold structure (full cession, partial retention, full retention). This structure reduces the reinsurer's problem to one-dimensional optimization over a compact interval; finite-player and mean-field Stackelberg equilibria are characterized, an efficient threshold-continuation algorithm is given for the finite case, and convergence of finite-player equilibria to mean-field equilibria is proved without requiring uniqueness of the mean-field premium. Numerical examples illustrate amplification of spillovers by relative-performance concerns.

Significance. If the fixed-point characterization, monotonicity, and convergence results hold, the work supplies a tractable endogenous-pricing framework that captures feedback absent from exogenous-premium models and yields concrete computational and comparative-static tools. The reduction to scalar fixed point and the convergence statement that does not presuppose uniqueness of the mean-field equilibrium are genuine technical strengths.

minor comments (3)
  1. The abstract and introduction should state the precise regularity conditions (e.g., moment bounds on claim sizes, Lipschitz constants on the relative-performance term) that guarantee existence and uniqueness of the scalar fixed point for each premium; without these the threshold-structure claim is difficult to verify from the high-level description alone.
  2. Numerical section: parameter values, grid sizes, and convergence tolerances used to generate the figures should be reported explicitly so that the spillover-amplification and non-uniqueness observations can be reproduced.
  3. Notation for the mean-field measure and the finite-N empirical measure should be introduced once and used consistently; occasional switches between P^N and the limiting P create minor ambiguity in the convergence argument.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading, positive summary, and recommendation of minor revision. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper's central steps—characterizing insurer equilibrium retention via a scalar fixed point for fixed premium, establishing monotone response and threshold structure from relative performance concerns plus common noise, then reducing the reinsurer's Stackelberg problem to 1D optimization over a compact interval—are presented as derived directly from the game setup. No quoted reduction equates a claimed prediction or result to its inputs by construction, no fitted parameters are relabeled as predictions, and no load-bearing self-citation or imported uniqueness theorem appears in the provided text. The finite-player to mean-field convergence is stated without requiring uniqueness of the mean-field premium. This aligns with standard mean-field game analysis and remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Review based on abstract only; no explicit free parameters, axioms, or invented entities are detailed in the provided text. Standard game-theoretic assumptions such as equilibrium existence are implicit but not enumerated.

axioms (2)
  • domain assumption Existence of Nash equilibrium in the insurers' retention game for any fixed premium
    Required to allow reduction to scalar fixed point as stated in abstract.
  • domain assumption Common insurance-claim and financial-market noise affecting all insurers
    Stated explicitly in abstract as part of the model setup enabling feedback loop.

pith-pipeline@v0.9.1-grok · 5814 in / 1518 out tokens · 29921 ms · 2026-06-26T01:29:01.997683+00:00 · methodology

discussion (0)

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