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Optimal Underreporting and Competitive Equilibrium

Zongxia Liang, Jiayu Zhang, Zhou Zhou, Bin Zou

TL;DR

This paper analyzes a dynamic insurance market with two competing insurers and a continuum of insureds under a finite N-class Bonus-Malus System. It proves the insureds' optimal reporting barrier b^* exists uniquely and depends continuously on BMS premiums, and, in the two-class case, establishes existence of Nash equilibrium premiums via a Stackelberg-Nash framework, using a symmetry result that the barrier is the same across insurers. The analysis combines Markov-chain dynamics for reporting and insurer switching via an exponential choice function, with a Brouwer fixed-point argument ensuring equilibrium existence under technical conditions. Numerical results for a two-class Gamma-loss model illustrate how price sensitivity and brand preference shape the equilibrium premiums and their gap, highlighting how competition intensifies as sensitivity rises. Overall, the work provides theoretical foundations for strategic reporting and competitive BMS pricing, with implications for actuarial pricing and regulatory policy; extensions to asymmetric information and heterogeneous losses are proposed.

Abstract

This paper develops a dynamic insurance market model comprising two competing insurance companies and a continuum of insureds, and examines the interaction between strategic underreporting by the insureds and competitive pricing between the insurance companies under a Bonus-Malus System (BMS) framework. For the first time in an oligopolistic setting, we establish the existence and uniqueness of the insureds' optimal reporting barrier, as well as its continuous dependence on the BMS premiums. For the 2-class BMS case, we prove the existence of Nash equilibrium premium strategies and conduct an extensive sensitivity analysis on the impact of the model parameters on the equilibrium premiums.

Optimal Underreporting and Competitive Equilibrium

TL;DR

This paper analyzes a dynamic insurance market with two competing insurers and a continuum of insureds under a finite N-class Bonus-Malus System. It proves the insureds' optimal reporting barrier b^* exists uniquely and depends continuously on BMS premiums, and, in the two-class case, establishes existence of Nash equilibrium premiums via a Stackelberg-Nash framework, using a symmetry result that the barrier is the same across insurers. The analysis combines Markov-chain dynamics for reporting and insurer switching via an exponential choice function, with a Brouwer fixed-point argument ensuring equilibrium existence under technical conditions. Numerical results for a two-class Gamma-loss model illustrate how price sensitivity and brand preference shape the equilibrium premiums and their gap, highlighting how competition intensifies as sensitivity rises. Overall, the work provides theoretical foundations for strategic reporting and competitive BMS pricing, with implications for actuarial pricing and regulatory policy; extensions to asymmetric information and heterogeneous losses are proposed.

Abstract

This paper develops a dynamic insurance market model comprising two competing insurance companies and a continuum of insureds, and examines the interaction between strategic underreporting by the insureds and competitive pricing between the insurance companies under a Bonus-Malus System (BMS) framework. For the first time in an oligopolistic setting, we establish the existence and uniqueness of the insureds' optimal reporting barrier, as well as its continuous dependence on the BMS premiums. For the 2-class BMS case, we prove the existence of Nash equilibrium premium strategies and conduct an extensive sensitivity analysis on the impact of the model parameters on the equilibrium premiums.
Paper Structure (13 sections, 8 theorems, 98 equations, 3 figures, 1 table)

This paper contains 13 sections, 8 theorems, 98 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Model
  3. The insureds' optimal reporting problem
  4. The insurance companies' pricing game
  5. Numerical analysis
  6. Conclusion
  7. Acknowledgements.
  8. Proofs
  9. Proof of Theorem \ref{['thm:b_op']}
  10. Proof of Theorem \ref{['thm:bstar']}
  11. Proof of Theorem \ref{['thm:c_op']}
  12. Proof of Proposition \ref{['p4.1']}
  13. An Example

Key Result

Theorem 3.1

For all $\mathbf{c}:= (\mathbf{c}^1, \mathbf{c}^2) \in \mathcal{C}^2$, there exists a unique optimal barrier strategy $b^* = \{ b^{*, i}_n(\mathbf{c})\}_{n \in \mathcal{N}, i \in \mathcal{I}}$ to the insured's reporting problem in eq:b_op_def, and the mapping, ${\bf c}\mapsto b^{*, i}_n(\mathbf{c} in which $j = 3 - i \in \mathcal{I}$, $n_{\downarrow}$ and $n_{\uparrow}$ (rating up or down by one

Figures (3)

  • Figure 1: Effect of the price sensitivity coefficient $k_1$ on the equilibrium premiums
  • Figure 2: Effect of the preference coefficient $k_2$ for Company 1 on the equilibrium premiums
  • Figure 3: Effect of the upper bound on the premium $M$ on the equilibrium premiums

Theorems & Definitions (22)

  • Theorem 3.1
  • Remark 3.1
  • Remark 3.2
  • Definition 4.1
  • Theorem 4.1
  • Theorem 4.2
  • Remark 4.1
  • Proposition 4.1
  • Remark 4.2
  • proof
  • ...and 12 more