Stackelberg Equilibrium Premium Strategies for Push-Pull Competition in a Non-Life Insurance Market with Product Differentiation

Two insurance companies <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>I</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>I</mi> <mn>2</mn> </msub> </mrow> </semantics> </math> </inline-formula> with reserves <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>R</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>,</mo> <msub> <mi>R</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> compete for customers, such that in a suitable differential game the smaller company <inline-formula> <math display="inline"> <semantics> <msub> <mi>I</mi> <mn>2</mn> </msub> </semantics> </math> </inline-formula> with <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>R</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> <mo>&lt;</mo> <msub> <mi>R</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> aims at minimizing <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>R</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>&#8722;</mo> <msub> <mi>R</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> by using the premium <inline-formula> <math display="inline"> <semantics> <msub> <mi>p</mi> <mn>2</mn> </msub> </semantics> </math> </inline-formula> as control and the larger <inline-formula> <math display="inline"> <semantics> <msub> <mi>I</mi> <mn>1</mn> </msub> </semantics> </math> </inline-formula> at maximizing by using <inline-formula> <math display="inline"> <semantics> <msub> <mi>p</mi> <mn>1</mn> </msub> </semantics> </math> </inline-formula>. Deductibles <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>K</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>K</mi> <mn>2</mn> </msub> </mrow> </semantics> </math> </inline-formula> are fixed but may be different. If <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>K</mi> <mn>1</mn> </msub> <mo>&gt;</mo> <msub> <mi>K</mi> <mn>2</mn> </msub> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <msub> <mi>I</mi> <mn>2</mn> </msub> </semantics> </math> </inline-formula> is the leader choosing its premium first, conditions for Stackelberg equilibrium are established. For gamma-distributed rates of claim arrivals, explicit equilibrium premiums are obtained, and shown to depend on the running reserve difference. The analysis is based on the diffusion approximation to a standard Cram&#233;r-Lundberg risk process extended to allow investment in a risk-free asset.

Abstract licence: CC BY

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