Arrow's Theorem of the Deductible with Heterogeneous Beliefs

Abstract

In Arrow's classical problem of demand for insurance indemnity schedules, it is well-known that the optimal insurance indemnification for an insurance buyer—or decision maker (DM)—is a deductible contract when the insurer is a risk-neutral Expected-Utility (EU) maximizer and when the DM is a risk-averse EU maximizer. In Arrow's framework, however, both parties share the same probabilistic beliefs about the realizations of the underlying insurable loss. This article reexamines Arrow's problem in a setting where the DM and the insurer have different subjective beliefs. Under a requirement of compatibility between the insurer's and the DM's subjective beliefs, we show the existence and monotonicity of optimal indemnity schedules for the DM. The belief compatibility condition is shown to be a weakening of the assumption of a monotone likelihood ratio. In the latter case, we show that the optimal indemnity schedule is a variable deductible schedule, with a state-contingent deductible that depends on the state of the world only through the likelihood ratio. Arrow's classical result is then obtained as a special case.

Notes

In Aumann's framework, an event is said to be common knowledge if both individuals know it, if each individual knows that the other individual knows it, if each individual knows that other individuals knows that the former knows it, and so on.

I am grateful to Rachel Huang for pointing out to me the analysis done in Huang et al. (2002). I came to be aware of the latter paper after the completion of this present work.

A finite measure η on a measurable space is said to be nonatomic if for any with η(A) > 0, there is some such that B⊊A and 0 < η(B) < η(A).

By the theory of equimeasurable rearrangements described in Appendix A.

For instance, if Q○X^{− 1} has atoms.

Let μ₁ and μ₂ be two probability measures on a measurable space . The probability measure μ₂ is said to be absolutely continuous with respect to the probably measure μ₁ (denoted by μ₂ ≪ μ₁) if for all with μ₁(C) = 0, one has μ₂(C) = 0. This does not rule out the existence of some such that μ₂(D) = 0 but μ₁(D) > 0.

Note that the optimal contract described in Theorem 4.6 is can be seen as a contract that combines a straight deductible and a (variable) co-insurance.

By two “changes of variable,” as in (Aliprantis and Border 2006, Theorem 13.46), and using the definition of f and g as Radon-Nikodým derivatives of P○X^{− 1} and Q○X^{− 1}, respectively, with respect to the Lebesgue measure.

This choice of x^⋆ is possible because if it were not, then that would contradict the fact that Y₁ and Y₂ are identically distributed for P.

For any , the Dirac delta function centered at a is zero everywhere except at a where it is infinite. Moreover, ∫^{a + ϵ}_{a − ϵ}φ(t)δ(t − a) dt = φ(a), for any ϵ > 0 and for any function φ. See, e.g., Bracewell (2000, Ch. 5).