# Game Theory and Life Insurance

4444 words 18 pages
Astln Bulletin 11 (198o) 1-16

A GAME T H E O R E T I C LOOK AT L I F E I N S U R A N C E UNDERWRITING* JEAN LEMAIRE Universit6 Libre de Bruxelles Tim decision problem o[ acceptance or rejection of life insurance proposals is formulated as a ~vo-person non cooperattve game between the insurer and the set of the proposers Using the mmtmax criterion or the Bayes criterion, ~t ~s shown how the value and the optunal stxateg~es can be computed, and how an optimal s e t of medina!, mformatmns can be selected and utlhzed 1. FORMULATIONOF THE GAME The purpose of this paper, whose m a t h e m a t i c a l level is elementary, is to d e m o n s t r a t e how g a m e t h e o r y can help the insurers to formulate a n d solve some of their underwriting

The detector can even be so imperfect that the line .FE passes below the intersection of B D and AC; then the medical information is so weak that it is useless.

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healthy

f~n heall hy

Fig. 2 2.3.

Optimal Deteclwn System

A detector is characterized by a pair (Ps, PFF) of probabilities. The underwriters can decide to render the standards of acceptation more severe, by rejecting more people, thereby incrcasing the success probabihty Ps. Unfortunately, the false alarm probability PF will then increase too. Can gaine theory help us to select an optimal detection system ? Must the company choose a "nervous" detector, with a high success probability, but also a high false alarm rate, or a "pldegmatic" or "slow" system with low probabilities Ps and PF ? Let us assume for sunplicity that all the medical information has been aggregated mto a single discriminating variable (for instance by using discrlminant- or regression analysis). The distribution of the discriminatmg variable for the healthy population will usually overlap the dastribution for the non healthy group. The choice of a particular detector can consist of selecting a critical value, any higher observed value leading to rejection, any lower value to acceptance (this procedure is optimal if the distributions are normal with equal

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