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 problems. The f r a m e w o r k a d o p t e d here is life insurance acceptance, but the concepts developed could be a p p h e d to a n y other branch. The decision problem of acceptance or rejection of life insurance proposals can be f o r m u l a t e d as a two-person non cooperative g a m e the following w a y : player 1, P~, is the insurer, while player 2, P2, is the set of all the potential pohcy-hotders. The g a m e is p l a y e d m a n y times, m fact each time a m e m b e r of P.- fills m a proposal. \Ve suppose t h a t tlfis person is either perfectly h e a l t h y (and should be accepted) or affected b y a disease which should be detected and cause rejection. We shall assume for the m o m e n t t h a t the players possess only two strategies each. acceptance a n d rejection for P~, health or disease for P2. To be more realistic we should introduce a third pure s t r a t e g y for P~: a c c e p t a n c e of the proposer with a surcharge. To keep the analysis as simple as possible we shall delay the introduction of surcharges until sectmn 4. Consequently we can define a 2 x 2 p a y o f f m a t r i x for the insurer.

.P~

•

P2

healthy proposer A B

ill proposer C D

acceptance rejection

I t iS evident t h a t the worst o u t c o m e for the insurer is to accept a b a d risk. I n t e r p r e t i n g the payoffs as utilities for P1, C should be the lowest figure. Clearly D > B : it is better for the insurer to reject a b a d risk than a good risk. Also A must be greater t h a n B. One anight argue a b o u t the relative * Presented at the 14th ASTIN Colloqumm, Taornuna, October x978.

2

JEAN LEMAIRE

values, A and D, of the good outcomes. We shall suppose in the examples and the figures that D > A, but the analysis does not rely on this assumptmn. In order to find the value of the game and the optimal strategy for P~, we can apply - - the minimax criterion, or - - the Bayes criterion. 2. THE MINIMAX CRITERION To apply the minimax criterion assimilates P2 to a malevolent opponent whose unique goal is to deceive the insurer and to reduce his payoff. This is of course an extremely conservative approach, to be used by a pessimistic insurer, concerned only by its security level.

2.1. Value and Optimal Strategies without information

Since P2's objective is to harm P~, the game becomes a 2 x 2 zero-sum twoperson game, which can be represented graphicaUy. The vertical axis of fig. 1 is the payoff to P1. His possible choices are represented by the two straight lines. The horizontal axis is P2's choice: he can always present an healthy proposer, or a non healthy, or pick any probability mix in between. The use of mixed strategies is fully justified here since the game is to be played m any times. Since P2's payoff is the negative of Pl's', his objective is to minimize the insurer's maximum gain, the heavy broken line. The ordinate of point M Payoff Io

p~

D

A

B

healthy

i'x~n

hi'Klllh

Fig. i

LIFE

INSURANCE

UNDERWRITING

3

is then the value of the game. The abscissa of M provides the optimal mixed strategy of P2 P~'s optimal strategy can be obtained similarly (for more details see for instance OWLN (1968, p. 29) ) Thus, by...

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