Game Theory and Life Insurance
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 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
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
References: AXELROD, 1~ (1978) Copzng wzth deception, International conference on applied game theory, Vmnna LEE, V,r. (1971) Dec~szon theory and human behaviour, J. Wiley, New York LuCE, R and H ]{AIFFA (1957). Games and deczszons, J Wiley, New York. OWEN, G. (1968) Game theory, ~V. Saunders, Philadelphia.