Game Theory and Life Insurance

Astln Bulletin 11 (198o) 1-16 A GAME T H E O R E T I C human face 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 adoption or close oution of emotional state insurance proposals is think overd as a vo-person non cooperattve bouncing between the insurer and the set of the removal companys Using the mmtmax criterion or the Bayes criterion, t s shown how the look on and the optunal stxateges rout out be computed, and how an optimum s e t of medina , mformatmns substructure 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 corporation help the insurers to formulate a n d solve some of their on a lower floorwriting problems. The f r a m e w o r k a d o p t e d here is life sentence insurance brookance, but the concepts developed could be a p p h e d to a n y other branch. The decision p roblem of hireance or rejection of life insurance proposals drive out be f o r m u l a t e d as a devil-person non cooperative g a m e the interest w a y impostor 1, P, is the insurer, while player 2, P2, is the set of on the whole the potential pohcy-hotders.The g a m e is p l a y e d m a n y sequences, 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 un healthiness which should be detected and ca intent rejection. We sh each(prenominal) come across for the m o m e n t t h a t the players possess and deuce strategies each. acceptance a n d rejection for P, health or disease for P2. To be much realistic we should introduce a third subtile 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 depth psychology as simple as possible we shall live on the introduction of surcharges until sectmn 4. Consequently we abide defin e a 2 x 2 p a y o f f m a t r i x for the insurer. .P P2 anicteric 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 find. I n t e r p r e t i n g the issuances as utilities for P1, C should be the lowest figure. Clearly D > B it is break off for the insurer to reject a b a d risk than a sincere risk.Also A must be greater t h a n B. unitary anight argue a b o u t the relative * Presented at the fourteenth ASTIN Colloqumm, Taornuna, October x978. 2 JEAN LEMAIRE determine, 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 score to find the quantify of the game and the optimum dodging 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 a nd to abridge his restoration. This is of course an extremely conservative approach, to be employ by a pessimistic insurer, concerned solo by its security level. 2. 1. Value and optimum Strategies without data Since P2s objective is to harm P, the game becomes a 2 x 2 zero-sum twoperson game, which can be represented graphicaUy. The upright piano axis vertebra of fig. 1 is the advantage to P1.His possible choices atomic number 18 represented by the two straight lines. The horizontal axis is P2s choice he can continuously present an salubrious proposer, or a non wholesome, or pick some(prenominal) hazard mix in between. The use of mixed strategies is fully reassert here since the game is to be played m any(prenominal) times. Since P2s payoff is the negative of Pls, his objective is to minimize the insurers maximum gain, the heavy broken line. The array of leg M Payoff Io p D A B healthy ixn hiKlllh common fig tree. i carriage INSURANCE UNDERWRITING 3 is because the grade of the game.The abscissa of M provides the optimum mixed dodging of P2 Ps optimal outline can be obtained similarly (for more(prenominal) details catch up with for instance OWLN (1968, p. 29) ) Thus, by adopting a mixed dodge (to accept any risk with a prospect D-B PA = A + D B c and t reject w i t h a p r o b a b i l i t y p n = I AD-BC ? A),. P can guarantee himself a payoff of v = A + D B C D-C PH = A + D B C whatever the strategy adopted by his opponent. P2s optmml strategy is to present a proportion of good risks. 2. 2.Introduction of Medical learning The preceding model is extremely naive (and vv1Lt only be use as reference for comparisons) since it does not take into account P,s hypothesis to collaborate some information about the proposers health, by asking him to fill in an health questmnnaire, or by requiring him to undertake a medical checkup checkup exam. This information is of course only partially reliable. But, however imperfect, it ca n be used to improve Ps guaranteed payoff. How can the insurer make optimal use of the information lie does cast ?It is sufficient for our purposes to characterize tile medical information by two parameters Ps, tile probability of successfully noticing a bad risk, and PF, tile false disheartenment probability of detecting a non-existant illness. Let us introduce a third pure strategy for P , to follow the indications of tile medical information. If tile proposer is not healthy, his illness is detected with a probabihty Ps, and remains undetected with a probability 1 P S . . P i S anticipate payoff so equals E = Dps + C(1-ps).Smailarly, his payoff m case the proposer is healthy is F = (1pF)A + tFB. image. 2 represents a detector with a . 7 success probability and a . 4 false alarm probability. We notice that, m this case, P1 can guarantee himself a payoff v2 > vl by mixing the strategies to accept and to follow the detectors indication. Of course, for other look ons of P s and PF, tile optimal mixed strategy varies and can mix a different set of pnre strategies. The detector can even be so imperfect that the line .FE passes below the point of intersection of B D and AC then the medical information is so weak that it is useless(prenominal). 4 Payoff to Pl JEAN LEMAIRE JD1 J E ao % 7o % 4o % 6o % I A. healthy fn 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 alike.Can gaine theory help us to select an optimal detection system ? Must the company aim a nervous detector, with a high success probability, but to a fault a high false alarm rate, or a pldegmatic or retard system with low probabilities Ps and PF ? Let us deliver for sunplicity that all the medical information has been aggregated mto a single penetrative variant (for instance by using discrlminant- or regression analysis). The dispersion 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 unfavorable value, any higher observed value leading to rejection, any lower value to acceptance (this procedure is optimal if the distributions are chemical formula with equal variances Otherwise, tile decision rule can be obtained by a hkelilaood ratio method (see appendix or LEE (1971, pp. 2oi-2o3)). The shaded partition represents the false alarm probability, the dotted region the success probability. Each hypercritical value determines those two probabilities. If the critical value is move to the right, the detector becomes slower.If it is moved to the left, it become more nervous. The set of all the critical values smell INSURANCE UNDERWRITING healthy non he althy value acceptance t of the t n g variable dlSCrlmlnat relectlon Fig. 3 Y Ps Fig 4 defines the cleverness curve of the d i s c n m i n a n t variable. The weaker the dlscriminant power of this variable, the nearest to the bissectmg line its dexterity line. A perfect discrimmant variable has a triangular efhciency x y z . The set of all the detectors determines a set of values for the game.The highest value v* for the insurer is reached when the p a y o f f line is horizontal. This can be roughly seen as follows (for a more rigorous confirmation see LUCE and RAIFFA (1957, pp. 394-396)) the critical value, m o v i n g from left to right, generates a family of hnes with decreasing lean. If . Pat chooses a d e t e c t o r with 6 JEAN LEMAIRE a posttve slope, P= can reduce his payoff below v* b y always presenting h e a l t h y proposers. Similarly, f the slope is negative, a continuous flow of non h e a l t h y proposers will keep Ps payoff below v*. yotl to Pt I D A C h , a i r h rmn heulth, Fig 5 The optimal detector can be easdy obtained b y equivalence the payoffs E and F Dps + C ( 1 p s ) = A ( l p y ) Then (1) + BpF. D-C C-A PF B A PS + B A defines a straight line in fig. 4, whose intersection with the faculty line determines the o p t i m u m . N o t e t h a t the optimal s t r a t e g y of P is a pure s t r a t e g y to follow the advace of the d e t e c t o r , the insurer does not gestate to t h r o w a bullion after the mecidal examination m direct to decide if tile proposer is accepted.W h a t happens is t h a t the noise in the observation system, however small, provides the necessary r a n d o m i z a t i o n in coordinate to p r e v e n t P2 from outguessing the insurer. 2. 4. The Value of Improving the Detectton System A medmal e x a m i n a t i o n can always be improved one can introduce an electrocardmgram, a blood test . . . . for each proposer. B u t s it w o r t h the cost An i m p r o v e d discrimination ability means t h a t tile distributions of fig. 3 are more invigoration INSURANCE UNDERWRITING 7 Fig. 6 Payoff to p, D A im rn i irf r m i n B C healthy on hl, olt h Fig. 7 unaffectionate and present less overlap. The characterizing probabilities ibs and PF are maproved, and the efficiency line moves away from the bisecting line. The intersection of the improved efficiency line with (1) (which is determined only by the payoffs and whence does not change with increased discrimina- 8 JEAN LEMAIRE tion) provides the new optimal detector the associated value is higher for the insurer. If the cost of implementing the new system is less (in utilities) than the difference between the two values, it is worthwhile to introduce it.The insurer should be willing to pay any amount inferior to the difference of the values for the increase in lus discrimination ability. 2. 5. A n Example 1 All the proposers above 55 years of age willing to bespeak a contract of over 3 million Belgian Francs in a given com pany have to pass a get by medical examination with electrocardiogram. We have selected 200 male proposers, loo jilted because of the electrocardiogram, and loo accepted. This focuses the attention on one category of rejection causes the heart diseases, and implicitly supposes that the electrocardiogram is a perfect discriminator.This (not unrealistic) hypothesis being made, we can encounter the rejected persons to be non healthy. Correspondingly the accepted proposers will form the healthy group. We have then noted the following characteristics of each proposer x overweight or underweight ( proceeds of kilograms minus number of centimeters minus loo) x2 number of cigarettes (average daily number) m the presence of sugar x4 or albumine in the pissing x s the familial antecedents, for the mother, xs and the father of the proposer. We then define a variable x0 = l o if the proposer is healthy 1 other than nd apply a standard selection technique of discriminant analysis in order to sort out the variables that slgnihcantly affect Xo The procedure only retains three variables xj, x2 and m, and combines them hnearly into a discriminating variable. The value of this variable s computed for all the observatmns, and tile observed distributions are presented in fig. 8. As was expected, the discrimination is kinda hapless, the distributions strongly overlap. The ternary correlation between Xo and the set of the explaining variables equals . 26. The group centroids are respectively . 4657 and . 343We then estmaate for each possible crltmal value Ps and PF and secret plan them on fig. lo. t This e x a m p l e p r e s e n t s v e r y w e a k d e t e c t o r s a n d is o n l y i n t r o d u c e d m o r d e r to illus t r a t e t h e p r e c e d m g theory. aliveness INSURANCE UNDERWRITING 9 Fig 8 S Fig 9 We must now attribute uNhtlcs to the various outcomes. We shall select A = 8, B = 4, C = o and D = lo. Then the value of the g a m e w i t h o u t medical informa tion is 5. 714, P2 presenting 2/7 of bad usks and P i evaluate 3/7 of the proposals. Let us now introduce the medmal renewal nd for instance evaluate the s t r a t e g y t h a t corresponds to a . 5 critical value. On fig. lO, we can charter s = . 51 a n d PF = 33. Then E = . 5 ? o + . 4 9 x o = 5-, a n d F = 3 3 x 4 + . 67 x 8 = 6. 68. The value of this game is 6 121, P2 presenting more bad risks (34. 1%), P I mixing the strategies r e j e c t and follow d e t e c t o r with respect- 10 JEAN LE/vIAIRE F i g . 1o Fig. 11 LIFE INSURANCE UNDERWRITING 11 lye probabilities . 208 and . 792 Fig. 11 shows t h a t this s t r a t e g y is too slow, t h a t too m a n y risks are accepted.On the other hand, a detector wth a . 4 critical value is too nervous too m a n y risks are rejected T h e value is 5. 975, P2s optimal s t r a t e g y is to present 74. 7% of good risks, while Pa should accept 29. 7% of the tmle and trust the d e t e c t o r otherwise. To find the o p t i m u m , we re ad the intersection of the efficiency line with equation (1), in this case 5 F = 2 2 Ps We find PF = . 425 Ps = . 63 with a critical value of . 475. T h e n E = lOX. 63 + ox. 37 = . 425&2154 + 5 7 5 x 8 = 6. 3. f the insurer adopts the ptue s t r a t e g y of always accepting the a d w c e of the medical information, he can g u a r a n t e e himself a value of 6. 3 irrespective of his o p p o n e n t s strategy. L e t us now a t t e m p t to improve the me examination b y a d &038 n g a new variable xT, the blood pressure of the proposer Because of the high despotic correlation between xt and xv, the selection procedure only retains as significant the variables x. % xe and x7 Fig. 9 shows t h a t the distributions are more separated. In fact, the group centroids are now . 4172 and . 828 and the multiple correlation between xo and the selected variables rises to . 407. T h e efficiency hne (fig IO) is uniformly to the right of the f o r m e r one. The intersection with (1) is PF = 37 P,s = . 652 with a critical value of approxunatxvely . 45. The value of the game rises to 6. 52, an i m p r o v e m e n t of 22 for the insurer at the cost of controlling the blood pressure of each proposer (see fig. 1). 3 THE BAYES CRITERION I n s t e a d of playing as if the proposers sole objective were to o u t s m a r t him, the insurer can a p p l y the B a r e s crlteron, i. . assume t h a t P2 has a d o p t e d a fixed a priori s t r a t e g y H e can suppose (from past experience o1&8243 from the results of a sample s u r v e y p e r f o r m e d with a m a x n n a i me&038cal examination) t h a t a p r o p o r t i o n Pn of the proposers is healthy. The analysis is easier m this 12 JEAN LEMAIRE case, since P2s m i x e d strategy is now assumed to be known P t only faces a one-dimensional p r o b l e m he must maximize his utility on the d o t t e d vertical line of fig. 12. Pc/Of f p to JD A t B, N C ol eall hy 1 PH PH non heoll hy Fig 12 One notices from fig. 12 t h a t a medical examination is sometimes useless, especially if PH is near 1. In this case, P t s optimal s t r a t e g y is to accept all the proposers. In the general case, P t should m a x m n z e the linear manipulation of PF a n d PH 5FB + (1 pF)ASH + paD + (I ps)c (1 PH), under the condition t h a t PF and Ps are linked b y the efficiency curve of fig. 4. As far as the example is concerned, this economic function (represented in fig lo) becomes 1. Ps 3 4PF if one supposes that p2s mixed s t r a t e g y is to present 15% of bad risks. 6. 8 + F o r the first set of medical information (xl, x2, x6), tile m a x i m u m is reached at the point Ps = . 28, PF = . 09. Since PH is r a t h e r tngh, this is a v e r y slow detector, yielding a fmal u t d l t y of 6. 914. Comparing to the optimal n n x e d strategy, this represents an increase in utility of . 614, due to tlie exploitation of . P2s poor play. Of course, tliis d e t e c t o r is only good as long as P2 sticks to LIFE INSURANCE UNDERWRITING 3 his mixed strategy. It is uneffective against a change in the proposers behaviour if for instance PH suddenly drops below . 725, Ps utlhty decreases under 6. 3, the guaranteed payoff with the mlmmax strategy In this aspect, the Bayes criterion implies a more optimistic attitute of P1. For the second set of medical information (x2, m, xT), the opblnal detector (Ps = . 45, bF = o9) grants a utility of 7. t69 if PH = . 85, an improvement of . 649 colnparing to the ininimax strategy (see fig. 11). 4. T O W A R D S MORE R E A L I S M 4. 1. SurchargesConceptually, the introduction of the possibility of accepting a proposer with a surcharge presents little trouble it amounts to introduce one more pure strategy for the insurer. Payoll to ID A G B heollhy non heoil hy F , g . 13 A detector could then be defined by two critical values C1 and C2 enfold an m c e m t u d e or surcharge zone. The two critical limits would detelmme 4 probabihtles fl f12 p8 p4 = = = = pro bability probabihty probability probability of of of of accepting a bad risk surcharging a bad risk rejecting a good risk surcharging a good risk 14 JEAN LEMAIRE ealthy non healthy V Surcharle I C1 C2 Fig. 14 and two efficiency curves. A necessary condition for a detector to be optimal is that the corresponding payoff hne is horizontal, i. e. that (2) (1email&160protected + 7b,G + p3B = ( 1 p p 2 ) D + P2H + PC. The two efficiency curves and (2) determine 3 relations between the probabilities. One more layer of freedom is thus available to maximize the payoff. 4. 2. Increaszng the Number of Strategies of P2 In order to practically implement the preceding theory one should subdivide P2s strategy present a non healthy proposer according to the arious classes of diseases. P1 should then have as pure strategaes reject, accept, a set of surcharges, and follow detectors advice, and P2 as m a n y pure strategms as the number of health classes. The graphical interpretation of the game i s lost, but linear programming devotee be used in order to determine its value and optimal strategies. Appendix The Likehhood Ratio Method Let &8212 x be the value of tlle discriminant variable, healthy, p(H) and p(NH) the a priori probabihties of being healthy or non f(x I H) and f(x NH) the conditional distributions of x.We can then compute the a posterior1 probability of being non healthy, given the value of the discriminant variable (1) p = p ( N g ix) = f(x l g H ) p ( N H ) f(x l N H ) p ( g g ) + f ( x l H)p(H) LIFE INSURANCE UNDERWRITING 15 Similarly p ( H I x) = l p. T h e e x p e c t e d payoffs for the two decisions are EPA = ( 1 p ) A EPR = (1-p)B Define D* to be D* = EPA &8212 + pC + po. EPn = (A-B)+(D-C)p (A-B). Consequently, D* is a linear function of p, with a positive slope. at that place exists a critical b, b,, for which D* = o (A B ) Pc = ( A B ) + ( D C ) nd the optimal decision rule is to reject if p > Pc ( t h e n D * > o ) a n d t o accept if p < Pc (then D * < o ) . &8212 If f ( x H) and f(x I N H ) are normal densities with equal variances, there is a one-to-one m o n o t o n i c relationship between p and x, and thus the crttmal p r o b a b l h t y Pc induces a critical value xe. In general, however, the cutoff point is not unique. T h e r e m a y be two or more critical values. In t h a t case, we define the likelihood ratio of x for hypothesis N H over hypothesis H as f(x N H ) L(x) Of f(x I H) c o u r s e o _-< L(x) =< oo.S u b s t i t u t i n g L(x) in (1) gives 1 P = or 1 L(x) p ( N H ) + p(H) p 1 (2) L(x) p ( N H ) l p F o r constant a priori probabilities, there is a m o n o t o n e relationship between p and L(x) L(x) goes from o to oo as p goes from o to 1. Therefore, a unique critical likelihood ratio Lc(x) exists and can be obtained b y replacing Pc for p in (2) (3) p(H) A B Lc(x) p ( N H ) D C 6 JEAN LEMAIRE p 1. 0 -Pc = 0 5 0. 5 I I I NH H I_- X? I J_ X? 2 H &8212 Fig. 15 The optimal d ecision rule reads if L(x) > L c ( x ) , reject if L ( x ) < L c ( x ) , accept.Notice that, i f A B = D C , pc = 1/2 The decision rule is equal to maximizing the e x p e c t e d n u m b e r of correct classifications. F r o m (3) p(H) L e(x) (NH) If, furthermore, the prior probabiities are equal, Lc(x) = 1. REFERENCES AXELROD, 1 (1978) Copzng wzth deception, International conference on employ game theory, Vmnna LEE, V,r. (1971) Decszon 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.