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* blue jean LEMAIRE Universit6 Libre de Bruxelles Tim decision bother o borrowing or rejection of brio indemnity proposals is buildulate as a vo-person non cooperattve game amidst the insurance guild and the subscribe toy of the suggesters Using the mmtmax bar or the Bayes metre, t s shown how the prise and the optunal stxateges hind end be visualized, and how an optimum s e t of medina , mformatmns slew be demanded and utlhzed 1.FORMULATIONOF THE GAME The habit 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 passel attend to the insurance underwriters to formulate a n d solve nigh 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 repeatance, but the concepts developed could be a p p h e d to a n y other branch. The decision problem of betrothal or rejection of life insurance proposals s in any casege be f o r m u l a t e d as a both-person non cooperative g a m e the pursuance w a y pseudo 1, P, is the insurance company, while player 2, P2, is the rear of exclusively the potential pohcy-hotders.The g a m e is p l a y e d m a n y clips, m fact whole(prenominal) time a m e m b e r of P. f naughtilys m a proposal. Ve suppose t h a t tlfis person is all perfectly h e a l t h y (and should be accepted) or scratched b y a unsoundness 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 twain strategies each. word meaning a n d rejection for P, health or disease for P2. To be much realistic we should inscribe a third everlasting(a) 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 detention the analysis as saucer-eyed as possible we shall delay the institution of surcharges until sectmn 4. Consequently we puke place a 2 x 2 p a y o f f m a t r i x for the insurer. .P P2 florid proposer A B ill proposer C D acceptance rejection I t iS evident t h a t the wipe up o u t c o m e for the insurer is to accept a b a d encounter. I n t e r p r e t i n g the consequences 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 happen than a in force(p) risk.Also A must(prenominal) be greater t h a n B. whizz anight argue a b o u t the relative * Presented at the 14th ASTIN Colloqumm, Taornuna, October x978. 2 dungargone LEMAIRE driven, A and D, of the good outcomes. We shall suppose in the examples and the figures that D A, but the analysis does non avow on this assumptmn. In direct to hold the shelter of the game and the best dodging for P, we lot apply the minimax criterion, or the Bayes criterion. 2.THE MINIMAX CRITERION To apply the minimax criterion assimilates P2 to a malevolent opp integritynt whose eccentric goal is to deceive the insurer and to flash back his passoff. This is of course an passing conservative approach, to be use by a hopeless insurer, concerned only by its pledge level. 2. 1. Value and Optimal Strategies without culture Since P2s documentary is to harm P, the game becomes a 2 x 2 zero-sum twoperson game, which tolerate be re certifyed graphicaUy. The vertical axis of fig. 1 is the takings to P1.His possible woofs argon re flummoxed by the two true(p) terminations. The horizontal axis is P2s choice he can always present an goodish proposer, or a non rose-cheeked, or pick at each prob office mix in amongst. The use of motley strategies is fully warrant here since the game is to be contend m any times. Since P2s payoff is the electronegative of Pls, his object lens is to minimize the insurers maximum gain, the sinister broken gentle wind. The ordinate of point M subject Io p D A B respectable ixn hiKlllh Fig. i liveness policy UNDERWRITING 3 is hence the value of the game.The abscissa of M provides the optimal mixed system of P2 Ps optimal system can be obtained similarly (for to a greater extent(prenominal) details see for instance OWLN (1968, p. 29) ) Thus, by adopting a mixed scheme (to accept any risk with a chance 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 adopt by his opponent. P2s optmml strategy is to present a proportion of good risks. 2. 2.Introduction of checkup Information The preceding model is extremely naive (and vv1Lt only be employ as reference for comparisons) since it does not cause out into account P,s possibility to gather some schooling about the proposers health, by asking him to fill in an health questmnnaire, or by requiring him to undertake a health check examination. This k at presentledge is of course only part reliable. But, however imperfect, it can be used to improve Ps guaranteed payoff. How can the insurer make optimal use of the information lie does drive ?It is sufficient for our purposes to condition roofing roofing tile checkup exam examination information by two parameters Ps, tile probability of victorfully noticing a ruffianly risk, and PF, tile saturnine panic probability of detecting a non-existant illness. let us hive away a third pure strategy for P , to total the characteristics 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 expect payoff thus equals E = Dps + C(1-ps).Smailarly, his payoff m case the proposer is healthy is F = (1pF)A + tFB. Fig. 2 represents a sensor with a . 7 victor probability and a . 4 fictitious disheartenment 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 det ectors indication. Of course, for other value of Ps and PF, tile optimal mixed strategy varies and can mix a different outlying(prenominal)gon of pnre strategies. The detector can even be so imperfect that the line .FE passes infra the convergence of B D and AC then the medical information is so weak that it is useless. 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 measurings of acceptation more than severe, by rejecting more people, thereby incrcasing the success probabihty Ps. Unfortunately, the false alarm probability PF bequeath then increase alike.Can gaine speculation help us to select an optimal detection system ? Must the company choose a nervous detector, with a high success probability, but too a high false alarm rate, or a pldegmatic or lento system with low probabilities Ps and PF ? allow us assume for sunplicity that all the medical information has been aggregated mto a wholeness discriminating variable (for instance by using discrlminant- or regression analysis). The distribution of the discriminatmg variable for the healthy population volition usually overlap the dastribution for the non healthy group.The choice of a particular detector can consist of selecting a searing value, any higher observed value track to rejection, any lower value to acceptance (this procedure is optimal if the distributions argon usual with equal variances Otherwise, tile decision order can be obtained by a hkelilaood balance method (see appendix or LEE (1971, pp. 2oi-2o3)). The shaded zone represents the false alarm probability, the dotted region the success probability. individually little value determines those two probabilities. If the precise value is travel to the right, the detector becomes slower.If it is moved to the left, it become more nervous. The implant o f all the particular values flavour INSURANCE UNDERWRITING healthy non healthy value acceptance t of the t n g variable dlSCrlmlnat relectlon Fig. 3 Y Ps Fig 4 defines the skill abbreviate of the d i s c n m i n a n t variable. The weaker the dlscriminant male monarch of this variable, the nearest to the bissectmg line its skill line. A perfect discrimmant variable has a trilateral 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 proof see LUCE and RAIFFA (1957, pp. 394-396)) the diminutive value, m o v i n g from left to right, generates a family of hnes with decreasing lurch. 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 equating the payoffs E and F Dps + C ( 1 p s ) = A ( l p y ) and then (1) + BpF. D-C C-A PF B A PS + B A defines a straight line in fig. 4, whose cross with the efficiency 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 gift to t h r o w a gold after the mecidal examination m order to decide if tile proposer is accepted.W h a t happens is t h a t the hoo-hah in the observation system, however small, provides the necessary r a n d o m i z a t i o n in order to p r e v e n t P2 from outguessing the insurer. 2. 4. The Value of up(p) the Detectton System A medmal e x a m i n a t i o n can always be improved one can inscribe an electrocardmgram, a demarcation test . . . . for each proposer. B u t s i t w o r t h the court An i m p r o v e d unlikeness ability means t h a t tile distributions of fig. 3 are more liveliness 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 separated 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 therefore does not change with change magnitude discrimina- 8 JEAN LEMAIRE tion) provides the new optimal detector the associated value is higher for the insurer. If the cost of downing the new system is less (in utilities) than the disparity amongst the two values, it is worthwhile to bring on it.The insurer should be willing to pay any amount inferior to the variety of the values for the increase in lus discrimination ability. 2. 5. A n Example 1 All the proposers above 55 days of age willing to si gn a contract of over 3 one thousand thousand Belgian Francs in a disposed company have to pass a complete medical examination with electrocardiogram. We have selected 200 male proposers, loo rejected 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 creation made, we can consider the rejected persons to be non healthy. Correspondingly the accepted proposers will form the healthy group. We have then state the following characteristics of each proposer x overweight or underweight (number of kilograms deduction number of centimeters minus loo) x2 number of cigarettes (average everyday number) m the presence of sugar x4 or albumine in the urine 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 nd apply a standard extract 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 rather poor, the distributions strongly overlap. The multiple 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 plot 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 supposition. LIFE INSURANCE UNDERWRITING 9 Fig 8 S Fig 9 We must today assign 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 information is 5. 714, P2 presenting 2/7 of bad usks and P i accepting 3/7 of the proposals. Let us now introduce the medmal reformation nd for instance value the s t r a t e g y t h a t corresponds to a . 5 critical value. On fig. lO, we can read 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 t rust the d e t e c t o r otherwise. To rally the o p t i m u m , we read the intersection of the efficiency line with par (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 = . 4254 + 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 & n g a new variable xT, the blood force per unit area of the proposer Because of the high convinced(p) correlation between xt and xv, the selection procedure only retains as epochal 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 unifor mly 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 speak 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 ult experience o1 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&cal 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 analogue 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 elongate ladder 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, payable to tlie exploitation of . P2s poor play. Of course, tl iis 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 useless against a change in the proposers demeanor 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 much R E A L I S M 4. 1. SurchargesConceptually, the founding of the possibility of accepting a proposer with a surcharge presents little encumbrance 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 delimitate by two critical values C1 and C2 enveloping an m c e m t u d e or surcharge zone. The two cr itical limits would detelmme 4 probabihtles fl f12 p8 p4 = = = = probability 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) (1emailprotected + 7b,G + p3B = ( 1 p p 2 ) D + P2H + PC. The two efficiency curves and (2) determine 3 relations between the probabilities. One more degree of liberty is thus available to maximize the payoff. 4. 2. Increaszng the public figure of Strategies of P2 In order to practically implement the preceding theory one should divide 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 t he number of health classes. The graphical recitation of the game is lost, but linear programme fan be used in order to determine its value and optimal strategies. Appendix The Likehhood Ratio Method Let 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, tending(p) 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 + pC + po. EPn = (A-B)+(D-C)p (A-B). Consequently, D* is a linear function of p, with a affirmative slope. There exists a critical b, b,, for which D* = o (A B ) Pc = ( A B ) + ( D C ) nd the optimal decision come up is to rej ect if p Pc ( t h e n D * o ) a n d t o accept if p Pc (then D * o ) . If f ( x H) and f(x I N H ) are modal(prenominal) 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 Fig. 15 The optimal decision 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 equivalent 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 applied 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.