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 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 adopting a mixed strategy (to accept any risk with a probability 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. P2's optmml strategy is to present a proportion of good risks. 2. 2. Introduction of Medical Information The preceding model is extremely naive (and vv1Lt only be used as reference for comparisons) since it does not take into account P,'s possibility to gather some information about the proposer's health, by asking him to fill in an health questmnnaire, or by requiring him to undertake a medical examination. This information is of course only partially reliable. But, however imperfect, it can be used to improve P~'s guaranteed payoff. How can the insurer make optimal use of the information lie does have ? 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 alarm 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 expected payoff thus equals E = Dps + C(1-ps). Smailarly, his payoff m case the proposer is healthy is F = (1–pF)A + t~FB. Fig. 2 represents a â€Å"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 â€Å"to accept† and â€Å"to follow the detector's indication†. Of course, for other values of Ps 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 intersection 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 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 â€Å"nervous† detector, with a high success probability, but also a high false alarm rate, or a â€Å"pldegmatic† or â€Å"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 variances Otherwise, tile decision rule can be obtained by a hkelilaood ratio method (see appendix or LEE (1971, pp. 2oi-2o3)). The shaded zone represents the false alarm probability, the dotted region the success probability. Each critical value determines those two probabilities. If the critical value is moved 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 LIFE INSURANCE UNDERWRITING healthy [ non healthy value acceptance t | of the t n g variable dlSCrlmlnat relectlon Fig. 3 Y Ps Fig 4 defines the efficiency 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 efficiency 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 proof 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 slope. If . Pat chooses a d e t e c t o r with 6 JEAN LEMAIRE a pos~ttve 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 P~'s 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 ) Then (1) + BpF. D-C C-A PF – B – A PS + B – A defines a straight line in fig. 4, whose intersection 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 have to t h r o w a coin after the mecidal examination m order to decide if tile proposer is accepted. W h a t happens is t h a t the â€Å"noise† 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 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 LIFE INSURANCE UNDERWRITING 7 Fig. 6 Payoff to p, D A im rn i ir~f 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 increased discrimina- 8 JEAN LEMAIRE tion) provides the new optimal detector; the ass ociated 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 sign a contract of over 3 million Belgian Francs in a given 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 being made, we can consider 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 (number 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 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 otherwise 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 i s quite 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 theory. LIFE INSURANCE UNDERWRITING 9 Fig 8 S Fig 9 We must now 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 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 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 â€Å"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 â€Å"slow†, t h a t too m a n y risks are accepted. On the other hand, a detector w~th a . 4 critical value is too nervous: too m a n y risks are rejected T h e value is 5. 975, P2's 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 read 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 = . 425Ãâ€"4 + – 5 7 5 x 8 = 6. 3. f the insurer adopts the ptu'e 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 pressure of the proposer Because of the high positive correlation between xt and xv, the selection procedure only retains as significant the varia bles 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 proposer's 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 crlter~on, 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†³ 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 hea lthy. The analysis is easier m this 12 JEAN LEMAIRE case, since P2's 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 function of PF a n d PH [~5FB + (1 – – pF)A]~SH + [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 p2's 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 informatio n (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 . P2's 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, P~'s 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. Surcharges Conceptually, the introduction of the possibility of accepting a proposer with a surcharge presents little difficulty: 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 enveloping an m c e m t u d e or surcharge zone. The two critical 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 Surchar~le 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) (1–[email  protected] + 7b,G + p3B = ( 1 – p ~ – p 2 ) D + P2H + P~C. The two efficiency curves and (2) determine 3 relations between the probabilities. One more degree of freedom is thus avai lable to maximize the payoff. 4. 2. Increaszng the Number of Strategies of P2 In order to practically implement the preceding theory one should subdivide P2's strategy â€Å"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 detector's advice, and P2 as m a n y pure strategms as the number of health classes. The graphical interpretation of the game is lost, but linear programming 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, 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 — + pC + po. EPn = [(A-B)+(D-C)]p – (A-B). Consequently, D* is a linear function of p, with a positive slope. There 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 ) . — 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, h owever, 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 probabii]ties are equal, Lc(x) = 1. 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.