DEPARTMENT OF ECONOMICS INSURANCE AND PROBABILITY WEIGHTING FUNCTIONS

DEPARTMENT OF ECONOMICS INSURANCE AND PROBABILITY WEIGHTING FUNCTIONS Ali al-nowaihi, University of Leicester, UK Sanjit Dhami, University of Leicester, UK Working Paer No. 05/19 July 2005 as Insurance, Gambling and Probability Weighting Functions Udated Setember 2006

Insurance and Probability Weighting Functions Ali al-nowaihi Sanjit Dhami Setember 2006 Abstract Evidence shows that (i) eole overweight low robabilities and underweight high robabilities, but (ii) ignore events of extremely low robability and treat extremely high robability events as certain. Decision models, such as rank deendent utility (RDU) and cumulative rosect theory (CP), use robability weighting functions. Existing robability weighting functions incororate (i) but not (ii). Our contribution is threefold. First, we show that this would lead eole, even in the resence of fixed costs and actuarially unfair remiums, to insure fully against losses of sufficiently low robability. This is contrary to the evidence. Second, we introduce a new class of robability weighting functions, which we call higher order Prelec robability weighting functions, that incororate (i) and (ii). Third, we show that if RDU or CP are combined with our new robability weighting function, then a decision maker will not buy insurance against a loss of sufficiently low robability; in agreement with the evidence. We also show that our weighting function solves the St. Petersburg aradox that reemerges under RDU and CP. Keywords: Decision making under risk; Prelec s robability weighting function; Higher order Prelec robability weighting functions; Behavioral economics; Rank deendent utility theory; Prosect theory; Insurance; St. Petersburg aradox. JEL Classification: C60(General: Mathematical methods and rogramming), D81(Criteria for decision making under risk and uncertainty). We would like to thank David Peel for his valuable comments. The usual disclaimers aly. Deartment of Economics, University of Leicester, University Road, Leicester. LE1 7RH, UK. Phone: +44-116-2522898. Fax: +44-116-2522908. E-mail: aa10@le.ac.uk. Deartment of Economics, University of Leicester, University Road, Leicester. LE1 7RH, UK. Phone: +44-116-2522086. Fax: +44-116-2522908. E-mail: Sanjit.Dhami@le.ac.uk.

1. Introduction The exlanation of insurance is, rightly, regarded as one of the triumhs of Exected Utility Theory (EU). For examle, it is a standard theorem of EU that eole will insure fully if, and only if, they face actuarially fair remiums. Since insurance firms have to at least cover their costs, market remiums have to be above the actuarially fair ones. Thus EU rovides a comletely rational exlanation of the widely observed henomenon of under-insurance. This has the olicy imlication that if full-insurance is deemed necessary (because of strong externalities for examle), then it has to be encouraged through subsidy or stiulated by law. However, roblems remain. The following are three well known examles. First, it is difficult for EU to exlain the fact that many eole simultaneously gamble and insure; see, for examle, Peel, Cain and Law (2005). The gambling and insurance industries are too large and imortant for such behavior to be dismissed as quirky. Second, EU redicts that a risk averse decision maker will always buy some ositive level of insurance, even when remiums are unfair. What is observed is that many eole do not buy any insurance, even when available. Indeed for several tyes of risk, the government has to legislate the mandatory urchase of insurance. Third, when faced with an actuarially unfair remium, EU redicts that a decision maker, who is indifferent between full-insurance and not insuring, would strictly refer robabilistic insurance to either. However, the exerimental evidence is the reverse; see Kahneman and Tversky (1979, 269-271). These and other anomalies have motivated a number of alternatives to EU. The most imortant of these are rank deendent utility theory (RDU), see Quiggin (1982, 1993), and cumulative rosect theory (CP), see Tversky and Kahneman (1992). Both RDU and CP use robability weighting functions to overweight low robabilities and underweight high robabilities. However, the standard robability weighting functions infinitely overweight infinitesimal robabilities, in the sense that the ratio between the weight and the robability goes to infinity, as the robability goes to zero. Our first contribution is to show that this would lead eole, when faced with an exected loss, to insure fully against that loss if it is of sufficiently low robability. This result holds even when remiums are actuarially unfair and there are fixed costs of insurance, rovided a articiation constraint is satisfied. Simulations suggest that this articiation constraint is quite mild. This behavior is contrary to evidence, as we shall now see. Two articularly striking examles are given in the seminal work of Kunreuther et al. (1978). These are the unoularity of flood and earthquake insurance, desite heavy government subsidy to overcome transaction costs, reduce remiums to below their actuarially fair rates, to rovide reinsurance for firms and rovide relevant information. This 1

not only contradicts the rediction of EU 1, it also contradicts the redictions of RDU and CP when standard robability weighting functions are used. While there is considerable evidence that eole overweight low robabilities and underweight high robabilities, it is also a common observation that they ignore events of extremely low robability and treat extremely high robability events as certain. We could follow Kahneman and Tversky (1979) and rely on an initial editing hase, where the decision maker chooses which imrobable events to treat as imossible and which robable events to treat as certain. No doubt such editing does occur. However, as yet, there is no general theory of the editing hase. Instead, in our second contribution, we roose a new class of robability weighting functions that combine the editing hase with the robability weighting hase. While our roosed functions overweight low robabilities and underweight high robabilities, they also have the feature that the ratio between the weight and the robability goes to zero as the robability goes to zero. We call these higher order Prelec robability weighting functions because they are generalizations of Prelec s (1998) robability weighting function. Our third contribution is to show that when RDU or CP is combined with any one of these new robability weighting functions, then a decision maker will not buy insurance against an exected loss of sufficiently low robability; in agreement with the evidence. To quote from Kunreuther et al. (1978, 248) This brings us to the key finding of our study. The rincial reason for a failure of the market is that most individuals do not use insurance as a means of transferring risk from themselves to others. This behavior is caused by eole s refusal to worry about losses whose robability is below some threshold. 2 Blavatskyy (2004) and Rieger and Wang (2006) show that the St. Petersburg aradox reemerges under CP, even with a strictly concave value function. Rieger and Wang (2006) derive a new robability weighting function that solves this aradox. We show that the higher order Prelec robability weighting functions also resolve this aradox. The aer is structured as follows. In Section 2, we define the concets of infiniteoverweighting and zero-underweighting. In Section 3 we derive some roerties of the Prelec weighting function. In Section 4, we resent our roosed weighting function and derive its roerties. Sections 5 and 6 derive the imlications of the robability weighting functions for insurance when an individual uses, resectively, RDU and CP. Section 7 1 To quote from Kunreuther et al. (1978, 240)... a substantial number of those who have sufficient information for making decisions on the basis of the exected utility model frequently behave in a manner inconsistent with what would be redicted by the theory. Other studies that reach a similar conclusion, reviewed by Kunreuther et al. (1978, section 1.4), cover the decisions to wear seat belts, to obtain breast examinations, to sto smoking, to urchase subsidized crime insurance and to urchase flight insurance. The last of these, however, shows that eole urchase too much flight insurance, comared totheredictionofeu. 2 Kunreuther et al. (1978) is a major study, involving samles of thousands, survey data, econometric analysis and exerimental evidence. All three methodologies give this same conclusion. 2

briefly considers the St. Petersburg aradox. Section 8 gives the conclusions. 2. Weighting of robabilities In this section, we introduce the concets of infinite-overweighting and zero-underweighting of infinitesimal robabilities. These will be crucial for the rest of the aer. Definition 1 :By a robability weighting function we mean a strictly increasing function w :[0, 1] [0, 1],w(0) = 0,w(1) = 1. Definition 2 :We say that the robability weighting function, w, infinitely-overweights w() 1 w() infinitesimal robabilities, if(a)lim = and (b) lim =. 0 1 1 Definition 3 : We say that the robability weighting function, w, zero-underweights w() 1 w() infinitesimal robabilities, if(a)lim =0and (b) lim =0. 3 0 1 1 3. Prelec s robability weighting function The Prelec (1998) robability weighting function has the attraction that it is arsimonious and is consistent with much of the available emirical evidence 4. Therefore, we chose it as our starting oint. Definition 4 : (Prelec, 1998). By the Prelec function we mean the robability weighting function w :[0, 1] [0, 1] given by w (0) = 0 (3.1) w () =e β( ln )α ;0< 1, 0 <α<1, β>0 (3.2) We roduce below, a grah of the Prelec function for α =0.35, β=1. 3 w() We could introduce the further concets, finite-overweighting : lim 0, lim 1 w() 1 1 (1, ) and finite-underweighting : lim, lim 1 w() (0, 1). But we will not use these in this aer. w() 0 1 1 4 Prelec (1998) gives a derivation based on comound invariance, Luce (2001) gives a derivation based on reduction invariance and al-nowaihi and Dhami (2005) give a derivation based on ower invariance. Since the Prelec function satisfies all three, comound invariance, reduction invariance and ower invariance are all equivalent. 3

w() 1 0.75 0.5 0.25 0 0 0.25 0.5 0.75 1 A Prelec function w() Proosition 1 : (Prelec, 1998, 505). For Prelec s function (Definition 2): lim 0 1 w() and lim =, i.e.,w, infinitely-overweights infinitesimal robabilities. 5 1 1 = Proof: For 0 <<1, (3.2) gives ln w() =lnw () ln = β ( ln ) α ln = ( ln ) α ( ln ) 1 α β and since 0 <α<1, wegetlim ln w() 0. Thisrovesthefirst art. To rove the second art, note that, as 1, 1 w () 0 and 1 0. Hence, we evaluate lim lim 1 d(1 w()) d(1 ) dw() /lim = lim d 1 d 1 d 1 w() 1 1 αβw() 1 ( ln ) 1 α =lim w() =. Hence, lim = 0 using L Hoital s rule. This gives lim =. 1 w() = 1 1 According to Prelec (1998, 505), these infinite limits cature the qualitative change as we move from certainty to robability and from imossibility to imrobability. On the other hand, they contradict the observed behavior that eole ignore events of very low robability and treat very high robability events as certain. In sections 5 and 6, below, we show that this leads to eole fully insuring against all losses of sufficiently low robability, even with actuarially unfair remiums and fixed costs to insurance. This is 5 Tversky and Kahneman (1992) roose the following robability weighting function, where the lower bound on γ comes from Rieger and Wang (2006), γ w () =, 0.5 γ<1 [ γ +(1 ) γ γ ] 1 w() Clearly, this function also has the roerty lim 0 =. It can be shown that other robability weighting functions that have been roosed, for examle, Gonzalez and Wu (1999) and Lattimore, Baker and Witte (1992), also have this feature. A notable excetion is the robability weighting function of Rieger and Wang (2006). 4

contrary to observation. See, for examle, Kunreuther et al. (1978). Following Kahneman and Tversky (1979), we could rely on an initial editing hase, where the decision maker chooses which imrobable events to treat as imossible and which robable events to treat as certain. While we are ersuaded by this choice heuristic, as yet, there is no general theory of the editing hase. In the next section, we roose a class of robability weighting functions that combine the editing hase with the robability weighting hase. 4. Higher order Prelec robability weighting functions Lemma 1 : (Prelec, 1998, 507, footnote). Prelec s function (Definition 2) can be written as w (0) = 0,w(1) = 1 (4.1) ln ( ln w) =( ln β)+α ( ln ( ln )), 0 <<1 (4.2) Lemma 1 motivates the following develoment. exanded as a ower series in ln ( ln ), i.e., Assume that ln ( ln w) can be P ln ( ln w) = a k ( ln ( ln )) k, 0 <<1 (4.3) k=0 P Lemma 2 :If a k ( ln ( ln )) k is convergent, then (4.3) defines a function w :(0, 1) (0, 1). k=0 Proof: The result follows from the fact that (4.3) is equivalent to µ µ P w () =ex ex a k ( ln ( ln )) k, 0 <<1 (4.4) k=0 Definition 5 :Byahigher order Prelec robability weighting function, wemeanarobability weighting function (Definition 1) given by (4.1) and (4.4). Note that a Prelec function (Definition 4) is a higher order Prelec robability weighting function. Hence, the class of functions defined by Definition 5 is not emty. In what follows, we will show that it is a rich class with interesting members. Suose that a n 6=0but a k =0, for all k>n. Hence, (4.3) becomes P ln ( ln w) = n a k ( ln ( ln )) k, 0 <<1,a n 6=0 (4.5) k=0 Definition 6 : Let w be a robability weighting function that satisfies (4.1) and (4.5). We call w a Prelec robability weighting function of order n. 5

Clearly, a Prelec function is a Prelec robability weighting function of order 1. We roduce some grahs below. First we lot a third order Prelec function (i) over the entire robability range and (ii) for low robabilities; we use the following arameter values: a 1 =0.35, a 3 =0.25, a 0 = a 2 =0. Then we lot a fifth order Prelec function using the values: a 1 =0.35, a 3 =0.25, a 5 =0.20, a 0 = a 2 = a 4 =0. w() 1 0.75 0.5 0.25 0 0 0.25 0.5 0.75 1 A third order Prelec function w() 0.02 0.015 0.01 0.005 0 0 0.005 0.01 0.015 0.02 0.02 The third order Prelec function for low robabilities 6

w() 1 0.75 0.5 0.25 0 0 0.25 0.5 0.75 1 A fifth order Prelec function The following lemma states, without roof, a few useful mathematical facts. Lemma 3 :Lety (x) = ln ( ln x),x (0, 1).Then (i) y :(0, 1) onto (, ) (ii) y is a strictly increasing function of x. (iii) y (e 1 )=0, (iv) y (x) < 0 x (0,e 1 ), (v) y (x) > 0 x (e 1, 1). (vi) y (x) x 0. (vii) y (x) x 1. Lemma 4 :Letw () be given by (4.5). Then, for near 0 and near 1, the behavior of w () is dominated by the behavior of the leading term, a n ( ln ( ln )) n.secifically, ln ( ln w) a n ( ln ( ln )) n,as 0 or as 1. Proof: (4.5) can be written as follows ln ( ln w) =a n ( ln ( ln )) n 1+ n 1 P k=0 a k ( ln ( ln )) (n k), 0 <<1,a n 6=0 a n (4.6) Let 1. ByLemma3(vii), ln ( ln ). Hence, ( ln ( ln )) (n k) 0 for k<n. It follows, from (4.6), that ln ( ln w) a n ( ln ( ln )) n. Let 0. By Lemma 3 (vi) ln ( ln ). Hence, ( ln ( ln )) (n k) 0 for k<n. It follows, from (4.6), that ln ( ln w) a n ( ln ( ln )) n. 7

Proosition 2 : The following defines a Prelec robability weighting function of order 2n +1: w (0) = 0,w(1) = 1 (4.7) P ln ( ln w) =a 0 + n a 2k+1 ( ln ( ln )) 2k+1, 0 <<1,a 2k+1 0,a 2n+1 > 0 (4.8) k=0 Proof: Since ln ( ln ) is a strictly increasing function of, 2k +1is odd, a 2k+1 np 0,a 2n+1 > 0, it follows that a 2k+1 ( ln ( ln )) 2k+1 is a strictly increasing function k=0 of. Hence, from (4.8), ln ( ln w) is a strictly increasing function of. Hence, w is a strictly increasing function of. Therefore, (4.7), (4.8) define a Prelec robability weighting function of order 2n +1. Proosition 2 gives sufficient, but not necessary, conditions for (4.5) to reresent a Prelec function of order n. On the other hand, Proosition 3 (a), below, gives necessary, but not sufficient, conditions. Proosition 3 :Letw () be a Prelec function of order n (Definition 6). Then (a) n is odd and a n > 0. (b) limw () =0and limw () =1, i.e., w() is continuous at zero and at one. 0 1 w() (c) If n>1, thenlim 0 robabilities. 1 w() =0and lim 1 1 =0, i.e., w, zero-underweights infinitesimal Proof: Since a n 6= 0,we have either a n > 0 or a n < 0. Suose a n < 0. By Lemma 3 (ii), (v), ln ( ln ) is a ositive and strictly increasing function of (e 1, 1). Hence, ( ln ( ln )) n is also a strictly increasing function for (e 1, 1). Thus, a n ( ln ( ln )) n is a strictly decreasing function for (e 1, 1). From Lemma 4, it then follows that ln ( ln w), and hence w, is a strictly decreasing function for sufficiently close to 1. But this cannot be, because a robability weighting function is strictly increasing. Hence, a n > 0. Suose n is even. By Lemma 3 (ii), (iv), ln ( ln ) is a negative and strictly increasing function of (0,e 1 ).Then( ln ( ln )) n, and hence also a n ( ln ( ln )) n, is a strictly decreasing function for (0,e 1 ). From Lemma 4, it then follows that ln ( ln w), and hence w, is a strictly decreasing function for sufficiently close to 0. But this cannot be, because a robability weighting function is strictly increasing. Hence, n is odd. This roves art (a). From art (a), Lemma 3 (vi) and (vii), and Lemma 4 we get 1 ln ( ln w) a n ( ln ( ln )) n w 1. Wealsoget 0 ln ( ln w) a n ( ln ( ln )) n w 0. Thisrovesart(b). To rove the first art of (c), write (4.6) as a n( ln( ln )) n ln w = e P n 1 1+ k=0 a k an ( ln( ln )) (n k) 8, 0 <<1,a n > 0,n>1,n odd (4.9)

Since ln w =lnw ln =lnw + eln( ln ), (4.9) gives for 0 <<1, a n > 0, n>1, n odd: n 1 P ln w ln( ln ) 1 a n( ln( ln )) ln ) n 1 a 1+ k ( ln( ln )) (n k) an k=0 = eln( 1 e (4.10) Let 0. ByLemma3(vi) ln ( ln ). Hence, ( ln ( ln )) (n k) 0 for k<n. Furthermore, since n 1 is a ositive even number, and a n > 0, it follows that a n ( ln ( ln )) n 1. In the light of these facts, (4.10) gives ln w,as 0. Hence, w() 0,as 0. Thisrovesthefirstartof(c). Torovethesecondartof (c), write (4.5) as lim 1 lim 1 np ln w = e a k ( ln( ln )) k k=0, 0 <<1,a n > 0,n>1,n odd (4.11) In the light of art (b), we use L Hoital s rule to evaluate lim 1 w() 1 d ln w() d. = lim d(1 w()) d(1 ) /lim 1 d 1 d dw() = lim 1 d d ln w() = limw () 1 d 1 w() 1 1. This gives = lim 1 w () lim 1 d ln w() d = From this, the fact that ln = e [ ln( ln )], from (4.11), and since n>1, weget 1 w () lim 1 1 = nlim 1 ex n 1 a n ( ln ( ln )) 1+ n 1 P (n k) ka k na n ( ln ( ln )) k=1 µa n ( ln ( ln )) n 1 1 a n ( ln ( ln )) (n 1) + n 1 P k=0 (n k) a k a n ( ln ( ln )) a n ( ln ( ln )) n 1 = nlim = n(a 1 ean( ln( ln ))n n ) 1 x n n lim =0, x xe xn where x =(a n ) 1 n ( ln ( ln )). Note that it follows from Definition 6 and Proosition 3(b), that a Prelec function of order n is continuous. Proosition 3(c) formalizes the exact sense in which imrobable events are ignored and robable events are treated as certain. Of course, how robable or imrobable, deends on the arameters a k. Comaring the grahs of the (1st order) Prelec function with the 3rd and 5rd order Prelec functions, we see that they are similar for robabilities in the middle range. However, the higher order functions allow imrobable events to be ignored and robable events to 9

be treated as certain. In rincile, the order, n, andthearameters,a k, can be chosen to fit the data. We now give an examle of an infinite order Prelec function. Taking a 1 = a>0 and, for k 1, a 2k =0, a 2k+1 0, we get that (4.1) and (4.3) define a strictly increasing function, rovided the series is convergent. An easy way to guarantee convergence is to take a 2k+1 = a. Then, for any (0, 1), the series in (4.3) converges (absolutely (2k+1)! and uniformly) to a 0 + 1a e ln( ln ) e ln( ln ) = a 2 0 + a sinh ( ln ( ln )). 6 To get the right shae, we take a (0, 1). For robabilities in the middle range, sinh ( ln ( ln )) ' ln ( ln ), hence this function is a good aroximation to the (first order) Prelec function for such robabilities. Using arguments similar to those in the roof of Proosition 3, it is straightforward to rove: Proosition 4 :Letw () be defined by: w (0) = 0,w(1) = 1 (4.12) ln ( ln w) =a 0 + a sinh ( ln ( ln )), 0 <<1, 0 <a<1 (4.13) Then (a) w :[0, 1] [0, 1] is continuous and strictly increasing; w is C on (0, 1). w() 1 w() (b) lim =0and lim =0, i.e., w, zero-underweights infinitesimal robabilities. 0 1 1 Definition 7 : By the hyerbolic Prelec function (HP), we mean the robability weighting function defined by (4.12) and (4.13). Note that (4.12) and (4.13) define a two-arameter family of functions. Hence a hyerbolic Prelec function is just as arsimonious as the (1st order) Prelec function (Definition 4). The following is a grah of (4.12) and (4.13) for a 0 =0and a = 1 2 : In the next two sections we comare the behavior of the decision maker when she uses, resectively, a robability weighting function of order 1 (which is just the standard Prelec function) and of order greater than one. We show that her behavior differs significantly between the two cases. 5. Rank deendent utility theory and insurance In this section, we model the behavior of an individual using rank deendent utility theory (RDU), which we may regard as a conservative extension of exected utility theory (EU) to 6 The hyerbolic sin function is defined as sinh x = 1 e x e x. 2 10

w() 1 0.75 0.5 0.25 0.25 0.5 0.75 Figure 4.1: A Hyerbolic Prelec Function w() 5e-5 3.75e-5 2.5e-5 1.25e-5 0 2.5e-5 5e-5 7.5e-5 Figure 4.2: A Hyerbolic Prelec Function For Low Probabilities 11

the case where robabilities are transformed. In the next section, we consider cumulative rosect theory (CP), which is a more radical dearture from exected utility theory. Consider a decision maker with initial wealth, W, robability weighting function, w, and utility function, u, whereu is strictly concave, differentiable, u 0 > 0 and u 0 is bounded above 7 by (say) u 0. Assume that she faces the lottery: win x 1 with robability 1 or x 2 with robability 2, x 1 x 2, 0 i 1, 1 + 2 =1(if x i < 0, thenx i is, in fact, a loss). According to rank deendent utility theory, her exected utility will be U = w ( 2 ) u (x 2 )+ [1 w ( 2 )] u (x 1 ). For w () =, rank deendent utility theory reduces to standard exected utility theory. Note that the higher outcome, x 2, receives weight w ( 2 ), while the lower outcome, x 1, receives weight w ( 1 + 2 ) w ( 2 )=w (1) w ( 2 )=1 w ( 2 );see Quiggin (1982, 1993) for the details. Suose a decision maker can suffer the loss, L>0, with robability. Shecanbuy coverage, C, atthecostrc + f, where0 C L, 0 <<1, 0 <r<1, f 0 and f is a fixed cost. We allow deartures from the actuarially fair condition. We do so in a simle way by setting the insurance remium rate r =(1+θ). Thus, θ =0corresonds to the actuarially fair condition, θ>0 to the actuarially unfair and θ<0 to the actuarially over-fair condition. With robability 1, her wealth will become W rc f. With robability, her wealth will become W rc f L + C W rc f. If she buys insurance, her exected utility under RDU will then be: U I (C) =w (1 ) u (W rc f)+[1 w (1 )] u (W rc f L + C) (5.1) Since U I (C) is a continuous function on the non-emty comact interval [0,L], anotimal level of coverage, C, exists. For full insurance, C = L, (5.1) gives: U I (L) =u (W rl f) (5.2) On the other hand, if she does not buy insurance, her exected utility will be: U NI = w (1 ) u (W )+[1 w (1 )] u (W L) (5.3) For the decision maker to buy insurance, the following articiation constraint must be satisfied: U NI U I (C ) (5.4) 7 The boundedness of u 0 is needed for art (b) of Proosition 5. This seems feasible on emirical grounds, since eole do undertake activities with a non-zero robability of comlete ruin, e.g., using the road, undertaking dangerous sorts, etc. However, the boundedness of u 0 excludes such tractable utility functions as ln x and x γ, 0 <γ<1. By contrast, the boundedness of u 0 is not a requirement in CP, as we shall see. 12

Proosition 5 : Under RDU, (a) If a robability weighting function infinitely-overweights infinitesimal robabilities (Definition 2) then, for a given exected loss, the decision maker will insure fully for all sufficiently small robabilities. (b) If a robability weighting function zero-underweights infinitesimal robabilities (Definition 3) then, for a given exected loss, a decision maker will not insure, for all sufficiently small robabilities. Proof: Consider an exected loss L = L (5.5) Differentiate (5.1) with resect to C to get UI 0 (C) = rw (1 ) u 0 (W rc f) (5.6) +(1 r)(1 w (1 )) u 0 (W +(1 r) C f L) (5.7) Since u is (strictly) concave, u 0 > 0 and 0 <r<1, it follows, from (5.6) that U 0 I (C) is adecreasingfunctionofc. Hence, UI 0 (L) U I 0 (C) U I 0 (0) for all C [0,L] (5.8) Relace r by (1 + θ) in (5.6), then divide both sides by, toget U 0 I (C) = (1 + θ) w (1 ) u 0 (W (1 + θ) C f) +(1 (1 + θ) ) 1 w (1 ) u 0 (W (1 + θ) C f L + C) (5.9) For C =0and C = L, (5.9) gives (using (5.5)): U 0 I (0) =[1 (1 + θ) ] µ 1 w (1 ) u 0 W f L (1 + θ) w (1 ) u 0 (W f) (5.10) UI 0 (L) 1 w (1 ) = (1 + θ) u 0 W (1 + θ) L f (5.11) Since 0 < (1 + θ) <1, 0 <<1, 0 <w(1 ) < 1, 0 <u 0 < u 0 we get, from (5.10), U 0 I (0) < 1 w (1 ) u 0 (1 + θ) w (1 ) u 0 (W f) (5.12) Let F () = 1 w (1 ) 13 (1 + θ) (5.13)

From (5.11) and (5.13) we get, From (5.14) we see that U 0 I (L) = F () u 0 W (1 + θ) L f (5.14) U 0 I (L) > 0 F () > 0 (5.15) From (5.2), (5.3), (5.5), (5.13) and the facts that u is strictly increasing and strictly concave, simle algebra leads to f<lf () U NI <U I (L) (5.16) Put q =1 (5.17) (a) Suose w () infinitely-overweights infinitesimal robabilities. Then, from (5.17) and 1 w(1 ) 1 w(q) Definition 2, lim = lim =. Hence, from (5.13), for given exected loss, 0 q 1 1 q L, wecanfind a 1 (0, 1) such that, for all (0, 1 ),wegetf < LF (). From (5.16) it follows that the articiation constraint (5.4) is satisfied for all (0, 1 ).From f<lf () we get that F () > 0 for all (0, 1 ). From(5.15)itfollowsthatU I 0 (L) > 0 for all such. From (5.8) it follows that UI 0 (C) > 0 for all such and, hence, the decision maker insures fully for all (0, 1 ). (b) Suose w () zero-underweights infinitesimal robabilities. Then, from (5.17) and 1 w(1 ) 1 w(q) Definition 3, lim = lim =0. Hence, from (5.12), there exists 0 q 1 1 q 2 (0, 1) such that for all (0, 2 ), UI 0 (0) < 0. Hence, from (5.8), U I 0 (C) < 0 for all C [0,L]. Hence the otimal level of coverage is 0. From Proosition 1 and Proosition 5(a), a decision maker using a Prelec robability weighting function (Definition 4), will fully insure against all losses of sufficiently small robability, rovided the articiation constraint (5.4) is satisfied. It is of interest to get a feel for how restrictive this articiation constraint is. Examle (1), below, suggests it is a weak restriction. Examle 1 : The the first row of the following table gives losses from 10 (Dollars, say) to 10, 000, 000, with corresonding robabilities (row 2) ranging from 0.1 to 0.000, 000, 1; so that the exected loss in each case is L =1. In row 3 are the corresonding values of 1 w(1 ) for the Prelec function w () =e ( ln )0.65, where the values α =0.65 and β =1 are suggested by Prelec (1998). loss 10 100 1000 10,000 100,000 1,000,000 10,000,000 robability of loss 0.1 0.01 0.001 0.000,1 0.000,01 0.000,001 0.000,000,1 1 w(1 ) 2. 067 4 4. 903 9 11. 161 25. 088 56. 219 125. 88 281. 83 14

From (5.16) we saw that the articiation constrain (5.4) is satisfied if the fixed cost, f, islessthanlf (), wheref () is given by (5.13) and, in Examle 1, L =1.Evenfor the high rofit rateof100% (θ =1),sothatF () = 1 w(1 ) 2, wesee,fromtheabove table, that the uer bound on the fixed comonent of the cost of ensuring against an exected loss of one unit (e.g. one Dollar), so that the articiation constraint is satisfied, is hardly restrictive for low robabilities. Thus, from Proosition 5(a), we see that using RDU in combination with the Prelec function of order 1, is likely to lead to misleading results, in that it would redict too much insurance. On the other hand, from Proositions 3(c), 4(b) and 5(b), a decision maker using a Prelec robability weighting function of order n>1 (Definitions 6 and 7) will not insure against any loss of sufficiently small robability, in agreement with observation. 6. Cumulative rosect theory and insurance Several anomalies, among them the ones mentioned in the introduction, motivated the develoment of rosect theory (Kahneman and Tversky (1979), Tversky and Kahneman (1992)). In rosect theory, the carriers of utility are not levels of wealth, assets or goods, but differences between these and a reference oint (reference deendence). The reference oint is usually (but not necessarily) taken to be the status quo. The value function, as the utility function is called in rosect theory, is concave for gains but convex for losses (declining sensitivity). The disutility of a loss is greater than the utility of a gain of the same magnitude (loss aversion) 8. Probabilities are transformed, so that small robabilities are overweighted but high robabilities are underweighted. A commonly used value function in rosect theory is v (x) =x γ, 0 <γ<1 (6.1) Consider a decision maker whose behavior is described by cumulative rosect theory (CP, Tversky and Kahneman, 1992). Let her initial wealth be W. Suose she can suffer the loss, L>0, with robability. She can buy coverage, C, at the cost rc + f, where 0 C L, 0 <<1, 0 <r<1, f 0 and f is a fixed cost. As in Section 5, r =(1+θ). Take her current wealth, W, to be her reference oint. With robability 1, her wealth will become W rc f; whichshecodesasthelossrc + f, relativeto her reference wealth, W. With robability, her wealth will become W rc f L + C; whichshecodesasthelossl C + rc + f rc, relative to her reference wealth W.Let v be her value function for the domain of losses, v :[0, ) [0, ) where v is strictly 8 Loss aversion is very imortant when the decision maker is in the domain of gains in one state of the world but in the domain of losses for another. However, loss aversion will not be imortant here because, as we shall see, the decision maker will always be in the domain of losses. 15

concave, v (0) = 0, v is differentiable on (0, ) with v 0 > 0. 9 cumulative rosect theory will then be: Her utility function under V I (C) = w () v (L C + rc + f) (1 w ()) v (rc + f) (6.2) Since V I (C) is a continuous function on the non-emty comact interval [0,L], anotimal level of coverage, C, exists. For full insurance, C = L, (6.2) gives: V I (L) = v (rl + f) (6.3) On the other hand, if she does not buy insurance, her exected utility will be (recall that v (0) = 0): V NI = w () v (L) (6.4) For the decision maker to buy insurance, the following articiation constraint (the analogue here of (5.4)) must be satisfied: V NI V I (C ) (6.5) Proosition 6 : Under CP, (a) A decision maker will insure fully against any loss, rovided the articiation constraint is satisfied. (b) For Prelec s robability weighting function (Definition 4), for the value function (6.1) and for a given exected loss, the articiation constraint (6.5) is satisfied for all sufficiently small robabilities. (c) If a robability weighting function zero-underweights infinitesimal robabilities (Definition 3) then, for a given exected loss, a decision maker will not insure against any loss of sufficiently small robability. Proof: (a) Since v is strictly concave, v is strictly convex. Hence, from (6.2), it follows that V I is strictly convex. Since 0 C L, itfollowsthatv I (C) is maximized either at C =0or at C = L. Hence, if the articiation constraint is satisfied, then the decision maker will fully insure against the loss. (b) Consider the Prelec function (3.2) and the value function (6.1). Let F () = e Consider an exected loss β ( ln )α γ (1 + θ),(recall0 <α<1, β>0, γ>0) (6.6) L = L (6.7) 9 These assumtions are satisfied by, for examle, v (x) =1 e x and v (x) =x γ, 0 <γ<1. Butthey are not satisfied by v (x) =lnx, sinceln 0 is not defined. 16

From (3.2), (6.1), (6.3), (6.4), (6.6) and (6.7), simle algebra leads to f<lf () V NI <V I (L) (6.8) From (6.6) and Proosition 1, limf () =. Hence, for given exected loss, L, we 0 get f < LF (), forallsufficiently small. From (6.8) it follows that the articiation constraint is satisfied for all such small. (c) From (6.3) and (6.4) we get the following lim 0 V I (L) V NI V I (L) V NI lim 0 = v (L) w () = v (L)lim 0 w () v (1 + θ) L + f 1 v (1 + θ) L + f 1 lim 0 (6.9) (6.10) Suose w () zero-underweights infinitesimal robabilities. Then, from Definition 3, =0. Hence, the first term in (6.10) goes to 0 as goes to 0. The second term in w() (6.10), however, goes to as goes to 0. Hence, there exists 2 (0, 1) such that for all (0, 2 ),V NI >V I (L). By Proosition 6(a), a decision maker will insure fully against any loss, rovided the articiation constraint (6.5) is satisfied. By Proosition 6(b), for Prelec s robability weighting function (Definition 4), for the value function (6.1) and for a given exected loss, the articiation constraint (6.5) is satisfied for all sufficiently small robabilities. It is of interest to get a feel for how restrictive this articiation constraint is. Examle (2), below, suggests it is a weak restriction. Examle 2 : Thethefirst row of the following table gives losses from 10 (Dollars, say) to 10, 000, 000, with corresonding robabilities (row 2) ranging from 0.1 to 0.000, 000, 1; so that the exected loss in each case is L =1. In row 3 are the corresonding values β of e γ ( ln )α, where the values α =0.65 and β =1are suggested by Prelec (1998) and γ =0.88 is suggested by Tversky and Kahneman (1992). loss 10 100 1000 10,000 100,000 1,000,000 10,000,000 robability of loss 0.1 0.01 0.001 0.000,1 0.000,01 0.000,001 0.000,000,1 e 0.88 1 ( ln )0.65 1. 416 9 4. 658 9 18. 48 81. 342 383. 83 1906. 3 9852. 3 From (6.8) we saw that the articiation constrain (6.5) is satisfied if the fixed cost, f, is less than LF (), wheref () is given by (6.6) and, in Examle 2, L =1.Evenforthe β high rofit rateof100% (θ =1),sothatF () = e γ ( ln )α 2, wesee,fromtheabove table, that the uer bound on the fixed comonent of the cost of ensuring against an exected loss of one unit (e.g. one Dollar), so that the articiation constraint is satisfied, is hardly restrictive for low robabilities. Thus, from Proosition 6(a) and (b), we see that 17

using CP in combination with the Prelec function of order 1, is likely to lead to misleading results, in that it would redict too much insurance. On the other hand, from Proositions 3(c),4(b) and 6(c), a decision maker using a Prelec robability weighting function of order n>1 (Definition 6 and 7) will not insure against any loss of sufficiently small robability, in agreement with observation. However, by Proosition 6(a), CP redicts that, if a decision maker decides to insure, she will insure fully, even with a fixed cost of entry and an actuarially unfair remium. 7. St. Petersburg aradox The St. Petersburg aradox occuies an imortant lace in the history of economic thought, as it motivated von Neumann and Morgenstern (1947) to introduce exected utility into economics. Exected utility has remained ever since the main tool for analyzing decision making under risk. A simle version of the aradox runs as follows. If the first realization of heads in a sequence of random throws of a fair coin occurs on the n-th throw, then the game ends and the reciient receives a ayoff of 2 n monetary units. This game has an infinite exected ayoff, yet exerimental evidence suggests that subjects will ay only a modest finite sum to lay this game. Bernoulli (1738) suggested that a decision maker maximized the exected utility of a lottery rather than the exected monetary value. Blavatskyy (2004) and Rieger and Wang (2006) 10 have shown that this aradox reemerges under CP. They rove that, even with a strictly concave value function, the Bernoulli lottery will have an infinite exected utility. Rieger and Wang (2006) go on to show that the robability weighting function 11 : 3(1 b) w () = + a (1 + a) 2 + 3 µ 2, a 1 a + a 2 9, 1,b (0, 1) (7.1) solves the aradox by generating a finite exected utility under CP. 12 By direct calculation, or by alying Theorem 1 of Rieger and Wang (2006), it is straightforward to show that any of the generalized Prelec robability weighting functions (of order n>1), will also generate a finite exected utility for the St. Petersburg aradox. Thus, the generalized Prelec functions solve both the insurance aradox and the St. Petersburg aradox. 10 We cannot do justice to Rieger and Wang (2006) in this brief section. The reader is urged to read their aer. 11 Rieger and Wang state, incorrectly, that a (0, 1). For sufficiently low a and ' 1 3, w (), asgiven by (7.1), is decreasing in. The lower bound of 2 9 on a is sufficient, but not necessary, for w () to be strictly increasing. 12 w() 1 w() From (7.1): lim 0 (1, ) and lim 1 1 (1, ). Hence, these functions finitely-overweight infinitesimal robabilities, in the sense of footnote 3. Hence, unlike the higher order Prelec functions, they do not cature the emirical fact that eole ignore extremely low robabilities and code extremely likely events as certain. 18

8. Conclusion Decision models that rely on non-linear transformations of robabilities, for instance, rank deendent utility (RDU) and cumulative rosect theory (CP) require using a robability weighting function. However, the standard robability weighting functions infinitely overweight infinitesimal robabilities, in the sense that the ratio between the weight and the robability goes to infinity, as the robability goes to zero. In actual ractice, individuals code very small robabilities as zero and very large ones as one. Given that many imortant decisions under uncertainty involve small robabilities, we show that the infinite overweighting of small robabilities feature of existing robability weighting functions leads to redictions that contradict observed behavior. Thus, individuals should insure fully even for ridiculously low robability natural hazards. This is at odds with the evidence. Indeed, governments often have to legislate comulsory insurance of several kinds and mortgage lenders have mandatory building and contents insurance requirements as a re-condition to their lending. Kahneman and Tversky (1979) roosed a two-ste heuristic rocedure to deal with these sorts of roblems. In the first ste, events associated with robabilities coded as zero are ignored. In the second ste, a robability weighting function is used to make a choice from the surviving alternatives. We roose a class of robability weighting functions- the higher order Prelec robability weighting functions- that allow us to combine the two-ste heuristic choice rocess of Kahneman and Tversky into one. Furthermore, while our roosed functions overweight low robabilities and underweight high robabilities, they also have the feature that the ratio between the weight and the robability goes to zero as the robability goes to zero. This enables us to show that when RDU or CP is combined with any one of these new robability weighting functions, then a decision maker would not buy insurance against an exected loss of sufficiently low robability; in agreement with the evidence. One attractive feature of the Prelec (1998) robability weighting function is that it has an axiomatic derivation. An interesting question, that lies beyond the scoe of this aer, is what behavioral assumtions lead to the higher order Prelec robability weighting functions? This could be a fruitful line of inquiry. Another toic for future research could be to try to fit the roosed weighting function to data and estimate its arameters. This could otentially reveal imortant information about individual choice under uncertainty. References [1] al-nowaihi, A., Dhami, S., 2005. A simle derivation of Prelec s robability weighting function. University of Leicester Deartment of Economics Discussion Paer 05/20. 19

[2] Bernoulli, D., 1738. Exosition of a New Theory of the Measurement of Risk. English translation in Econometrica 22, 1954, 23-36. [3] Blavatskyy, P.R., 2004. Back to the St. Petersburg aradox? CERGE-EI working aer series No. 227. [4] Gonzalez, R., Wu, G., 1999. On the shae of the robability weighting function. Cognitive Psychology 38, 129-166. [5] Kahneman D., Tversky A., 1979. Prosect theory : An analysis of decision under risk. Econometrica 47, 263-291. [6] Kunreuther, H., Ginsberg, R., Miller, L., Sagi, P., Slovic, P., Borkan, B., Katz, N., 1978. Disaster insurance rotection: Public olicy lessons, Wiley, New York. [7] Lattimore, J. R., Baker, J. K., Witte, A. D.,1992. The influence of robability on risky choice: A arametric investigation. Journal of Economic Behavior and Organization 17, 377-400. [8] Luce, R. D., 2001. Reduction invariance and Prelec s weighting functions. Journal of Mathematical Psychology 45, 167-179. [9] Peel, D., Cain, M., Law, D., 2005. Cumulative rosect theory and gambling. Lancaster University Management School Working Paer 2005/034. [10] Prelec, D., 1998. The robability weighting function. Econometrica 60, 497-528. [11] Quiggin, J., 1982. A theory of anticiated utility. Journal of Economic Behavior and Organization 3, 323-343. [12] Quiggin, J., 1993. Generalized Exected Utility Theory, Kluwer Academic Publishers. [13] Rieger, M.O., Wang, M., 2006. Cumulative rosect theory and the St. Petersburg aradox. Economic Theory 28, 665-679. [14] Tversky, A., Kahneman D., 1992. Advances in rosect theory : Cumulative reresentation of uncertainty. Journal of Risk and Uncertainty 5, 297-323. [15] von Neumann, J., Morgenstern, O., 1947. Theory of Games and Economic Behavior, Princeton University Press, Princeton. 20