Journal of Health Economics

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1 Journal of Health Eonomis 32 (2013) Contents lists available at SiVerse SieneDiret Journal of Health Eonomis ourna l h omepa ge: reatment deisions under ambiguity Loï Berger a, Han Bleihrodt b,, Louis Eekhoudt a FNRS researh fellow, Université Libre de Bruxelles, Belgium and oulouse Shool of Eonomis, Frane b Erasmus Shool of Eonomis, Netherlands Ieseg, Lille, Frane and CORE, Louvain-la-Neuve, Belgium a r t i l e i n f o Artile history: Reeived 28 August 2012 Reeived in revised form 11 February 2013 Aepted 12 February 2013 Available online 27 February 2013 JEL lassifiation: D81 I10 Keywords: Ambiguity aversion Diagnosti ambiguity herapeuti ambiguity Smooth ambiguity model Prospet theory a b s t r a t Many health risks are ambiguous in the sense that reliable and redible information about these risks is unavailable. In health eonomis, ambiguity is usually handled through sensitivity analysis, whih impliitly assumes that people are neutral towards ambiguity. However, empirial evidene suggests that people are averse to ambiguity and reat strongly to it. his paper studies the effets of ambiguity aversion on two lassial medial deision problems. If there is ambiguity regarding the diagnosis of a patient, ambiguity aversion inreases the deision maker s propensity to opt for treatment. On the other hand, in the ase of ambiguity regarding the effets of treatment, ambiguity aversion leads to a redution in the propensity to hoose treatment Elsevier B.V. All rights reserved. 1. Introdution wo seminal artiles in the seond half of the 1970s showed how aounting for people s attitudes towards risk an improve the pratie of health eonomis and medial deision making (MNeil et al., 1978; Pauker and Kassirer, 1975). hese studies were based on expeted utility, whih was the dominant desriptive theory of deision under risk at the time. Later studies showed how their reommendations ould be improved using new insights from deision theory, in partiular prospet theory (Wakker, 2008, 2010). Studies of medial deision making typially assume that probabilities are known. In real-life situations, however, these are often unknown and the available information is given with different degrees of preision. here is onsiderable unertainty about the risks to publi health that we fae. Examples are the reent debates about the threats of mad ow disease, limate hange, and the avian and swine flu. Similarly, dotors fae unertainty in making treatment reommendations. Data on the prevalene of disease and on the suess rate of treatment are inomplete or unknown and the available data do not allow extrating a single probability Corresponding author. el.: ; fax: address: bleihrodt@ese.eur.nl (H. Bleihrodt). distribution of the possible outomes (see Arad and Gayer, 2012 for an example involving medial deisions). In health eonomis, ambiguity has usually been addressed by sensitivity analysis (Briggs et al., 1994; Manski, 2011) or by metaanalyses in whih the different probabilities found in the literature are ombined using a weighted average of the available estimates. hese approahes impliitly assume that the deision maker is neutral towards ambiguity. However, an extensive amount of empirial work, originating from Ellsberg s (1961) famous thought experiment, show that people are not neutral towards ambiguity, but dislike ambiguity and are ambiguity-averse (for examples regarding medial deisions see Curley et al., 1989; Han et al., 2009; Portnoy et al., 2011). Ignoring this ambiguity aversion may distort treatment reommendations and may hinder the understanding of variations in treatment pratie. Many new models have been developed to apture ambiguity aversion (Ghirardato et al., 2004; Gilboa and Shmeidler, 1989; Klibanoff et al., 2005; Maheroni et al., 2006; Shmeidler, 1989; versky and Kahneman, 1992). hese models have found only limited appliation in health eonomis. he purpose of this paper is to explore the impliations of ambiguity aversion for treatment deisions. We study the impat of ambiguity aversion on two lassial problems in medial deision making under the smooth ambiguity model of Klibanoff et al. (2005). his model introdues a simple and easily interpretable way to apture ambiguity aversion /$ see front matter 2013 Elsevier B.V. All rights reserved.

2 560 L. Berger et al. / Journal of Health Eonomis 32 (2013) and it is popular in eonomis today (Gollier, 2011; reih, 2010). However, our main results also hold under other ambiguity models and we will briefly disuss these in the onluding setion. he main text will present the results in an intuitive, graphial way to explain the main ideas and onepts involved. Extensions and formal proofs of these intuitive results are provided in the appendix. In Setion 2, we start by inorporating ambiguity into Pauker and Kassirer s (1975) model of diagnosti hoie where the prevalene of the disease is unknown (diagnosti ambiguity). We show that in this model ambiguity aversion leads to an inrease in the propensity to treat. It has been argued that ambiguity aversion is irrational and a bias in human deision making (Wakker, 2010). 1 If so, ambiguity aversion leads to a welfare loss. We onsider these welfare osts in Setion 3. In Setions 4 and 5, we study the ase where the effets of treatment are ambiguous (therapeuti ambiguity). In this ase, the effets of ambiguity aversion are reversed and ambiguity aversion redues the propensity to treat. Setion 6 onludes the paper and disusses our main findings. N p1 p 1 p = p N p 2 N p 2 p N p 2. Diagnosti ambiguity Consider a patient who displays partiular symptoms. he deision maker, who ould be the dotor, the patient, a poliy maker or someone else, has to deide whether the patient should undergo treatment. he treatment deision has to be made before the deision maker knows the true health state of the patient. We assume that there are two possible health states: either the patient is sik (s) or he is healthy (h). he deision maker an only deide between treatment () and no treatment (N). he ase where the deision maker an also deide on the intensity of treatment is onsidered in Appendix A.2. Let H h [H s ] denote the patient s health when he is treated and he turns out to be healthy [sik]. Likewise, H N h [HN s ] is the patient s health when he is not treated and turns out to be healthy [sik]. We assume that health an be quantified, for example as the number of remaining (quality-adusted) life-years, and that H N h > H h > Hs > Hs N. In other words, treatment is benefiial when sik, but detrimental when healthy, and it is always better to be healthy than to be sik. he outomes of the treatment are known. he ase of therapeuti hazard is onsidered in Setions 4 and 5. he probability that the patient is sik is ambiguous. o explain the intuition underlying our general result, we assume that this probability an take two values, p 1 and p 2 with p 1 < p 2. he ase where the probability an take on more than two values is treated in Appendix A.1, whih also ontains a formal proof of the intuitive results derived in this setion. Based on the information at his disposal, the deision maker assigns beliefs to the probability of illness. Let denotes the deision maker s subetive probability that p 1 is the true probability of illness. Consequently, 1 is his belief that the true probability of illness is p 2. o study the impat of ambiguous beliefs, we assume that the deision maker behaves aording to the smooth ambiguity model of Klibanoff et al. (2005) (KMM). hen, his utility of treatment is equal to V = ϕ(p 1 U(H s ) + (1 p 1 )U(H h )) + (1 ) ϕ(p 2 U(H s ) + (1 p 2 )U(H h )). (1) And his utility of no treatment is 1 his view is by no means universally shared. For example, Gilboa and Marinai (forthoming) argue that ambiguity aversion is rational. 0 p 1 p p 2 Fig. 1. Expeted utility as a funtion of the probability of illness p. he light line (p N) shows the expeted utility of no treatment as a funtion of p. he dark line (p ) shows the expeted utility of treatment as a funtion of p. For probability p, an ambiguity neutral deision maker is indifferent between treatment and no treatment. For lower probabilities he prefers no treatment, for higher probabilities he prefers treatment. V N = ϕ(p 1 U(Hs N ) + (1 p 1 )U(H N h )) + (1 ) ϕ(p 2 U(Hs N ) + (1 p 2 )U(H N h )). (2) In Eqs. (1) and (2), U is a von Neumann Morgenstern utility funtion over health. We assume that the deision maker prefers more health to less, i.e., U is stritly inreasing (U > 0) and that he is averse to risk, i.e., U is stritly onave (U < 0). An attrative feature of the smooth ambiguity model is that it separates a deision maker s ambiguity, measured through the probabilities and 1, and his ambiguity aversion, measured through the funtion ϕ. he deision maker is ambiguity averse if ϕ is onave and ambiguity seeking if ϕ is onvex. Ambiguity neutrality, the ase usually assumed in medial deision making, orresponds with linearity of ϕ. he smooth ambiguity model an be interpreted as a two-stage model in whih the first stage determines the probability of illness (p 1 or p 2 ) and the seond stage determines whether the patient is healthy or sik. In eah stage, the deision maker uses an expeted utility evaluation, but the utility funtion that is used in the two stages differs. In the first stage, the deision maker uses ϕ, whih reflets his ambiguity aversion, whereas in the seond stage, he uses U, whih reflets his risk aversion. Let p N denote the expeted utility of treatment [no P treatment] when the expeted probability of illness is equal to p. Whatever the deision taken, expeted utility is a dereasing funtion of the expeted probability of illness. It dereases linearly beause the expeted utility model is linear in probability. he behavior of expeted utility as a funtion of p is illustrated in Fig. 1. he P line is less steep than the N line. his feature P will be important in the sequel and it results from the fat that the utilities of the potential outomes under treatment (H h and Hs ) are loser together than the utilities of the potential outomes under no treatment (H N h and Hs N ). Consequently, for a given level of ambiguity around the probability of illness (p 1 and p 2 in Fig. 1), the spread of the expeted utilities is less under treatment than under no treatment: P1 P2 is smaller than N p1 N p2. For p 1

3 L. Berger et al. / Journal of Health Eonomis 32 (2013) Fig. 2. he effets of ambiguity aversion when there is diagnosti ambiguity. he x-axis shows the expeted utilities () of treatment () and no treatment (N) for the possible probabilities of illness p 1 and p 2. he y-axis shows ϕ(). Ambiguity aversion (ϕ onave), illustrated by the dark line, inreases the outomes of treatment ( and ) relative to the outomes of no treatment ( N and N) p1 p2 p1 p2 ompared with the ase of ambiguity neutrality (the light line). low probabilities of illness, no treatment is better than treatment and for high probabilities of illness, treatment is better than no treatment. P and N ross at p. Beause an ambiguity-neutral P deision maker has linear ϕ and behaves aording to expeted utility, p is the probability of illness at whih an ambiguity neutral deision maker is indifferent between treatment and no treatment. Let us now onsider what happens under ambiguity aversion. Fig. 2 illustrates the funtion ϕ whih is defined over the different expeted utilities. he straight line depits the situation under ambiguity neutrality (ϕ linear). Without loss of generality, we have saled ϕ suh that ϕ( N p2 ) = N p2 and ϕ( N p1 ) = N p1.2 Fig. 2 shows that ambiguity aversion (onavity of ϕ) implies that ϕ( p1 ) > p1 and that ϕ( p2 ) > p2. his means, in turn, that ϕ( p2 ) ϕ( N p2 ) exeeds p2 N p2 and that ϕ( N p1 ) ϕ( p1 ) falls short of N p1 p1. Compared with ambiguity neutrality, ambiguity aversion makes the advantage of treatment over no treatment (the pereived differene between p2 and N p2 ) more salient relative to the advantage of no treatment over treatment (the pereived differene between N p1 and p1 ). Consequently, ambiguity aversion inreases the likelihood that treatment is preferred to no treatment. We an now reprodue Fig. 1 under ambiguity aversion (see Fig. 3). he effet of ambiguity aversion is to shift the line displaying the benefits of treatment upwards, while the line displaying the benefits of no treatment is unaffeted (by the hosen saling of ϕ). Fig. 3 shows that an ambiguity averse deision maker prefers treatment at p, the expeted probability of illness at whih an ambiguity neutral deision maker is indifferent between treatment and no treatment. Indifferene between treatment and no treatment is restored at the lower probability of illness ˆp. An ambiguity averse deision maker will sooner opt for treatment, hene, ambiguity aversion inreases the propensity to treat. 2 his saling is allowed by the uniqueness properties of ϕ. Fig. 3. Diagnosti ambiguity aversion makes treatment more attrative and the line showing the utility of treatment (ϕ( p )) shifts upwards. he probability of illness for whih the deision maker is indifferent between treatment and no treatment shifts from p to ˆp. 3. he welfare osts of diagnosti ambiguity aversion It has been argued that ambiguity aversion is irrational and that a rational deision maker should be ambiguity neutral (Raiffa, 1961; Wakker, 2010). If so, ambiguity aversion leads to a welfare loss, whih we will illustrate in this setion. Fig. 4 shows the demand for treatment when all agents in the market are homogeneous, i.e., they have the same preferenes. Let us start with the ase Fig. 4. he welfare osts of diagnosti ambiguity aversion when agents are assumed to be homogeneous. he solid urve shows the demand for treatment under ambiguity neutrality. he interrupted urve shows the demand for treatment under ambiguity aversion. Under ambiguity aversion agents are more treatment-prone and this is refleted in the higher ost (ĉ) that they are willing to pay for treatment. he shaded area illustrates the welfare osts of ambiguity aversion when the ost of treatment equals.

4 562 L. Berger et al. / Journal of Health Eonomis 32 (2013) A B ~ Fig. 5. he welfare osts of diagnosti ambiguity aversion when agents have heterogeneous ambiguity aversion. he solid urve shows the demand for treatment under ambiguity neutrality. he interrupted urve shows the demand for treatment under ambiguity aversion. ĉ is the ost that homogeneously ambiguity averse agents are willing to pay for treatment. he shaded area in Panel A illustrates the welfare osts of ambiguity aversion when the ost of treatment equals. he dotted area in Panel B illustrates the welfare osts of ambiguity aversion when the ost of treatment equals. in whih there is no ambiguity aversion and agents are ambiguity neutral. In our analyses, the ost of treatment played no role and was impliitly assumed to be zero. As illustrated by Fig. 1, agents will hoose treatment if the expeted probability of illness is suffiiently large. Moreover, beause the utility of treatment exeeds the utility of no treatment, agents will still hoose treatment for a positive ost. However, if the ost of treatment beomes too high, in Fig. 4, agents will no longer hoose treatment and prefer to go untreated. Hene, the demand for treatment as a funtion of the ost of treatment is a step funtion as illustrated by the solid line in Fig. 4. Suppose next that agents are ambiguity averse. As we saw in Setion 2, ambiguity aversion inreases the propensity to hoose treatment and the maximum ost of treatment for whih ambiguity averse agents opt for treatment will rise to ĉ. he demand urve (whih is still a step funtion) will shift upwards as illustrated by the interrupted urve in Fig. 4. Now, if the true ost of treatment exeeds ĉ or falls short of ambiguity aversion does not lead to a welfare loss. In the first ase, ambiguity aversion is not strong enough to entie agents to opt for treatment. In the seond ase, ambiguity neutral agents already hose treatment and the introdution of ambiguity aversion only reinfores their preferene. However, if the true ost of treatment is between ĉ and, a welfare loss equal to the shaded area in the figure ours. Ambiguity averse agents are willing to pay this ost of treatment, but the true (normative) valuation of treatment is given by the ambiguity neutral demand urve and this is less than the ost of treatment. In real life, agents are not homogeneous and their attitudes towards ambiguity will vary. Fig. 5 illustrates the effet of heterogeneous ambiguity aversion on the demand for treatment. If there is heterogeneity in ambiguity aversion, it will no longer be true that all agents swith from treatment to no treatment at the same ost ĉ. Instead, the swith from treatment to no treatment will be a gradual proess. First the least ambiguity averse agents will opt for no treatment and, as the ost of treatment rises further, more and more agents will opt for no treatment until the ost of treatment is so high that only extremely ambiguity averse agents still opt for treatment. Hene, the demand urve will be downward sloping up till point. For simpliity, we have drawn the demand for treatment as a linear funtion of the ost of treatment although in reality it will probably have a more agged harater. As in Fig. 4, if the true ost of treatment falls short of, heterogeneous ambiguity aversion leads to no welfare loss, beause ambiguity neutral agents would have hosen the same level of treatment. If the true ost of treatment is between ĉ (the ost of treatment that agents with homogeneous ambiguity aversion (see Fig. 4) are willing to pay) and, the welfare loss is equal to the shaded area in Panel A of Fig. 5, whih is smaller than the welfare loss in the ase of homogeneous ambiguity aversion. his is so beause some agents (the least ambiguity averse) do not opt for treatment at this ost whereas all homogeneous agents opted for treatment. On the other hand, if the ost of treatment exeeds ĉ, there is still a welfare loss in the heterogeneous ase (equal to the dotted area in Panel B) beause the most ambiguity averse agents will still hoose treatment. 4. herapeuti ambiguity In the previous two setions, the only soure of ambiguity was the probability of illness. he effets of treatment were known with ertainty. In this setion, we will analyze the ase where the effets of treatment are ambiguous. We onsider a model that was introdued by Eekhoudt (2002). Assume that there is no diagnosti ambiguity. he deision maker knows for sure that the patient is ill so that in the absene of treatment, the patient s health is Hs N. he effets of treatment are, however, ambiguous. An example is the situation in whih a physiian is unsure about the mortality risk of a speifi kind of surgery. Different medial studies may have reported different mortality rates and the physiian is unsure about the orret rate. Or, alternatively, the mortality rate may depend on patient harateristis, whih are unobservable for the physiian. Let Hs + and Hs denote the patient s health if treatment is suessful and not suessful, respetively. We will assume throughout that Hs < Hs N < Hs +. In words, suessful treatment is benefiial for the patient, but if treatment fails, he ends up in worse health than if he were left untreated. he deision maker is unsure about the probability that treatment will fail and believes that it an take two values, p 1 and p 2 with p 1 < p 2. he more general ase where the set of possible failure rates p onsists of more than two values will

5 L. Berger et al. / Journal of Health Eonomis 32 (2013) U s + p p 1 = p 1 p1 ( N N = p N = U H s N N p2 p2 = p2 U s 0 p 1 p p 2 1 p Fig. 6. Expeted utility as a funtion of the failure rate of treatment p. he light line ( N ) shows the expeted utility of no treatment. Beause the outome of no treatment is ertain, N is onstant. he dark line ( p ) shows the expeted utility of treatment, whih dereases as a funtion of p. For probability p, an ambiguity neutral deision maker is indifferent between treatment and no treatment. For lower failure rates he prefers treatment, for higher rates he prefers no treatment. be onsidered in Appendix A.3 where we also give a formal proof of the results presented intuitively in this setion. By we denote the deision maker s subetive belief that the probability of treatment failure is p 1 and thus 1 is his belief that the probability of treatment failure is p 2. Aording to the smooth ambiguity model, the utility of treatment is equal to: V = ϕ(p 1 U(H h ) + (1 p 1 )U(Hs + )) + (1 )ϕ(p 2 U(H h ) + (1 p 2 )U(Hs + )). (3) And the utility of no treatment is equal to: V N = ϕ(u(hs N )). (4) Assume that an ambiguity neutral deision maker is indifferent between treatment and no treatment at expeted probability p: pu(h s ) + (1 p)u(h + s ) = U(H N s ). (5) Fig. 6 illustrates the ase of an ambiguity-neutral deision maker. Let p [N p ] denote the expeted utility of treatment [no treatment] when the failure rate of treatment is p. p N is equal to U(Hs N ) and, hene, it is onstant and does not depend on p. his is illustrated by the horizontal light line in Fig. 6. Beause p N is onstant, we will simply write N from now on. he expeted utility of treatment p dereases with the failure rate of treatment and is equal to U(Hs ) when treatment fails for sure (p = 1) and to U(Hs + ) when there is no risk of treatment failure (p = 0). Fig. 7 shows the effet of ambiguity aversion (ϕ onave). Ambiguity aversion inreases the attrativeness of no treatment relative to treatment. Without loss of generality, we saled ϕ suh that ϕ( p1 ) = p1 and ϕ( p2 ) = p2. Conavity of ϕ then leads to an inrease in the benefits of no treatment, ϕ( N ) > N, while the benefits of treatment remain onstant. Beause ambiguity aversion inreases the attrativeness of no treatment, the line depiting the value of no treatment shifts upwards and the deision maker now prefers no treatment to treatment at the failure rate p (see Fig. 8). he ambiguity averse deision maker is less willing to aept treatment failure. He will now be p 2 N p 1 Fig. 7. he effets of ambiguity aversion when there is therapeuti ambiguity. he x- axis shows the expeted utilities () of treatment () and no treatment (N) for the possible probabilities of treatment failure p 1 and p 2. he y-axis shows ϕ(). Ambiguity aversion ( onave), illustrated by the dark line, inreases the outome of no treatment ( N ) relative to the outomes of treatment ( p1 and p2 ) ompared with the ase of ambiguity neutrality (the light line). indifferent between treatment and no treatment for a lower failure rate (ˆp). 5. he welfare osts of therapeuti ambiguity aversion Fig. 9 shows the welfare osts of therapeuti ambiguity aversion when all agents are homogeneous. he solid urve shows the demand for treatment under ambiguity neutrality. Like in Fig. 4, if the failure rate of treatment is suffiiently low, agents will be willing to pay a ost of for treatment. If the ost of treatment exeeds, the demand for treatment drops to zero. As we saw in Setion 4, ambiguity aversion makes deision makers less inlined to hoose treatment. Hene, the effet of ambiguity aversion is to shift the demand urve for treatment downwards: the maximum ost agents are willing to pay drops from to ĉ. here p p1 = N p p2 = N p = p N N 0 p 1 p p p 2 1 p Fig. 8. herapeuti ambiguity aversion makes no treatment more attrative and the line showing the utility of no treatment (ϕ( N )) shifts upwards. he failure rate of treatment for whih the deision maker is indifferent between treatment and no treatment shifts from p to ˆp.

6 564 L. Berger et al. / Journal of Health Eonomis 32 (2013) Fig. 9. he welfare osts of therapeuti ambiguity aversion when agents are assumed to be homogeneous. he solid urve shows the demand for treatment under ambiguity neutrality. he interrupted urve shows the demand for treatment under ambiguity aversion. Under ambiguity aversion, agents are less treatmentprone and this is refleted in the lower ost (ĉ) that they are willing to pay for treatment. he shaded area illustrates the welfare osts of ambiguity aversion when the ost of treatment equals. will be no welfare osts of ambiguity aversion if the ost of treatment exeeds (then ambiguity neutral agents would not hoose treatment either) or when it falls below ĉ (then ambiguity averse agents would also hoose treatment). However, when the ost of treatment lies between ĉ and, there will be a welfare ost equal to the shaded area in Fig. 9, whih reflets the net value ambiguity neutral agents derive from treatment. Fig. 10 shows the welfare osts of therapeuti ambiguity aversion when agents have heterogeneous ambiguity attitudes. he demand for treatment is now downward sloping for osts lower than, with first the least ambiguity averse agents willing to pay the ost of treatment. here are no welfare osts of therapeuti ambiguity aversion when the ost of treatment is above, beause then ambiguity neutral agents would not opt for treatment either. If the ost of treatment is between ĉ and, the welfare osts of treatment are equal to the shaded area in Panel A. hese osts are smaller than the welfare osts in the ase of homogeneous ambiguity aversion beause the least ambiguity averse agents will opt for treatment at this ost and, hene, they will not inur a welfare loss. On the other hand, if the ost falls below ĉ, there are no welfare osts in the homogeneous ase but in the heterogeneous ase there are, beause the most ambiguity averse agents are still not willing to hoose treatment. hese welfare osts are illustrated by the dotted area in Panel B. aversion ontribute to observed differenes in treatment pratie. Many new theories of ambiguity aversion have been proposed and the study of ambiguity is urrently a entral topi in eonomis and deision theory. In spite of this, ambiguity has largely been ignored in health eonomis and medial deision making, whih still rely on the tools of evidene-based mediine and sensitivity analysis. 3 We have shown in two medial deision problems that an inrease in ambiguity entails a ost for an ambiguity averse deision maker in the sense that he deviates from his optimal hoie in the absene of ambiguity. his is muh like an inrease in risk entails a ost for a risk averse deision maker. In the problems we onsidered, ambiguity aversion reinfores risk aversion and leads to an inrease in the propensity to treat in the ase of diagnosti risk and to a derease in the propensity to treat in the ase of therapeuti risk. Ambiguity aversion does not always reinfore risk aversion. Gollier (2011) showed that there are situations in whih ambiguity aversion and risk aversion go in opposite diretions. In our models, this did not our beause we analyzed the ase where there are two states of the world (sik versus healthy in the ase of diagnosti ambiguity and treatment suess versus treatment failure in the ase of therapeuti ambiguity). It is of interest to explore the effets of ambiguity aversion when there are more than two states of nature. he impliations of our results depend on whether ambiguity aversion is seen as rational or not. he literature is divided on this. If ambiguity aversion is rational, our analyses show the distortion in treatment reommendations when it is ignored. On the other hand, if ambiguity aversion is onsidered irrational, our analyses (and in partiular Setions 3 and 5) show the welfare loss resulting from ambiguity aversion. We assumed that ambiguity aversion ould be modeled by the smooth ambiguity model. his model, while inreasingly popular in eonomis, has reently been ritiized (Baillon et al., 2012b; Epstein, 2010). herefore, we have also studied the effets of diagnosti ambiguity and therapeuti ambiguity under other ambiguity models. 4 Our onlusions about the effets of ambiguity aversion under diagnosti and therapeuti ambiguity held in all these models. 5 o illustrate, the analysis under prospet theory (versky and Kahneman, 1992), the main desriptive alternative for the smooth ambiguity model, is presented in Appendix A.4. Our paper an be extended in several diretions. One obvious extension is to simultaneously study the effets of diagnosti ambiguity and therapeuti ambiguity. In real-world deisions these two types of ambiguity often our ointly. o model this, the utility of treatment for a speifi probability of illness as expressed by Eq. (3) has to be substituted into Eq. (1). he resulting expression is omplex and diffiult to analyze beause it involves two types of ambiguity whih go in opposite diretions. Nevertheless, our analysis provides some guidane on the overall effets of diagnosti ambiguity and therapeuti ambiguity. In general, the outome of these opposing fores will depend on three fators. he first is the degree of diagnosti and therapeuti ambiguity. Our analysis reveals that the more ambiguity there is (refleted by the differene between p 1 and p 2 in Figs. 1 and 6) the stronger the impat of ambiguity aversion is. So if, for instane, 6. Disussion In many medial deisions reliable information about the risks involved is laking. Empirial studies have shown that people dislike suh ambiguity and often reat strongly to it. An example from publi health is the overreation to the risks posed by the swine flu (a ase of diagnosti ambiguity) and to the vaine used against it (a ase of therapeuti ambiguity). It is plausible that similar overreations take plae in linial pratie and that differenes in ambiguity 3 An exeption is the reent work by Paul Han and o-authors (Han et al., 2009, 2011). 4 In partiular, we studied the impat of diagnosti ambiguity and therapeuti ambiguity under maxmin expeted utility (Gilboa and Shmeidler, 1989), -maxmin expeted utility (Eekhoudt and Jeleva, 2004; Ghirardato et al., 2004; Jaffray, 1989), ontration expeted utility (Gados et al., 2008), and prospet theory (versky and Kahneman, 1992). 5 he results of these analyses an be found at bleihrodt/results under other ambiguity models.pdf.

7 L. Berger et al. / Journal of Health Eonomis 32 (2013) A B ~ Fig. 10. he welfare osts of therapeuti ambiguity aversion when agents have heterogeneous ambiguity attitudes. he solid urve shows the demand for treatment under ambiguity neutrality. he interrupted urve shows the demand for treatment under ambiguity aversion. ĉ is the ost that homogeneously ambiguity averse agents are willing to pay for treatment. he shaded area in Panel A illustrates the welfare osts of ambiguity aversion when the ost of treatment equals. he dotted area in Panel B illustrates the welfare osts of ambiguity aversion when the ost of treatment equals. there is more diagnosti ambiguity than therapeuti ambiguity, ambiguity aversion will tend to inrease the deision maker s propensity to opt for treatment. Likewise, if therapeuti ambiguity dominates diagnosti ambiguity, the deision maker will be inlined not to treat. he seond fator that plays a role is the dispersion of the outomes of treatment and no treatment. If, for example, in the ase of therapeuti ambiguity, the negative effets of treatment failure are small, the slope of the treatment line in Fig. 6 will be relatively flat and ambiguity has little effet on the utility of treatment, and the deision maker will be more likely to opt for treatment. Finally, the deision maker might have a different attitude to diagnosti ambiguity than to therapeuti ambiguity. If the deision maker is more averse to diagnosti ambiguity than to therapeuti ambiguity, he will be more likely to treat. Conversely, if he is more averse to therapeuti ambiguity, the deision maker is more likely not to treat. Empirial evidene suggests that people have different attitudes to different soures of unertainty (Abdellaoui et al., 2011; Kilka and Weber, 2001; versky and Wakker, 1995). he smooth model does not allow for suh soure preferene as it uses one funtion ϕ whih applies to all soures of unertainty. In the literature, there are similar models to the smooth model that do allow for suh soure preferene (Chew et al., 2008; Ergin and Gul, 2009). Likewise, prospet theory an aount for soure preferene (Fox and versky, 1998). Whether soure-dependene also applies to medial deisions is a topi worthy of future researh. o summarize the above, if there is both diagnosti and therapeuti ambiguity, treatment is more likely if (a) there is more diagnosti ambiguity than therapeuti ambiguity; (b) the spread in the outomes of treatment is small relative to the spread in the outomes of no treatment; () the deision maker is more averse to diagnosti ambiguity than to therapeuti ambiguity. Another potentially fruitful area of further exploration is to study the impat of bakground soures of unertainty. It is known from the theory of deision under risk that the presene of bakground risks an substantially affet optimal behavior and has led to the important notion of prudene (Bui et al., 2005; Courbage and Rey, 2006, 2012; Kimball, 1990). It is of interest to explore whether similar effets our under ambiguity. Ambiguity is a rih field and its impliations have only been partially understood. We hope that our paper will prove useful in showing how its impat on medial deisions an be modeled and that it will pave the way for further studies of ambiguity in health eonomis. Aknowledgements Aurélien Baillon, the editor Rihard Frank, and an anonymous referee provided helpful omments. Han Bleihrodt s researh was made possible through a grant from the Netherlands Organization for Sientifi Researh (NWO). Appendix A. Extensions and proofs A.1. Diagnosti ambiguity with more than two beliefs In this appendix, we show that the onlusion that ambiguity aversion leads to an inrease in the deision maker s propensity to treat ompared with ambiguity neutrality holds for any finite set of possible probabilities of illness. Suppose that there are n probabilities in the set of possible beliefs and assume, without loss of generality, that they are suh that p 1 < < p n. Let denote the deision maker s subetive probability distribution over the probabilities of illness. Let denote the support of, the (finite) set of probabilities of illness that the deision maker onsiders possible (i.e., the probabilities p for whih (p ) > 0). Eqs. (1) and (2) then beome: V = (p )ϕ(p U(Hs ) + (1 p )U(H h )), (A1) V N = (p )ϕ(p U(Hs N ) + (1 p )U(H N h )). (A2) he deision maker deides to treat when V V N. An important advantage of the smooth ambiguity model is that it permits appliation of the mahinery of expeted utility. Hene, similar to the Arrow Pratt definition of risk aversion, we an define deision maker 2 to be more ambiguity averse than deision maker 1 if ϕ 2 /ϕ 2 ϕ 1 /ϕ 1. Result 1. Suppose that deision maker 2 is more ambiguity averse than deision maker 1. Suppose also that there is diagnosti ambiguity and that the two deision makers share the same beliefs.

8 566 L. Berger et al. / Journal of Health Eonomis 32 (2013) hen deision maker 2 is more inlined to hoose treatment than deision maker 1 in the sense that i. If deision maker 1 deides to treat a patient, deision maker 2 will also deide to treat this patient. ii. If deision maker 1 is indifferent between treating and not treating a patient, deision maker 2 will deide to treat this patient. iii. Assuming that there is a probability of illness p 1 at whih both deision makers deide not to treat and a probability of illness p 2 at whih both deision makers deide to treat, there is a probability of illness p at whih deision maker 2 deides to treat and deision maker 1 deides not to treat. Proof of Result 1. ϕ(p ) ϕ(p N) is negative for p = 0, positive for p = 1, and inreases in p. Beause U and ϕ are differentiable, they are ontinuous. By the ontinuity of U and ϕ, there exists a probability of illness ˆp suh that ϕ( Ṱ p ) = ϕ( N ˆp ) ˆϕ. Let k be an inreasing and onave funtion. By the intermediate value theorem for derivatives (Apostol, 1974, heorem 5.16), for eah p in there exists a real number between ϕ(p ) and ϕ(p N) suh that k(ϕ(p )) k(ϕ(p N)) = k ()(ϕ(p ) ϕ(p N)). For p < ˆp, ϕ(p ) < ϕ(p N) and thus, by the onavity of k, k (ϕ(p )) k (). Beause ϕ(p ) ϕ(p N) is negative for p < ˆp, it follows that k(ϕ(p )) k(ϕ(p N)) k (ϕ(p ))(ϕ( p ) ϕ(p N)). For p > ˆp, k (ϕ(p )) k () and beause ϕ(p ) ϕ(p N) is positive for p > ˆp, it follows that k(ϕ(p )) k(ϕ(p N)) k (ϕ(p ))(ϕ( p ) ϕ(p N)) for p > ˆp also. Hene, for all p, k(ϕ(p )) k(ϕ(p N)) k (ϕ(p ))(ϕ( p ) ϕ(p N)). It is also true by the onavity of k that for p < ˆp, k (ϕ( p )) k ( ˆϕ) and for p > ˆp, k (ϕ( p )) k ( ˆϕ). Hene, for all p, k(ϕ(p )) k(ϕ(p N)) k ( ˆϕ)(ϕ(p ) ϕ(p N)). Consequently, (p )(k(ϕ(p )) k(ϕ(p N ))) k ( ˆϕ) (p )(ϕ(p ) ϕ(p N )). (A3) Beause k is inreasing, k > 0 and it follows from (A1) that if (p i)(ϕ(p ) ϕ(p N )) is positive then (p )(k(ϕ(p )) k(ϕ(p N )) is also positive. Consequently, if a deision maker whose ambiguity aversion is aptured by ϕ hooses treatment (i.e. (p )(ϕ(p ) ϕ(p N )) is positive), a more ambiguity averse deision maker whose ambiguity aversion is aptured by an inreasing and onave transformation of ϕ will also hoose treatment. his proves Statements i and ii. Statement iii follows from the ontinuity of U. A.2. Diagnosti ambiguity with ontinuous treatment In Setion 2, the deision maker ould only hoose between treatment and no treatment. his ase is not always realisti as in many real-life situations a deision maker an also hoose the intensity of treatment. For example, a dotor not only deides on whether to presribe mediation, but also on the appropriate dose. hen the hoie variable is ontinuous instead of dihotomous. We will now show that our onlusion that an inrease in ambiguity aversion generates an inrease in the propensity to treat is unaffeted when the deision maker an selet the intensity of treatment. he proof that an inrease in ambiguity aversion redues the propensity to treat in the ase of therapeuti ambiguity aversion is largely similar and we will not present it separately. he treatment variable t is ontinuous and bounded. As before, there are two states of the world, sik and healthy. he levels of health reahed in the two states depend on the seleted intensity of treatment and we denote them as H s (t) and H h (t), respetively. We assume that treatment is benefiial when the patient is sik, H s 0, and detrimental when the patient is healthy, H h 0. However, for any treatment intensity, the patient is always in better health when healthy than when sik: for all t, H s (t) H h (t). We assume that the marginal effets of treatment are dereasing both when healthy and when sik: H s 0 and H h 0. As in Appendix A.1, the number of probabilities of illness that the deision maker onsiders possible an be any finite number and his beliefs regarding the probability of illness are aptured through the funtion. he deision problem now beomes: max t (p )ϕ(p U(H s (t)) + (1 p )U(H h (t))). (A4) Result 2. Suppose that deision maker 2 is more ambiguity averse than deision maker 1. Suppose also that there is diagnosti ambiguity and the two deision makers share the same beliefs. 6 hen the optimal level of treatment hosen by deision maker 2 will be at least as large as the optimal level of treatment hosen by deision maker 1. Proof of Result 2. Denote p t = p U(Hs t) + (1 p )U(H t h ). he first order ondition an be written as 7 : (p )ϕ (p t )[p U (H s (t)) H s (t) + (1 p )U (H h (t)) H h (t)] = (p )ϕ (p t ) t p = 0 (A5) t p an be written as U (H h (t)) H h (t) + p [U (H s (t)) H s (t) U (H h (t)) H h (t)]. (A6) Beause H s (t) 0 and H h (t) 0, it follows that t p is inreasing in p. Beause U > 0 and H s (t) 0 it follows that t 1 0. Beause U > 0 and H h (t) 0, t 0 0. On the other hand, p t is dereasing in p by the assumption that for all treatment levels t, Hs t H t h. Beause ϕ > 0, it follows from (A5), t p inreasing in p, and t 1 0 t 0 that for eah t there must exist a ˆp in suh that t p 0 for those p for whih p < ˆp and t p > 0 for those p for whih p ˆp. ˆp will depend on t, but to keep the notation manageable, we will suppress this dependene in what follows. Beause p t is dereasing in p, it must be true for eah treatment level t that for all p, t p t p ṱ p t p. (A7) If p < ˆp, then p t > ṱ p and (A7) follows from t p 0. If p > ˆp, then p t < ṱ p and (A7) follows from t p > 0. Let ϕ 2 = k(ϕ 1 ) with k inreasing (k > 0) and onave (k < 0) so that deision maker 2 is more ambiguity averse than deision maker 1. Let t 1 be the optimal treatment level of deision maker 1. Beause ϕ 1 is inreasing, it follows from (A7) that for all p, ϕ 1 ( t 1 p ) t 1 p ϕ 1 ( t 1 ˆp ) t 1 p. Beause k is onave, for all p, k (ϕ 1 ( t 1 p )) t 1 p k (ϕ 1 ( t 1 ˆp )) t 1 p. (A8) 6 One extension would be to drop the requirement that the two deision makers share the same beliefs. Baillon et al. (2012b) provide tools to analyze this ase. 7 Beause the marginal effets of treatment are dereasing and U and ϕ are onave, (A4) is onave in t and, hene, the optimal level of treatment resulting from the maximization of (A4) is unique.

9 L. Berger et al. / Journal of Health Eonomis 32 (2013) If p < ˆp then t 1 p > t 1 ˆp, hene, ϕ 1 ( t 1 p ) > ϕ 1 ( t 1 ˆp ), hene k (ϕ 1 ( t 1 p )) < k (ϕ 1 ( t 1 ˆp )) and (A8) follows from t 1 p 0. If p > ˆp then t 1 p < t 1 ˆp, hene, ϕ 1 ( t 1 p ) < ϕ 1 ( t 1 ˆp ), hene k (ϕ 1 ( t 1 p )) > k (ϕ 1 ( t 1 ˆp )) and (A8) follows from t 1 p > 0. Beause ϕ 2 = k (ϕ 1 )ϕ 1 > 0, multiplying both sides of (A8) by (p )ϕ 1 (t 1 p ) and summing over all p yields (p ) ϕ 2 (t 1 p ) t 1 p k (ϕ 1 ( t 1 ˆp )) (p ) ϕ 1 (t 1 p ) t 1 p. (A9) Beause (p ) ϕ 1 (t 1 p ) t 1 p = 0 by the first order ondition, it follows that (p ) ϕ 2 (t 1 p ) t 1 p 0. (A10) Hene, at t 1 the marginal benefits of treatment are positive for deision maker 2 and thus he will inrease the level of treatment. hus t 2 t 1, whih is the desired result. A.3. herapeuti ambiguity with more than two beliefs In this appendix we show that the onlusion that therapeuti ambiguity aversion makes the deision maker less prone to hoose treatment holds for any (finite) set of beliefs = {p 1,...,p n }. We assume without loss of generality that p 1 < p 2 < < p n. Result 3. Suppose that deision maker 2 is more ambiguity averse than deision maker 1. Suppose also that there is therapeuti ambiguity and that the two deision makers share the same beliefs. hen deision maker 2 is less inlined to hoose treatment than deision maker 1 in the sense that i. If deision maker 1 deides not to treat a patient, deision maker 2 will not treat this patient either. ii. If deision maker 1 is indifferent between treating and not treating a patient, deision maker 2 will deide not to treat this patient. iii. Assuming that there is a failure rate p 1 at whih both deision makers deide to treat and a failure rate p 2 at whih both deision makers deide not to treat, there is a failure rate p at whih deision maker 1 deides to treat and deision maker 2 deides not to treat. Proof of Result 3. he proof of Result 3 is similar to that of Result 1. reatment will be hosen if (p ) ϕ(p ) ϕ(u(hs N )). ϕ(p ) ϕ(u(hs N )) is negative for p = 0, positive for p = 1, and inreases in p. By the ontinuity of U, there exists a failure rate ˆp suh that ϕ( Ṱ p ) = ϕ(u(hs N )) ˆϕ. Let k be an inreasing and onave funtion. By the intermediate value theorem, for eah p in there exists a real number between ϕ(p ) and ϕ(u(hs N )) suh that k(ϕ(p )) k(ϕ(u(hs N ))) = k ()(ϕ(p ) ϕ(u(hs N ))). For p < ˆp, ϕ(p ) > ϕ(u(hs N )) and thus, by the onavity of k, k (ϕ(p )) k (). Beause ϕ(p ) ϕ(u(hs N )) is positive for p < ˆp, it follows that k(ϕ(p )) k(ϕ(u(hs N ))) k (ϕ(p ))(ϕ( p ) ϕ(u(hs N ))). For p > ˆp, k (ϕ(p )) k () and beause ϕ(p ) ϕ(u(hs N )) is negative for p > ˆp, it follows that k(ϕ(p )) k(ϕ(u(hs N ))) k (ϕ(p ))(ϕ( p ) ϕ(u(hs N ))) for p > ˆp also. Hene, for all p, k(ϕ(p )) k(ϕ(u(hs N ))) k (ϕ(p ))(ϕ( p ) ϕ(u(hs N ))). It is also true, by the onavity of k, that for p < ˆp, k (ϕ(p )) k ( ˆϕ) and for p > ˆp, k (ϕ(p )) k ( ˆϕ). Hene, for all p, k(ϕ(p )) k(ϕ(u(hs N ))) k ( ˆϕ)(ϕ(p ) ϕ(u(hs N ))). Consequently, (p )(k(ϕ(p )) k(ϕ(u(hs N ))) k ( ˆϕ) (p ))(ϕ(p )) ϕ(u(hs N ))). (A11) Beause k is stritly inreasing k > 0 and parts i and ii follow from (A11). Beause U is differentiable, it is ontinuous and Statement iii follows. A.4. Results under prospet theory In this appendix, we will show that the onlusions derived under the smooth model also hold under prospet theory. Start with the ase of diagnosti ambiguity. Let E( p) denote the expeted value of the probabilities of illness that the deision maker onsiders possible (i.e., the expetation of the p in ). Under prospet theory, treatment and no treatment are evaluated as 8 V = W()U(H s ) + (1 W())U(H h ), (A12) V N = W()U(Hs N ) + (1 W())U(H N h ). (A13) W is an event-weighting funtion that takes values between 0 and 1. Let m(e( p)) denote the unambiguous probability suh that W()U(Hs ) + (1 W())U(H h ) = m(e( p))u(hs ) + (1 m(e( p)))u(h h ).9 he deision maker is ambiguity neutral if m(e( p)) = E( p) for all E( p), whih implies that m is linear. Ambiguity aversion orresponds with m onave (Baillon et al., 2012a). Conavity of m means that the deision maker overweights the higher probabilities of illness, i.e., he behaves in a pessimisti manner. Extreme ambiguity aversion means that he only onsiders p n the highest probability of illness. Fig. A1 illustrates the impat of diagnosti ambiguity aversion under prospet theory. Panel A shows that under ambiguity neutrality, P p [PN p ] the prospet theory value of treatment [no treatment] when E( p) = p is linear in p. he slope of P N p is steeper than the slope of P p beause the spread of the outomes of no treatment exeeds the spread of the outomes of treatment. he deision maker is indifferent between treatment and no treatment for E( p) = p. Panel B shows that under ambiguity aversion (m onave), the relationship between the prospet theory value and E( p) beomes onvex. his makes treatment more attrative and the indifferene value falls from p to ˆp. Intuitively, ambiguity aversion has the effet of putting extra weight on the higher probabilities of illness. his favors the treatment option and, thus, generates an inrease in the propensity to treat. he ase of therapeuti ambiguity is similar. We have V = W()U(Hs ) + (1 W())U(Hs + ) = m(e( p))u(hs )) + (1 m(e( p)))u(hs + ), (A14) V N = U(Hs N ). (A15) 8 Often the prospet theory formulas are presented in a dual way with the weight W() applied to the best outome. his formulation is equivalent to the one we use here but it is more umbersome in terms of the notation used in this paper. 9 Prospet theory also assumes that the deision maker weights unambiguous probabilities to reflet his attitudes towards risk. We will, without loss of generality, abstrat from this to study the pure effet of ambiguity aversion.

10 568 L. Berger et al. / Journal of Health Eonomis 32 (2013) P P p = P p^n P p P p N 0 p^ p E p ~ 1 Fig. A1. Panel A shows that under diagnosti ambiguity neutrality, the prospet theory value of no treatment (P N p ) and of treatment (P p ) dereases linearly with the expeted probability of illness E( p). For probability p, the ambiguity-neutral deision maker is indifferent between treatment and no treatment. Panel B shows that under ambiguity aversion, P N p and P p beome onvex funtions of E( p) and the indifferene probability shifts to ˆp, implying a higher propensity to treat. U s P U s + P N = P p P N = U s N 0 p p E p ~ 1 P p Fig. A2. Ambiguity aversion redues the propensity to treat in the ase of therapeuti ambiguity. 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