SOCIO-Econ. Pkmn. Sci. Vol. 24. No. 4. pp. 285-294. 1990 Pnnted I” Great Bntarn. All rights reserved

0038-0121 90 53.00 + 0.00 CopyrIght 5 I990 Pcrgamon Press plc

Resource Allocation, Equity and Public

Risk: Dying One at a Time vs Dying all Together

ABRAHAM MEHREZ Department of Industrial Engineering and Management. Ben Gurion University of the Negev,

Beer Sheva. Israel

and

AMIRAM GAFNIt Centre for Health Economics and Policy Analysis, Department of Clinical Epidemiology & Biostaristics.

McMaster University, Hamilton. Ontario, Canada LSN 325

Abstract-This paper focuses on the evaluation, from an individual and societal perspective. of risk in terms of possible loss of life due lo an exposure to two different types of events over a period of lime. The two types are: risk of death from a catastrophic event (a sudden death of many people in a disaster at a yet unknown point in time) expecled to occur during a planning period. or risk of death from another event (e.g. disease, road accident, etc) which claims fewer lives each year, but for which the expected total number of deaths over the planning period is equal lo the expected number of deaths from the catastrophic event. Our analysis considers the extreme case in which these two lypes of events have the same probabilities of death every year and the same expected number of fatalities over the planning period. The individual’s decision problem is described using a von-Neumann Morgenstcrn (vNM) utility function. The model suggests that the choice between these types of events depends on the value of the following variables: the probability of death over Ihe planning period, the length of Ihe planning period, the individual’s time preference pattern. and the utility of being in diNerent anxiety staks. Stochasric extensions that may direct the public decision making process (involving aggregated preferences) are discussed. We also discuss issues of implementation.

INTRODUCTION

During their lifetimes, members of society are exposed to various risks to their lives. Almost every activity in which an individual participates, by choice or by external conditions, results in some chance that the individual will lose his/her life. Many of these activities or projects are readily accepted by individuals and the public in general because it is felt that the advantages outweigh the disadvantages, one of which is the risk of death (e.g. recreational activities such as skiing, swimming, rock climbing). In many other cases, individuals are looking for an intervention from the government which will result in the reduction of the risk of death (e.g. actions to deal with road accidents, actions to deal with potential natural disasters such as floods, earthquakes, etc.). The decision problem is to determine how to best allocate resources in order to lower the risk of death to individuals from exposure to different types of events. Such decisions, if taken from a societal perspective, involve concentration on both efficiency (the use of limited resources in a manner that will provide maximal output) and equity dimensions.

In this paper, we provide a framework to analyze an individual’s (using a prescriptive behavioral model) and society’s attitudes (using a majority rule) toward two different types of risky events. We assume that over a given period of time (the planning period) an individual is exposed to risk of death from either a catastrophic event (which results in the sudden death of many people at an unknown point in time) or another event (disease, road accident, etc.) which claims fewer lives each year, but for which the expected number of deaths over the planning period equals the expected death toll from the catastrophic event. Our analysis thus relates to the comparison of two

tTo whom all reprint requests should be addressed.

SEPs 214-o 285

286 ABRAHAM MEHREZ and AMIRAM GAFNI

types of events involving the same probabilities of death each year and the same expected number of fatalities over the planning period. However. these two types of events will manifest themselves differently-in one case. many people wilt die at once at a yet unknown point in time. while in the other case the same number of people will die through an accumulative process. These two types of risky events can be labelled as catastrophic and chronic risks, respectiveIy.

The individual’s decision problem is to rank the two events according to his/her preferences, assuming, for the sake of simplicity, that these are two mutually exclusive types of events. This assumption holds if individuals are willing to move from one place where they are exposed to one type of event to another place where they are exposed to the other type. For a decision maker in places where exposure to both types of events exists (e.g. in large urban areas), this assumption does not hold. The decision problem is thus how to best allocate resources in order to lower the risk of death from exposure to one type of event or the other.

The question of equity and public risk in a context of utility functions has already been addressed by Keeney (26,271. Keeney analyzes the problem of ranking projects that involve the same expected number of fatalities but which generate different probability distributions over the number of fatalities. In doing so, he develops the concept of an equitable distribution of risk. Hammerton rf ul. @OJ report empiricat results suggesting that the majority of individuals display “catastrophe aversion” rather than “equity proneness” for the kind of alternatives described in their study. Broome [8] claims that van-Neumann Morgenstern (VNM j utility functions are incompatible with valuing fairness and lead to an invalid conclusion about risk proneness. Further attempts to examine the ~ippropriat~ness of using utility functions to represent preferences regarding equity can be found in Fishburn [ 141, Sarin [33], Harvey [21], and Fishburn and Straffin [15].

An additional conceptualization of equity and public risk discussed in the literature involves the notion of cx-post risk equity (the equity associated with the fatalities that actually occur) vs ex-ante risk equity (the equity of the process and risks which eventually lead to the fatalities). A detailed discussion of these two concepts and their implications for public policy can be found in Pauly and Willet [32]. More recent papers by Kecney and Winkler [28] and Sarin [33] and Sounderpandian [37] pursue the notion of modelling equity in a utility framework, distinguishing between ex-post and ex-ante equity, and analyzing problems in terms of decision strategies.

In this paper, we use the utility approach to help a public decision-maker investigate the implications of different value judgments in examining the alternatives described earlier. Our problem ditTers from the one dealt with in the literature in the following way. The papers noted above deal with the equity dimension following the case where individuals are exposed to different levels of risk, Herein, we deal with the case where the differences between the various types of exposure are not considered in the risk of death to individuals but rather in the nature of the exposure (catastrophic vs chronic risk).

The use of vNM utility functions might seem inappropriate in our case as the situations being compared are a compound lottery (the catastrophic event) and a simple lottery with both having the same (equal) annual probability of death and expected number of fatalities. One of the basic axioms of vNM utility theory regarding the reduction of compound lotteries (lotteries composed of several stages) into simple lotteries [29, p 231, suggests that the decision-maker should be indi~erent between these two cases. We, however, claim that in a real-life situation, individuals are not necessarily indifferent between these two cases. The question thus becomes: Is this another example in which individuals do not behave according to the utility axioms?

Bell and Farquhar [6] have classified the various attempts to explain this discrepancy into three scenarios:

1. Decision-makers who violate the axioms are being fooled by cognitive illusions, but unlike visual illusions, there is no way to demonstate, convincingly, that the preference kinds really are illusions.

2. Decision analysts have not considered all relevant dimensions of the risk-taking problem, Unless all features are accounted for, there is no way to reflect accurately a decision-maker’s preferences.

3. Decision analysts need to develop more flexible theories to accommodate actual decision behavior.

Resource allocation. equity and public risk 287

In the current study. different levels of anxiety for an individual resulting from exposure to simple

and compound lotteries are explicitly incorporated into the analysis. We show that this can explain why those individuals may not be indifferent to the same level of risk to life if it stems from different sources. Our approach to explaining the discrepancy. noted above. thus falls into the second category/scenario described by Bell and Farquhar [6]. Anxiety is clearly an important element in an individual’s decision-making process. In particular. it enables us to explain, in the context of a vNM utility function, potential differences in preferences across the alternatives. (The importance of anxiety (or fear of its unpleasantness) in the decision-making process as it relates to the death question is suggested in a paper by Shubik [36].)

Following the school of thought which claims that social decision-making criteria should be fundamentally individualistic and utilitarian. and thus reflective of the interests and preferences of individual members of society. the current analysis begins from the point of view of a representative individual. A sensitivity analysis is then performed to relate the parameters of the individual’s problem to his/her preferences. Following this analysis, stochastic extensions that may direct the public decision-making process by aggregating individuals’ preferences are presented. Finally. we discuss issues of implementation and use of the proposed model for public decision-making.

THE INDIVIDUAL’S UTILITY MODEL

Assume a rcprcsentative individual who is a vNM expected utility maximizer and who has to rank two mutually exclusive types of events. The first one is exposure to the risk of death from a catastrophic event (cvcnt A); the second is cxposurc to the chronic risk of death from another source (cvcnt B). The model dcscribcd rcfcrs only to possible death due to cxposurc to cvcnts A or B. and ignores all other causes of death. Wc assume that the ranking of the two events is done at the beginning of a planning period and that it cannot bc revised during the period. The length of the planning period (horizon) is assumcd to equal the cycle of the catastrophic event (i.e. a flood every IO yr). Although we first assume that this cycle is known. we later relax this assumption. In addition, without loss of gcncrality, WC assume that event B occurs at a finite number of points in time. equally spaced, during the cycle (dcnotcd as I. I being an integer).

We use the following notations, definitions and assumptions:

T = length of the planning horizon. We assume that there arc at least two periods in the planning horizon (i.e. T >, 2).

t = a point in time during the planning horizon (1 = I,. . . T),

P = ;L uniformly-distributed, random variable describing the time of occurrence of the catastrophic event type A. Specifically, P( P = I) = I/T for I = I,. . . , T. To simplify the mathematical treatment, WC will restrict ourselves to the case where the catastrophic event occurs only at discrete. equally spaced, points in time,

Z = a random variable equal to I if the individual dies as a result of being exposed to event type

A. and equal to 0 otherwise. We denote P(Z = I) = P (the probability of death caused by event A during the planning horizon),

Q = a random variable equal to 1 if the individual dies as a result of being exposed to event type B each period, and equal to 0 otherwise. We denote P(Q = 1) = P. For the sake of simplicity, we denote Q as the event of Q = I and 0 as the complementary event, Q = 0,

Z, = a random variable equal to I if the individual dies at time I as a result of being exposed to event type B, and equal to 0 otherwise.

We assume that B occurs at each point of time, r(t = 1,. . . , T). Furthermore we assume that:

and

P(Q =Z, = I)= P(Q = I)P(Z, = IlQ = I)= P/T,

= P[(T - I)/T][I/(T - I)] = P/T.

288 ABRAHAM MEHREZ and AWRAM GAFSI

This can be easily extended to the general case wherein the individual faces the same level of risk of death from event B at each point in time given that he survives to that point in time. Note that this probability is also equal to the probability of death from a catastrophic event at each point in time (again. given that the individual survives to that point in time).

Because of the risk of death stemming from exposure to either A or B during the planning period, the individual is in one of the following “psychological states”. If the individual is exposed to A, he/she may be in either a state of anxiety before the catastrophic event occurs (denoted as Sl) or a state of relief after the event occurs (denoted as S3). If the individual is exposed to B, he/she suffers from a constant level of anxiety during the entire planning period (denoted as S2). To simplify the model, we assume time-independent states “before-after” the events. For example, a study by Barnett and Lofaso [3] indicates passenger responses to an aircraft crash might depend on the time that has passed since the accident.

Denote by U a vNhI utility function over the different possible “psychological states”. We then assume that the following relationships exist: U(SI) < U(S2) < U(S3). Also, as becomes obvious later in this paper, the choice between A and B for the case where U(S2) 6 U(SI) < U(S3) is a trivial one. Since vNM utility functions are unique up to a linear monotonic transformation, we assume that U(S1) = 0 and U(S3) = 1. U( *) can be determined using classical methods for measuring cardinal preferences (see, for example, Keeney and Raiffa [25] and Farquhar [12]).

Define EA(U) and EB(U) as the inidividual’s expected utility from being exposed to A and B, respectively, measured at the beginning of the planning period. We assume E(U) to be additively separable over time. (For a similar formulation, see Yaari [39] and Barro and Friedman [2]. In a health context, see Gafni and Peled [ 171 and Mehrcz and Gafni [3 I].) In order to compare the two types of events, we calculate the difference between the individual’s expected utility in each case:

EA(U)-ER(U)= P(Z =O)E’(UlZ =0)+ P(Z = I)EA(UjZ = I)

- N?)EAW’l@ - f’(QEB(~IQ). (1)

Following our definitions and the assumption made about the equality of the probability ofdeath due to exposure to either A or B, we can rewrite (1) as:

EA(II)-EB(I/)=(1-P)[EA(U]Z=0)-E”(I/f~)]+P~EA(II~Z=l)-EB(LI~Q)]. (2)

Denote the first expression on the right-hand side of eqn (2) as A’ and the second as A’. A’ can be further rewritten as:

A’=(1 -P) [

i EA(UIZ=O, P=f)P(P=f)-P(U@) . (3) I-l 1

Based on the relationships assumed between the different ‘“psychological states”, we can rewrite (3) as:

A’=(1 -P) i i ((l+r)-‘j/T [ ( >

- i (1 +r)-‘U(S2) t-1 I-l+1 1-I 1 , (4)

where r is the individual’s discount rate, assumed to be constant, which takes into account his/her time-preference pattern. (The concept of time preference relates to the timing of an outcome.~

In general. individuals can exhibit three patterns of time preference: a positive rate of time preference (r > 0), no time preference (r = 0), and a negative rate of time preference (r < 0). Based on evidence from the health care literature (e.g. Fuchs 1161). and the common assumption that people prefer their health benefits sooner than later (e.g. Shepard and Thompson (351, Drummond ef al. [IO]), we assume a positive rate of time preference (r > 0). (Further discussion of time preference and its application to “goods” such as years of life can be found in Gafni and Torrance

[181.)

Let L(1) = i (1 + ~)-j for t =l,..., T-l, and L(T)=O.

This expression describes the present value, at point t, of the remaining years of life, until the end of the planning period. For the case of positive time preference (r > 0), L(t) is a monotonically

Resource allocation. equity and public risk 289

decreasing function. Thus, the following relationship exists:

E[L(P) - L(l)] < 0. (5a)

From eqns (4) and (5a) it is easy to see that if U(S2) is sufficiently close to one (that is, if the fear of situation B does not cause significant anxiety), then A’ < 0. This implies that the representative individual, if faced with the decision to be exposed to event A or B, prefers to face B rather than A. The genera1 relation is:

A’>0 if E[L(~)]>~(SZ)[L(l)+(l+r)-‘1. (5b)

Thus. A is preferred to B if the expected utility from relaxation following the catastrophic event is sufficiently large to justify the anxiety before it occurs, compared to the expected utility of being constantly under a lower level of anxiety stemming from the risk to life due to exposure to B. Note that in both cases, the individual is alive at the end of the planning period.

AZ, which evaluates the difference between the conditional expected utilities for realizations resulting in the individual’s death, can be rewritten as:

A’= -PI/(SZ) i i (1 +r)-‘,‘r [ 1 . (6) 1-I f-1 It is clear that A’ < 0. Thus. if the individual knows for certain that he/she wili die (from A or B).

then event B is preferred to event A. This is due to our assumption that at each point in time the probabilities of death are equal, but that the utility from being in a state of anxiety before a catastrophic cvcnt is lower than that from being anxious due to exposure to B. Changing the order of summation in (6). and using the definition of L(p), we can rewrite equation (6) as:

A’= -~~~S2)[-~(~~~))+~(1)+(~ +t) ‘1. f7)

From expressions (2). (5) and (7) WC can thus rewrite (I) as (8):

E”(u)-E”(LI)=A=A’fA’=(I -P)[E(L(P))-(/(S2)(L(l)+(l -t-r)-‘)]

-P[LI(SZ)(-E(L(F))+f.(l)+(l +r)-‘)I. (8)

SENSITIVITY ANALYSIS

The model described in this paper suggests that the choice between the two events (A and B) depends on the value of the following variables: the probability of death over the planning period (horizon) due to exposure to one of the events, P; the length of the planning period T; the individual’s time preference pattern, r; and the utility of being in a constant state of anxiety due to exposure to event 8, Li(S2). In this section, we examine the effects of changes in these parameters on the relative ranking of events A and B.

We start by analyzing the effect of changes in the probability of death, P, over the planning period from exposure to A or B, on the relative ranking of A and B.

L~rtrr~~ 1-A necessary condition for A to be preferred to B is that:

E(t( P,,- ~(s2)(~(l)+{i +r)_‘)>O.

Proof-Since A? < 0, it implies that, for A to be positive, A’ must be positive. From eqn (Sb), it follows that A’ > 0 if and only if the above-mentioned condition is met. Lenrnzu Z--sAjctP = - E(L( I’))( I - U(SZ)) < 0. Proof-This follows directly from expression (8). Using Lemmas I and 2, one can prove the following proposition:

Proposition f-Above a cutoff level of P, denoted as P* (0 < P* z$ I), B is always preferred to A. Proof-If the condition stated in Lemma 1 does not hold, B is always preferred to A and thus P* = 0. If the condition holds. then, following Lemma 2, .?A/aP < 0. In addition, given eqn (8)

290 ABRAHAM MEHREZ and AMIRAM C&w

and the fact that f.( 9) is a monot~~i~aliy decreasing function. we observe that P = I impIies that AC 0.

Note that the results of Proposition I are ~ntu~tj~ely appealing. Thus, the higher the probability of death from a catastrophic event. the Lower the probability of enjoying the relaxation period following that event. Furthermore, from Proposition 1 it follows that A does not dominate B. In particular, A is not afways preferred to B since, for P sufficiently close to 1, 3 is preferred to A.

Also, there is a cutoff level. P*. above which B is preferred to A. Under certain conditions (see Lemma I) it cm thus be shown that B dominates A. An important result, FoIlowin~ from Proposition I, is that a monotonicatiy decreasing relationship exists between A and P.

We now examine the effect of changes in length of the planning horizon (the number of periods, T) on the relative ranking of A and B. From eqn (8). we calculate the effect of changes in T as follows;

ZAjW = [(I - P) -F- ~(S2)~~~~~~(~)~~~~ - ~(S2)~~(l)~~~ (9)

In general, a monotonic r~~atjonsb~~ between A and T does not exist. However, we can derive four propositions about the relationship between A and T that do exist.

~~~~~~~~~ff~ 2--P, > I exists such that for every P. P, < P G I, the fottowing relationship holds:

Pmof--For P = t. eqn (9) becomes:

?A/?T = -Lr(S2)fifL(i)/r’T-(?E(L(~))ldT]. (10)

it is easy to show that ~~~~(~)~~~T < JL(f)/dT. The claim made by the proposition is thus derived from A being a continuous function of P.

An increase in the p~~~nnj~~ horizon thus makes it more likely for B to be preferred to A, given that P is greater than a certain value,

~~u~~.sj~j~~~ f-For P = 0, fl*(S2) exists such thot for every U(S2):

0 6 U(S2) < U*(S2), i?AjZT > 0. (11)

Proof-For P = 0, eqn (9) becomes:

c?&t’T = t,7EfL( ~)],GT - ~~S~~~~~~~/~T~ (121

It can be shown that ZE[li( P)]/ST > 0 and 4L(l))2T r 0. Since JA/dT at P = 0 is a linear dmxasing function of U(Z) and negativety valued for U(S2) = f, the proposition follows,

~~ff~u~~i~~~ 4---dA,QT is a decreasing function of B. Proof-This follows directly from Lemma 2 and eqn (9). P~~~~~~r~~n S-Z&/U changes signs at most once as a fu~~tjon of P. ff this occurs, the change is from positive to negative. Proof-aAjt7T is a linear decreasing function of P. From Proposition 2, when P = 1 then adfaT c 0. From Proposition 3, when P = 0 if U(S2) G U*(S2). then JA/iJT r 0 and there is one change of sign from positive to negative. if U(S2) > ti*(SZ), then ZA/aT -C 0. Further, CtA/ZT is a linear function of P. It follows that there is no change of sign in ~A/~T.

Examining the tradeoff between T and P on an indifference curve reveals the following: the dope of such a curve is evaluated by

c?T 8 A/G?P -= t’P -zJ$

From Lemma 2, the sign of ZA,‘SP is negative. Thus, ZT/L?iP is positive if and only if aAJr?T k positive. Since, by Proposition 5. this sign is not uniquely determined, the sign of dT/SP depends on the value of P. For P su~ciently large. the sign wilt be negative. Thus, for such P, increasing the probabitity of the catastrophic event requires a decrease in the life cycle of event A in order to maintain A at the same level of utility.

Rrsource allocation, equity and public risk 191

We now examine the tradeoff between P and z/‘(SZ) using an indifference curve. The slope of such a curve is evaluated by

SV(S2) S AISP -3 - SP 2 A/i:U(SZ) ’

From Lemma 2, the sign of SAISP is negative. From expression (8) it is easy to see that the sign of (?A/./dU(SZ) is negative. The sign of ~~(SZ)~~P is therefore always negative. Thus, an increase in the “objective” risk of death should result in an increase in the anxiety level (S2). (Note that an increase in the anxiety level decreases the utility of that psychological state.)

A parameter that can also affect the relative ranking of A and B is r, the individual’s discount rate (r represents the individ~al’s time preference pattern.) Recall that we have assumed a positive rate of time preferences (r > 0) in the current study.

We do not present a formal sensitively analysis for r since it is very similar to the analysis for T. Furthermore, since both T and r affect only t(P), the effect of changes in r on the relative ranking of A and B depends on the specific corresponding values of P and li(S2).

Finally, we examine the effect of a decrease in the anxiety Ievef due to exposure to B (as measured by fili(S2)]) on the relative ranking of the two events. From eqo (8) it is easy to see that ~A/~~~S2) < 0. Thus. a decrease in the anxiety level due to exposure to B [which implies an increase in U(S2)J makes A less attractive retative to B. (Note that this coincides with intuition.)

THE STOCHASTIC CASE

As discussed above, an individual’s ranking of situations A and B depends on the values of the parameters P, T, r, and LI(S2). Up to this point, we have assumed that these factors are known with certainty.

In reality, however, for a given P and T, individual preferences, calculated using U(S2) and T, may vary from one individual to another. Thus, a public decision-maker, who believes that social d~ision-eking criteria should reflect the interests and preferences of individual members of society, would wish to incorporate these differing preferences into his/her analysis. This requires an examination of the pubficchoice problem in a stochastic setting. From the impossibility theorem (see Arrow [I]), we know that a general normative welfare function to deal with this problem does not exist. Any aggregation across a population of N individuals calls for some rather strong assumptions (see, for example, Keeney and Raiffa [25]). H owever, for purposes of exposition we suggest a natural measure that incorporates individuals* choices using a majority rule within the framework of a social analysis, This measure computes the percentage of individuals who prefer A to Il.

In the subsequent analysis we describe a procedure for computing the percentage of individuals who prefer A to B in a general stochastic setting. More precisely, we assume a more realistic situation where P. T, r, and V(S2) are random variables denoted by I’, p, ?, and 0@2), respectively.

Let P, T, r and ti(S2) be possibte realization values of i3, F, r’ and i?[S2) [where: 0 6 P 6 1, T is integer, r 2 0, and 0 < UfS2) < 11. Note that J and o(S2) depend on the nature of the preference pattern of the different individuals while P and i” depend on the nature of the phenomena (situations) compared.

The percentage of individuals who prefer A to B is evaluated by P&S,) where:

S, = (fP, T, r, U(SZ)]l A[P, T, rr tr(S2)] 2 0) 11%

S, is the event that corresponds to all realizations of the individuals’ problem for which A is preferred to B or A 2 0. The computation of P&S,) involves solution of a stochastic problem which partitions the set of realizations into two subsets: S,, and its complementary set. Alternatively, P(S,) is equal to:

PG) = E f [P. t, f. &!m~ (14)

292 ABRAHAM MEHREZ and Awk0i GA~I

where the expected value is taken with respect to an indicator random variable defined as:

The distribution of the indicator random variable is evaluated via the joint distribution of P, T, I and U(S2). To facilitate the computations, we note that f is a discrete random variable. Without loss of generality, the remaining random variables may be assumed continuous. We now proceed to evaluate P(S,J.

For each T, PfS, 1 iz = T) can be computed using a conditional expectation argument as follows:

Expression (15) can be rewritten as:

wheref(brfi”= T) denotes the conditional density function of 7 given is [f(i = r 1 i?; = 7’) is assumed to exist.]

The computation of P(S,) as described by (16) can be further simplified in light of the relationships between P(i”) and o(S2) for given realizations of F and E Via Proposition 3, given T and r. for each P there exists iJ*(P, T, r) such that for all U(S2) < U*(P, T, r) A is preferred to B. Furthermore, by the sign of ~~(SZ~/~~ on the indifference curve between A and B, there exists P* such that for every P > P*, event B will be preferred to event A for every value of o(S2). Combining these two arguments leads to the second computational step of P(S,,) which depends on the foilowing refationship:

S(P = P, 0(S2) = U(S2)I F = T, i = r) dU(S2) 1 dp, (17)

whereJ’[p = P, 0(S2) = U(S2)t F = T, i = I] is the ~ond~t~ona~ density function of P and o(S2)

given realizations of 7 and ?. (f[P = P, o(S2) = tr(SZ)l? = T, 7 = r] is assumed to exist.) We note that (16) and (17) provide a computationa procedure by which the majority rule can

be dete~ined. Although the analysis might not be easily implemented, it can serve as a guide for public decision-makers. In this regard, an important problem is the ailocation of resources in reducing exposure to both situations, A and B. To be sure, the percentage of individuals who prefer one alternative over the other can be taken into account in determining the level of resources allocated to each alternative.

SUMMARY AND CONCLUSIONS

This paper has focused on the evaluation, from an individual and societal perspective, of the risk of loss of life from the exposure to two different events during a given period of time. One event is catastrophic and results in the sudden death of many people; the other claims fewer lives every year, but the deaths accumulate to a large number over the period. In spite of the lack of scientific evidence to support our claim,? it appears that society treats the two events differently, devoting more attention (and. perhaps, more resources) to possible catastrophic events. This view is shared by many researchers. Urquhart and Heilmann f38], for example, claim that people find greater misfortune in an accident which befalls a group than if the same number of individuals suffer the same kind of accident individually. They also claim that the larger the group of people who die, the worse the accident is perceived to be, This is itlustrated by the way traffic fatalities are reported: if ten people die in individual accidents there are short notes in the paper but if all ten die in a single accident. it merits a frong page headline and gets television coverage.

tSome indirect evidence dots exist in the literature on “ambiguous probabilities*’ (See Segal [34] for a discussion of theoretical issues and Ellsberg [I 1 j, Becker and Brownson [S] and Hogarth and Kunreuther 1231 for empirical results).

Resource allocation. equity and public risk 293

To test our hypothesis that people treat the two events differently, even in the extreme case

described in this paper. we conducted a small experiment. Using a convenient sample of 30 people, we asked each if faced with the choice of being able to avoid only one of the two events, which

one would they seek to avoid more. The majority of the participants (about 75%) preferred to avoid the catastrophic event. The rest were either indifferent between the events (15%). or preferred to avoid the other “bad event” (about 10%).

We also examined here whether such a pattern of social preferences is consistent with an individual’s preferences for the extreme case when the exposure to both events results in the same probabiiities of death each year (given that the individual survives), and that the expected number of fatalities over the planning period is the same. The individual’s decision problem was described using the vNM utility function. We showed that this case does not contradict one of the basic axioms of vNM utility theory-the reduction of compound lotteries into simple lotteries. Furthermore, Brookshire et al. [7] showed, using data from California on earthquake hazards and the prices of houses, that the expected utility hypothesis is a reasonable description of behavior of consumers who face a low-probability, high-loss natural hazard event, given that they have adequate information. The advantage of using a vNM utility function is that techniques for measuring such functions are well advanced. Thus, a public decision-maker could use the proposed framework to prevent events of types A and B through effective allocation of resources. It should

be noted, however, that more “flexible” theories exist that do not require the reduction of compound lotteries. An example of such theory, and a good description of the literature can be found in recent papers by Becker and Sarin [4], Machina [30], and Fishburn [13].

WC have suggcstcd that the choice between events A and B depends upon: the length of the planning period T. the probability of death over the planning period due to exposure to one of the events P. the individual’s time preference r. and the utility of being in a constant state of anxiety due to exposure to B. (/(Q). The first two depend on the nature of the phenomena and can be mcasurcd using statistical techniques for T (e.g. renewal theory [9]), and epidemiological [22,24] and probabilistic [ 191 tcchniqucs for P. (I(S2) and r depend on the prcfcrence pattern of the individual. Measuring the individual’s discount rate (time prcfcrcnce) can bc accomplished by asking conventional time-prcfcrcnce questions cast in a health as opposed to a financial, domain [IS]. Finally, assessing LJ(S2) can be effected using standard techniques described by Keeney and Raiffa [25] or Farquhar [12].

Our model considers the extreme case of an equal probability of death every year and the same expected number of fatalities over the planning period in both events. This is not, however, the typical case. The risk of death due to exposure to a catastrophic event appears smaller than the risk of death due to other. more common events such as diseases, and road accidents. Yet society devotes more attention to catastrophic events. The proposed model can be extended to incorporate this asymmetry. Once again T, r. and u(S2) would play important roles in determining the ranking of the two events. However, instead of having one parameter to describe the risk of death we would have two (one for each event), with the difference between them being an important determinant of the alternative ranking. We also assume that the occurrence of the catastrophic event is uniformly distributed. Again, this is not the typical case. We have used this assumption to simplify the mathematical treatment in examining the extreme case of equal probabilities of death every year from exposure to the two events. When one examines a more general case where, for example, the probabilities of death are not equal, one should use the distributions appropriate for that situation.

Finally. to simplify the mathematical treatment, we have employed some strong asumptions. For example, the ranking of the two events, which is done at the beginning of the planning period, cannot bc revised during that period. This might not be such a strong assumption when the analysis takes the decision-maker’s perspective (e.g. the building of a dam to protect a city from a flood that occurs. on average. once in 10 yr. is not a reversible decision). However, from an individual’s point of view it might be easier to revise the ranking of the two events in future years. This would require a dynamic model to describe the individual’s behavior. A dynamic model might also better handle other phenomena such as the learning effect resulting from surviving a catastrophic event (e.g. experienced sailors don’t fear storms as much as landlubbers). Our approach assumes that individuals make their ranking decision also at the beginning of the period without the ability to

294 Anlwtil~ MEHREZ and Awtru~ GAFNI

revise it during the period.This assumption is clearly reasonable under certain circumstances, e.g. citizens are asked to reveal their preferences through voting at the beginning of a planning period.

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