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ON RISK AVERSION IN RISK A C C E P T A N C E CRITERIA

J. S. Wu-CHIEN t~ G. APOSqOLAKIS

School oJ Engineering and Applied Science, University of CaliJbrnia, Los Angeles, California 90024, USA (Received: 6 October, 1980)

ABSTRACT

The risk averse attitude that is included in some proposed risk acceptance criteria is examined. It is shown that it is a weaker attitude than risk aversion, as is" commonly defined in decision theory. Consequently, the boundary curve separating acceptable and unacceptable regions does not haw'. to be a straight line on the logarithmic Ji'equenc)~consequence space. A curve o f variable slope would express the same attitude as long as the slope is less than - 1.

1.

INTRODUCTION

The study of'acceptable risk' or 'quantitative safety goals' has been an active area of research in recent years. The two classic papers by Farmer t and Starr 2 have addressed the problem in the contexts of reactor siting and acceptability of a new technology, respectively. Even though one may question the wisdom of trying to define acceptable levels of risk while ignoring the other elements (e.g. benefits and costs) of the decision problem that a decision theoretic approach would include, there is merit in establishing quantitative safely goals mainly for administrative reasons. The proposed criteria use the term 'risk aversion' to reflect a conservative attitude toward risk. A review of the literature reveals that risk aversion is used in a broad sense. In decision theory 3,4 a risk averse decision maker finds uncertain consequences less preferable than a consequence with certainty, when the latter is the mean value of the uncertain consequences. Farmer, 1 and Okrent and Whipple s express their risk aversion in a different sense, as the following quotation from reference 5 shows: 'Risk aversion refers to the 45 Reliability Engineering 0143-8174/81/0002-0045/$02.50 © Applied Science Publishers Ltd, England, 1981 Printed in Great Britain

46

J.S.

WU-('ttlIIN.

{;. A I ) O S T O L A K I S

weighting of severe accidents in such a way as to predict a higher social cost for scverc accidents than for less severe accidents, even for cases in which thc expected loss (frequency × magnitude) is equal. (As an example, if one accident in which 100 lixc> were lost was considered more severe than 100 smaller accidents, each causing one fatality, this would bc considered to represent risk aversion.)' In this paper we examine the risk aversion attitude that is reflected in some of the proposed risk acceptance criteria in lhe context of utility theory.

2.

A('('EIYIAN{'I(

('RITERIA

2.1 Mcahematica/ hack~rmmd In gencral, an accident has several consequences, e.g. acute deaths, injurics. delayed deaths, etc. Therefore, we can define a vector .\ of consequences, the c o m p o n e n t s of lhe vector being the various kinds of consequences of interest. However, most of the criteria are formulated in terms of one or two measures of consequences and, for simplicity, we will takc .\- to be a scalar quantity. We define the impact function r(x) as the negative utility of x, i.e. r ( x ) =_ --u{x)

{li

The impact function is a measure of how undesirable x is. Because ofeqn. (1), r{x) has the properties of a utility function'* properly adjusted. In particular, consequence x A is less desirable than consequence -\B if. and only ill r(XA) > r(.vB)

(2)

Furthermore, we make the reasonable assumption that r{x) is monotonically increasing over the consequence space. Finally, we can always define r(0) = 0

(3)

because of the properties of the utility f u n c t i o n ) The criteria that have been proposed are usually based on the assumption of a constant rate of occurrence, 2,., o f consequence x(strictly speaking, of consequences in the interval (x, x + Ax)). This assumption is actually valid for a period of time T and not for all times, because the occurrence of the first event which is perceived as an accident, may change the impact function and the rate 2, for later times. In general, there are large uncertainties associated with the rate 2,.. For present purposes, however, it will suffice to assume that 2., is known. The frequency of occurrence of accidents of consequence x in (t, t + dt), where t < T, is given by the exponential distribution (constant rate 2,), i.e. p ( t ) d t = 2~ e "~' dt

{4)

ON RISK AVERSION IN RISK A C C E P T A N C E CRITERIA

47

Therefore, the expected impact of that accident is R(x, 2~, • T) =

d t z"~ e - ~~t r (x) = r ( x ) ( 1 - e

~,T)

(5)

) or

R ( x , 2~, T) = r(x)X~T

f o r 2 x T < 0-10

(6)

Equation (6) is the basis for setting up criteria in the frequency-consequence space. This space is divided into the acceptable and unacceptable regions by the indifference line r(x)2,T = c

(7)

where c is a constant determined by external constraints. Obtaining this constant is by no means an easy task, since it requires risk-benefit evaluations at a higher level (e.g. comparisons among different technologies). The criteria are usually expressed on logarithmic scales. Taking logarithms (base 10) of eqn. (7) we get log r ( x ) + log 2.,. = log (c/T)

(8)

We can now examine the existing criteria in the context of eqn. (8). 2.2 F a r m e r ' s c r i t e r i o n

Farmer I considers as consequence x the release of 1311 measured in curies. The indifference curve is a straight line (on logarithmic scales) with slope - 1.5 which assigns an acceptable rate of 0.66 x 10 4 per reactor year to a release of 104Ci, i.e. 1.51ogx + log2.~ = 1.82

x > 104Ci

(9)

The unacceptable region consists of all points above this line. Comparing eqns. (8) and (9) we conclude that the impact function is r(x)=x

l'~

x > 104Ci

(10)

and that C --=

10 T M = 6 6

T

(11)

The time interval T t h a t Farmer considers is 1000 reactor years. He argues that one release of a few thousand curies during this period (on the average) should be acceptable, if it is to be compared with the risks from other industrial activities. The constant c is, therefore, determined from risk considerations, although the benefits to be derived from nuclear power plants must have implicitly influenced this choice. From eqn. (11) we get c = 6'6 x 104

(12)

48

J. S. W U - C H I E N , G . A P O S T O L A K I S

The risk function of eqn. (10) is of the form r(x)=x ~

~> 1

(13)

with ~ -- 1.5. The fact that ~ > 1 expresses risk aversion. Farmer chooses this value because 'it is that most people would apply a heavier penalty against the possibility of large release than a smaller release...'. An important theoretical point requires clarification. We have defined ).,. as the rate (per reactor year) of the accidents that lead to consequences in (x, x + Ax). This means that when the criterion is used, all (or at least, the dominant) accident sequences with consequences in the stated interval, must be considered together. In risk analysis, however, one usually begins by obtaining accident sequences through event tree analysis. It would be inconsistent with the derivation of the criterion to apply it to one of the sequences thus obtained in isolation. In fact, as Kaplan points out, 6 it is conceivable that a sequence which is unacceptable could be divided into sub-sequences, each of which would be acceptable. 2.3 Kinchin's criteria

Kinchin 7"8 proposes criteria for early and delayed deaths. For individual risks he accepts the average frequencies of 10 ~ reactor year- 1 for early death and 3 x 10 s reactor year l for delayed death. For societal risks Kinchin chooses to work with the frequency of x or more deaths, and he does not specify the period T. The frequency of at least x deaths is t

Ax=- j . ' 2~.d.'('

(14)

where x* is the maximum possible number. From eqn. (7) we get l o g A , = log

Tr(x') d.\"

15)

Kinchin proposes the following two indifference straight lines log A., + log x = - 3

for delayed deaths

(16)

for early deaths

(17)

and IogA, + l o g x = - 4 . 4 8

Comparing with eqn. (15) we conclude that, for large x*, his impact function is r(x) = x e

(18)

which is also of the form of eqn. (13) with ~ = 2. Therefore, Kinchin is more risk averse than Farmer. This finding appears to contradict Kinchin's statement in reference 7 that "it seems not unreasonable that the probability of an accident should

ON RISK AVERSION IN RISK ACCEPTANCE CRITERIA

49

be inversely proportional to the n u m b e r o f deaths in suggesting a c r i t e r i o n . . . ' . This would imply a value o f ~ equal to unity, i.e. no risk aversion. The reason for this inconsistency is that Kinchin works with A x rather than 2 x.

2.4 Consequences of the straight line The impact function for the preceding criteria is of the form of eqn. (13), which yields a straight line on logarithmic scales. Griesmeyer et al. 9 examine the implications o f a constant ~ for high consequences and they suggest that 'facilities with the potential for extremely large events m a y never be able to satisfy criteria of this type, ifa large value o f c0is chosen'. F o r example, a technological system that has the potential o f killing l0 s people would be acceptable, if such an accident had a frequency of occurrence much smaller than 3 x 10- 7 per year (if~ = 1.5), a n u m b e r that would be extremely difficult to verify. It turns out that the constancy o f ~ is not necessary even if one wishes to retain the risk averse attitude that we have described. We now turn our attention to the mathematical definitions of risk aversion.

3.

RISK AVERSION IN RISK ANALYSIS

3.1 Definition A decision maker is risk averse if her impact function r(x) satisfies

r(px) < pr(x)

(19)

for all 0 < p < 1. A mathematical function r(x) that satisfies eqn. (19) is called star-shaped.l° We have the following criterion: A function r(x) defined on 0 < x < m is starshaped if, and only if, r(x)/x is a non-decreasing function of x. The following is an example of a star-shaped function (Fig. 1): {i~ 3 r(x) =

0< x < 1 1< x

(20)

0_

(21)

Indeed, we have

r(x)_ {12 x

a function which is non-decreasing in x.

3.2 Comparison with decision theot 3' In decision theory the concept of risk aversion is defined as follows. 4 The decision maker is faced with an uncertain situation which m a y result in consequence x I with

50

J. S. WU-CHIEN, G. APOSTOLAK1S

r(x)

3

2 x3

,

O

rCx) = 1

< x

1

0

i

i

J

I

2

3

Fig. l.

x

A star-shaped function.

probability p or consequence _v2 with probability ( 1 - p). The expected consequence is, therefore

.v =px~ + (1 - p ) x 2

(22)

The decision maker is risk averse if she prefers consequence .f for certain to the uncertain outcomes-\1 and -\2, Formally, the impact function that expresses such an attitude must satisfy r(px t + ( I - p ) . v 2

(23)

for a l l 0 < p < l . A mathematical function r(x) that satisfies cqn. (23) is called convex. W h e n r(.¥) is differentiable, a useful criterion for convexity is that the first derivative, r'(x), be non-decreasing. The function ofeqn. (20) is not convex, as an easy calculation shows. However, the function of eqn. (18) is convex, since r'(.\ ) = 2x, which is a non-decreasing function o f x . Similarly, for eqn. (13) we get r'(.v) - ~.v~ 1 which, for ~ > I. is also a n o n decreasing function of x. When x2 is zero eqn. (23) becomes eqn. (19). This means that if wc restrict ourselves to situations where the n o n - o c c u r r e n c e of the event with consequencc .\ implies zero consequence, the two risk aversion attitudes coincide. This, of course, is the case in risk assessments, where the n o n - o c c u r r e n c e of an accident has no consequences. A formal property of convex and star-shaped functions that expresses the same thing is the following. A convex function (satisfying eqn. (23)) is star-shaped, but the converse need not be true. ~° For convenience, we will call a risk attitude thai is

ON RISK AVERSION IN RISK ACCEPTANCE CRITERIA

51

expressed by a convex r(x) strong risk aversion, and that which is expressed by a star-shaped r(x) weak risk aversion. Obviously, then, strong risk aversion implies weak risk aversion but not vice versa. 3.3 Acceptance criteria revisited

We can now investigate the acceptance criteria that have been proposed in the context of the concepts that we have introduced. Given two consequences x 1 > x z such that (24)

21x 1 = 2zX 2

risk aversion is expressed by the requirement 21r(xl) > 22r(x2)

(25)

From eqns. (24) and (25) we get (26)

r(xl) > r(x2) X 1

X 2

Equation (26) shows that, according to the criterion stated in Section 3.1, the impact function r(x) is a star-shaped function. Therefore, the risk aversion attitude that eqns. (24) and (25) imply is that of weak risk aversion. The mathematical expression, however, for the impact function is a power law (eqn. (13)), and we have shown that this impact function is convex; therefore it expresses strong risk aversion.

~x

\

\

\ -|

i

Fig. 2.

Some boundary lines that include risk aversion.

52

J . s . WU-CHIEN, G. APOSTOLAKIS

We c o n c l u d e that an a c c e p t a n c e criterion o f the form o f e q n . (13) is stronger than the attitude expressed by eqns. (24) a n d (25) a n d could be relaxed, Figure 2 shows three straight lines o f slopes - 1 , - 1 . 5 and - 2 and a curve expressing weak risk aversion. N o t e that the slope of this curve c a n n o t be greater than - I. 4.

('ONCLUI)ING REMARKS

We have e x a m i n e d the n o t i o n o f risk aversion as applied to p r o p o s e d risk acceptance criteria. W e have f o u n d that this a t t i t u d e is w e a k e r than the risk aversion a t t i t u d e defined in decision theory. C o n s e q u e n t l y , the i m p a c t function does not have 1o be convex and the risk acceptance b o u n d a r y line does not have to be a straight line (on l o g a r i t h m i c scales). However, the slope o f the b o u n d a r y curve, even in the weak sense, c a n n o t exceed - 1 . This m a y create p r o b l e m s with verifiability, when accidents with very high consequences are considered, which w o u l d be required 1o have very low frequency o f occurrence. The whole analysis rests on eqns. (5 7), which assume that the irnpact function d e p e n d s o n l y on the consequences x and is i n d e p e n d e n t o f the frequency ,:,~. This a p p e a r s to be counter intuitive, since o u r attitudes t o w a r d accidents d o d e p e n d on their perceived frequency o f occurrence. In fact, F a r m e r is influenced by the frequency in the high frequency low c o n s e q u e n c e region a n d he states~ : 'there is a transition in the slope in the range 10 curies to 1000 curies which serves to minimize the frequency o f small releases, largely on the basis o f their nuisance value'. P e r h a p s the frequency should also affect the criteria in the low frequency high c o n s e q u e n c e region. This subject is c u r r e n t l y being investigated.

REFERENCES 1. 2. 3. 4.

FARMER, F. R. Reactor safety and siting: A proposed risk criterion, Nuel. SaJbly, 8 (1967), p. 539. STARR,C. Social benefit versus technological risk, Science, 165 (1969), p. 1232. PRATT, J. W. Risk aversion in the small and in the large, Econometrica, 32 (1964), p. 122. KEENEY,R. L. and RAIFFA, H. Decisions with Multiple Ot?jectit'es." Preferences and Value Tradeo[L~,

John Wiley & Sons Inc., New York, 1976. 5. OKRENT, D. and WHIPPLE, C. An Approach to Societal Risk Acceptance Criteria and Risk Management, School of Engineering and Applied Science, University of California, Los Angelcs.

UCLA-ENG-7746, 1977. 6. KAPLAN, S. Notes ]or a workshop on risk analysLs and decision under uneertainty, Presented at

Lawrence Livermore Laboratory, Kaplan and Associates, Inc., Irvine, California, 1979. 7. KINCmN, G. H. Assessment of hazards in engineering work, Proc. Instn. Cir. Engrs., Part I. 64 (1978), p.431. 8. KINCHIN, G. H. Design criteria, concepts andJeatures important to sajety and licensing, ENS/ANS International Meeting on Fast Reactor Safety Technology, Seattle, August 1979. 9. GRIESMEYER,J. M., SIMPSON,M, and OKRENt,D. The use o f risk aversion in risk acceptance criteria ~' Technical Report UCLA-ENG-7970, University of California, Los Angeles, October 1979. 10. BARLOW,R. E. and PROSCHAr~,F. Statistical Theo D, of Reliability and Life Testing, Holt, Rinehart and Winston, Inc., London, 1975.