ON COMPARING RISKS WHEN THE PROBABILITY

DISTRIBUTIONS ARE NOT FULLY KNOWN.*

NGO VAN LONG

Australian National University

One often encounters in popular debates the following type of statement: “The risk of being killed when operating a nuclear plant is considerably smaller than the risk of being killed in a car accident or in a coal-mine”. It seems legitimate to ask whether it is meaningful to compare these risks when the number of observations relating to nuclear plant accidents is very small compared with the number of observations relating to car accidents or accidents in coal-mines. This note attempts to pose the problem and to find some preliminary answers to some questions relating to comparing projects with different degrees of riskiness.

Suppose that the probability that at least x people are killed in project a (say, coal- mining) in a given year is given by

G(x)=e-OUC, a>o, X > O . (1)

F ( x ) = e - P X , P > O , x > o (2)

where the parameter a is known with certainty, Suppose that the corresponding prob- ability for project f l (say, uranium processing) is also known to be of the form

but the constant f l is notAknown with certainty. Suppose that p” is an “unbiased” estimate of fi (in the sense that E ( f l ) = P ) , and that

b > % A (3)

so that e-0‘ < e-OUc for all x > 0.

At first sight, (3) seems to indicate that nuclear plants are less risky than coal-mines,

(a) e-0’ < e--OOc for all x > 0; and

(b) given any disutility function D (x) with D (x) > 0 and D‘ (x) > 0, the expected

because it implies A

disutility of coal-mining accidents

*I would like to thank Brian Ferguson for discussion.

364

1980 ON COMPARING RISKS 365

00

W (a) I W - ~ D (x) d x 0

is greater’ than 0 0 . 8

W (ri> -f 6eTpxD ( x ) dx . 0

(4)

(5)

A

Upon reflection, matters are not so unambiguous, because is after alI only an estimate. Given the above information, can one justifiably say that nuclear plantsare less dangerous? The answer seem to depend on whether one accepts as meaningful and useful the con- cept of a subjective probability di$ribution of 0. If one does (i.e. if one is a Bayesian’), then one should not compare e - b with eParx, nor _should one compare the expected disutility W (a) with the pseudo expected disutility W (0).

A Bayesian would argue that the decision maker should make use of his “s~bjective”~ probability distribution of the constant 0, and calculate the expected disutility

0 2

P1

E [W(0)1 = I s (PI W ( P ) d O , (6)

where s (0) is the subjective probability distribution of 0, assuming s (0) > 0 only over the interval [PI , P2 ] and where

A

‘The proof is straightforward. The two curves Ole-& and oe-‘ meet at a unique pointf , defined by

(W = & + Qn(a/$) (3 > 0 since 0 >a). Hence

h

w ( a ) - w ( f l ) = I , + I 2 , where

II =

l2 =

- X

A

J ((Yepaw - fie-0x) D ( x ) dx < 0 0 00 n

(ae-aor - fie-OX)D ( x ) d x > 0 . - X

m A * 00

Now since I (Ye-&dx = .f 0 0

This proof also establishes that W (a) is a decreasing function.

= 1, I I + 1 2 > 0.

2For a classic in Bayesian decision theory. see Raiffa and Schlaifer [ 31. For some applications of 3Bayesian decision theory to economic analysis, see for example Long [ 11 and Turnovsky 141. Of course the subjective distribution incorporates all available information, and is constantly up- dated in the light of new information. Bayes’s rule gives a systematic updating procedure.

366 AUSTRALIAN ECONOMIC PAPERS DECEMBER

m

W (0) ./ PeCPXD (x) dx . 0

What is the relationship between the Bayesian expected disutility E [W ( P ) ] and the pseudo expected disutility W (j)? Denoting the mean of the subjective distribution of

P by E ( P ) , 0 2

E ( P ) = I Ps(P)dP (8) P I

one can prove the following proposition

Proposition E [ W ( P ) ] > ~ ( 2 ) if (i) 8 2 E ( @ , and

(ii) W (0) is a strictly convex function.

Pro0 f The strict convexity of W ( P ) implies that

E [W(P)1 > W [E(P>I (This result is known as Jensen’s ineq~a l i ty .~ )

Now from footnote 1, W is a decreasing function, hence (i) implies

W [E(P)1 Wd). This completes the proof,

As an example of a convex W ( P ) , let D ( x ) = eyx , where PI >’Y > 0. Then for all P within the interval [ P I , P 2 ] ,

which is positive, convex and decreasing in 0. The above proposition and example show that there are circumstances where thf non-

Bayesian and the Bayesian rankings of projects are opposite, even when E (0) = 0. This result carries over to the case where a is also unknown, and to more general cases, with quite general probability distributions. I have chosen a very simple and specific example in order to make the point succinctly without bothering the readers with too much abstraction.

4See, for example, J. J. McCall [2, p. 4121.

1980 ON COMPARING RISKS 367

REFERENCES 1. N. V. Long, “Expectation Revision and Optimal Learning a Foreign Investment Model”, Inter-

national Economic Review, vol. 17, No. 2, 1976. 2. J. J. McCall, “Probabilistic Economics”, Bell Journal of Economics and Mananement Science,

vol. 2, No. 2,. 1971. 3. H. Raiffa and S. Schlaifer, Applied Statistical Decision Theory (Cambridge: Harvard University

Press, 1961). 4. S. J. Turnovsky, “A Bayesian Approach to the Theory of Expectations”, Journal of Economic

Theory, vol. 1, 1969.

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