risk analysis || foundations of risk measurement. i. risk as probable loss

Foundations of Risk Measurement. I. Risk As Probable LossAuthor(s): Peter C. FishburnSource: Management Science, Vol. 30, No. 4, Risk Analysis (Apr., 1984), pp. 396-406Published by: INFORMSStable URL: http://www.jstor.org/stable/2631428 .

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MANAGEMENT SCIENCE Vol. 30, No. 4, April 1984

Printed in U.S.A.

FOUNDATIONS OF RISK MEASUREMENT. I. RISK AS PROBABLE LOSS*

PETER C. FISHBURN

Bell Telephone Laboratories, Inc., Murray Hill, New Jersey 07974

This paper seeks to get behind specific contextual referents of risky situations to consider characteristics of risk that apply to many situations. It is guided by previous theoretical and empirical research in perceived risk, and focuses on the joint effects on risk of loss probability and the distribution of losses. The approach taken follows modern axiomatic theory by proposing conditions on a relation "is at least as risky as" between pairs of probability distributions over an outcome variable. Several sets of axioms for risk that characterize different forms for risk measurement are presented. (RISK MEASUREMENT; PROBABILITY OF LOSS)

1. Introduction

Conventional wisdom in medicine, insurance, safety, strategic planning, personal relations, diplomacy, banking, and other fields says that risk is a chance of something bad happening (Krewski and Brown 1981, Ricci and Molton 1981, Slovic et al. 1981).

Although numerous context-specific measures of risk have been proposed, my aim here is to go behind specific contextual referents and to explore risk measurement from an axiomatic perspective. The study will be guided by three things, namely previous theoretical work on axiomatic models of risk and preference, empirical studies of perceived risk, and the conventional notion that risk arises from a possibility of undesirable outcomes.

It will be presumed that outcomes can be preferentially ordered and assigned numerical values that preserve the preference ordering. It is assumed further that a 'target' outcome can be identified so that every outcome whose value is smaller than the value of the target is viewed as 'undesirable' or 'risky', while outcomes with values as large as the target's are nonrisky. Depending on context, risky outcomes may involve rates of heart disease or accidents, loss of life, loss of face, defaults on loans, and loss of capital. For convenience, the value of the target outcome will be set at zero. Values of outcomes will often correspond to a natural index (lives lost, dollar gains), and the zero value may have a natural interpretation (0 lives lost, no loss-no gain point).

The axioms for risk measurement will be based on a binary relation -, "is at least as risky as," defined on a set of probability distributions over outcome values. The distributions can be viewed as hypothetical; the ubiquitous and often knotty problem of estimating probabilities for actual decision alternatives will not be considered here. The asymmetric and symmetric parts of ? will be denoted >- ("is riskier than") and - ("is equally risky as").

The preceding formulation implies that, apart from targets and probabilities, the only risk-relevant aspect of outcomes is their relative preferences. I assume in effect that risk makes no distinction among equally-valued outcomes. Other commitments that apply throughout the study include (i) ? is a weak order (complete, transitive), (ii) the risks of probability distributions over outcome values are numbers that preserve X, (iii) the numerical measure of risk is nonnegative, and (iv) a distribution has zero risk if and only if it has no chance of producing a bad outcome.

*Accepted by Rakesh K. Sarin, Special Editor.

396 0025- 1909/84/3004/0396$O1 1.25

Copyright (C 1984, The Institute of Management Sciences



RISK AS PROBABLE LOSS 397

The present paper concentrates on distributions that assign probability 1 to values less than or equal to 0, the target value. Distributions admitting favorable as well as undesirable outcomes will be considered in a sequel (Fishburn 1982). The numerical measure of risk used here is denoted as p. The forms of p to be axiomatized are outlined at the end of the next section.

A word about preference, uncertainty and risk as they apply to the present study seems advisable in light of semantic confusion about "risk" in the literature of decision theory (Arrow 1965, Coombs 1974, Hertz 1964, Luce and Raiffa 1957, Markowitz 1959, Pratt 1964). I use preference in the traditional mode (Chipman et al. 1971, Fishburn 1970, Krantz et al. 1971). Uncertainty refers to probabilities strictly between 0 and 1, and to distributions with such probabilities, or to decision alternatives with several possible outcome values. Risk is intended in a conventional manner. It is based in part on outcome preferences and targets. I generally assume that risk increases as bad outcomes become more probable, and as probable bad outcomes get worse. While favorable outcomes are not associated with risk by themselves, their presence in a distribution that has positive probability for bad outcomes might decrease the risk of the distribution. This will be discussed further in Fishburn (1982).

Uncertainty is not taken as an inherent part of risk. In the present conceptualization, a distribution with a large variance but probability 1 over favorable outcomes has no risk, while a sure-thing distribution with probability 1 of $10,000 loss is regarded as risky.

Finally, although preferences between probability distributions are presumably associated with the distributions' risks in ways that go beyond our bare connection with outcome preferences, I shall not speculate on such associations in this study. A discussion that touches on this point is provided by Schaefer (1978): also see (Fishburn 1977, Libby and Fishburn 1977, Payne et al. 1981).

2. Notations and Definitions

Outcomes are assumed ordered by preference and represented numerically so that one outcome is preferred to another if and only if the former has a larger value (Fishburn 1970, Krantz et al. 1971). Let X denote the set of outcome values. By convention, a designated nonrisky target outcome has value 0. Nonzero outcome values are partitioned into unfavorable and favorable subsets as

X- = {x E X: x < 0} and X+ = {x E X: x > 0}.

I shall refer to outcome values simply as outcomes, to outcomes in X - as losses, and to outcomes in X + as gains. It is assumed that X - is not empty. If lives lost is the sole concern, X - = - 1, -2, . . ., - N) and X+ = 0. A monetary context could have X - = (-oo,0) and X + = (0, oo).

Probability measures are defined on a Borel algebra of subsets of X that includes singleton subsets and 'intervals' of outcomes, and are assumed to be countably additive (Fishburn 1970, 1975). We let P - denote a set of probability measures with p(X - ) = 1 for each p E P -, so that each such p is certain to result in a loss. In the sequel I shall assume also that X + is not empty and incorporate a second set P + of probability measures that are certain to yield gains. Combinations of measures in P-

and P + will then be used to characterize probability measures over X. All relevant one-point measures on outcomes in X - are assumed to be in P -, which

is taken as closed under convex combinations. Thus Xp + (1 - X)p' E P- when p, p' E P - and 0 < X < 1. Integration is Lebesgue-Stieltjes integration.

Experiments with monetary gambles have convincingly demonstrated the salience of loss probability and amount on perceived risk (Aschenbrenner 1978, Coombs and Lehner 1981, Payne 1975, Slovic 1967). Moreover, preference attitudes toward gambles



398 PETER C. FISHBURN

in the loss and gain regions exhibit striking differences (Fishburn and Kochenberger 1979, Hershey et al. 1981, Kahneman and Tversky 1979, Laughhunn et al. 1980, Payne et al. 1980, 1981). These observations, coupled with the usual conception of risk, suggest a formulation for risk in the P - context that takes explicit account of loss probability as well as the distribution of losses. Accordingly, I shall apply > to

A = [0, 1] X P - = {(a, p) :0 < a < l,p E P - },

where (a, p) is interpreted as a measure that yields an x E Y for Y C X - with probability ap(Y), and gives outcome 0 with probability 1 - a. The probability of loss is a and, given a loss, p(Y) is the probability that the loss will be in Y.

I shall sometimes abbreviate (1, p) as p, (0, p) as 0, and (a, p) as (a, x) when p({x}) = 1. By interpretation, (0, p) = (O,q) = 0 for allp,q E P. In addition, (1,x) is sometimes written simply as x.

Set A applies when X + is empty, or when X + is not empty and gains have the same effect on risk as outcome 0. Numerical risk measures p that represent > on A will adhere to the following:

DEFINITION. p is a nonnegative real valued function on A that has p(a, p) 0 if and only if a = 0. When p({x}) = 1, p(a, p) is also written as p(a,x).

The basic risk representation for A has

(a, p) : (a', p') )p(a, p) > p(a', p'). (1)

I then discuss axioms for : on A that yield the following special forms of p:

p(a, p) = pj(a)P2(^

p(a, p) = _p(a, x) dp(x),

p(a, p) = p(a)_P2(x)dp(x),

p(a, p) = a p2(x) dp(x),

p(ap) = P1(a)f xI'dp(x), 9 > 0.

The first form posits separability between total loss probability and the distribution of losses, so that a and p act as independent factors whose measurements combine multiplicatively in the overall measurement of risk. The second form, which is logically independent of the first, says that for each a, risk can be written in expected-value form for the various measures in P -.

The third form essentially combines the first two in a separable-expectations model. While Pi is increasing in a with Pj(0) = 0, this form allows Pi to be nonlinear in the loss probability a. Linearity in a appears in the fourth form, which is tantamount to Huang's (1971) expected-risk model for A.

The final form specializes the third form in a different way by taking p2(x) as the power function lxl with parameter 9 > 0. When pl(a) = a,

p(a,p)= a I xIdp(x).

This form is used in a mean-risk analysis by Fishburn (1977) and has been axiomatized by Luce (1981). When 9 = 2, it gives Markowitz's (1959) below-target semi- variance risk measure; when 9 = 1, we get the weighted-losses measure of Domar and Musgrave (1944). These risk measures are palatable only when outcomes have a natural meaning (accidents per month, dollars lost).

Parameter 9 reflects judgments of the risks incurred by losses of different magni-




tudes. If risk escalates rapidly as losses increase, 9 > 1. If the likelihood of loss is paramount without special regard for magnitude, 9 < 1. At 9 = 0, risk is simply the probability of loss.

3. Basic Axioms and Separability

Five axioms are used for the basic representation (1). They apply to all a, /3, y E

[0, 1], all x, y E X-, and allp,q E P. Al. > on A is a weak order; A2. If a > / then (a, p) >-( /3,p); A3. If x <y then x >-y; A4. If (a, p) >- (,8, q) >- (yp ), then (Xa + (1 - X)y, p)-(/3, q) for some 0 < X < 1; A5. w :p for some w E X -. Axiom A2 says that risk increases as the loss probability increases (p fixed), and A3

says that worse outcomes entail greater risk. A4 is a continuity condition in which X is unique according to Al and A2. The fifth axiom says that for every measure in P -, some outcome is at least as risky as the measure. This seems sensible if there is a worst outcome, but might be less obvious if outcomes are unbounded below.

THEOREM 1. Axioms Al through A5 imply that there is a p on A that satisfies (1) for all (a, p), (a', p') E A and is continuous and increasing in a.

PROOF. The proof when X - has a worst outcome is subsumed under the following analysis. Suppose X - has no worst outcome. Let wI, w2, ... be a decreasing sequence of outcomes in X - such that, for all x E X -, wk < x for some k. By A2, A3 and A5, every (a, p) E A has some wk : (a, p).

Construct p on [0, 1] x {wIw2, .... 4 as follows. Let p(a, w1) = a. By Al and A2, (1) holds at w1 with p continuously increasing in a and p(O) = 0. Take w2 next. By AI-A4, there is a unique X2 at which (X2, w2)-wW, and for every a E [0, X2] there is a unique f(a) where (a, w2)-(f(a), wI). It is easily checked thatf is continuous and increasing in a with f(O) = 0 and f(X2) = 1. To satisfy (1), set p(a, w2) equal to f(a) for 0 < a < X2. For a > X2 take p(a,,w2) = 1 + (a - X2)/(1 - X2). Then (1) holds on [0, 1] x {w,w2) with p(-, w2) continuously increasing in a and p(l, w2) = 2. Next, W3 is traded off against w2 in a similar manner, extending p to [0, 1] x { WI, W2, W34 with p(l, w3) = 3. The process continues through the Wk.

Suppose p X {ww2, .. . 4. Take wk-p, and, by Al-A4, let g(a) be the unique number at which (a,p)-(g(a),wk). The axioms imply that g is continuous and increasing with g(O) = 0 and g(l) < 1. To satisfy (1), take p(a, p) = p(g(a),wk). Since g is continuously increasing in a, and p( I, wk) is continuously increasing in g, p(-, p) is continuously increasing in a. This is done for each such p. It follows from Al that (1) holds everywhere. Q.E.D.

Three more axioms are used for a - p separability. They apply to all a, /3, y E [0, 1] and allp,q,r E P-.

A6. If (a, p) >- (3, q) >- (a, r), then (a, Xp + (I -X)r) - (/3, q) for some 0 < X < 1; A7. If a, 3 >0, then (a,p) (a,q) implies (/P,p) (/P,q); A8. If (a, p) - ( /3, q) and (/,, r)>-(y, p), then (a, r) - (y, q). Axiom A6 is a continuity axiom for the p component: if (/3, q) is riskier than (a, r)

but less risky than (a, p), then it is equally risky as some convex combination of (a, r) and (a, p). Independence condition A7 says that the risk order on P - is the same for each positive a. This is necessary for separability, but we also need Thomsen's condition (Krantz et al. 1971), A8, which seems less obvious for risk than preceding axioms. Its necessity for separability when a, /, y > 0 follows directly from the separable form p = P1P2* Its plausibility for risk may be debatable.




THEOREM 2. Axioms A1-A8 imply that there are real valued functions pI on [0, 1] and P2 on P - such that (1) holds for all (a, p), (a', p') E A when p(a, p) = pI(a)PAA with:

(i) Pi continuous and increasing in a; (ii) PI(O) = 0; (iii) P2(P ) an interval of positive numbers; (iV) p2(X) increasing as x decreases.

Moreover, if X - contains two or more outcomes, then Pi and P2 are unique up to similar power transformations, i.e., p' and p'2 also satisfy the representation if and only if there are positive numbers a, al and a2 such that p' = a pa and p'2 = a2p .

PROOF. Let p satisfy Theorem 1 and, to avoid the trivial case, assume that IX - I > 2. Then, using A2 and A6, and with h(p) = p(l, p), h(P - ) is a nondegenerate interval of positive numbers. A6 will not be used further. Using A7, represent (a, p) by

(a, h(p)) and apply ? to (0, 1] X h(P - ), with risk strictly increasing in each component. Standard theory for conjoint measurement, e.g. (Fishburn 1970, pp. 65-71) or

(Krantz et al. 1971, Chapter 6), implies that there are real valued functions r, on (0, 1] and r2 on h(P -) such that, with r2(p) = r2(h(p)),

(a, p) (I3,q)i=rI(a) + r2(p) > r (/3) + r2(q)

whenever a, /8 > 0. In this additive form, rl((0, 1]) and r2(P - ) are nondegenerate intervals, the two functions are unique up to simultaneous affine transformations of

the form {r' = ar, + bl, r' = ar2 + b2, a > 0), and it follows easily from A4, A7, and

other axioms, that r (a) approaches - co as a ->0. With p (a)= eri(a) and P2(P) = er2(P), we get (a, p) (/3,q) if and only if p1(a)P2(P) > pI(3)p2(q), a, / > 0, with PI and P2 having properties (i) through (iv) of the theorem when PI is extended to 0 by

taking Pl(O) = 0. The uniqueness part of Theorem 2 follows from that for r, and r2.

Q.E.D.

4. Linear Forms for A

This section axiomatizes three special p forms that are based on linearity and

independence conditions used in utility theory (Fishburn 1970, Herstein and Milnor

1953, Keeney and Raiffa 1976). The first form is linear in p, i.e.,

p(a,Xp + (1 - X)q) = Xp(a,p) + (1 - X)p(a,q),

but it does not presume separability between a andp as in Theorem 2. The second and

third forms entail separability. One technical point that is relevant only when P - contains nonsimple measures

(pr. < 1 on every finite subset of X -) requires attention. In addition to earlier

conditions on P - (one-point measures, convex closure), I shall assume that P - is

closed under conditioning on intervals. If Y is an interval in (- xo, 0) and p( Y n x - )

> 0, let py be defined by py(Z) = p( Y n Z)/p( Y n x - ) for all Z in the algebra. When p E P -, I assume that all such py are also in P -.

Special axioms are needed for integrability, or expectation forms, when P - contains

nonsimple measures. Since these are redundant when P - has only simple measures,

they will be denoted as Cl, C2, and so on to distinguish them from the other axioms.

Nonseparable Expectation

I shall axiomatize p(a, p) = fp(a, x) dp(x) in three steps. The first deals with each a > 0 separately. The following axioms apply to all a, X e (0, 1], all p, q, r E P -, all

simple (pr. 1 on some finite subset of X ) s E P -, all Y C X - in the algebra, and all

xE X-.




A9. If (a,p)>-(a,q), then (a,Xp+(1 -X)r)>-(a,Xq+(1 -X)r); C1. If p ( Y) = 1, then (a, p) ? (a, x) if (a, y) : (a, x) for all y E Y, and (a, x) > (a, p)

if (a, x) : (a, y) for all y E Y; C2. If (a, p) >- (a,s) then (a, p[Yo)) (a,s) for some y E X-; if (a,s) >- (a, p) then

(a, s) 7 (a, p( - ? ) for some y E X -. Axiom A9 is a typical linearity-independence axiom that is essential for the linear

form

fa(p + ( -X)q) = Xfa(p) + (1 -X)fa(q)

implicit in Lemma 1. The special conditions Cl and C2 are used to extend this to

fa (P) = ffa (x) dp (x) when p is nonsimple. Cl is a reasonable dominance axiom, and C2 is a truncation-continuity axiom.

LEMMA 1. Axioms Al, A3, A6, A9, Cl and C2 imply that for each aE (O, 1] there is a real valued function fa on X - such that fx -fa (x) dp(x) is well defined and finite for every p E P- and, for all p,q E& P

(a p ~ (,q) <JX fa (x) dp (x) > Jfa (x) dq (x)

Moreover, fa is unique up to positive affine transformations fa = afa + b (a > 0).

PROOF. See Theorem 3 in Fishburn (1975). Q.E.D. The second step towards fp(a, x) dp(x) aligns the fa so that an expression like (1)

holds on (0, 1] x P -. The following apply to all a, /3, y E (0, 1] and all p, q, r, s E P -. AO0. If (a, p) -( /3, r) and (a, q) -( /3, s), then (a, yp + (1 - y)q) -( /3, yr + (1 -)s); All. If ao< I then there exist n>2, a =a1< a2< ... < an= I and pi,qiEP

such that (ai, pi) >- (ai+ 1, qi) for i = 1, . . , n -1. Axiom A10, which is necessary for the form in Lemma 2, says that equally risky a

and /8 comparisons remain equally risky under similar convex combinations in their second components. The second new axiom is a version of standard-sequence axiom (Krantz et al. 1971) that ensures overlaps in risk for different a.

LEMMA 2. Axioms Al-A3, A6, A9-AI 1, Cl and C2 imply that there is a real valued function f on (0, l]>< X - such that fx-f(a, x) dp(x) is well defined and finite for all (a, p) E A\{O} and, for all (a, p),(13,q) E A\{O),

(a, p)i(83, q)j _f( a,x)dp(x) > f(8/3,x)dq(x).

Moreover, f is unique up to positive affine transformations (af + b, a > 0).

PROOF. Let the fa be as specified in Lemma 1, and let fa(p) = ffa (x) dp(x). Set f(l, p) = f1(p). Let Em = {0 < a < 1: minimum n for a in All is m). We work with E2 first. Given a E E2, let

P(a) = {p E P (a, p):(I,q) for some q E P}.

It is easily seen that P(a) is closed under convex combinations and, for every p E P(a), there is q E P - such that (a, p)-(l, q). Given the latter- statement, set f(a, p) equal to f(l, q). Suppose (a, p)>(l, q) and (a, p')>(l, q'). Then, by A10, (a, yp + (1-y)p')-(l, yq + (1 - y)q'). Therefore

f(a,yp + (1 - y)p') =f(l,yq + (1 - y)q')

= yf(l,q) + (1 - y)f(l,q')

= yf(l, p) + (1 - y)f(a, pI),




sof(la, ) is linear on P (a). Since f(a,p) )f(la, q)(a,p)>(a, q)<>fa(p) )fa(q) on P(a), f(a,.) must be a positive affine transformation of fa(.) on P(a), say f(a,.) - a(a)fa ) + b (a). Extend to all of P - by settingf(la, p) = a (a)fa(p) + b (a).

Suppose a, 8 E E2, a #/3 8, and (a, p)>(/3,q) but (1,r) >- (a, ) for all r E P. Then pi and qi can be chosen so that Pi, P2 E P (a), q1, q2 E P(3)a, Pi) (3,qi) (a, P2)-( /3, q2), (a, Pl) >- (a, P2) >- (a, p) and (/3, qj) >- (/3, q2) >- (/,, q). By A6 and AlO, get y E (0, 1) with

(a, P2)-(a, yp1 + (1 - y)p)'8( /3,yql + (1 - y)q)-( /, q2).

By linearity and Lemma 1,

f(a, P2) = yf(a, Pi) + (1 - y)f(a, p),

f(/3,q2) = -yf(/3,qi) + (1 - f(,q),

and, since f(a, pi) = f( /3, qi) for i = 1, 2, it follows that f(a, p) = f( 3, q). Because of the order and continuity axioms, it then follows that the representation of Lemma 2 holds on ({1} U E2) x P- whenf(la,x) = a (a)fa(x) + b (a) on X-. Moreover,f as thus far constructed is unique up to a positive affine transformation since this is true of f(l, .).

If E3 is not empty, the next step integrates the a E& E3 into the representation by suitable positive affine transformations of the fa. The process continues through all nonempty Em. Details will be omitted. Q.E.D.

Lemma 2 leads to p(a, p) = fp(a, x) dp(x) with p(O) = 0 if f is bounded below, but examples show that its axioms do not imply lower boundedness (even when X - has a maximum). The third step therefore requires conditions that bound f below. This can be done in various ways, but I shall identify only one here.

A12. There are 8 E (0, 1) and p E P - such that for every a E& (0, 1] there is a pa E P -

for which (a, p + (I - 8)t) > (1, pa) for every t E P -. This is a uniform boundedness condition that is substantially stronger than All

since it implies that A 1I always holds for n = 2. It is most plausible if sup X - = 0, for then pa can be chosen as xa near 0. However, Al12 can hold when X - is bounded away from 0, as will be shown in the next subsection.

THEOREM 3. The axioms of Lemma 2 with All replaced by A 12 imply that there is a real valued function p on [0, 1] X X - such that p(O, x) = 0, fp(a, x) dp(x) is well defined and positive for all (a, p) E A \ {O} and, for all (a, p), (/3, q) E A,

(a, p) >:(/,8q) <= p(a x) dp (x) > Jxp(/,8,x) dq(x).

Moreover, we can require inf p((O, 1] X X ) = 0 and, when this is true, p is unique up to similarity transformations (ap, a > 0).

PROOF. Let f be specified by Lemma 2, and fix 8, p and the Pa as assured by A1 2. For a = 1, (1,3p + ( - 8)t) >- (1, p) gives

f(,t)> f(l, PI)- f(l, p) for all t E P

where f(a, q) = ff(a, x) dq (x). Hence f(l, * ) is bounded below, say f(l, t) > K for all t E P-. For a < I get

f(a, t)> f(l, pa)3-f(a, p) for all t E P.

Since f(l, pox) > K, and -f(a, p) > -f(l, p) by A2 and Lemma 2,

f(at,t) >(K- 3f(l, p))/(l 1-3) for every 0 < af K 1 and every t E P-.




Hence f is bounded below. Since f decreases as a decreases for fixed x or p, it has no minimum so, by adding an appropriate constant to f to get p we have inf p((O, 1] X X - ) = 0 with p(a, x) > 0 whenever a > 0. Setting p(O, x) = 0, Theorem 3 then follows from Lemma 2. Q.E.D.

Separable Expectation

Axiom A7- on P - is the same for each a > 0-is now added to the analysis of the preceding subsection. In the context of Lemma 2, A7 implies (Fishburn 1970, Theorem 8.4) that there are real valued functions f1 and g1 on (0, 1] with f1(l) = 1,

gl(l) = 0 and fi > 0 such that, with f2(x) = f(l, x),

f(a, x) = f1(a) f2(x) + gI (a) for all (a, x) E(0, 1] x X.

In the context of Theorem 3, with p2(x) = p(l, x), we get

p(a, x) = p I (a)p2(x) + a1 (a) for all (a, x) E (0, 1] x X,

where pI(1) = 1, aI(1) = 0 and PI > 0. Moreover, to have p(0,x) = 0, we need PI(O) = a1(0) = 0. If a 0, this gives the form p(a, p) = pI (a)fp2(x)dp(x) mentioned earlier.

However, the axioms of Theorem 3 plus A7 do not imply a _ 0. For example, if P2(P - ) = [1, 4], p (a) = (a + 1)/2 and a1(a) = (a - 1)/2 for a > 0, then

p(ap) =J p(ax)dp(x) = [(a + 1)fp2(x)dp(x) + a - 1]/2 (a > 0)

along with p(O) = 0 implies A7 and all axioms of Theorem 3, including A12, when we take ? defined by

(a, p) ( q) -*p(a, p) >, p(,8q)

Moreover, since min{p(a, p): p E P - } = a for each a > 0, inf p((0, 1] X X -) = 0. On the other hand, if P2 approaches 0 as x increases towards sup X -, then a 0.

The following companion of A12 suffices for this. A13. If (a, x) E (0, 1] x X -, then (a, x) >- (1, y) for some y E X.

THEOREM 4. Axioms A1-A3, A6, A7, A9, A10, A12, A13, Cl, and C2 imply that there are real valued functions PI on [0, 1] and P2 on X - such that, for all (a, p), (/3, q) EA,

(a, p) (3q),> XpI (at) _PO) dp (x) > pl ) I P2(x) dq (x),

with pl(1)= 1, pi(0)= 0, Pi increasing in a, P2 positive and decreasing in x, and inf p2(X- ) =0. Given these aspects of PI and P2' PI is unique and P2 is unique up to similarity transformations.

PROOF. Begin with p of Theorem 3, including inf p((O, 1] X X) = 0. Define Pi, P2 and aI as before:

p(a,x) = pI (a)p2(X) + a1(a), P2(x) = p(1,X),

Pi(l) = 1, Pi() = a(') = a1(0) =0.

By A13, pI(a)P2(X) + a1(a) > P2(Y) for some y, given a > 0 and x. Let inf P2(Y) = K > 0. Then pl(a)K + a1(a) > K for all a > 0, hence 0 > K, hence K = 0. Thus a1(a) > 0 for all a > 0. If a1(a) > 0, p2(x) near 0 gives pl(a)p2(x) + a1(a) > p2(x), or

(a,x) >- (1,x), contrary to A2. Therefore a1((a) = 0 for all a. The rest of Theorem 4 follows readily from A2, A3 and the uniqueness part of Theorem 3. Q.E.D.




Linearity in Loss Probability

The following strengthenings of A6 and A9, applied to all A E (0, 1] and p *, q *, r* E A, will be used to get p (a) = a. One can view p*, and so forth, as measures with probability 1 on X- U {0}.

A6*. If p* > q* > r* then Ap* + (1-A)r*,q* for some 0 < A < 1; A9*. If p* > q* then ,p* + (1 - ,)r* > puq* + (1 - )r*. New versions of CI and C2 are also needed. The following apply to all p* E A, all

simple s* E A, all Y c X- U {0} in the algebra, and all x E X- U {0}. C1*. If p*(Y) = 1, then p* >x ify x for ally E Y, and x cp* if x y for ally E Y; C2*. If p* > s* then p[*O] > s* for some y E X - U {0}. The conclusions of the next theorem follow readily from Theorem 3 in Fishburn

(1975) when p* E A is written as ap + (1 - a)O, where 1 - a = p*({0}) and p =

p(- _O)* The form for p is tantamount to Assumption 3 in Luce (1981).

THEOREM 5. Axioms A1-A3, A6*, A9*, C1* and C2* imply that there is a nonnegative, decreasing function P2 on X - U {0} with P2(O) = 0 such that, for all (a, p),(1, q) e A,

(a, p)> (/3, q) < af P2(X) dp(x) > X fP2(x dq(x),

and P2 which satisfies this is unique up to similarity transformations.

5. A Power Function for A

For analytical simplicity, the power form p2(x) = I x19 is considered only when X --(-so, 0), and then within the context of Theorem 4, where p2(x) approaches 0 as x approaches 0 from below. One can axiomatize the power form for the setting of Theorem 5 in a slightly simpler way, but what I do here will suffice for Theorem 5.

A new axiom, patterned after Luce (1981), is based on scale changes in outcomes. For simple p E P -, let p(X) be the simple measure in P - whose point probabilities are defined by

p(X)(x) = p(Xx) for all x E X (X > 0).

For example, if p is an even-chance gamble for - 1 or -5, then p(2) is an even-chance gamble for - 1/2 or - 5/2, and p( 1/2) is an even-chance gamble for -2 or - 10.

The following applies to all simple p, q E P - and all X > 0. A14. If p.- q then p(X) , q(X). This says that , is preserved under uniform rescaling of outcomes. Like other

axioms, its plausibility is subject to empirical investigation.

THEOREM 6. The axioms and representation of Theorem 4 in conjunction with A14, X - = (- o, 0), and P2( - 1) = 1 imply that there is a real 9 > 0 such that p2(x) = Ix19 for all x E X-.

PROOF. It suffices to work with a = 1 in the context of Theorem 4. Let x denote the measure that has probability 1 for x E X -. Given x < y < z < 0, consider x andy as fixed, and let A(z) be the unique number in (0, 1) for which y-A(z)x + [1 -(z)]Z'. Then

P2(Y) = p'(Z)P2(X) + [ - (Z) ] P2(Z)

As z approaches 0, P2(Z) -*0 and A(z) increases. Let A* = sup{ A(z): 0 < z <y}. Then

P2(Y) = p'*P2(X), with M* < 1 since P2(X) > P2(Y). It also follows from A14 that




P2(AXY) = p*P2(>X) for every X > 0. Therefore

P2(Y)P2(AX) = P2(XY)P2(X) for x <y < O. > O.

Fix a scale for P2 by setting P2(- 1) = 1. Then y = - 1 in the preceding equation gives P2(XX) = P2( - )p2(X) for X > 0 and x < - 1, and x = - 1 in the same equation gives P2(AY) = P2( W2(Y) for X > 0 and - 1 < y < 0. It then follows from Theorem 3 (p. 41) in Aczel (1966) that p2(x) = Jx10 for some 9 > 0. Q.E.D.

As noted earlier, the power function of Theorem 6 in conjunction with the linear Pi form of Theorem 5 gives p(a, p) = af IIx 1dp(x), a form that has been used in a generic or specific (9 = 1, 2) way by several investigators. The more general form implicit in Theorem 6, namely p(a, p) = pI(a)fIx19 dp(x), adds flexibility by not requiring Pi to be linear. For example, limited evidence from Coombs and Lehner (1981) suggests that Pi increases at a decreasing rate for many individuals, as would be the case if p(a, p) + p(a + 2A, p) <. 2p(a + A, p), or p(a + 2A,p) - p(a + A, p) < p(a + A, p) - p(a, p) for A > 0.

6. Summary and Acknowledgments

Although risk is pervasive in society, few attempts have been made to systematically characterize perceived risk by an axiomatic formulation based on "is at least as risky as" applied to a set of probability distributions on nonfavorable outcomes. My purpose here has been to consider general conditions that may govern perceived risk that are based on a conventional understanding of risk and are influenced by empirical studies on perceived risk for gambles. The numerical representations obtained from these conditions were guided by prior axiomatizations of risk and preference, common-sense models of risk that have appeared in the literature, and empirical findings on the effects of loss probability and loss amounts on risk. Effects of gains on perceived risk will be examined in a sequel.'

'I am indebted to Duncan Luce for suggesting this study, and to Clyde Coombs and associates for their pioneering research on risk measurement.

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risk analysis || foundations of risk measurement. i. risk as probable loss

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