Journal of Risk and Uncertainty, 1: 333-347 (1988)
© 1988 Kluwer Academic Publishers
UZI SEGAL
University of Toronto
AVIA SPIVAK
Ben Gurion University
Key words: anticipated utility theory, conditional risk premium, ambiguous probabilities.
Abstract
This paper discus.ses two problems, (a) What happens to the conditional risk premium that a decision makeris willing to pay out of the middle prize in a lottery to avoid uncertainty concerning the middle prize outcome, when the probabilities of other prizes change? (b) What happens to the increase that a decision maker is willing to accept in the probability of an unpleasant outcome in order to avoid ambiguity concerning this probability, when this probability increases? We discuss both problems by using anticipated utility theory, and show that the same conditions on this functional predict behavioral patterns that arc consistent both with a natural extension ofthe concept of diminishing risk aversion and with some experimental findings.
Expected utility theory used to be the most widely used theory of decision making under uncertainty. It owes much of its popularity to its simplicity. Every contingent outcome is evaluated by the utility indicator and then multiplied by its own probability, without any dependence on other parts ofthe lottery. The separability of outcomes and linearity in the probabilities are at the heart of this theory.
However, ifthe theory, due to the linearity or separability, fails to account for empirical phenomenan, an alternative theory must be used.
In this paper we consider two cases. The first one cannot be dealt with by expected utility theory because ofthe separability of outcomes, while the other one contradicts the linearity assumption. These two cases seem to be totally independent, but as is shown by our analysis, they are correlated. In the first case the following question is asked: How will the attitude of decision makers towards one prize in a lottery depend upon the levels and probabilities ofthe other prizes in the lottery?
We are grateful to Larry Epstein and Mark Maehina for their comments and suggestions. Support from
SSHRC Grant #410-87-1375 is gratefully acknowledged.

334 SEGAL AND SPIVAK
More specifically, we consider a lottery with three payments, and make the middle prize riskier by transforming it into a lottery. The original lottery is x with probability/?,>> with probability^, and 2 with probability r, where/?
+ q + r = 1, andx
<y <z.
The riskier lottery is the same, but (j',^) is replaced by j ' - e with probability q/2, andy + e with probability ql2 (see figure 1), If the decision maker is riskaverse, then he is willing to pay some positive conditional risk premium out of this middle prize to avoid this extra risk. This conditional risk premium n is the number that makes him indifferent between the two lotteries at the upper and lower panels of figure 1, We then inquire into the question of how this conditional risk premium n depends on the probabilities ofthe other prizes. Intuitively, if the decision maker is richer, or, equivalently, if the probability ofthe larger prize is higher, he will be willing to pay a lower conditional risk premium. However, by its structure, the way in which expected utility theory treats the middle prize and hence the conditional risk premium associated with the extra risk is independent ofthe other prizes. Indeed, by expected utility theory, the two panels of figure 1 are equally attractive iff pu{x) + qu{y - n) + ru{z) = pu{x) + | WCF - e) + |wO; + e) + ru{z)
» 2u{y - n) = u{y - e) + u{y + e).
In other words, according to this theory, n is a function ofy and e, but not ofx, z,p, and r (as long as the sum/? + r does not change). This seems to be too strong a restriction on decision makers' behavior. For a more detailed discussion of this separability property, see Machina (1982b, p, 1072-1073),
The second case we consider is the case of ambiguous probabilities. One possible way to handle such probabilities is to say that the probabilities themselves are uncertain and are obtained by lotteries on probabilities (see Segal, 1987a), The result is a two-stage lottery. In the first stage, the probabilities are chosen, while in the second stage the prize itself is determined, using the probabilities that have been obtained in the first stage. Intuition, as well as empirical results (Ellsberg,
1961; Becker and Brownson, 1964) suggest that decision makers are willing to pay to avoid the extra risk imposed by ambiguity. Furthermore, this extra risk premium should depend on the level of the ambiguous probability. Indeed, such dependence was found empirically by Hogarth and Kunreuther (1985), Expected utility theory, however, does not distinguish ambiguous from nonambiguous probabilities. To see this, consider a person who buys insurance against a car accident that may result in a loss of Z,, The probability ofthe loss is uncertain. It is 0.5% with probability Vi, and 1.5% with probability Vi.
By expected utility theory, this lottery is reduced to a 1% chance of losing L,'
The inadequacy of expected utility theory in treating those cases calls for an alternative approach. Indeed, recent years have seen the emergence of several new theories for evaluating uncertain prospects, one of which, called anticipated utility

NON-EXPECTED UTILITY y-Ti y
335
X y-e y y+E Z
Fig. I.
Original lottery X = (x,p;y,q;z,r) contrasted with lottery Y = (x,p\y — e,q/2:y + e.q/2;z,r), showing conditional risk premium n.
theory, generalizes expected utility theory in a very natural way. According to this theory, decision makers no longer weight uncertain prospects by their probabilities, but rather distort the cumulative distribution function, and use the new function to rank uncertain prospects. This theory was first suggested by Quiggin
(1982), who named it anticipated utility,^ Yaari (1987) gave an independent axiomatization of anticipated utility for the case of linear utility (but see also
Segal, 1984),
It turns out that anticipated utility theory allows for more general types of behavior, and thus can handle both of our cases according to the abovementioned intuitive lines. The main result of this paper, however, is deeper than

336 SEGAL AND SPIVAK that. As mentioned above, diminishing risk aversion (with respect to wealth) suggests that the conditional risk premium out ofthe middle prize in the first case will decrease as a probability mass is shifted from the lower prize to the higher one.
Empirical evidence suggests that the extra risk premium that decision makers are willing to pay in order to avoid the ambiguity concerning the probability of a certain loss is relatively decreasing when this probability increases. These two phenomena seem, on their faces, to be independent and uncorrelated. It turns out that the same conditions on the anticipated utility functional imply both of these results. This indicates that within the anticipated utility framework, the concept of risk aversion is wider and more useful than that of expected utility.
This paper is organized as follows. Section 1 is an introduction to anticipated utility theory. In sections 2 and 3, the first and the second cases are analyzed. The results are discussed in the light ofthe theoretical and experimental literature, the appendices contain formal counterparts and extensions of the main part of the paper.
1. Definitions and notations for anticipated utility
We consider here only discrete and finite random variables. A random variable is denoted by a vector of the form (A'I,/?,; . . .
^,x„,p„) where X| < . . . < Xn, and Z/?, = 1.
Such a vector represents a lottery yieldingx, dollars with probability/?,,/ = 1,...,«.
We denote the set ofall such lotteries by Li,Li = j(X|,/?i;... ;Xn,/?«):X|, ... ,x„E.
R,X|
< . . . < x „ , / ? , > 0, / = 1,...,«, Z/?, = 1|.
O n L , there exists a complete and transitive preference relation >-X>- YiffX>Y but not Y>-X, and X ~ Kiff A ' ^ T a n d Y>X.
The function V: L, ^ " R represents the order > if F(A^ > ViY) iffA'^ K T h e most celebrated ofall the possible representation functions is the expected utility functional, given by
K(x,,/?,;• • ••,x„,p„) = Y.Pi"iXi)(1)
This theory, despite its simplicity and its normative appeal, fails to explain some well-known evidence, such as the Allais paradox or the common ratio effect
(Allais, 1953; MacCrimmon and Larsson, 1979). In recent years, several alternatives to this theory have emerged (Kahneman and Tversky, 1979; Machina,
1982a; Chew, 1983). Quiggin (1982) suggested the following generalization of (1), presented in two equivalent forms:
I,/?,;- • ••,x„,p„) = w(X|) + 2^
(2)

NON-EXPECTED UTILITY 337 where «(0)=0, the transformation function /: [0,1] ^ [0,1] is continuous and increasing, and satisfies/(O) = 0 and/(I) = 1. Note that w h e n/ is linear, (2) is reduced to the expected utility operator. Quiggin's axioms imply in addition that
/('/2) = yi.
Others have claimed that if the decision maker dislikes risk, then/ must be convex. These issues are discussed in section 4.
To illustrate the difference between expected utility theory and anticipated utility theory, consider the lottery (x,,/?,;.. ;x«,/?J. According to expected utility theory, the value of this lottery is the sum ofthe utilities of its prizes, each of them being multiplied by its probability. Rewriting this expression, one obtains
+ • • • + u{Xi)pi + • • • + u{x„)p„
Y^Pj- Y.P)\ + • • • r "
+ U{XM Y.Pj-
" "I
Z / ' > + • • • + l*iXn)Pn-
Anticipated utility theory just takes a transformation/of these sums of probabilities. So, while the expected utility ofthe lottery (x,/?;>',^;z,/-) where x <j; < z is pu{x) + qu{y) + ru{z), its anticipated utility is u{x)[\ - f{q + r)\ + u(y)\f(ci + /•) f{f)\ + M(2)/(r). Note that the value of a prize does not depend only on its probability, but also on its relative rank within the lottery. Thus, the anticipated utility of
{y,q\z,r\w,p) where >» < z < w is u(y)\\ - f(r + /?)] + u{,z)\f{r + /?) -/(/?)] + u{w)f{p).
A special case of anticipated utility is when u is linear. This approach was developed by Yaari (1987).
2. Measuring risk aversion: the case of outcome uncertainty
In this section we suggest a new measure for aversion of Rothschild-Stiglitz
(1970) type changes in risk. Throughout this section we assume that/is convex, and following Yaari (1987) we assume a linear utility function u (though not a linear transformation function/).
We start with a lottery X,X = {x,p:,y,q;z,r) and transform it to the lottery Y,Y =
{x,p:,y - e,q/2;y -I- e,^/2;z,r).
Y is obtained from X by a mean-preserving spread, and hence we expect a risk-averse decision maker to preferZto Y.
It is indeed easy to verify that if/ is convex (and u linear), the decision maker is averse to meanpreserving spreads. By the continuity of/ there exists n > 0 such that Z =
{x,p;y — T\,q;z,r) ~ Y.
(See figure 1.) Following Machina (1982a), this n is called the conditional risk premium, and our goal in this section is to show that if ^ is sufficiently small, n is proportional to f"{r)/f'{r) (provided the utility function u is linear).^

338 SEGAL AND SPIVAK
By (2), x[l -fiq + r)] -Kj - e)[/(^ + r) - / ( | +
+ 0'
K(Z) =x[\-fiq and hence, efiq + /-)
+ r)]+iyT T ) [ / ( ^ + r) -
Following Pratt's (1964) procedure, we take Taylor's expansion with respect to q around 0 to obtain efir) + qef'ir) + ^ ef'ir) 2e/(r) qef'ir) - -^e/"(r)
+ efir) + oiq') = 7r/(r) + qi^f'iq) - nfir) + oiq).
Collecting terms, we obtain
Note that under the assumptions about the monotonicity and convexity off, n is positive. For a generalization of (3), see appendix B.
An immediate conclusion from (3) is that the conditional risk premium TT depends on the level ofthe probabilities of other prizes (i.e., r).
This is an obvious departure from the standard expected utility model, in which n is a function of e and;;, but not of r The natural question to follow is therefore what happens to TT when r increases.
One possible way to evaluate changes in r (and accordingly in/?) is by assuming that higher values ofr represent higher levels of wealth. Consider forexampie the case of uncertainty with respect to employment, and reinterpret prospects X, Y, and
Z accordingly./? is the probability of subsistence level of income (x), r is the probability of being fully employed (z), and q is the probability of partial employment with remuneration y.
Our analysis suggests that workers with better chances of full employment are less bothered by more uncertainty in the state of partial employment as represented by prospect Y.
This may be due to the fact that decision makers feel more secure when q is higher, or in the same vein, they feel richer because their expected wealth is higher.

NON-EXPECTED UTILITY 339
Given this interpretation of changes in r, it seems to be in line with the assumption of decreasing absolute risk aversion (see Arrow, 1974; Ross, 1981; Machina,
1982b; and Epstein, 1985) that TT is decreasing with r, and hence, thatf'/f is decreasing with r.
This result is in agreement with one ofthe major assumptions of
Segal (1987a), where decreasing/"//'' is essential to solve the Ellsberg paradox.
This analysis seems to contradict the well-known fact that people are less riskaverse in the higher prize than in the lower prize. Formally, if (x — n,,/?; w,^; z,p) ~
(x - e,/?/2; X + e,p/2; w,q:, z,p), and (x,/?; w,q:, z T\2,p) ~ (x,/?; w,q; z - e,/;/2; z + e,/?/2), where x < w < z and 2p -\- q = 1, then by decreasing risk aversion it follows that the conditional risk premium n, out ofx is larger than the conditional risk premium TI2 out of z."* (See figure 2.) However, in our case, the conditional risk premium n out of^" is larger when >> is the higher prize {r=0), and smaller when}' is the lower prize (/? = 0). These two observations do not contradict each other because the important factor in the first case is not whether a certain outcome is the highest possible outcome, but rather the decision maker's attitude towards it. TT, is greater than nj not because z is the highest outcome, but because z is greater than
X. According to expected utility theory, the same conditional premium will be paid even when z is the smallest possible outcome. In our case, on the other hand, we are interested in the cross-effect of changes in the probabilities of other outcomes on the conditional risk premium out of j , whose own probability did not change.
This analysis is impossible in the expected utility framework, where cross-effects do not exist.^
Another aspect of cross-effect was discussed by Machina (1982a). In theorem
5(iv) he proved that Hypothesis II implies that if (x,/? y;y,q + y:,z,r) ~ (x,/? — y + a:,y,q + y ~ ci ~ P^z,/" + P), then da/d^ is decreasing with y. By anticipated utility theory, da _ f'{r)[z-y] which is indeed decreasing with y provided/is convex. This question is of course different from our problem, since we are interested in shifting a probability mass from X to z (and not from ^^ to x and z), and in its effect on y.
The anticipated utility theory thus allows for an explicit treatment of a phenomenon lying in the background ofthe expected utility model but that cannot be fully formalized within its framework. Even though it is natural to assume that the point of reference changes as r moves from 0 to 1 — q (especially when q is small), there is no way to measure the effect of this change on the conditional risk premium n. This connection becomes apparent and explicit in the anticipated utility model.
3. Measuring risk aversion: the case of probability ambiguity
In this section we consider the simple lottery A" = (—x,/?; 0,1 — p),x > 0, and make it riskier by turning the probability/? itself into a random variable. One possible

340
P P
X - T T .
X
X+E
SEGAL AND SPIVAK z+e
Eig.
2 Decreasing risk aversion implies that conditional risk premium n, out ofx is greater than conditional risk premium ni out of z.
way to analyze this situation is by looking at the uncertain probability as a twostage lottery. (See Segal, 1987a. For a review of anticipated utility theory and twostage lotteries, see appendix A). From now on we assume that the decision maker satifies the strong independence axiom, but not the reduction of compound lotteries axiom. That is, a decision maker is not necessarily indifferent between a compound lottery and its actuarially equivalent one-stage lottery. The exclusion of this axiom is mandatory in order for the uncertain probability to be different from

NON-EXPECTED UTILITY 341 the certain one, since this axiom implies that A = {{-x,p + e;O,l -/? z),Vr.,
{~x,p e;O,l - /? + e),V2) ~ (-x,/?;O,l — p) = X.
There is some empirical evidence to support the exclusion ofthe reduction axiom. Schoemaker (1980) found that people are not indifferent between a distribution over probabilities and its average. For a discussion of his findings and anticipated utility theory, see Segal
(1987c).
The aim of this section is to find what increase in the probability/? a decision maker is willing to take to avoid the uncertainty concerning this probability. This risk premium, paid in terms of probability and therefore (following Arrow, 1974) called the probability premium, satisfies the following condition:
Y = {-x,p + TT;0,1 - /? - n) ~ {X^Jh-X^J/i) = A, (4) where X_^ = {-x,p e;O,l - /? + e) and Z+, = {-x,p -\ e;O,l - /? - e), e > 0. Let f(p) = 1 - / ( I -P} Obviously/(O) = 0,/(l) = 1,/'(/?) = / ' ( l -/?), and f'Xp) =
- / " ( ' ~P^ (Qu'ggip- 1982, used this function. See also Segal, 1987b, for a further discussion of/and/.) We compute the value of^ by using the certainty equivalent mechanism (see appendix A). It follows that
_J = u-\u{-x)f{p e)),
,J = u-\u{-x)f{p -H e)), and hence
V{A) = u{-x)f{j, e)[l -7(1/2)] -f ui-x)f{p
Equivalence (4) now implies
7(/' + n) =7(P - e)[l -fiV2)\ 7
Taking Taylor's expansion around /? we obtain
7 fiP) + T^f'{p) + o(n) and hence
The first term on the right-hand side of this equation is positive, a s / being con-

342 SEGAL AND SPIVAK vex implies that/is concave. It thus follows that for sufficiently small values of e,n is positive.* For a generalization of (5), see appendix B.
One might wish to measure the risk and the probability premium as proportions ofthe probability/?. Define n = n//?, e = e//?, and divide (5) by/? to obtain
Pf'XpVf'ip) is our measure of relative risk aversion.
In section 2 we saw that it is natural to assume f"(p)/f'{p) is decreasing with /?.
We now a_ssume, in addition, thatpf"{p)/f'{p) is increasing with/?. It thus follows thatboth7"(/')/7'(;^) 3indpf'\p)if'{p) are decreasing. (Note that/77' is negative.)
In other words, in absolute and relative terms, the decision maker is willing to pay less to avoid the uncertainty regarding the probability of an unpleasant event when this probability increases.
Hogarth and Kunreuther (1985) investigated the issue of ambiguous lotteries in a slightly different framework. They showed that decision makers are willing to pay extra positive amounts of money to avoid uncertain probabilities concerning losses, over and above the insurance premium they are willing to pay to avoid the basic risk itself Hogarth and Kunreuther (1985) further found that the ratio between this extra premium and the insurance fee declines as the probability ofthe unpleasant event increases. This empirical finding parallels ours, that the relative probability premium decreases when the probability of damage increases.
The main result of this paper is thus established. We proved that the empirical findings of Hogarth and Kunreuther concerning the extra premium decision makers are willing to pay to avoid probability ambiguity agree with the fundamental rule of decreasing risk aversion. Moreover, these results show that in the anticipated utility framework the concept of risk aversion covers a wider range of problems, and it can be extended to phenomena that cannot be analyzed by expected utility theory.
4. A note on the literature
The anticipated utility literature concerning the concept of risk aversion concentrated almost entirely on the conditions that guarantee a rejection of a Rothschild-
Stiglitz mean preserving increase in risk. It is agreed that sufficient conditions for this are concave w and convex/(Chew, Kami, and Safra, 1987; Segal 1987b). It is still an open question what are the necessary conditions for such a rejection.
The connection between the convexity of/ and risk aversion leads to the conjecture that if/2 is a convex transformation of/, then/2 is associated with greater aversion to risk. Let/2 = go/,. Obviously, /"//i = f\g"/g' + /"//I, and ifg is convex, fyfi > f"\/f'\.
Yaari (1986) shows that if two decision makers have linear utility functions and their decision weights functions are / and / respectively.

NON-EXPECTED UTILITY 343 then decision maker 2 is more risk-averse than decision maker 1 in the following sense. LetX= (X|,/?i;...
^,x„,p„).
For decision maker/,/ = 1,2, define 9, implicitly by
(0,1 — qi;Xn,qi) ~ XJ Yaari (1986) proves that if/ is a convex transformation of/,, then qi > q\.
Yaari also shows that in that case, decision maker I's certainty equivalent of each lottery X is greater than that of decision maker 2. However, the converse of this is not true: to make CE2(A^ < CE|(X) for all X, it is sufficient that/2
In contrast with other writers, we tried to derive the risk aversion measure from the conditional risk premium. Our main aim, however, was to investigate the connection between the conditional risk premium and other parameters, and to show the hidden connection between outcome uncertainty and probability ambiguity.
Notes
1, 0,5-0.005 -I- 0.5-0.015 = 0,01,
2, Chew. Kami, and Safra (1987) use the term expected utility with rank-dependent probabilities for the same theory,
3, This assumption that (--.y.q;-) is replaced by (-\y - &.ql2\y + e,q/2;-) is made for the sake of simplicity. We could get the same results with Y= (—-.y - e,S/2;y.q - ?>;y + e.6/2;-).i,e,, when only part of the probability mass of y is shifted.
4, This result remains true even without the expected utility hypothesis—see Machina (1982a),
5, Of course, this analysis is possible in anticipated utility theory only if the location of>' on the cumulative distribution function has been changed. In other words, it requires that a probability mass is shifted from the left of^; to the right of y, or vice versa,
6, By using an alternative reasoning one can show that for all e > 0, TT must be positive (Segal, 1987a),
This result does not contradict our expression for n, because the Taylor's approximation becomes less accurate as e increases,
7, Using Raiffa's (1968) terminology, (0,1 qi\Xn.q>) is a basic reference lottery,
8, Even if/| and/2 ^re both convex and for a\\p,fi{p) <f\(p), it is not necessarily true that/2 '^ ^ co"" vex transformation of/|, since there may be a point 0 < /) < 1 for which f\(p) = fjip).
References
Allais, M, Le compartenient de l'Honime Rationel devant le Risque, Critique des Postulates et Axiomes de l'Ecole Americaine, Econometrica (Vol, 21, 1953). pp 503-546,
Arrow, K.J, Essay.s in the Theory of Risk Bearing.
Amsterdam: North Holland, 1974,
Becker. S,W, & Brownson FO. What Price Ambiguity? On the Role of Ambiguity in Decision Making,
Journal of Political Economy (Vol, 72. 1964), pp 62-73,
Chew, S,H, A Generalization ofthe quasilinear Mean with Applications to the Measurement of Inequality and the Decision Theory Resolving the Allais Paradox, Econometrica (Vol, 51.1983) pp 1065-
1092,
Chew. S,H,. Kami. E,, & Safra. Z, Risk Aversion in the Theory of Expected Utility with Rank-
Dependent Probabilities yoMwa/ of Economic Theory (Vol, 42, 1987) pp 370-381,
Ellsberg. D. Risk, Ambiguity and the Savage Axioms.
Quarterly Journal of Economics (Vol, 75, 1961) pp
643-669,
Epstein. L. Decreasing Risk Aversion and Mean Variance Analysis, Econometrica (Vol, 53, 1985), pp
945-961,
Hogarth. R.M,, & Kunreuther, H,. Risk Ambiguity, and Insurance, Center for Risk and Decision Processes. The Wharton School. University of Pennsylvania. Philadelphia, 1985,

344 SEGAL AND SPIVAK
Kahneman, D.,&Tversky, A Prospect Theory: An analysis of Decision Under Kisk. Econometrica (Vol.
47, 1979), pp 263-291.
MacCrimmon, KR., & Larsson, S, Utility Theory: Axioms versus 'Paradoxes'. In: M. Allais and O.
Hagen eds.
Expected Utility Hypotheses and the Allais Paradox.
Holland: D. Reidel Publishing Company, 1979.
Machina, M.J. (1982a). 'Expected Utility' Analysis Without the Independence Axiom.
Econometrica
(Vol. 50, 1982), pp 211-m.
Machina, M.J. (1982b). A Stronger Characterization of Declining Risk Aversion.
Econometrica (Vol. 50,
1982), pp 1069-1079.
Pratt, J.N. Risk Aversion in the Small and in the Large.
Econometrica (Vol. 32, 1964), pp 122-136.
Quiggin, J. ATheory of Anticipated \JU\i\y. Journal of Economic Behavior and Organization (Vol. 3, 1982), pp 323-343.
Raiffa, H.
Decision Analysis.
Reading, MA: Addison-Wesley, 1968.
Ross, S.A. Some Stronger Measures of Risk Aversion in the Small and the Large with Applications.
Econometrica (Vol. 49, 1981), pp 621-638.
Rothschild, M., & Stiglitz, J. Increasing Risk, I. A Definition.
Journal of Economic Theory (Vol. 2, 1970) pp 225-243.
Schoemaker, P.J.H.
Experiments on Decision Under Risk.
Boston: Martinus Nijhoff, 1980.
Segal, U. Nonlinear Decision Weights With the Independence Axiom. UCLA working Paper #353,
1984.
Segal, U. (1987a). The Ellsberg Paradox and Risk Aversion: An Anticipated Utility Approach.
International Economic Review (Vol. 28, 1987), pp 175-202.
Segal, U. (1987b). Some Remarks on Quiggin's Anticipated Utility.
Journal of Economic Behavior and
Organization (Vol. 8, 1987), pp 145-154.
Segal, U. (1987c). Two-Stage Lotteries Without the Reduction Axiom. University of Toronto, Department of Economics, #8708, 1987.
Yaari, M.E. Univariate and Multivariate comparisons of Risk Aversion: A New Approach. In: W.H.
Heller, R. Starr, and D. Starrett, eds..
Essays in Honor of Kenneth J. Arrow.
Cambridge: Cambridge
University Press, 1986.
Yaari, M.E. The Dual Theory of Choice Under Risk.
Econometrica (Vol. 55, 1987), pp 95-115.
Appendix A
L e t L j = \ { X , , p , ; . . . ; X ^ , p J , ^ P i = 1 , P u - - , P n , > 0 , X , , . . . , X ^ e
L|j. Elements of L2, called two-stage lotteries, are denoted hy A, B, etc, A lottery
A G L2 yields a ticket to lottery Xi with probability /),, / = 1,,,.
,m.
More specifically, at time f, the decision maker faces the lottery (I,/?,;.., ;OT,/7J.
Upon winning the number /, he participates at time fj > t, in the lottery A',. It is assumed that the decision maker does not discount the prizes in the second-stage lotteries. Thus, once he knows that he won acertainamountof money, the actual time at which he receives this prize does not make any difference to him,
Let>;2 be a complete and transitive preference relation on L,. Elements of L; can be naturally written as elements of L,, where (x,,/?,;,,, ^,x„,p„) and ((x,,!),/;,;,,,;
{x„,\),p„) are equally attractive. The subscript 2 is therefore omitted and the preference relation over o n e - and two-stage lotteries is denoted by >, A similar discussion holds for mixed lotteries, where the set of prizes is R U L|.
This last discussion is relevant for lotteries of the form ((x,,!),/?,;...
•{x„,\),p„) only. So far nothing restricts the decision maker in comparing other lotteries in Lj with lotteries in L,. The following two axioms deal with such comparisons.

NQN-EXPECTED UTILITY 345
/.
Reduction of Compound Lotteries Axiom (RCLA): If the decision maker is indifferent to the resolution timing ofthe uncertainty and to the numberof lotteries, then he may assume both stages to be conducted at time f |, Thus, a two-stage lottery is reduced to a simple one-stage lottery. Formally, let A^ = {x^ulA', • • • '••^n^p'n),'
= 1,,,, ,m.
{x\,p,p\; • • • •,xl^,p,p\;, • • • •,xr,p^pT; • • • ;x:^.p^p';;j.
( A , 1 )
2. Strong Independence Axiom (SIA).
The relation >- on Lj induces several relations on L,. The strong independence axiom assumes that these relations coincide and are equal to > on L|, Formally,
>{X,,p,-- • --Z^pi-- • --X^^pJ^Y^Z (A.2)
The expected utility functional (1) is the only continuous function satisfying both (A,l) and (A,2), Anticipated utility is compatible with RCLA or SIA. Some empirical evidence concerning two-stage lotteries suggest that decision makers accept SIA, but not necessarily RCLA, (See Kahneman and Tversky, 1979; and
Segal, 1984).
Let CE(A^ be the certainty equivalent of A', given implicitly by (CE(AO,1) ~ X.
If
>; satisfies SIA, then
(A-,,/?,; • • • \X^,pJ ~ (CE(X,),/;,; • • • •CE{XJ,pJ.
(A.3)
If >; can be represented by the anticipated utility function of (2), then CE(A^ = u'\V{X)).
Let (X,,/?,;...
\X„„p„) £ Li and assume, without loss of generality, that
CE(A',) < , , , < CE(X^), (A.3) thus implies that
{X,,p,- • • • •X^,pJ ~ {u-\V{X,)},p,- • • • •u-\V{XJ),p„,).
(A.4)
(A.4) holds of course for expected utility theory, being a special case of anticipated utility theory. If > satisfies SIA and can be represented by (1) (the expected utility functional), then by (A.4)
On the other hand, if > satisfies SIA and can be represented by (2) (the anticipated utility functional), then by (A,4)

346 SEGAL AND SPIVAK
= u{x\)+ Z
=k
"/-I
Y
(A,5)
/=k
Appendix B
This appendix shows that the calculations ofthe risk premiums developed in sections 2 and 3 hold for more general random variable. As the set of discrete and finite random variables over [a,P] is dense in the set of all random variables over
[a,P], we restrict the discussion to finite random variables.
Let X = {x,p;y,q\z,r), Y = {x,p;y - ei,y^q\ ,.. ;y ^„.,'^„q\z,r) such that Zy/ = ',
Y,e, = 0, and let Z = (A,/;;y T\,q\zj) such that Y ~ Z.
Let YO = 0, By (2),
V{Y) = x[l - / ( I i
,) - f(r + q
V{Z) = x[\ - / ( I - / ; ) ] +{y- j^)\f{q + r) - f{r)] + zf{r).
Take Taylor's expansion with respect to q around 0 to obtain
-qf'{r) X Y,e, " ~f"{r)D + o{q'~) = -T^J where nf{r) + o{q).
Collecting terms we obtain

NON-EXPECTED UTILITY 347 which generalizes the expression of (3).
Consider now the model of section 3. Let Xj = {-x,p + Y,e;O,l - p Y/e), Y =
{-x,p + n;O,l - p - -nXA = (X,,^,;...
•,X„,q„) such that Z^, = 1, I:^,Y, = 0, and A ~
Y.
By (2) and (A.5), it follows that fip + Y,e) + Z Uip + Y,e) -f{p + Y
' = 2
Taking Taylor's expansion around p with respect to e and n, we obtain where and hence t i = 2 which generalizes the expression of (5).