order indifference and rank-dependent probabilities u. &gal, order indifference ... and z, and...

Journal of Mathematical Economics 22 (1993) 373-397. North-Holland

Order indifference and rank-dependent probabilities

Uzi Segal*

University of Toronto, Toronto, Ont., Canada

Submitted August 1989, accepted June 1992

This paper presents a new axiomatization of the anticipated utility (also known as the rank-dependent) model for decision making under uncertainty. This axiomatization is based on the analysis of two-stage lotteries with the compound independence axiom but without the reduction of compound lotteries axiom. The major new axiom assumes indifference to the order at which uncertainty is resolved in some simple two-stage lotteries.

1. Introduction

Recent years saw the emergence of several axiomatized generalizations of expected utility theory. One of the most widely used of these new theories is Quiggin’s (1982) anticipated utility theory, also known as expected utility with rank-dependent probabilities. Let the lottery X =(x1, pl;. . . ; x,, p,,) yield $xi with probability pi, 1 = 1,. . . , n and assume, without loss of generality, that 05x,5 ... sx,. Denote p. =O. According to anticipated utility theory, the value of the lottery X is given by

(1)

where the utility function u is strictly increasing, continuous, and unique up to positive linear transformations and the probability transformation function g: [0, l]-[0, l] is strictly increasing, continuous, onto, and unique.’ Since its introduction, its theoretic foundations and behavioral implications were intensively investigated [see, for example, Yaari (1987), Chew et al. (1987), R6ell (1987), Lute (1988), Segal (1984, 1987a, b, 1989, 1990), Chew

Correspondence to: Uzi Segal, Department of Economics, University of Toronto, 150 St. George Street, Toronto, Ontario M5S lA1, Canada.

*I am grateful to Chew Soo Hong, Larry Epstein, Kim Border, and two anonymous referees for suggestions and comments.

‘See section 2 for a discussion of the two different forms in which this functional appears in the literature. Note that when the function g is linear the anticipated utility functional reduces to expected utility: cpiu(xi).

03044068/93/$06.00 0 1993-Elsevier Science Publishers B.V. All rights reserved

374 U. &gal, Order indifference and rank-dependent probabilities

and Epstein (1990), Chew (1989), and Karni and Safra (1987)]. Several sets of axioms for this theory were suggested, and this paper is to offer another such set. Each of the different axiomatizations reveals another aspect of the theory, and usually leads to new and different applications.

Most axiomatizations of this theory, as those of other alternatives to expected utility theory, seek to weaken the mixture independence axiom, that for all three lotteries X, Y, and Z, and tl~(O, 11, CYX +( 1 - cc)Z is preferred to cry +(l -a)Z if and only if X is preferred to Y. [See Quiggin (1982) and Chew (1989).] Yaari (1987) and Roe11 (1987), on the other hand, replaced the mixture independence axiom by its dual presentation: For all lotteries X=(xr,~r;...;x,,,~~), Y=(Y,,P~;...;Y,,P,), and Z=(z,,p,;...;z,,p,) and a~(O,l], the lottery (ax,+(l-~)z,,p,;...;ax,+(l--)z,,p,) is preferred to the lottery (ay,+(l-cl)z,,p,;...; a~,,+(1 -c()z,,p,) if and only if X is preferred to Y. This axiom leads to the above functional with a linear utility function U.

A different approach is suggested in Segal (1989). Although the anticipated utility model is a generalization of expected utility theory, it is yet a special case of a wider set of functionals, those that assign to a lottery a measure of its epigraph. This family of functionals is defined as follows. Let 9 be a positive measure on [0, co) x [0, l] such that for a bounded set S, S(S) < co. Denote as before pO=O, let X=(x,,p,;...;x,,p,) where O~X,~*..IX,, and define

WV= i 9 [O .] i=l ( ‘xL x[~Pjy$oPj]).

It is easy to verify that anticipated utility is the most general case in which this measure can be decomposed as a product measure (see section 2 below). As proved in Segal (1984, 1989, 1991), Green and Jullien (1988) and Chew and Epstein (1990), measure representations can be obtained by axiomatizing preference relations over simple lotteries. For axioms that further guarantee that the preference relation can be represented by a product measure, and is therefore anticipated utility, I suggest here, as in Segal (1990), to enlarge the space of lotteries by letting in two-stage lotteries.

The rejection of the mixture independence axiom by most recent writers is justified, as experimental data shows that it has very little behavioral appeal - the paradox of Allais and the common ratio effect are just two of these examples.’ However, in an intertemporal model, where the set of lotteries includes lotteries over lotteries, this axiom can be obtained from two more

*By the paradox of allais, (5,000,000,0.1;0,0.9) is preferred to (1,000,000,0.11;0.0.89), but (l,OOO,OOO, 1) is preferred to (S,OOO,OOO,O.l; 1,000,000,0.89;0,0.01). By the common ratio effect, (3,000,l) is preferred to (4,000,0.8;0,0.2), but (4,000,0.2;0,0.8) is preferred to (3,000,0.25;0,0.75).

V. Segal, Order indifference and rank-dependent probabilities 375

fundamental axioms, the reduction of compound lotteries axiom and the compound independence axiom [see Lute and Narens (1985), Segal (1984, 1990)]. The reduction of compound lotteries axiom suggests that a two-stage lottery is equally attractive as the simple lottery yielding the same prizes with the compound probabilities. By the compound independence axiom, the two- stage lottery (X,p;Z, 1 -p) is preferred to the two-stage lottery (Yp;Z, 1 -p) if and only if the simple lottery X is preferred to the simple lottery Y. Lute (1988) offered an axiomatization of anticipated utility theory based on the distinction between one- and two-stage lotteries. In Segal (1990) I show how replacing the reduction axiom by a weaker dominance axiom, while keeping the compound independence axiom, still maintains expected utility theory, and how a weaker concept of this dominance axiom yields the anticipated utility model.3

According to the reduction of compound lotteries axiom, the decision maker is only interested in the compounded probabilities, rather than in the probabilities themselves. In particular, this implies that decision makers do not care for the order in which uncertainties are resolved. A weaker version of this idea is formalized in section 3 under the name of order indifference, and together with the compound independence axiom it is proved to imply that a measure representation is a product measure, hence anticipated utility. The order indifference axiom suggests that for Osx, s ... 5x,, the decision maker is indifferent between the two-stage lottery A, where with probability 4i he wins a ticket for the lottery (0, 1, -p; xi,p), i= 1,. . . , n and the two-stage lottery B, where with probability 1 -p he wins zero and with probability p he wins a ticket for the lottery (xi, ql;. . .; x,,,qJ. To a certain extent one may say that the difference between these two lotteries is the stage in which the decision maker knows that he did or did not win zero.

This axiom has some immediate applications for games in normal forms. Assume two players. Such games may be interpreted by each of the players in two different ways. Firstly, each pure strategy of a player results in a lottery, defined by the other player’s mixed strategy. The player now uses a mixed strategy, that is, he plays a lottery, the outcome of which determines which pure strategy he is going to play. He thus faces a two-stage lottery. Alternatively, he may consider his opponent as the first player. Each pure strategy of the other player results in a lottery, its probabilities being defined by the first player’s own mixed strategy. This too results in a (different) two- stage lottery. In an extensive form, these two interpretations differ in the order of the players in the tree. Of course, if both players employ the reduction of compound lotteries axiom, there is no difference in the lotteries resulting from these two interpretations. Without this axiom, two different

‘For a discussion of Lute’s (1988) axioms and some of the dominance axioms in Segal (1990), see section 5.

376 U. Segal, Order indifference and rank-dependent probabilities

games may be defined. [See Dekel et al. (1991) for more details on these two interpretations.] Indifference to the order of the two stages, even if only in some cases, may thus be of some interest.

The paper is organized as follows. Section 2 presents the definitions of one- and two-stage lotteries, and the axioms leading to a measure representation. The order indifference axiom and the main representation theorem appear in section 3, and several alternatives to this axiom are discussed in section 4. Section 5 concludes with some remarks on the literature.

2. Definitions and axioms

For a positive real number s, let L”, be the set of all the finite lotteries with outcomes in [O,s]. That is, LS,=((x,,p,;...;x,,p,): ~pi=l,pi~O, i= 1 ,..., n,O~x,~...~x,~s}. Let L1=U,,O L”,, which is the set of all the finite lotteries with non-negative outcomes. For X EL,, define the cumulative distribution function F, by F,(x) = Pr(X 5x), and let X0 be the epigraph of this function, that is, X0 = Cl {(x, p) E [0, co) x [0, 11: p > F,(x)}. The degenerate lottery (x, 1) is denoted 6,. Let D” = [0, s] x [0, l] c ‘%’ and ds = OS\

((0, l), (s, O)}. F or t>O, let L:’ be the set of lotteries {XE L”,: for x E [0, t), F,(x) 5 1 -t} and let Q’ be the square [0, t) x (1 -t, 11. Let L: = u L”i’. szo

Let 2 be a complete and transitive preference relation over L,. Define the relations > and - by X> Y if and only if X> Y but not YtX, and X - Y if and only if X2Y and YtX. We say that the function I/: L,+!K represents the preference relation 2 if V(X) 2 V( Y)oX> Y. Also, define the preference relation 2” on L”, by: t/X, YE L”,, X 2” Y if and only if Xt Y, and define the relations >” and -’ accordingly. Consider the following three axioms:

(a) Continuity. The preference relation 2” on L; is continuous in the topology of weak convergence. That is, let X, Y, Y1, Y,, . . . E L”, such that at each continuity point of z of F,, F,i(z)+F,(z). If, for every i, X>“x, then X t” Y. If, for every i, yi t”X, then Y 2” X.

(b) First-order stochastic dominance. Let X, YE L”,. If, for every z, F,(z) 5 Fy(z) and there exists z such that F,(z)< Fy(z), then X>“Y.

(c) Irrelevance. Let X, Y, X’, Y’ E L”, and let S be a finite union of segments in [O,s]. If VZES, F,(z) = F,(z) and F,.(z) = Fy,(z), and VZE [O,s]\S, F,(z) = F,.(z) and Fy(z) = FyG(z), then X 2” Y if and only if X’ 2” Y’.

U. Segal, Order indifference and rank-dependent probabilities 377

Definition. A curve Cc D” is the continuous image of a function f: [O, l] -+ D”. The curve C is increasing if (x, p) E C + C n {(y, q): y < x, q > p} = 8.

Let 9” be a countably additive measure on ds such that for every t ~(0, I], Q’ is a measurable set. For t > 0, define the measure ,Y~’ on D” as follows: For every P-measurable set B c D’, P’(B) = 9”(B\Q’).

Theorem 1. The following two conditions are equivalent:

(1)

(4

The preference relation &-” on Ls satisfies the continuity, first-order stochastic dominance, and irrelevance axioms. There is a (countably) additive measure 9” on D”, unique up to multiplication by a positive number, satisfying: (a) For B=[x,y] x [p,q]cD’ such that x<y and p<q, O<P(B)<co; (b) If Ccds is an increasing curve, then P(C) =O; and (c) For every t > 0, the preference relation 2” on L”;’ can be represented by

P’(X) = Isrs9’(X”). That is, for X, YE L;‘,X 2” Y if and only if P’(X”) 2 W( Y”).

For a proof of this theorem, see Segal (1991).4 Note that the measures 9 are not necessarily bounded. Next, I show how to extend Theorem 1 to the space Lt. As before, if 9 is a countably additive measure on [0, cc) x [O, 11, then for t > 0, 9’ is defined by ,9’(B) = 9(B\Q’).

Lemma I. The following two conditions about the preference relation 2 on L, are equivalent:

(I) For every s, the restriction of 2 to L”, satisfies axioms (a)+c). (2) There exists a measure 9 on [0, co) x [0, 11, unique up to multiplication by

a positive number, such that for every t >O, the functional Vt(X) = $‘(X”) represents the preference relation 2 on L:.

Proof. Clearly (2) a(1). To prove that (l)*(2), define the preference relation >-” on L”, as before. By Theorem 1, for every s and t there is a measure 9” on [O,s] x [O, 11, unique up to multiplication by a positive number, such that the order 2” on L?’ can be represented by the function P’(X)=P’(X’). Since the measure 9” is unique up to multiplication by a positive number, one may assume, without loss of generality, that for sz 1, P([:, l] x [0, 11) = 1. (Since s”([$, l] x [0, 1])=,!P0.25([i, l] x [O, 11) and p.o.25 represents the order 2” on LS30.25, it follows that P([$, l] x [0, 11) is

4Earlier versions of this theorem, with slightly different conditions, appear in Segal (1984, 1989), Green and Jullien (1988) and Chew and Epstein (1990). As pointed out by Wakker (1990, 1991), these theorems and proofs are erroneous. One of the problems with the earlier versions is that the measure 9” may be infinite on sets containing (0,l) or ($0).

378 V. Segal, Order indifference and rank-dependent probabilities

finite. Otherwise, since Vs,o.25 (X) =P”.25(Xo) represents the order on L”;“.25, it follows that d2 IV ‘Si, a contradiction to the first-order stochastic dominance axiom.) It thus follows that for r Is2 1 and for a given t, the functionals P’ and I/‘,’ coincide on L”;‘. Of course, this proves that 8,’ is a bounded measure. Let t 5x, 5 +*. sx,. For X=(x1, pi;. . . ; x,, p,), define V(X) = PqX) = JPqXO).

Define also a measure 9 on [0, co) x [0, l] by S([x, y] x [p, q]) = JP’([x, y] x [p, q]).5 Obviously, S’(Cx, ~1 x CP, 41) = 9y*‘(Cx, ~1 x Cp, 41). Let V’(X)=$‘(X’). We have still to prove that the functional I” represents the preference relation 2 on L\. Note first that as Li consists of finite lotteries only, V’(X) is always finite. Let X, YE L\. There exists therefore s >O such that X, YE L? *, hence

x~Yox~sY~vs”(X)~;V”~‘(Y)

0 P’(X”) 2 P’( YO) 0 $‘(X”) 2 $‘( Y”) 0 V’(X) 2 V’(Y).

Suppose now that there is another measure $* such that for every X and Y in L:, Xt Yo$*‘(X”)L9*‘(Y”). Since both 9’ and $*’ represent the order on L?’ for every s, it follows that on D”, 9* =K”$, and moreover, that K” does not depend on s. Q.E.D.

Let 9 be as in Lemma l(2) and suppose that it is finite on bounded sets containing (0,l). In that case, the order t can be represented by V(X) = 9(X”). Define G,(x,p) =$([O,x] x [l-p, 11). Obviously, 9 and V can be reconstructed from Gg, as

S(Cxv ~1 x Cl -P, 1 - 41) = WY, P) - Gs(x, P) - Gs(Y, 4) + G,(x, q),

and as, by the properties of distribution functions, X0 is the union of a finite set of rectangles {Qi> w h ere i # j* Int (Qi) n Int (Qj) = @. Different G functions thus define different representations of the preference relation 2. Consider the following four examples. As before, we assume Osx, 5 .*. 5x,:

Expected Value. G(x,p)=px, V(X)=CpiXi, and S([x,y]x[l-p,l--q])=

(Y-x)(P-q).

Expected Utility. G(x,p)=pu(x) for some (strictly) increasing function a. V(x) =C Pi”txi)3 and WX,YI x Cl -P, 1 -d)= CU(Y)-WI(P-~.

Dual Theory [Yaari (1987)]. G(x,p) =xf(p) for some strictly increasing, continuous, and onto function f: [0, l] -[O, 11.

‘The fact that 9 is a measure follows immediately from Theorem 12.5 in Billingsley (1979).


and WGYI x Cl -_p, 1 -ql)=(y--W(p)--f(q)].

Anticipated Utility Theory. [Quiggin (1982)16 G(x, p) =f(p)u(x) for some functions f and I* as in the last two cases.’

(2)

and &CX,YI x Cl -_p, l-41) =C~(Y)--(x)lCf(~)-_(s)l.

In all four examples, the functions G are multiplicative separable and the corresponding measures 9 are product measures. Anticipated utility theory is the most general form of a product measure. Indeed, let p be a positive measure on [0, co) such that the measure of bounded sets is finite, and let v be a positive bounded measure on [0, 11. Define U(X) =,u([O,x]) and f(p) = v([l --p, 11). Define a measure 9 on [O, co] x [O, l] as follows. For a box S = Cx, ~1 x CP, 41 = CO, ~0) x CO, 11, let

W) =P(cx?Yl). w% 41) = [U(Y) -u(x)lCf(l -PI - .I-(1 -4)l.

It follows that V(X) in (2) equals the measure 9 of the epigraph of X”. There is some confusion in the literature as to whether the proper

exposition of the anticipated utility functional should be (1) as in the introduction or (2). Let g(p) = 1 -f(l -p) and obtain (1) from (2). Note that in that case f(p)= 1 -g(l -p). For a further discussion of these two alternative representations, see Segal (1987a). In the sequel I use form (2) of the functional.

Later, I use the following Archimedean axiom:

(d) Archimedean. For every X E L, and for every E>O, there exists x >O such that (0,l -E; x, E) 2 X.

As stated in the Introduction, the aim of this paper is to axiomatize the

‘%ome authors refer to this theory as ‘expected utility with rank-dependent probabilities’. I use both names throughout this paper.

‘Quiggin suggested a less general form of this functional where J(f)=f. The general form of this functional is presented in Segal (1984, 1989). It is useful not to restrict the function f to satisfy f(p)=p for some p~(0, l), because risk aversion implies a convex probability transformation function / [see Chew et al. (1987)].


anticipated utility representation functional through two stage lotteries.* A two-stage lottery is a lottery at which the outcomes are tickets for lotteries in L,. Let

be the set of these lotteries. Natural isomorphisms occur between L, and two subsets of L,. Let ~={((x~,P~;...;x.,P~),~):(x~,P~;...;x~,P~)EL~) and let ~=(((X1,l),pl;...;(Xn,l),pn):(X1,pl;... ; x,,p,) E L,}. The set n should be interpreted as consisting of all the lotteries where all the uncertainty is resolved in the second stage, while the set r should be interpreted as the set of those lotteries where all the uncertainty is resolved in the first stage. For X=(x,,p,;...; x,,p,) EL,, let yx and 1, represent the corresponding lotteries in r and A, respectively. That is, yx=((xl, l),p,;. . . ;(x., l), p,) and ;1,=

((x,,p,;...; x,,pJ, 1). On L, assume the existence of a complete and transitive preference relation &. We say that A w2B if and only if A&B and B&A, and we say that A >213 if and only if A &B but not B&A. Representation functions of tZ are denoted by W. The relation tz induces preference relations tr and 2,, on L,, where X&,-Y if and only if yx t2yY and X 2 ,, Y if and only if 2, & Ar.

Definition. The preference relation k2 satisfies axioms (a)-(d) if both kr and t,, satisfy them.

To the previous four axioms (a)<d) add now the following one:

(e) Time neutrality. For X EL,, yx -*Ax.

This axiom states that if the decision maker has to participate in the lottery X E L,, then he does not care whether all the uncertainty is resolved at the first or at the second stage. If the two stages represent two different time periods, then this implies that he does not care whether all the uncertainty is resolved at the first or at the second time period, hence the term ‘time neutrality’. It is easy to prove [see Segal (1990)] that the preference relation tZ satisfies the time neutrality axiom if and only if for every X, YE L,,

A common assumption in decision theory, which this paper does not assume, is the reduction of compound lotteries axiom, According to this

‘The description of the space Lz and two-stage lotteries follows Segal (1990).

U. Segal, Order indifjkrence and rank-dependent probabilities 381

axiom, a decision maker is indifferent between a two-stage lottery and its actuarial equivalent one-stage lottery. Formally:

Reduction of compound lotteries. Let

Xi=(xl,~l;...;xk,,pt,), i=l,..., m,

and let A=(X,,q,;...; X,, q,,J. Define R(A) EL, by

R(A)=(x:,q,p:;...;~,‘~,q,p,‘~;...;x’;,q,p~;...;x~~:_,q,p~~).

Then A -2YR(A).

If the reduction of compound lotteries axiom is satisfied and the order tr can be represented by the functional 2,-, then the preference relation k2 can be represented by the functional W(A) = VdR(A)).9

The reduction of compound lotteries axiom finds for every two-stage lottery A EL, a corresponding lottery in r, an isomorphic set to L,. This axiom can therefore be interpreted as a mechanism reducing two-stage lotteries into one-stage lotteries. Alternatively, decision makers may reduce two-stage lotteries by using the compound independence axiom:

(f) Compound independence. Let A =(Z1, ql;. . . ; X, qi;. . . ; Z,, q,,,) and B= (Z,,q,;...;Y,qi;...; Z,, q,,,) be two lotteries in L,. Then A t2 B if and only if X&Y.

For X=(x,,p,;...; x,,,P”) E L,, let the certainty equivalent of X, denoted CE(X), be defined implicitly by (CE(X), 1) -X. By the first-order stochastic dominance axiom, 6 maxrxi12X>6,i,1,i1, and by the continuity axiom it thus follows that the certainty equivalent exists, and moreover, that it is between min (xi} and max {xi}. Again by the first-order stochastic dominance axiom it follows that it is unique.

Let CEAX) and CE,(X) be the certainty equivalents of the lottery X with respect to the orders tr and 2,,, respectively. That is, ((CEAX), l), 1) -2~X and ((CE,(X), l), 1) - 2;1x. By the conclusion from the time neutrality axiom it follows that for every X, CEJX)= C_!?,(X). If the preference relation k2 satisfies the compound independence axiom, then

(X,,q,;...; X,~4rn)~2((C~,4(XJ, l),q,;. . . ;(CE,W,), 1LafJ.

Note that the right-hand side of this equivalence is an element of r In other words, the compound independence axiom too can be used to find for each two-stage lottery an equally attractive lottery in r, an isomorphic set to L,.

‘Note that V,is defined on L1.


In that case, if the order t,- can be represented by the functional V,, then the preference relation & can be represented by the functional WX,,q,;...; x,, 4J = UCE,(X,), 41;. . . ; CE,(X,h qm).

If the preference relation & satisfies both the reduction of compound lotteries axiom and the compound independence axiom, then it can be represented by an expected utility functional [see Samuelson (1952)]. Some empirical evidence supports the claim that decision makers to not obey the reduction of compound lotteries axiom. [See, for example, Schoemaker (1980, 1987). See also Segal (1990) for further references.] This may be due to the fact that risk-averse decision makers prefer to take part in as few lotteries as possible. This conjecture is indeed proved to be correct within the anticipated utility model - see Segal (1990). In this paper I assume that the preference relations kr and t,, satisfy axioms (a)-(c) and can therefore be represented by a measure on the epigraphs of the distribution functions as in Lemma 1. I assume, furthermore, that the preference relation & satisfies the compound independence axiom, but not necessarily the reduction of compound lotteries axiom. As is proved in Segal (1990), this does not impose any new restrictions on the preference relations tr and kn, unless more axioms are assumed. In the next section I suggest one such additional axiom implying representability by an anticipated utility functional.

3. Order indifference

Consider the two two-stage lotteries A = ((0, l), l-q; (0, 1 - p; x, p), q) and B = ((0, l), 1 -p; (0,1-q; x, q), p). If the reduction of compound lotteries axiom is employed, both are equally as attractive as the lottery ((0, l), 1 -Pq; (x, l), Pq),

hence the decision maker is indifferent between them. As mentioned above, subjects do not necessarily follow the reduction axiom as they may care for the number of lotteries in which they are to participate. On the other hand, in both lotteries there are two stages of uncertainty, in one the probability of success is p and in the other q. Although these uncertainties are resolved in a different order, it may still be true that the decision maker is indifferent between A and B, even if he does not follow the reduction of compound lotteries axiom. Indeed, I suggest indifference to the order at which uncertainty is resolved in a slightly wider range of lotteries:”

(g) Order indijj’krence. For every (xl,q,;.. .;x,,q,)~L~ and p>O,

((O,l),l-p;(x,,q,;...; ~“,q”)~P)-,((~,~-P;~,,P),q,;...;(~,~-PP;~”,P),q,).

[See fig. 1.1

“‘For the possibility of replacing the common outcome ‘0’ by other common outcomes see Theorem 3 below and Lute (1988).


0 Xl 5, 0 Xl

Fig. 1

1-P

A

P

0 5,

Theorem 2. Let the preference relation tZ satisfy the compound independence axiom. Then the following two conditions are equivalent:

(1) The preference relation k2 satisfies axioms (a)-(e) and (g). (2) The orders kJ- and tn can be represented by the (same) continuous

anticipated utility functional (2) with an unbounded utility function u.

Proof. (2)=(I). Let 05x,5 . .. 5 x, and let the orders t,- and 2,, be represented by the same anticipated utility functional

By the first-order stochastic dominance axiom it follows that CE,(0, 1 -p;

x,,p)=CE,(O,l-p;x,,p)~... SCCEr(O,l-p;x,,p)=CE,(O,l-p;x,,p). We may assume without loss of generality, that u(O) =O, hence the anticipated utility of the lottery (0,l -p; x, p) is u(x) f (p). Suppose that W: L,-+% represents the order &. It thus follows by the compound independence axiom that

=f(p) (Ir:uxl[ (jzi J) (jzi+I J)] u xn n 1. c ( 3 f i 4. -f i 4. + ( )f(q )

384 U. Segal, Order ind$ereerence and rank-dependent probabilities

It is easy to verify that if the preference relation t2 can be represented by the functional W, then it satisfies axioms (a)-(c) and (e)-(g). Since the function u is unbounded, the relation & satisfies axiom (d) as well.” (l)*(2). It follows from Lemma 1 that there exists a measure 9 on [0, co) x [0, I] such that for every t >O, the two orders tr and tn can be represented on L: by the function V(X) = ,9*(X”). For t >O, let M’= {(x,,p,;...; x,,p,) E L,:min {xi> 2 t}. Suppose that for every such t, there are u’ and f’ such that when restricted to M’, the preference relations can be represented by an anticipated utility functional (2) with a utility function u’ and a probability transformation function f’. Denote this representation by “V. For 0 <t’ < t, 9’“” and -lr’ represent the same order on M’. Therefore f” E f’ and u” is an increasing linear transformation of u’. Without loss of generality assume that for t< 1, u’(x) =ul(x), x= 1,2, and for t> 1, uf(x) = u’(x), x = t, t+ 1. It thus follows that for all t’ < t, u” and U’ coincide on [t, 00). Define f =f’, and for x>O, define u(x) =ux(x). Obviously, for x2 t, U(X) = u’(x).

Next I show that lim x+O~(x) > - co. Otherwise, let pi+0 and xi-+0 such that for every i, u(x,)[l -f( 1 -pi)] <41)-u(2). It follows that for every i, b1>(xi,pi;2, 1 -pi). By continuity 6,2d2, a violation of the first-order stochastic dominance axiom. It remains to prove the existence of u’ and f’. To simplify the notations, assume that t = 0. That is, assume that there exists a measure 9 on [0, co) x [0, l] such that both krand t,, can be represented by V(X)=@X’). In particular, 9 is finite on sets of the form [0,x] x [0, 11.”

Let u(x) =G(x, 1) (as before, G(x, P) = S(CO, xl x Cl -P, 11)). BY the continuity axiom it follows that for every x and p there is y such that (0,l -p; x, p) N r(y, l), hence u(y) = G(x, p) and u- ‘[G(x, p)] is well defined. Note that uP’[G(x,p)]=CE~O,l-p;x,p)=CE,(O,l-p;x,p). As ((0, l), 1 - 4; (0,l -P; x, ~1, q) N &A U, 1 - P; CO,1 - 4; x, d, P), it follows that

G(u - 1 CW, p)l,q) = G(u - ’ CG(x, 41, P),

u-‘CG(u- ‘CG(x,p)l, dl =u- ‘CG(u- ‘CW, dl,p)l.

Let h:[O,co)x[O,l]+[O,oo) be given by h=u-‘oG, that is, h(x,p)=

u-‘[G(x,p)]. It follows that

M&k PI, 4) = we, 41, P). (3)

“The fact that the orders 2,. and >,, can be represented by the same anticipated utility functional does not imply that the order k2 satisfies the compound independence axiom. For a counterexample, see Segal (1990).

“The only assumption needing elaboration is the Archimedean axiom. Let t>O. Note that the Archimedean axiom implies that for every X E L: and for every E > 0, there exists x > t such that (t, 1 -E;x,E)>X.


Define H: (0, co) x {CO, 1) u (1, co)> -+(O, co) as follows. For p E [0, l), H(x,p) =h(x, 1 -p). For p> 1, define H(x,p) to be that number y such that x = h(y, 1 -(l/p)). (Note that by the Archimedean axiom there exists y* such that (0, (l/p); y*, 1 - (l/p))>,-6,. Therefore, by the continuity and first-order stochastic dominance axioms, H is well defined.) Let x > 0 and let p, q E [0, 1). It follows immediately by (3) that

H(W, P), q) = H(H(x, q), P). (4)

Next, let p,q> 1. Define y=H(x,p), z=H(y,q), y’=H(x,q), and z’=H(y’,p). By the definition of H and by (3) we obtain

and

It follows by first-order stochastic dominance that z=z’. Finally, let p> 1 and qE [0, 1). Define y,z, y', and z’ as before and obtain x=h(y, 1 -(l/p)). Therefore,

Y’ = H(x, q)

=h(h(yJ-$,1-q)

=s z = H( y’, p).

Hence z = z’, and (4) holds for all p, q E [O, 1) u (1, co).


Invoke now the following corollary from Aczil (1966, p. 273):

If x,9(x, w) E Z, w E P where Z is a real interval and 9(x, w) is continuous in x and transitive, that is, 9(x, w) =y has at least one solution w for all pairs x,y~Z, then

with arbitrary continuous and strictly monotonic 9 is the general solution of

sq2qx, w), a] =9[F(x, u), w].

In our case, Z=(O, co) and P= [0, 1) u (1, co). The function H is clearly continuous in x. It is also transitive. Indeed, let x, y > 0. If x 2 y, then by first- order stochastic dominance and continuity there exists pi [0, 1) such that

(O,P;x, l-P)- +5,,, hence y = h(x, 1 -p) =H(x, p). If y> x, then by the Archi- medean and continuity axioms it follows that there exists p> 1 such that (0, (l/p); y, 1 -(l/p)) N ,-SX, hence x = h(y, 1 -(l/p)) and by definition, y =

Wx, P). ’ 3 Therefore,

Wx, P) = 9 - 1 IX4 + WP)I.

Define &(O,co)-+% by &x)=3(x) and tj:(O,l]+% by $(p)=%(l-p), and obtain that on (0, co) x (0, 11,

The value h(x, p) is the certainty equivalent of the lottery (0,l -p; x, p),

hence, by the first-order stochastic dominance axiom, the functions C#J and II/ are either both increasing or both decreasing. If they are both decreasing, replace 4 by 4* = -4 and replace + by $* = -$. [Note that c$*-‘(x)=4-i(-x)]. Also,

h(x, 1)=x=$(1)=0.

By the first-order stochastic dominance axiom, the function Ic/ is strictly increasing, hence for p E (0, l), $(p) -C 0.

Let p, q ~(0,l). By the continuity and first-order stochastic dominance axioms, for every x*, y* >O there are positive x and y such that (O,l-p;x*,p)>(O,l-q;y,q) and (O,l-q;y*,q)>(O,1-~;~,~). Let q>p>O

and let {Xi} and { yi} go down to zero such that for every i, (0,l -p; Xi, p)> (0,l -q;y,, q). Since 4 is increasing, so is 4-l. Therefore,

13Note that because of the transitivity condition one cannot define H for x=0.


(‘3 l --Pi Xi, P)>(Q l -q; yt, q)

=> h(xi2P)>h(Yi3 4)

* +txi) + V+(P) ’ $(Yi) + $(4)

It thus follows by the continuity of 4 that lim,,, 4(x)= -co. Similarly, lim,,, $(p) = - co.

Consider now the lotteries of the order indifference axiom with 05x, 5 ... 5x,. By this axiom

((O,l),l-p;(x,,q,;...; ~“,q”)~P)-_,((~,~-P;~,,P),q,;...;(~,~-P;~,,P),q,),

hence

n-l

=G(u-‘CG(xn,P)l,qn)+ C U-‘CG(Xi,P)l, i qj i=l j=i

>

u-‘CG(xi,P)l, i qj . j=i+l

(5)

By the Archimedean axiom it follows that 9 is unbounded (see below). By the Archimedean and continuity axioms it follows that for every E>O, OI> 0 and n one can choose XT 5.. .s xx and qT, . . . , q.* such that x1= 1 qT = 1 and

and

i=l ,..., n,

G(x:,j-$lq+e. i=l,..., n-l.

Since this construction is certainly possible for some a> 0, it follows that 9 is unbounded. (See fig. 2.)

For every x: and p, u- ’ [G(xF,p)] 5x,!: hence


q; + . . . + g-1

0 =; .

=2 . . =;

Fig. 2

u-‘CGW, PII, i 4: 5 6, i=l ,.*.,n-1. j=i+l

Substituting into (5) one obtains

where Ial, 1~1 s(n- 1)~. This equation implies

44 - 1 C4(u - 1 Gnu + 01) + Ic/(P)l)

n

= I:( [ u qv’ 4b*)+$(P)+$ i 4; +z. i=l ( )1> j=i

Indeed,


=u(h(u_‘[na+a],p))

and

= 44 - 1 c+tu - l Cna + 01) + VVPU)

2 G(ulCGb:,p)l, i q:)+~ i=l j=i

n

=

= cc u h h(xT,p), 5 4; +r i=l j=i )I

”

=

= b- [ U 1 4(4-‘cddx:)+rl/(Pm+II/ i 4T +z i=l ( 11) j=i

”

zz z( u 6 i=l

+$(p)+$ Lc ( )I) +-z. j=i

By the definition of ~1,

~~‘(~)=u-‘[G(x:.~~~f)]

=h(x~,~~q:)=g-‘[$(xfl+B(~i4:)

*+(lC1(CL))=4(XT)+$ i 4; . ( > j=i Hence,

i u(~-l[m(xt)+~(~i4:)+~(P)]) i=l

=i~lu(i-1Cl(u-1(a))+9(P)1)

=dC ‘C4(u- ‘(4)++(P)l). (7)

390 V. Segal, Order indijizrence and rank-dependent probabilities

It follows therefore from (6) and (7) that

U(4-1c+(U-1Cn~+fll) + wJ)l)=~~(~-‘Cd4~- ‘(4 +vml) +z.

Since this equation holds for all E, it follows by continuity that

44- ‘C4(u-1Cfl~l) + em =n44- ‘Cd4u-‘W + 4wl). (8)

Let w=uo$-l and w-~=~ou-~ and obtain from (8) that

40- ‘(4+&4) =n4w-‘(a) + Ii/(P)).

This last equation holds for all positive integers n and c1 >O. Hence, for all positive integers n and m and for all cc>O,

u(w-l(a)+‘tl/(p))=w u-l ( (y)+*(P))

=nu(u-‘(g+*(P))

=Ao(w-l(a)+~(p))= n

u(u-(~)+*~P~)

By continuity it now follows that for all ~1, fl> 0,

e4+(B4 +Il/(p)) =/Wo- ‘(4 + ti(P)).

Let S(s,t)=w(w-‘(s)+IC/(t)),B(t)=S(l,t), and obtain

S(Pa, p) = BS(a, P)

=>S(s,t)=sfqt)

=>o(t)=o_‘(e(t))-o-‘(l)


The solution of this functional equation is w-‘(x) =alnx+b [see AczCl (1966, p. 39)], hence o(x) =ecx-b)‘a. By their definitions,

u - l CW, PII = &G P) = 4 - l C4(4 + +(P)I,

hence

G(x,p)=o(~(~)+Il/(p))=e[“‘““~(~‘-~~’~.

Define

U(x) = etW’ - bl/a

and

f(p) = eIJl(P)l/a

to obtain that G(x, p) =u(x)f(p). Note that lim,,, U(X) = 0, lim,,, f(p) =O, and f(l)= 1. Q.E.D.

4. Other versions of order indifference

This section is devoted to a discussion of several potential alternatives to axiom (g). First I show that if the order indifference axiom is replaced by the weaker axiom

then Theorem 2 ceases to hold. Indeed, suppose that the preference relation t can be represented by a measure of the epigraphs of the cumulative distribution functions with the generating function

:

1

G(x, P) = (l/x)-lnp’ xp’o

0, xp=o

Obviously, G(x, 1) =x and by the compound independence axiom

wo, 1191 - 4; to,1 - p; x, P), 4)

= Y(O, 11, 1 - p; (0,l - q; x, q), p) = 1

(l/x)-lnp-lnq’

One has still to show that the function G does not induce a product measure 9. Let


NCx, ~1 x Cl -P, 1 - d) = G(Y, PI - G(x, P) - G(Y, d + G(x, d

1 1 1 1

=(1/y)-lnp - (l/x)-lnp - (l/y)-lnq +(1/x)-lnq*

If 9 is a product measure, then for every x < y < z and r < q < p,

Let x = 1, y = 2, z = 3, p = 0.6, q = 0.5, and r = 0.3 and obtain that the two sides of this last equation do not equal each other.

Next, I discuss two possible stronger versions of the order indifference axiom.

(h) Order indifference with respect to every common outcome. For every

(x,,q,;...; x,,q,,)E&,p>O, and ~20,

((Y,1),1-P;(xl,q,;...;x,,q,),P)

(i) General order indifference. For every

xi=(xI; ,..., xi), i=l ,..., m,

p=(p1,..., PA and q=(ql,...,aA

((x:,ql;...;x,l,q,),P,;...;(x~,q,;...;x~,q”),P,)

Definition. The probability transformation function f is called symmetric if,

for everyp, f(p)+f(l-pp)=l.

Theorem 3. Let the preference relation tz satisfy the compound independence axiom. Then the following two conditions are equivalent:

(1) The preference relation tz satisfies axioms (a)<e) and (h). (2) The orders kr and t,, can be represented by the (same) continuous

anticipated utility functional (2) with an unbounded utility function u and a symmetric probability transformation function f.

Proof. (2) =>(I). Let W: L,+% represent an order tz and let 0 5x1 5 ...sxksysxk+lJ . ..sx.,. Assume first that ysCE(x,,q,;...;x,,q,). Then for a symmetric function f,


WY, 1 --Pi x,,p),q,; *. . ;(y, 1 --pi %P),4”)

n-l

+i=F+, {“(Y)C1--f(P)l+u(xi)f(P)} f ,I.% -f j=F+l % [ L ) ( n )I

+ {U(Y)Cl -“ml +aMP))f(4”)

n-l

= izI {4Y)C1 -f(P)1 + u(xi)f(P)l f j;i 4j -f j=g 1 4j Kl r 11

+ {4YW -f(P)1 +aJf(P)}f(q,)

=u(Y)cl -f(P)1

The proof for the case y> CE(x,, ql;. . . ; x,, q.) is similar. (l)=>(2). It follows from Theorem 2 that the orders t,- and k,., can be represented by (the same) anticipated utility functional with an unbounded utility function U. It remains to prove that f is symmetric. Assume, without loss of generality, that u(0) =O. First I show that f(f) =f. Let x* > 1 be the outcome for which u(x*)f(f) = u( 1).14 By axiom (h) it follows that

((0, $; l,M;(x*,k 1,t,,4, ~2((1,l),+;(O,+;x*,t),+).

By the compound independence axiom and the anticipated utility functional one obtains

2u(I)f(3CI -ml +u(x*hm)=41),

hence f(f) = $. Similarly,

((031 -_p; LP),+;(x*, l- p; LP),6)“2((1, aP;(Qf;x*,9,1 -PI,

hence

‘%ch an outcome exists by the continuity and the Archimedean axioms.

394 V. Segal, Order indlrerence and rank-dependent probabilities

wu-(P)+4Nl-f(l -PII +u(x*)fU -PII =u(l).

It thus follows that f(p)+f(l -p)= 1. Q.E.D.

This proof also shows why one cannot replace the common outcome 0 in axiom (g) by another common outcome y. Even if the preference relation can still be represented by an anticipated utility functional, it follows from the last proof that f(p) + f( 1 -p) = 1. This is so because in the anticipated utility model the value of an outcome depends not only on its probability, but also on its position relatively to other possible outcomes.

It is easy to verify that the general order indifference axiom implies axiom (h). To see this, let xf=y,j=l,..., n, and m=2. It now follows immediately from Theorem 3 that replacing axiom (g) by axiom (i) yields representability by an expected utility functional. Indeed, let x1 =(a, b,a) and x2 =(a,c,d) where u(a) =O, u(b) = 1, u(c) = 2, and u(d) = 3, and let p1 =pz =). It follows from (2) that f(p2+p3)=f(p2)+f(p3), hence f is linear [see Acztl (1966, p. 34)], and the anticipated utility functional reduces to expected utility.

5. Some remarks on the literature

Different versions of the anticipated utility (or rank-dependent) functional have received so far several axiomatic representations [see Quiggin (1982); Yaari (1987); Roe11 (1987); Lute (1988); Chew (1989); C hew and Epstein (1990); and Segal (1984, 1989, 1990)]. The approach presented in this paper obtains the anticipated utility form from axioms made on two-stage lotteries. In that it is similar to Lute (1988) and to Segal (1990). Instead of weakening the mixture independence axiom, these approaches suggest to replace the reduction of compound lotteries axiom by weaker axioms. In Segal (1990) the reduction axiom is replaced by several possible extensions of the concept of first-order stochastic dominance to two-stage lotteries. With a little abuse of notations, for X, YEI,,, let X n Y be the lottery whose epigraph consists of the intersection of the epigraphs of the cumulative distribution functions of X and Y, and let X u Y be the lottery whose epigraph consists of the union of those of the cumulative distribution functions of X and Y. For a lottery A =(X1,(1/m);. . .; X,,(l/m)) E L,, define

Aij= ( X,,l;...;XinXj,~;...;Xi"Xj,';...;X,,' . m m m >

The weak upper compound dominance axiom assumes that A tz Ai,, and the weak lower compound dominance axiom assumes that Aijtz A. It is then proved that under assumptions (a)<c) and (e)-(f), the preference relation satisfies the upper (lower) compound dominance axiom if and only if it can


be represented by an anticipated utility functional with a convex (concave) probability transformation function J Note that in Theorem 2 above there is no restriction on the possible shapes of the function J Of course, axiom (g) is entirely different from the upper and lower compound dominance axioms, as it deals with the order at which the uncertainty is resolved and not with other potential lotteries the decision maker could have won in the second stage.

Lute (1988) suggests an axiomatization of the rank-dependent model based on two-stage lotteries with events, rather than probabilities. Although it is in that respect more general, I prefer to present his axioms in terms of probabilities, to make it easier to compare them to the model presented in this paper. His central ‘accounting’ equation is that for every x, y, p and 4,

((Y, l), l-q;(y, 1-P;x,P),q)~Z((Y, 1171 -P;(Y, l-%X,&P).

(Compare this axiom with the beginning of section 4). This axiom is weaker than axiom (h) above, as it is restricted to only one possible outcome x. However, it is not clear whether it is stronger or weaker than the order indifference axiom. On the one hand it restricts the lottery (x,,p,; . . . ; x,, p,) to the degenerate lottery 6,. On the other hand, it requires indifference with respect to all y, not only y = 0.

Lute’s other axiom requires the following. For each vector of probabilities Pi,..., pn+ 1 such that ~~~~ pi= 1, there are ql,. . . ,qn and q such that Cy=1qi=l and such that for everyx,s...Sx,+,,

((xl,P,;*..; X,+1,Pn+1),l)N2((X1,41;...;Xn,4n),4;(Xn+1,1),1-4).

Of course, this second axiom is completely different from the invariance axiom used above.

Chew (1989) suggests the following axiom. Let x=(x1,. . .,x,) and y=

(Y i,...,yJ where xiS...Sx, and yis... 5 y,. The vectors x and y are called rank preserving if, for every 1< i < II, yi _ 1 5 xi 5 yi + 1 and xi _ i 5 yi 5 xi + I. The pair (w,z) is a rank preserving rearrangement of (x, y) if w and z are rank preserving and, for every i, {wi,zi} = {xi, y,}. I5 Chew’s commutative substitution axiom assumes that there exists cr~(O, 1) such that for every vector of probabilities (pl,. . . , p.) and rank preserving x and y,

C~(C~h, a; Y,, 1 --a),Pl,. . .; cm,, KY,, 1 -f$,P,)

for any rank preserving rearrangement (w, z) of (x, y). This axiom weakens

“For example, x=(1,3,6), y=(2,4,5), w=(1,4,6), and z=(2,3,5).

396 U. &gal, Order indifference and rank-dependent probabilities

Quiggin’s (1982) major axiom by assuming that such a just exists, while Quiggin assumes it for a=$. Together with some other technical assumptions, Chew proves that this axiom implies representability by the anticipated utility functional. He does not assume that the preference relation can be represented by a measure of the epigraph of the distribution function, but his commutative substitution axiom is strictly stronger than the order indifference axiom of this paper. To prove this I show that Theorem 2 does not hold true without axiom (c). For X=(X~,P~;...;X~,~,,), let V(X)=max

{v,(X), v,(X)1 h w ere for i = 1,2, y is an anticipated utility functional of the form (2) with the linear utility function u(x) =x and the probability transformation function fi.16 It is easy to verify that unless fi and fi do not intersect each other, this functional is not a transformation of a measure of the epigraphs of the cumulative distribution functions.” It is therefore not an anticipated utility functional, hence it does not satisfy the commutative substitution axiom. This functional satisfies the order indifference axiom. Let

X=(x,,q,;...;%l,q,).

W(O> 11, 1 --Pi x, p) = w, 1 - p; V(X), p)

On the other hand,

= V(x,~max(f~(~),f~(~)},q~;...;x,~max(f~(p),f~(P)},q,)

V,(X, . max {f,(p), f2(p)), ql;. . . ; x, * max {_fl(P), f2(P)13 4.))

Of course, Chew (and Quiggin) do not consider their axioms as decision rules for comparing two-stage lotteries, but as weaker versions of the mixture independence axiom.

16VI and V, are Yaari’s (1987) dual theory functionals. 17V is not an anticipated utility functional with the probability transformation function

f(p)=max(f,(p),f,(p)).


References

A&l, J., 1966, Lectures on functional equations and their applications (Academic Press, New York).

Billingsley, P., 1979, Probability and measure (Wiley, New York). Chew, S.H., 1989, Axiomatic generalization of the quasilinear mean and the Gini mean, Working

paper (Tulane University, New Orleans, LA). Chew, S.H. and L.G. Epstein, 1990, A unifying approach to axiomatic non-expected utility

theories, Journal of Economic Theory 49, 2077240. Chew, S.H., E. Karni and Z. Safra, 1987, Risk aversion in theory of expected utility with rank

dependent probabilities, Journal of Economic Theory 42, 370-381. Dekel, E., Z. Safra and U. Segal, 1991, Existence and dynamic consistency of Nash equilibrium

with non-expected utility preferences, Journal of Economic Theory 55, 229-246. Green, J.R. and B. Jullien, 1988, Ordinal independence in nonlinear utility theory, Journal of

Risk and Uncertainty 1, 355-387. Karni, E. and Z. Safra, 1987, ‘Preference reversal’ and the theory of choice under risk,

Econometrica 55, 675-685. Lute, R.D., 1988, Rank-dependent, subjective expected utility representations (with an appendix

by K.L. Manders), Journal of Risk and Uncertainty 1, 3055332. Lute, R.D. and L. Narens, 1985, Classification of concatenation measurement structures

according to scale type, Journal of Mathematical Psychology 29, 1-72. Quiggin, J., 1982, A theory of anticipated utility, Journal of Economic Behavior and Organiza-

tion 3, 323-343. Roell, E., 1987, Risk aversion in Quiggin and Yaari’s rank order model of choice under

uncertainty, Economic Journal (Supp.) 97, 143-159. Samuelson, P.A., 1952, Probability, utility, and the independence axiom, Econometrica 20,

670-678. Schoemaker, P., 1980, Experiments on decision under risk: The expected utility hypothesis

(Martinus Nijhoff, Boston, MA). Schoemaker, P., 1987, Preferences for information of probabilities versus payoffs: Expected

utility violations involving two-stage lotteries (Center for Decision Research, University of Chicago, Chicago, IL).

Segal, U., 1984, Nonlinear decision weights with the independence axiom, Working paper no. 353 (University of California, Los Angeles, CA).

Segal, U., 1987a, Some remarks on Quiggin’s anticipated utilty, Journal of Economic Behavior and Organization 8, 145-154.

Segal, U., 1987b, The Ellsberg paradox and risk aversion: An anticipated utility approach, International Economic Review 28, 175-202.

Segal, U., 1989, Anticipated utility: A measure representation approach, Annals of Operation Research 19, 359-373.

Segal, U., 1990, Two-stage lotteries without the reduction axiom, Econometrica 58, 3499377. Segal, U., 1991, The meausre representation: A correction, Journal of Risk and Uncertainty,

forthcoming. Wakker, P., 1990, Counterexamples to additive representation on non-full Cartesian products,

Mimeo. (Duke University, Fuqua School of Business, Durham, NC). Wakker, P., 1991, A criticism of Segal (1989), ‘Anticipated utility: A measure representation

approach, Journal of Risk and Uncertainty, forthcoming. Yaari, M.E., 1987, The dual theory of choice under risk, Econometrica 55, 95-115.

order indifference and rank-dependent probabilities u. &gal, order indifference ... and z, and...

Documents