E den from those who sometimes violate ihem ill cwbjecrs refused tti believe that they cstilc! have

natized) by Mais ( 1952).

ought to derive weights a funct;on OT :!ndividual ~~rc~bab~it~es, ~~~d~ate difkulties w prDbabi!ities 3f the forx l/n. Severe

unless the weight, ~(l/n) is set equal

with art&nty, and compare it with ed by a srn~ 11 but pssi tive random rent va’iues 3; each with probability

‘chat i:~, ttle certain event is preferred, even thou@ the random variable has a superior outcome with probability *.

s anomaly is to be avoided, a similar argument will es the ent that for i 5 ks AT. Consider the choice na

with ~rababii~t~ k,‘n twith asthi,xg received otherwise), and k sums of the form X-+x1 (or X-q), each with probability l/n. Thus, if

t of y, denoted c= CE(y). If two distin@shed.

II= construction of a function I/ on Y such )‘. The ??M ‘theory involves constructing a

on X md setting V(y) ==E ptU(xi). anda offers the seemingly symmetrioai proposal V(y) = >z, w(pJx,, where the i are assumed to represent

ntity of a given good md IV is a real-vaiued function on the unit interval such that w(O)=O. Kahneman and Tversky combine the two, setting V(y) =& w(p&&,), where the x1 are assumed to represent changes from some initial situation. Karmarkar suggests setting

,)W(XJ I C w(pi), where

i

mentioned above. Kahneman and Tversky themselves pain+ to the difIiculties which arise with their approach (and c1 jhrtt& that of Handa) when w is non-linear. Further discussion is given by Karmarkar (1979).

In general, if w is non&tear it is possible to find an ‘overweighted’ pair of prohabiities p and 1 -I) such that

w@)+wp-p)> 1. (2)

If xi and x2 are chosen so that x1 is very slightly preferred to x2, then

Yet the prospect xi clearly domin&s ((xl,,xz); @? 1 -p)j irr the strong sense that its outcome will be preferred (or at least indserent) with probability 1. Kahneman and Tversky avoid this implication by assuming that dominated prosmts are ‘edited out’ but this leads to the undesirable result that pain&e choices we iritransitive. Note again that, if w is linear, so that w(p) -6 w(l -B)== w(l) the &ory reduces to that of NM.

A more cornplcx example shows that the problem of dominance applies to Kqrnarkar’s theory. Suppose thatt, for some p, w(p)<2w(p,Q). Then we can find‘x,Pxp#kJ !pch that >.

- ~W~pi~l~~-~3~w(~--P)tJ(~3)~/~w(P)I ++ --PI)

329

fpticw of the function k. wuuld at first sight appear to be an DO bonnd has been imposed on the number of different

ct, yv However, it can be shown that a 1~ [Q, 1-j is sulfkient to dekzmine h(p) for

regardless of its Ir:ngtid. The apptoach used is very used io show that a non-linear weighting function

ttres will generat: violations of dominance. The central p&n% is that the ~VhdiO%l Of il prospect, in Which two very simliar outcomes x1 arrd x2 occur with the probabilities p2 and p2, must be ‘close’ to that of a prospect which is identical except that x1 replaces x2, occurring v&h a t&al probasitity pl +pz, This imposes constraints on the function h. If it is surncd, w in previous approaches, that h,@) depends only on pr, then these constraints are satisfied only by the NM expected utility function.

For each pc [Q, 13, write f@) = h,(p, I-- p). Thus, f(p) defmes the behaviour Of h on p&s fp, 1 -p). ne exknSiOn cJf h to triples f&, p2,p3) Itill n3W be

deskbed in detail. The approach used can be shown by induction to apply

Consider the two prospects y, = ((x,, x2); (pil, 1 - p,)> and y2 = ((x,, x3, x& (pz,pl -p2, 1 -p,)). As U(x,) approaches U(x,), V(y2) must approach V(yr). At the limit, x1 .=x3 and y, =y2, so that

+MP29 P-h:, I- Pl))u(X,) + h3& pi --P2,1- pl)U(X,).

(7)

Since x1 and x2 are chosen arbitrkly and h(p) is independent of x the coe$‘kients on U(x,) on the right-hand and left-hand sides of (7) must be equal. Hence.

M72Jv-P2~ 1 -PA= 1 --a 1‘1. (8)

Conlvzsely, by setting U(x,) close to W(x,) we can show

h@2r PI - ~2% 1 - PI) =,fkd, ad hence (9)

331

~.Mean Ahlw Theorem. Define y$*=(O,. . .,N- l/N;p*) and y,l;= CW* 6 * * , G&J*

The distributions 9f yft and y$” are given by step functions which lie, mspective:ly, abovy and below D. It is clear that y$ Py PJ$*~ while

g/(yjt;*) = l/N if,l g(x&jU((i - 1)/N), so that

3. Axioms

In order to generate a theory more general than NM expected utility theory, :lt is necfi.ssary to begir, with a weaker set of axioms. In particdar, it seems appropriate to modify the controversial independence axiom.

The notation adopted here creates some difliculties in comparisons of the two sets of axiaqs, First, it must be noted that acceptance of the NM complexity axiom (that the value of a compound lottelj depends only en the uSma& probability of elach outcome) is implicit in the use of this notation. If this <axiom did not hold, two prospects, both expressed as (x,&, need not bc of equir-aleqt value. Seqond, the independence and continuity axioms in NM theoqy rtre signi&aoty weakened if th$y apply only to outcomes x E X.

For Lsta~,ce~ the $dependzace3 axiom:

becomes a simple statement that preferences preserve dominance if yn and y, are rep&&_ by q@qWs of X (9 Axiom 2).

In ad&ion to ‘the complexity axiom, the usual completeness axiom will be adopted.

Axiqn .a... ./ /’ ? is co?plete, reflexive and transitive (completeness).

J. Qpiggiq ..4 tbeory qf anti@m&d wtility 333

The major resuh of this paper is:

Proposit ion 1. Suppose X and Y satisJy R.1 and R.2 and P satisJie,n Axioms 1-4, Then thera exists a function V: Y -+R such that

(i) : V(y) 2 V(y3 ifhIm ogy if yPy’,

(ii) V(x;& k& h&)U_(xJ for some functions U and h,

where h(l)= I and h&4)=($,$. If two functions V and V’ sati& (i) and (ii) there exists constants a altd b, a > 0, such that

(iii) V’(y)=aV(y)-t b.

The proof, which is not very illumin7:rng, is included in the appendix. It is clear that Axioms l-3 are .~eak forms of the corresponding NM

axioms, Axiom 4 can be derived from t!~ NM independence axiom blat it is easier to show that any expected utility maximizer must satisfy it, sink

=c*(u(x,) + u(x:))p, =ECU(GP)J.

i

4 Coaditionaii on the aticipated utiiity fumdons

Under the anticipated utility theory, an individual’s attitudes to prospects are determined both by their attitudes to the possible outcome and by their attitudes tq the probabilities. These are m&ted in the utility functioh 11, and tl re, +eighting fur&ion, h, respectively..

@th regard tc the first, the con&pts of tisk aversion, risk neutrality and r:i& ’ prtheticx are still &evant though their interpretation is somewhat d.@&nt. A $sk neutral in&vi&a! tiil be ind%ercnt between a W-50 bet a@ its expbcted outcome, while a risk averter will prefer the certain outcome ai;d a risk preferrer the &et. %&, as in expected utihti theory, risk aversion is

?!I8 , uivprfc~$ ty a co*ve utiliiy ‘fu&on.

xg@st c#$qyls pi&&n df @obabiity distortion relates to the treatment o&&s with !Lr&ll $robabihticS and extreme outcomes. We may say that an i.n$vid,u$ ove?ights extreme events if

‘*,

335

individual is pessimistic, since the worst outcomes are, ted. If f(p) s 1 -J‘(l - p), the individual is optimistic,

dividual is neutral and h is symmetric. is rather difficult to distinguish empirically

on. It would thus be possible to require U to be linear, as uid imply dropping the plausible condition

imply coWant relative risk aversion for any odds. The theory p nted here is more general and flexible.

P&M useful a.pphcations of the Nhrl theory have been based oh the assumgltion that X is the set of real numbers (Its ek:ments are generally taken to correspond to levels of real wealth.) The preference ordering P on X is as~med to correspond to the natural ordering so that individuals always prefer rrrore wealth to less. These preferencss may be represented by a monotcotic increasing function U: R-+R. One of the most important tools 01 analysis under these conditions is the concqt of stochastic dominance [Fishbum (E&54)]. Madar and Russell (1969) have devel,.ped scucbastic dominance r&s for the NM theory. Given random variables jO and y:,, with cumukrtive distribution fuxtions D, and D, respectively, t first stochastically dominate y,, (yl FSD y,) if

Q&)&D,(x) for all x

and to mend StochasticaXy dominate y, (1, SSD yO) if

(1’7)

(The tmm3 first degree artd second degree stochastic dominance are also wklely used for these condi%ons.) Th:y prove4

Propitiofl2. Let yr and y, be as above. Then

(i) yy FSD y. g and only if E[U(y’)] 2 E[U(y,)] fir all monotonic increasing , u’.

(ii) yl SSQ yQ jf and only if ECUQ&‘J ~E[U(yJ for all concave increasing U.

on (1976).

tit se&a,). Then by the mean value theorem,

?I(B,(~)~-ftD,(s))lds~~‘(Do(ai~~ 4W-W?d~. (26) ’ Ql ‘)l

Combining. (23) and’ (25) yields the desired result (22) for a’= 1. Suppose (22) holds for i=1_2 ,..., k. Then

I (We(s)) -s(W))) ds Wf’(Do(a.,, - ,)I=~ UU4 - &)I ds. (27) 40 ‘0

‘kpplication of an argument similar ts that for i= 1 to the interval (azt, fc2k + 2) yie:ds

4, Quiggin, A tltmy oj. antkiputd wility 33’9

(34). However these do not fit the intuitive conception of skewness as well as distributions satisfying (34).] A converse argument can be applied if yO is s&w& to the left, arid yr is either symmetric or skewed to the right.

Just as the expected utility theory permits the analysis of behaviour which would be excluded as irrational under a profit-maximization hypothesis, the AU theory permits the analysis of sqticipations which are not mathematical expectations.

It may be argued that the adoption of a more general theory makes it harder to ,obtain useful predictions. ‘The results of section 4 show that much of the dominance analysis which is useful in NM theory can be extended to AU theory. In particular, Proposition 3 shows that no pairvise preference rankiti:g whiGh is inadmissible under NM theory can be admissible under AU theory, Further work is needed in developing conditions under which other results can be extended.

Areas in which AU theory might usefully be ‘applied include the problem of individual’s apparent propensity to ‘over-insure’, and tbts economic analysis of gambling behavior. The theory is likely tr, be particu.drl! valuable in the analysis of decisions involving catastrophic (or extremely t:~vourable outcomes) which occur with low probability.

Appendix

The proof of Proposition 1 has two parts. In part (a) an arbitrary choice of values for [J for two elements of X is shown to imply that t‘{v) can have only one possible value of any YE Z In part (b) it is shown that the function V constructed in this way satisfies conditions (i) and (ii) oi the proposition. In both parts of the proof the special status of SO-50 bets is trsed extensively.

(a) C’hoose x’, X” E X, x”Px’ and set V(x’) =O, U(x”3= 1. pn the trivial w_ when x”l;le’ for $1 x’, Y, V must be a constant function and will satisfy conditions (i) and (ii).]

If x = XE((x’,x”); (i,&)> then by (ii) and Axiom 2, V(x) =3. More generally, q U&r) = u/2n, I&c& =(” + 1)/2”, x = CE((x, :, x2); (3, $)I, U(x) = (24 + n)Pa + l. Note ihat ~+%cP~; sri that the” function U preserves the preference ordering. The set of numbers, ilf the form u,Qk is dense in LO, l] since every real number has a binary expsinsion. We may thus coimplete the construction of U on (x;x”PxP$) as6 follows: Let ,.‘i , I. ‘

The density properties of X* mean that it will be sufikieat to prove the desired resu! t for y E: Y*

TJle result will first be established for the class, of B-50 bets. Suppose x, qr ;G~ B Xg and xzPxl. Then we wish to prove

Prrq$ We may note that it is suficient to prove the ‘only if” part of the proposition since, for any x1, 3~~15 X,*, there exists x3 such that U(q)= ~~.(w2M~d~>* ?-II’ IS can be seen by examikg the construction prmdure. Hence the ‘loaly Z’ proposition implies that xJ = CE(jx,, x,);&$)$ asltd that if x==CE{( xI,x2);(+,$)) then U(x)= U(.Y,). Since x,x1 EX~,

U(x) = u/? for some Q, and

U(x,) = I a I . Q/2& for some kj > 0.

The proposition holds for b= 1 by virtue of the construction may ‘be proved inductively for arbitrary b by showing that

procedure. It

(ii) Let b -= (bt -t bJ2 and

U(x&=(u-b&2, ,u(x‘&=(u+b,)/2k,

343

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