Non-transitive measurable utility for decision under uncertainty

ter AT&T Bell Laboratories,

way Hill, NJ 07974, USA

Submitted November 1987, accepted June 1988 e axiomatize a representation for preference between acts in Savage’s fo under uncertainty that is based on expectation of a non-separable utility consequences with respect to the decision maker’s subjective probabihty The representation has been discussed previously by Graham Loomes David Bell, and the present author. The representation follows from subjective utility when his ordering axiom is weakened and his conditional is added as an explicit axiom. All axioms except the Archimedean condit the non-separable representation.

Let S, X and F be non-empty se&s of states, consequences and acts, respectively, with F c Xs. ary is preferred to (1954) axiomatizes a relation > on F to obtain Vf,gEF:f

>g~SCu(f(s))-u~~s))ldx(s)rQ, S

where ICis a proba here is to present a Savage-type representation vf9lC-f

~g~S~(f(s),g(s))da(s)>O S

03

8/89/%3.500 1989,

ort

188

PC. Fishbum, Non-transitive measurable utility for decision under uncertainty

approach is similar to the view asureinent

f

W

X

2

t?Y

Fig. 1. x>y and z>w.

y

skew-symmetry,

over consequences

P.C. Fishburn, Non-transitive measurable utility for decision under uncertainty

procedure provi es a direct way to scale tne t$ values ther, and it plays a central role in our derivation of the tation from the axioms for > on F. axioms are noted in section 3, following brief review of Savage’ heory in the next se we make one concession to math licity simple acts, i.e., those with only er of possible enceforth, F=(fEXS:f(S)

189

consequen~s.

is finite}.

ecause of this, Savage’s final axiom ( SSA context and is used only to extend his representation from simple acts in Xs when X is infinite, will play no role in the S Additional remarks for Savage’s other axiom in part because al our SSA axioms. .l itself is weake .l.* With this weakening, w Archimedean axiom ( axiom. These additions are necessitated by the loss of analytical power from . proofs of Savage’s is .6-g =_-__:,:~YPv~~~ not assuming full transitivity, 4), Fishburn (1970, ch. 14)]. representation theorem [Sava The SSA representation theorem is stated in section 3 after its axioms are lied. Its proof appears in section 4. ecause of systematic violations of von ms uncovered independence (cancellati acCrimmon and Larsson (197961, of Savage’s s a&rimmon

an

tt ( 1979), Slavic and utility theory have alternatives either

1%

’ PC. Fishbum, Non-trunsitiue measurable utility for decision under uncertainty

Theories for our present concern of decision under uncertai

.

basic formulation of the pre tation for the utility of act $ monotonic but not defined in Choquet’ subsequent paper b events,. i.e., v(A) + @\A) = 1, on the basis of. a symmetry analysis, and suggests that a consistent account for conditional bability is possible only when v is additive, in which case the Schmeidle ilboa theory reduces to Savage’s, Alternative theo additive prob(1982). Loomes , Fishburn and LaValle (1987), and ssume transitivity but eschew the others avoid transitivity but adopt the SSA a consequence, the latter group endorses Savage’s and does not accommodate sberg’s violations of that gen and Savage are disisons between Allaise preceding paragraph that adopt Savage’s axioc in the present paper and that in Eishburn (19 ;e (1987). The latter two papers axiomatiie version of the SS resentation in which j’(s),g(s), . . . are ability distributions and &f(s),g(s)) is skew-symmetric and ilinear, i.e., linear in each argument with respect to convex combinations of robability distributions [e.g.9

(1987) (basic Savage acts).

ility distributions over acts,

PC Fishburn, Non-transitive measurable utility fir decision under uncertainty

191

random devices for convexification into the descriptions of states, a

probability distribcltions defined on a set of entities pure strategies, commodity bundles, Savage acts, etc.). bilinear functional C/Ion B x P such that

with 4 unique up to multiplication by a positive constant. This SS stern when 4 is representation reduces to that of von Neumann and theor assumes separable as &I, q) = (p) - u(q). Unlike their theory, neither transitivity nor independence [p>q,O
l~~,p+(l-I2)r>lq+(1_3,)rl.

SSA is through the lottery-acts The first way that S axiomatization of SSA in and Fishburn and LaValIe (1987). axioms are applied to > on F+ and a few more axioms en the three

probability distributions on

192

P.C. Fishburn, Non-transitive measurable utility for decision under uncertainty

Under Savage’s characterization probability distributions on are stochastically independent if : ~{SES:(f

of n, {nf:

m(s))

=(x,Yuj=Pwt

.

from > on F by

efine

p > q if f >g whenever nf = I_‘, Z~= q, and f and g are stochastically independent, ’ and for such f and

g

define

&(p,q)

by

where 4(x, y)= &.I‘, 4)) when p’(x) =q’(y) = 1. Then 4 on P x P is an f I ;tional with p > q-&p, q) >Q, for all p, q E P. Hence the SSB rep tation emerges as the specialization of the SS representation for pairs of stochastica!?y independent acts. A conclusion of these relationships betwee SSA and SSB is that the SSB theory is both a parent and a child of we begin with 5SB for > on SA representation emerges as , the cial case of the S form conju with a few other assumptions. we begin with SSA representation and a suitably rich YC,t ars as a specialization of SSA.

Several additional notations and definitions will be useful in this and ensuing sections. ith F the set of simple acts in X”, also let Fx,, denote the acts T--Ihoseonly outcomes are x and y:

F,={f

EF:f(S)gx,Y}l.

let @ be the empty event and denote the hen f21c A c S, {A, A”} is a two-part partition a finite set of non-empty, mutually disjoint

fzg

t f(s) =g(s) for every s E s that f(s) =x for every s E A,

burn, Non-transitive measurable utility for decision under uncertainty

1193

and xAy is the act f with f z x and ,f z y. Given > on F, define N (indifference) and >N(preference-or-indifference) in the usual ways: STg if -(f>g) f 2g if’f>g

and -(g>f),

or f Ng.

Also, x > y means that f >g when f 7 x and g 7 y, and similarly for x-y and xzym e say that > on a designated set is a weak order if it is asymmetric and both > and N are transitive on that set. The preference and indifference relations are extended from 6: to events in 2’ by the definitions Ad3

if for all x,y~X,x>y*xAy>xBy;

A-B

if

l(A>B)

and -l(B>A).

Although it is perfectly sensible to read A> usually think of > on 2” as a co A> B as ‘A is more probable than

referred Bs E, we

reMion an5 read

AEJV if, for all f,gd,fsg*f-g. Events in JV are referred to as null events, and it will turn out that A E N-A - $+a( A) = 0. Finally, for each event A we define a conditional preference rdatiolt 3 on F by f >gif,forall

f’,~%F,(f’~,f,g’~g9f’~g’)*f’>g’.

A

This reflects the basic feature of Savage’s representation representation that for which f(s) #g(s). f ~g if, for all f’,g’EF,(f’~ f >Ng if, for all f ‘)g’ E ,(f’ A

f,g’ 2g9 f' yx'bf'-

and t

194

P.C. Fishburn, Non-transitive measurable utility for decision under uncertainty

>=> and N=N, c?wever,this iced no g if not (f 3 g) and not(g 3 f). e's axioms are, for all f,g, f’, g’ E F, all X,y, 2, w E

RI.

> on F is a weak order.

P.3.

(A&M, f 7 x,g 7 y)*(f

P.4.

(x > y, z > w)+Ay

P.5.

x’ > y’ for sume x’, y’ E X.

3 g--y).

>x

zAw > zBw).

P.6. f >g+given x) there is a finite partition of S such that, for every event E in the partition, (f’z x,f’ E f)-f’>g, and (g’ F x,g’gg)*f >g’].

.3 as his sure-thing principle Savage refers to the conjunction of anC proves that, in th f zg whenever f(s) &(s) for all s. depends heavily on transitivity, and it does not follow fro s’lcne. provides consistency for the extension of > to 2’, which can on any pair of consequences that are not indifferent, and ence of such a pair. Savage’s rchimedean axiom, and leads to arbitrarily all positive value throughout S. J-P.6 hold. Then there is a probability measure CS, all O
‘) = h.(A) for some A’ c

nique and u is utile

P.C. Fishburro,Nowtransithe measurableutility for decision under uncertainty

axioms weaken only Saw > on F is asymmetric and, for all x, yE

shown by the fo f Savage’s theorem.

ing lemma, this we&e

195

axio is a weak order,,

affects only conclusion

Lemma 1. Suppose RI .6 hold. Then there is a unique probability measure II on 2’ that satisfiesconclusions (I), (2) and (3) of Savage’s theore oreover, for all x, y E X for which x > y, Vf,gEF,,:

f >goa({s:f(s)=x})>n((s:g(s)=x}).

when = {x, y}: the conclusions of Lemma 1 follow immediately from Savage’s theorem when ={x,y} and x>y. the same conclusions apply for each pa of non-indifferent consequences with identical z’s by Given that II satisfies the conclusions of Lemma 1, it is easily chec .6, are implied by the basic now consider an enrichment of yields the SSA representation in the absence of transitivity.

rY

theory makes just two strengthens the Archimedean axiom P.6”. (f >g; x, yEX)*[there event E in the partition, (f’s f’ >g’].

is a $nite partition of S such that, for every f,g’,=,g,f’z forf’zx,g’zg or g’z y)*

says that if f >g ability events such preserved when they are modified as indicate

P.C. Fishburn, Non-transitive measurable utility for decision u

196

s.1.

(

>

z g,f 2B g)q-

=0,f

.

A

A

part

~~ce~t~-~ty

fromthe than

dropping full transivit

oreover, n is unique and t$ is unique up to mu2 constant (++a4, a > 0). otheses of the d by the SS ted in the next representation. section. eople like Allais and Ellsberg who find fault with rinciple will find fault with SS theory for the same reason. n the other accommodate systematic violations of s, including versions stated for the von stochast . 3, 1987),

of the following lemma, which are easy consequences of e definitions in section 2, will be used extensively without out this section.

etr~c;

2

is sy

etric an

exive,

guide our definition of 4 and show that it is-unique in the For convenience, define TC*by a*(E)=n(E)/n(E’)

for all E E 2’ with n(E’) >O.

The following theorems apply to all x,y,z,w,t,~EX. Theorem I. AEON,

If x > y and z > w then there is a unique il~(O,l) such that,

n(A) > &xAw

> yAz,

n(A)=bxAw

- yAz,

R(A) < &yAz

> xAw.

Theorem 2.

Suppose x > y, z > w, t > v, A,

xAw - yAz, zBv - wBt, tcy - vex. Then n*(A)n*(B)n*(C)=

1.

airs of constant acts ir, the SS x>y,4(x,y)=O if x-y, and $(x,y) tation requires

n require 4(x, addition, the r

foralI

1%

PC Fishbum, Non-transitive measurable utility for decisim underuncertainty

when

x0 > vo,x1 iyl,

W>Z.

hat the same #I is obtained regardless of which scaling process. Suppose, for example, that that $&yI) is obtained from #(x0, yo) by the equation

that &, w) for z> w is scaled against each follows:

of(XOVYO)and @bYl)

as

(44(x0, YO), xoAw -YO&

)&p,y,),

or n*(A)=

2 with a rearrangement of terms. using at the appropri d 2. Complete validation of the S noted at the end of the section. ur first

ma 3.

(f(s) Z&) for all s~S)*f

zg. If, in addition, (s: f (s) >g(s)) $JV,

lit

s of S.1 to

aril y with x>y

a

z>w

P.C. Fishhrn,

Non-transitive ~as~~a~~e utility for decision, under uncertaihty

199

terests for the momen a 5, and the in equivalent events asserted in Lemma 7.

Proof.- Given the hypotheses, define h, k E F in part by z w,

h F y, h

E

z,

By Lemma 4, L Act k and h szD k. ence S.1 implies hAuBcCuDka S. 1, h Aze k-k czD h (else h 5c k or k z h), which is precisely f’ &)g’*

f AzBg*

c3

For the next lemma and later use we define /I, as the following subset of U&11: A,={k/2”:k=l,..., Lemma 6. f

jf

wg 7

Proof.

B=(~~,CEA,DEB;

'(An Yd

2”;n=l,2 ,... >.

5 @4f

*&J-f

Awl,,

?~(c)=ht(A),

a(D)=h(

&s)*

.

Suppose first that iz= l/2. Then, by emIDa 5 and S.4

uccesske bisections of C and implicF.tions for ‘CI, lead to the concksi f cz,g-f

‘s that

AzBg whenever k{:,$Q

,... >.

200

P.C. Fishbum, Non-trmsitive measurable utility for decision under uncertainty

The same conclusion for every A~ll, then follows from conclusion (2) of 0 Lemma 1 and successive applications of S.1. Our next lemma is similar to Lemma 5 except that A, JS,C and D are not all assumed to be mutually disjoint. (A-C, B-D, AnB=CnD=@; f TX, f ZW, g?y, gsz; IZ I3 wg'7 Y,g'3 z)*(f> g-f' > 8'). f c'Kf AWB CUD Lemma7.

PRX$. According to Lemma 6, it sufkes to prove Lemma 7 under the assumption that x(.4) + z(B) 54, since otherwise A, B, C and D in the present case can be ‘reduced’ by the same faclor under successive bisection. Given let E=S\(Au BuCuD) so that n(A)+n(E)&, hence n(C)+n(D)$i, a(E) 24. By (2) of Lemma 1, choose El, Ez c E, El n Ez = 0, so that and

n(&)=n(A)=~r(C)

a(&)=~r(B)=n(D).

Then, by Lemma 5 with

*p,g*z Y,g*E& f *=El xf ’ f

>

AUB

g-f’

> 8’9

EtuEz

and therefore f > g-f' > 8). AUB

CUD

Our final lemma prior to the proof of Theorem 1 is a monotonicity result.

ProoJ. Let C = S\(A u B), f = xAw and g = yAz. Assume f zg, as in the hypotheses. If g 2 AUcf then, since g g f(z>w,B&M, P.3), S.l implies g> f, a contradiction. Hence f >AUCg. Let

f'&f' g&g,

f'zx and

i!tzY,

201

P.C. Fishbum, Namtnmsitiue measurable utility fop decision wader uncertainty

so that f’=x(A u B)w and g’=y(A u B)z. Since f&g’

by f Azcg

and

f’ 3 g’ by x>y and P.3,

S.I implies f ’ >g’, i.e., x(A u B)w > y(A u B)z.

Iz]

Proof of Theorem 1. Assume that x >y and z > w, and consider xAw versus yAz. According to Lemma 7, preference between xAw and yAz depends only on n(A) and not on the specific identity of A. Hence, when n(A)=4 we -tite XAW and yR.zin the place of xAw and yAz, respectively. According to Lemma 89

Moreover, xl w 3,ylz and yOz>xOw, i.e., x > y and z > w. According to P.6*, with the consequence pair (WJ) in the hypotheses when f =x and g=y, and the consequence pair (y, x) when f -2 and g- w, xlw > ydz for some I < 1, and yk >x;lw for some 1~0. It follows that there is a unique X strictly between 0 and 1 such that A>I’-xlw>yAz, Axlw.

If either xA’w> yA’z or yi2’z> xXw, then a similar application of P.6* yields a contradiction of Lemma 8 with {f,g) = (xd’w,yd’z} in the hypotheses of P.6*. WenceXXW- yrl'z. 0 One further lemma, which generalizes Lemma 6, will be proved before we prove Theorem 2. Remark. Henceforth we use the conclusion of Lemma 7 freely along with the notation xay for xAy when a(A) = a. Lmma9. g F

Y, g s

(CnD=@,

0~14,

z)=dxAw>yA=+f

n(C)=dn(A), Ir(D)=iZn(AC),f F

X,

f 3

W,

&g)*

Proof. Lemma 9 follows from Lemmas 6 and 7 for all de A0 and it holds for all 0~ L< 1 by Lemma 3 and the definitions unless (x> y,z> wj or (y>x,w>z). Assume henceforth that x>y, z>w, 044 and iZ@l,. Also let a=x(A). Suppose that xaw > yaz. Then, by Theorem 1, there is a positive j!J y/k. Given such a fi, choose il0EA0 so that

202

P.C. Fishburn, Non-transitive measurable utility for

I
and

decision under uncertainty

i(l-a)<&(l-/I).

Let Co and D,, be disjoint events with COc C, D c DO, n( Co)= &/3 and

a(&) = A,(1 -/I).

Also let f *,g* E F have

f’ EX, f*~~w,

g*zy

and g*zz.

Then, since x/Iw > y/32 and since Lemma 9 holds for A,,E A,, f’ &og+~ Two applications of Lemma 8 (first replacing De by 0, then Co by C) for its

straightforward modification to the conditional case then yield f >cJDg. A converse proof that uses the same basic method shows that f >cuDg=* xaw>yaz.

0

Proof of Theorem 2.

x>y,

Assume that t > 0, xaw - yaz, Z/?VN wflt and fyym uyx,

z>w,

with a,/?, and y strictly between 0 and 1 by Theorem 1. Also let

~=B/(B+1-a), and let (A, B, C, D) be a four-part partition of S with n(A)=jZa, n(B)=A(l-aj,

n(C)=(l-A)/?,

a(D)=(l-1)(1-b).

Let (f,g) =(x, y), (w, z)? (z, w) and (u,t) on A, B, C and D respectively. by S.l. Since d(l-a)=(l-A)#? f *zBg and f cz,, g, so f-g by the definition of 1, f B;cg, so again by S. 1, f AzDg. Hence, by Lemma 9, BY Lemma

9,

x[la/(la+(l

-A)(1 -fi))]o-y[(l

-A)(1 -/I)/(Aa+(l

-A)(1 -/?))]t.

Then, by Theorem 1, na/(na+(l-A)(l-&)=1-y, which reduces to a/Iy/[( 1 - a)( 1 - /?)(1 - y)] = 1, the desired conclusion.

P.C. Fishburn,Non-transitivemeas~~abkutility for decisionunder uncertainty

203

We now complete the proof that the SSA representation holds for all f,ge F. Let a be given by Lemma I and 4:X x X-R be defined in the manner described earlier in this section. Also fix CEX. Although validation of the representation is routine at this point, we detail the essential steps to show that everything is in place. Proof of the SSA representation. Given f,g c F, let {A,, . . . , A,) be the smallest partition of S into events on each of which (f(s)&)) is invariant with

(f(S), l!(s))=(Xi, J’i) ai=n(AJ

for all S E Ai,

and zai=l.

Successively replace each (xi, yJ by (t, t) for the A*EJT and each Ai # ,Y for which Xi- yi. By S.1, part (4) of Lemma 2, and P.3, this does not change the preference or indifference between f and g. With f and g thus modified, assume for definiteness that Al,. . . , A,,,,m d n, are the events in the partition for which a,>0 and l(xi-yJ. If m=O, f =g, hence f-g and ) ti(f,g)ddn=O. If rnz 1 and Xi>y, for each ism, Lemma 3 gives f >g, and clearly I#(f,g)drr>O. Suppose there are i, j srn for which Xi> yi and yj> X~ Take (i, j) =( 1,2) for definiteness. By Theorems 1 and 2, and Lemma 9, there is a unique r >O such that

and

f’ czDg’

whenever

z(c)/I@)=~,

f’

F x1,

f’

E x2,

g’ 7 yI

and

g’ 5 y2.

If al/a2 =r, then f A,zA2g and, by S.l, we can replace both (x,,y,) and (x2,y2) by (t, t) without changing the preference or indifference between f and g thus modified. Moreover, al~(x*,yl)+a2~(x2,y2)=O=(a~+a2)~(f,f)

on A, u A2.

On the other hand, if al/a2 > r, by Lemma 1 there is a Bi c Ai with a(&) = a2r, hence f B,zA2g9 so we an replace (xr,yi) on B1 and (x2,y2) on A2 by (t,t) in a similar manner without changing preference or inditrerence between f and g. In this case (f(s),g(s)) remains at (xr,y,) on A*\&, and, on B1 u A2,

204

P.C. Fishburn, Non-transWve measurable utility for de&on waler mcertahty

Similar changes with the roles of Al and A2 reversed are made when al/a2 cr, The applicable change of the preceding paragraph eliminates at least one of (xl, y,) and (x2,yz) completely, replacing it by (t, t). So long as there remains x1>yi and yj> x/ with positive probabilities for the modified f and g, we repeat the elimination-replacement procedure. At the conclusion of the process either both f and g have been modified to t on all of S, with f-g and j&f,g)dn=O for the original and modified f and g, or positiveprobability events remain that all have xi>yl or all have yi>xi. If xi>yi in the latter case, we get f >g and I#(f,g)dz>o for both the original and final versions of $ and g. Similarly, if events with only y+q remain, g>$ and j@(f,g)dnO by skew-symmetry. Thus, for all f,ge F, f >g-l#( f,g)da>O. Appendix

This appendix comments briefly on the extension of the SSA representation from F to non-simple acts in Xs. ” We note first that Savage’s extension axiom, P.7. (f z g(s) for all SE

A)*f

z g;(f(s) A

2 g

jiw all

s~A)=+f

;L g, A

is unsuitable for the SSA context. Suppose n(B) = l/2, f = xBw, g= yBz and 4 satisfies

Then, with A = S in P.7, the SSA representation gives $ >g(s) for all s E S, but g> f since y>x and z>w. The SSA extension is more complex than extensions of separable expectation representations [Fishburn (1970, chs. 10 and 14)J because of the non-separability of 4 and the fact that 4(X xX) can be any subset of IR commensurate with the cardinality of X x X, skew-symmetry, and boundedness (see below). In addition, unlike the lottery-acts SSA extension [Fishbum and LaValle (1987)], 4 is not first defined on pairs of acts, so 4 for f >g-#( f,g) > 0 is not available for use in the non-lottery extension.

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A similar argument does not work for nondiscrete acts. In this case it appears necessary to consider the definition of I4(f,g)drc in terms of limits of expectations of simple real valued functions on S and to turn to bounding arguments involving pairs of simple acts. An additional axiom that may be needed here [cf. B.4* in Fishbum and LaValle (1987)J is

Although this uses 4, it can be deleted by observing that 4(x, y)2 #(z, w) is tantamount to xAw ZyAz with a(A)=i. In any event, the general extension is furthercomplicated by the arbitrarynature of 4(X,X) within its bounding interval such as [ - 1,11.I do not presently have a satisfactory resolution of the general case. References Allais, M., 1953, Le comportement de l’homme rationne! devant le risque: Critique des postulats et axiomes de !‘&o!e am&ante, Econometrica 21.503-546. Allais, M., 1979a, The foundations of a positive theory of choice involving risk and a criticsm of the postulates and axioms of the American school, in: M. Allais and 0. Hagen, eds., Expected utility hypotheses and the Allais paradox (Reide!, Do&e&t) 27-145. Allais, M., 1979b, The so-called Allais paradox and rational decisions under uncertainty, in: M. Allais and 0. Hagen, eds., Expected utility hypotheses and the Allais paradox (Reide!, Dordrecht) 437681. Anscombe, F.J. and RJ. Aumann, 1963, A definition of subjective probability, Annals of Mathematical Statistics 34, 199-205. Be!!, D.E., 1982, Regret in decision making under uncertainty, Operations Research 30, %l-981. Bernoulli, D., 1738, Specimen theoriae novae de mensura sortis, Commentarii Academiae Scientiarum Imperialis Petropolitanae 5, 175-192. English translation by L, Sommer, Econometrica 22 (1954), 23-36. Chew, S.H., 1982, A mixture set axicsatixation of weighted utility theory, Discussion paper 82-4 (revised), (College of Business and Public Administration, University of Arizona, Tucson, AZ). Chew, S.H., 1983, A generalization of the quasilinear mean with applications to the measurement of income ‘inequality and decision theory, resolving the Allais paradox, Econometrica 51, 1065-1092. Choquet, G., 1953, Theory of capacities, Annales de 1’InstitutFourier 5, 131-295. de Finetti, B., 1937, Le prevision: Ses lois logiques, ses sources subjectives, Annales de 1’Institut Henri Poincari 7, l-68. English translation by H.E. Kyburg in: H.E. Kyburg and H.E. Smokier, eds., Studies in subjective probability (Wiley, New York, 1964) 93-158. Ellsberg, D., 1961, Risk, ambiguity and the Savage axioms, Quarterly Journa!,of Economics 75, 643669. Fishbum, P.C., 1970, Utility theory for decision making (Wiley, New York). Fishbum, P.C., 1982, Nontransitive measurable utility, Journal of Mathematical Psychology 26, 31-67. Fishbum, P.C., 1983, Transitive measurable utility, Journal of Economic Theory 31,293-317. Fishbum, P.C., 1984. SSB utility theory and decision-making under uncertainty, Mathematical Social Sciences 8.253-285. Fishburn, P.C., 1987, Reconsiderations in the foundatidns of decision under uncertainty, Economic Journal 97. 825-841. Fishbum, P.C. anr: LH. LaValIe, 1987, A nonlinear, nontransitive and additive-probability mode! for decisions under uncertainty, Annals of Statistics 15,830-844.

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measurable utility for decision under mcertuimty

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