Dominance conditions in non-additive expected utility theory

Journal

of Mathematical

Economics

21 (1992)

173-184.

North-Holland

Dominance conditions in non-additive expected utility theory* Marco Scarsini Universitci LYAnnunzio, Submitted

65127 Pescara, Italy

April 1990, accepted

May 1991

In order to explain the Ellsberg paradox, a non-additive expected utility has recently been proposed. In this paper we determine dominance conditions for non-additive expected utilities. In the univariate case all the well-known stochastic dominance theorems can be extended to the non-additive framework, whereas in the multivariate case the generalization applies only to those results that assume a weak preference ordering or do not require extension to linear combinations of utilities.

1. Introduction

Several paradoxes arise in the context of expected utility theory g la von Neumann-Morgenstern or $ la Savage, and several alternative decision making paradigms have been proposed in the recent years to obviate those paradoxes. In particular, Ellsberg paradox [Ellsberg (1961)] can be formulated as follows: Two urns contain balls which are either black or red. Urn I contains 50 black balls and 50 red balls. Urn II contains 100 balls with an unknown proportion of red and black. An individual is allowed to choose an urn and a color. (S)he will win a positive prize if a ball of the chosen color is drawn from the chosen urn; (s)he will win nothing otherwise. If (s)he strictly prefers choosing urn II, then (s)he will behave in a way that violates Savage’s sure thing principle and therefore cannot be represented by maximization of expected utility with respect to any additive probability measure. In an attempt to justify Ellsberg paradox, Schmeidler (1986, 1989) and Gilboa (1987, 1989) have proposed different sets of axioms that are slightly different from Savage’s axioms and generate a non-additive expected utility theory, i.e. a theory where capacities assume the role of probability measures. One set of axioms leads to define the expectation of a function with respect to a capacity in terms of upper Choquet integral [Choquet (1955)], whereas a *Work performed while the author was visiting the University of Arizona, Tucson. Support CNR is gratefully acknowledged. The author thanks two referees for their helpful comments. 03044068/92/%05.00

0

1992-Elsevier

Science Publishers

B.V. All rights reserved

of

174

M. Scarsini, Dominance conditions in non-additive expected utility theory

dual set of axioms leads to the use of lower Choquet integral. As Gilboa (1989) suggests, ‘it is not clear why, if at all, the original [set of axioms] should be preferred to the dual’. Nevertheless the use of one integral rather than the other (i.e. of one set of axioms, rather than the other) produces different consequences, unless symmetric capacities are used, In this paper we establish dominance conditions for different classes of utility functions both in the univariate and in the multivariate case. In the case of weakly ordered spaces, the problem of finding dominance conditions for capacities is reduced to the same problem for probability measures, by showing that for every capacity there exists a (finitely additive) probability measure that coincides with the capacity on the class of upper sets (or on the class of lower sets, if the lower Choquet integral is used). All the well-known univariate stochastic dominance results fall under this umbrella, as well as the multivariate results for utility functions representing a same preference ordering [Levy and Levy (1984)]. The other multivariate cases, where only a partial order structure is assumed, have to be treated in a different way, since, in general, there does not exist a probability measure coinciding with a capacity on the class of upper (or lower) sets. The multivariate stochastic dominance results that we are able to obtain in the non-additive case are somehow weaker than the corresponding results for the additive case, just because of the lack of additivity. In the additive case the typical structure of a stochastic dominance theorem is the following: we have a convex cone F of functions 4 for which

Choquet’s theorem guarantees that each 4 can be written as a convex combination of extreme points of a convex set containing 4 and generating the cone. The additivity and homogeneity of the integral imply that (1.1) is true for all 4 E B iff it is true for these extreme points. In the non-additive case the Choquet integral is positively homogeneous and only comonotonically additive, therefore if inequality (1.1) holds for r$ in a certain set A, then it can be extended only to the convex combinations of functions 4 EA that are co-monotonic and to the functions in the cone that they generate. A necessary and sufficient condition for two functions to be comonotonic is that. their level curves never intersect. For this reason the multivariate results that can be extended to the non-additive framework either involve utility functions that represent the same preference ordering [e.g. Levy and Levy (1984)] or do not require the extension to a convex combination of functions in a cone [e.g. Levhari, Paroush and Peleg (1973, Scarsini (1986) Muliere and Scarsini (1989)J For the results that involve extension to linear combinations of functions [e.g. Mosler (1984), Scarsini (1985, 1988)] only a weaker version holds in the non-additive case.

M. Scarsini,

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conditions

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expected

utility

175

theory

Gilboa (1989) showed that any reasonable definition of conditional probability requires additivity. This accounts for the fact that the coupling characterization of first degree stochastic dominance does not hold for capacities. In section 2 we will consider dominance conditions for weak preference orders, whereas in section 3 we will study the case of partial preference orders.

2. Weak preference orders Let T be a set and let C be an algebra is a set function P: C+R, such that

of subsets

of T. A capacity

on (T, C)

(i) EcF* P(E)SP(F), (ii) P(o) = 0, P(T) = 1. Call %?(T, C) the set of capacities on (T, C). Call Y(T, C) the set of (finitely additive) probability measures on (T, Z). In the sequel we will often omit the argument and just write %?,.Y, unless the specification is necessary. Obviously PC’%. If 4 is a measurable real valued function on T, the upper Choquet integral of 4 with respect to g E 5%is defined as follows [see Choquet (195.Q Schmeidler (1986)]: jQdP=[P(s.T:&s)Za)da-

7 Cl-P(seT:$(s)~a)]da, -cc

whenever the Riemann integrals on the right-hand defined the lower Choquet integral analogously as

!4dP=;j[l-P

(s E T: 4(s) 2 a)] da -

0

side exist. Gilboa

(1989)

7 P(s E T: 4(s) 2 a) da. -CC

The Choquet integral satisfies the following properties of co-monotonic additivity and positive homogeneity. Two functions 4,$ are called comonotonic if Vs,t~ T,(+(s)-$(t))($(s)-$(t))ZO. (a) For any two co-monotonic

j($+$)dP=

j4dP+

!(++i)dP=i4dP+SIlrdP.

functions

j+dP,

*

4, $,

176

M. Scarsini, Dominance conditions in non-additive expected utility theory

(b) For anyL>O,

From now on, we will restrict our attention to spaces endowed with an order structure, which will be either a partial order (reflexive and transitive) or a weak order (linear and transitive). We will denote by S a partially ordered Polish space, and by W a weakly ordered Polish space. These spaces will be endowed with the corresponding Bore1 o-fields. We call pi”“(S) the class of real valued non-decreasing functions on S and Fdec(S) the class of real valued non-increasing functions on S. When no confusion is possible, we will omit the argument and just write 2ti”c,9dec. A subset A of a partially ordered set S is called upper if XEA and x
If P E %T(W, Bor ( W)), then there exist P*, P, E 9 such that,

(a) vi#IE @“C( W), ;ddP=@dP*, (b) VCpE 9d=(

W),

!4dP=@dP,.

Proof. (a) For every set AE%(W), let P*(A)=P(A). The definition of P* is clearly coherent, and therefore can be extended (non-uniquely) to Bor( W). Since (SE W: &s)za) is an upper set, then

54dPATP(sEW:$(s)za)da0

y [l-P(sEW:gb(s)za)]da -Kl

M. Scarsini. Dominance conditions in non-additive expected utility theory

=~P*(H’:+(s)&)dx-

177

; [l-P*(sEW:&)zCo]da -m

=@dP*. The proof of(b) is similar. There exist several well-known stochastic dominance results for classes of utility functions representing the same preference ordering, in the univariate and in the multivariate case. All of them can be extended to the non-additive case by using Lemma 2.1. Some care is needed in order to determine which Choquet integral has to be used. Basically all the results involving only the probability of lower sets [like the usual nth degree downward stochastic dominances or the multivariate ordering of Levy and Levy (1984)] can be extended to capacities, when the lower Choquet integral is used, whereas the results involving only the probability of upper sets (like the nth degree upward stochastic dominances) can be extended to capacities, when the upper Choquet integral is used. The distinction becomes unnecessary when symmetric capacities are used. A capacity P is called symmetric if P(A) = 1 -P(k) for every set A. If P is symmetric, then p $dP=s* 4 dP. Gilboa (1989) provides normative arguments for symmetry of capacities in decision making. We will now state the downward and upward nth degree stochastic dominance theorem for capacities. Let dSyzX&=$(y)-@(x) and Ak#= AAk-’ C#J,with A’ = A. Define R,, as the class of functions 4: R+R such that (Ak):_,q5(s))=0

Vk~{l,...,n)

Vxiy,

and gn as the class of functions 4: R-+R such that

FI =Y, is the class of non-decreasing functions, Fz is the class of nondecreasing and convex functions, Yz is the class of non-decreasing and concave functions, and so on. For PE% define G;(L)=~-~ G”p-r(x)dx and G;(t)=iyU G”p-l(~)dx, with G:(t) = P(( - co, t]) and G;(t) = P([t, co)). Theorem 2.2.

Let P,,P1~%(R,Bor(R)).

Then

178

(4

M. Scarsini, Dominance conditions in non-additive expected utility theory

jf#JdP,&llJ,

VCpEF”

lf

G&(t)

s

G&(t)

Vt E Ft.

Proof. Combine Lemma 2.1 and the well-known results about nth degree (downward and upward) stochastic dominance [see e.g. Rolski (1976), Fishburn (1976, 1980), Ekern (1980), Jean (1980), Stoyan (1983)]. 17 Levy and Levy (1984) provided first and second degree stochastic dominance conditions for multivariate utility functions that represent the same preference ordering. Given a positive, continuous, quasi-concave, increasing function u: Rd+R, we define 5$={w: [wd+lw:w(x)=&(x)),

k: [W-R increasing),

and sbf”‘= {w: w E FU, w concave}. Since k is an increasing function has the same level sets {x: 4(x) 2 a} is quasi-concave. The class ,y can If it is not empty, then it has a least Debreu, shows.

of a real variable, any function 4 in gU as u. These level sets are convex, since u be empty, as shown by de Finetti (1949). element as the following theorem, due to

Theorem 2.3 [Debreu (1976)]. If Py is not empty, then there exists u* E SF;=“,such that, t/w E 9;“‘, w(x) = k(u*(x)), with k increasing and concave. Let &us A,iff A,= {x: u*(x) 5 t}. The least element U* of Yr plays a role in the second degree stochastic dominance results of Levy and Levy (1984). The following theorem is analogous for capacities of their result. Theorem 2.4.

Let P,, P, E %?(Rd, Bor ( Rd)). Then

(a)

!4dP,+dP,

WE%

(b)

?dP,+$dP,

V~EF?

iff

P,(A) 5 P,(A) zj-

v, E d”.

jP,(A,)dtjT 0

P,(A,)dt 0

Vx E R, VA, E&,.

Proof. The proof stems from our Lemma 2.1 and Theorems 3 and 4 of Levy 0 and Levy (1984) since [Wdis weakly ordered by gU(xiyiff u(x)Iu(y)).

M. Scarsini,

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179

3. Partial preference orders

When we deal with a space that is only partially ordered (e.g. S = Rd, with 2.1 generally does not hold. In particular, it is not true such that P(A)= P*(A) for all A upper, as the following counter-example shows. Let the support of P be (0, l}‘, and let d > l), then Lemma that VPE%‘,~P*EY

P{(O,O)}

P((0,

=0.2,

l)} = P{( 1,O)) =O.l,

P{(

1, l,} =0.3,

If there existed a probability measure PEEL coinciding with P on 99, then we would have P*{(O, 1)) = P*{(l,O)} =0.2 and therefore P*((O, l),(l,O),(l, l)}= 0.7: a contradiction. Nevertheless, also in the multivariate case, it is possible to provide some stochastic dominance results, which are somehow analogous to the known results for the additive case. Let & be a class of subsets of S such that d c@(S). Let F& be the class of functions 4 whose level sets are in ~4. As the following theorem shows, if an inequality for the probability of all the sets in z? is established, then the same inequality holds for the expectations of the functions whose level sets are in d. This is an immediate consequence of the structure of the (upper) Choquet integral. Theorem

3.1.

Let PI, P, E V(S,

Bor (S)). Then

iff P,(A) 2 P,(A) Proof.

‘If’ part.

VA E Jlf.

‘Only if’ part. Choose 4 = I,, with A E ~4. {s~S:~(~)~t(}~~iff~~~~. Therefore, if (3.1) holds, then

[ P,(s:$(s)>a)daZ%

and

P2(s:$(s)2a)da,

(3.1)

180

M. Scarsini, Dominance conditions

_~~lP,(s:m(r)Lrr)lldat

in non-additive

expected

utility theory

; [P&$(s)ZCI)-l]dcr, -m

that is

It is clear that Theorem 3.1 holds Theorem 3.1 generalizes many other results to the case of capacities. Call

Fi”c~9a={~:

f#l(x)=u((a,x)),

when F& =Finc and multivariate stochastic

with u non-decreasing

Fine 3 F,r = {4: 4 is non-decreasing Fine ~9~ = (4: 4(x) = u(min (six,, and azO>. In the first two cases S is assumed Rd. Call

Corollary

. . . , q,x,,)), with u non-decreasing

to be linear,

(A: A is an upper hyperplane

& =I%,=

{A: A is an upper convex

3.2.

interval

and a?O},

and quasi-concave},

@ I@~=

ozdXI%i, = {A: A is an upper

& =%. But dominance

and in the last one, S is just

in S},

subset

of S>,

of S}.

For i = a, /3, y,

In the framework of additive probability measures, Scarsini (1986) and Muliere and Scarsini (1989) examined the class FE in the infinite and finite dimensional case, respectively. The case of flp has been studied by Levhari, Paroush and Peleg (1975). Mosler (1984) and Scarsini (1985), (1988) dealt with the case of d-antitone utility functions. This class contains FY, but it is too large for Theorem 3.1 to be used, as the following remark explains. Remark

3.3.

The classes %e,%B,%Y are not convex.

Call ST the convex

hull

M. Scarsini, Dominance conditions in non-additive expected utility theory

181

of pi (i=u,B, y). In the additive framework the conditions holding for functions Fi can be extended to 9: (by linearity of the expectation). This is not true in the non-additive context. Then (3.1) does not imply

This is due to the fact that Choquet integral is not additive, but only co-monotonically additive and 4, $ E& does not imply that they be co-monotonic, in fact their level curves generally intersect. Therefore the dominance condition does not hold, as the following counter-examples show. Example 3.4.

Let

P,((L1))=0.2,

P,((0,3)}=P,{(3,0)}=0.4, P,{(O,3),(3,0))=0.8,

P,{(O,3),(1,1)}=P,{(3,O),(l,1))=0.7;

P,{(O,3)} =P2{(3,O)j =0.3,

Example 3.5.

P,{(L 1)) =0.4,

Let

P, ((0,3)} =P, ((3,O)) =P, ((Z2)) =O.L

p, ((353)) =O.Z

PI {(0,3), (390)) = 0.5, P, {(0,3), (3,3)) = p, ((3, O),(3,3)) = P, {(2,2), (3,3)) =0.7, p,{(0,3),(3,3),(~,~)}=p,{(3,0),(3,3),(~,~)}=p,{(0,3)~(3,0),(3,3)~=0.7~ P*{(O, 3)} =P2{(3,0)}

=P,{(2,2))

=P,((3,3))

=O.L

P,{(O, 3), (3,O)) = 0.7, p, ((0,3), (393)) = p, ((39 O),(393)) = p, {(T 2)1(3,3)) = 0.5,

M. Scarsini,

182

Dominance conditions in non-additive expected utility theory

P2{(0,3),(3,3),(2,2)}

=P,{(3,0),(3,3),(2,2))

=O.%

P~{(O,3),(3,0),(3,3)1=O.g.

Example 3.6.

Let

P,{(0,0)}=0.1,

P,{(0,1))=P,((1,0)}=0.3,

P,{(0,1),(1,0)}=0.5,

P,{(l,1)}=0.2,

P,((O,1),(1,1)}=P,{(1,0),(1,1)}=0.7,

P,{(O, l),(l,O),(l,

1)) =0.7;

P2{(0,0)}=O.L

P2{(0,1,>=P2{(LO)) =P,{(l, 1,) =oz

Pz{(O,

=0.6,

l),(L 1)) =P,((l,O),(L

1))=0.5,

P*{(O,l),(l,O),(L 1))=m. We

have

VA E@~,P~(~ZP~(~.

If

jqM’,=P,1(4 1)4,0))+2p,{U,

l))=l.

In the additive case there exists a coupling characterization of first degree stochastic dominance for probability measures on partially ordered Polish spaces [see Kamae, Krengel and O’Brien (1977)]. Let P,, P, E 9(S, Bor (S)). Then f 4 dP, 2 s 4 dP, VC$E Fine iff 3P E B(S x S, Bor (S) 0 Bor (S)), such that P(A x S) = P,(A),

P(S x A) = P,(A)

VA E Bor (S),

and P{(x,y):xZy}=l.

(3.2)

M. Scarsini,

Dominance conditions in non-additive expected utility theory

183

In the non-additive case, no such characterization holds. It is even possible to have (3.2) and nevertheless s* $dP, ?ZJ* C$dP, Vqt ELF’“‘. We provide a counter-example. Example 3.7.

Let PE %?have support { 1, 2j2 with

P((1, 1)) =P((2,2)}

=0.2,

P((2, l)} =0.3,

P{(L 1)7(2,2)>(2,1,> = 1; P,{l}=O.2,

P,{2}=0.5,

P,{ I} =0.5,

P,(2)

Expression This has probability not behave

=0.6.

(3.2) holds, but P,(A) 5 P,(A) for every upper set A. to do with the fact that any reasonable definition of conditional leads back to the additive case [Gilboa (1989)] and capacities do nicely, when defined on joint or conditional events.

References Choquet, G., 1955, Theory of capacities, Annales de 1’Institut Fourier 5, 131-295. Debreu, G., 1976, Least concave utility functions, Journal of Mathematical Economics 3, 121-129. de Finetti, B., 1949, Sulle stratiticazioni convesse, Annali di Matematica Pura e Applicata 30, 173-183. Ekern, S., 1980, Increasing N-th degree risk, Economics Letters 6, 329-333. Ellsberg, D., 1961, Risk, ambiguity and the Savage axioms, Quarterly Journal of Economics 75, 643-669. Fishburn, P.C., 1976, Continua of stochastic dominance relations for bounded probability distributions, Journal of Mathematical Economics 3, 295-311. Fishburn, PC., 1980, Continua of stochastic dominance relations for unbounded probability distributions, Journal of Mathematical Economics 7, 271-285. Gilboa, I., 1987, Expected utility with purely subjective non-additive probabilities, Journal of Mathematical Economics 16, 65-88. Gilboa, I., 1989, Duality in non-additive expected utility theory, Annals of Operations Research 19,405-414. Jean, W.H., 1980, The geometric mean and stochastic dominance, Journal of Finance 35, 151-158. Kamae, T., U. Krengel and G.L. O’Brien, 1977, Stochastic inequalities on partially ordered spaces, Annals of Probability 5, 899-912. Levhari, D., J. Paroush and B. Peleg, 1975, Efficiency analysis for multivariate distributions, Review of Economic Studies 42, 87-91. Levy, H. and A. Levy, 1984, Multivariate decision making, Journal of Economic Theory 32, 36-51. Mosler, K.C., 1984, Stochastic dominance decision rules when the attributes are utility independent, Management Science 30, 1311-1322. Muliere, P. and M. Scarsini, 1989, Multivariate decisions with unknown price vector, Economics Letters 29. 13-19.

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expected

utility theory

Rolski, T., 1976, Order relations in the set of probability distribution functions and their applications to queueing theory, Dissertationes Mathema&ae 132. Scarsini, M., 1985, Stochastic dominance with pair-wise risk aversion, Journal of Mathematical Economics 14, 187-201. Scarsini. M.. 1986. Comparison of random cash flows. IMA Journal of Mathematics in Management 1,25-32. _ Scarsini, M., 1988, Dominance conditions for multivariate utility functions, Management Science 34,4.54460. Schmeidler, D., 1986, Integral representation without additivity, Proceedings of the American Mathematical Society 97, 255-261. Schmeidler, D., 1989, Subjective probability and expected utility without additivity, Econometrica 57, 571-587. Stoyan, D., 1983, Comparison methods for queues and other stochastic models (Wiley, Chichester and New York).