On preference relations which satisfy weak independence property

Journal of Mathematical Economics 18 (1989) 291-300. North-Holland

ON PREFERENCE RELATIONS WHICH SATISFY WEAK INDEPENDENCE PROPERTY*

Ezra EINY

Ben Gurion University, Beer-Sheva 84105, Israel

State University of New York, Stony Brook, NY 11790, USA

Received March 1988, accepted December 1988

We prove some representation theorems for two classes of preference relations on a convex subset of a topological vector space: continuous preferences which satisfy weak independence property with respect to convex combinations, and continuous preferences which are consistent with respect to translations.

1. Inlroduction

In this work we study two kinds of preference relations on convex subsets of topological vector spaces: preference relations which satisfy weak independence property with respect to convex combinations and preference relations which satisfy some consistency property with respect to translations which can also be interpreted as a weak independence property (with respect to translations). As we shall see for continuous preferences which are defined on some convex sets, these two notions are equivalent.

A preference relation t on a convex set K is independent (with respect to convex combinations), if for each x,y,z~ K and 0~11~ 1, if x>y then Ix +( 1 -A)z>3,y +( 1-2)~. This is one of the versions of the independence axiom of von Neumann-Morgenstern utility theory [see Fishburn (1982) for more details]. The relation 2 is weakly independent if for each x, y,z~K and O<A<l, if x>y then 2x+(1--A)z>ly+(l-_)z. Let K be a closed convex subset of a topological vector space X. We show that if K has non- empty interior (this assumption is not needed if X has a finite dimension), then each continuous and weakly independent preference relation on K can be represented by a utility function which is a composition of a continuous, monotonic (in the weak sense) function with a continuous linear functional

*This work was done while the author was a fellow at CORE. The support of CORE and helpful suggestions of an anonymous referee are gratefully acknowledged.

0304-4068/89/$3,50 0 1989, Elsevier Science Publishers B.V. (North-Holland)

292 E. Einy, Preference relations and weak independence property

on X (see Theorem A). It implies that any continuous and independent preference relation on K can be represented by the restriction to K of some continuous linear functional on X (see Corollary 3.4). These results are used to provide expected utility representations for preferences over measures (see Corollaries 3.5 and 3.7).

In Trockel (1987) a representation theorem for a preference relation which is continuous and translation invariant over a Euclidean space is proved and various economic applications are given (especially for demand and social choice theory). Here we study preferences which satisfy invariance property (with respect to translations) over convex subsets of topological vector spaces. We prove an analog of Theorem A (which is mentioned above) for continuous preferences which satisfy a consistency property with respect to translations (see Theorem B). This implies that on some convex sets translation invariance and independence are equivalent for continuous preferences (see Corollaries 3.4 and 4.3).

2. Preliminaries

Let X be a set. A binary relation 2 on X is a preference relation if it is complete and transitive. If t is a preference relation on a set X, we denote by > the strict relation induced by t (i.e. x>y if xty, but not y&-x) and

by - the indifference relation induced by 2. Let X be a convex subset of real vector space. A preference relation > on X is weakly convex if for each x E X the set {y 1 ~2x1 is convex. The relation 2 is weakly concave if the set (y ylx) is convex. 2 is weakly ufjne if it is weakly convex and weakly concave. Let X be a topological space. A preference relation 2 on X is continuous if for each x E X the sets {y E X 1 y>x} and {y E X 1 yix} are closed. Finally, a real valued function u on a set X is a utility representation for a preference relation 2 on X if for each x, y~x, xky iff u(x) zu(y).

3. Weakly independent preference relations

In this section we formulate and prove some results on the structure of a weakly independent and independent preference relations on a convex subset of a topological vector space.

Let K be a convex subset of a vector space. A preference relation t on K is independent if for each x,y,z~K and O<E+< 1, if x>y, then ix+( 1 - l)z>Ay+( 1-2)~. The relation 2 is weakly independent if for each x,y,z~K and 0~1~~1, if x>y then 2x+(1-+t1y+(l--2)~. We are now ready to state the main result of this section.

Theorem A. Let K be a closed convex subset of a topological vector space X. Assume that int K # Qr or dim X < co, and let 2 be a preference relation on K.

E. Einy, Preference relations and weak independence property 293

Then 2 is continuous and weakly independent if and only if there exist a continuous linear functional C$ on X and a monotonic and continuous function f on I = C/J(K) such that u = f o I$ is a utility representation of 2 on K.

For the proof of Theorem A we need some preparation.

Lemma 3.1. Let K be a closed convex subset of a topological vector space and let 2 be a continuous weakly independent preference relation on K. Then &- is weakly affine.

Proof Let a E K. We will show that the set K,= {XE K 1 x2a) is convex (the proof that the set (x E K 1 x<a} is convex is similar). Since t is continuous

and K is closed, by the denseness of the dyadic numbers in [0, l] it is sufficient to show that x, YE K,, implies 3x-t fy~ K,. Assume, on the contrary, that there are x, YE K, such that +x+-rye K,. Then fx++y<x and )x+_ty<y. We will show that for each natural number n we have

-)x+$y2r ( 1

l-& x+;y.

The proof by induction. For n= 1 it is obvious. Let n> 1. We assume that it is true for n- 1 and show for n. Then

1 ( 1 1 I--- __ 2”-’ x+2”_‘Y+Y.

Since 2 is weakly independent,

+x++(( 1 - ~)“+~Y)~!“+:r~

Thus

As 2 is continuous, we obtain that x+Jx+)y, which is a contradiction.

Lemma 3.2. Let K be a closed convex subset of a topological vector space X. Assume that int K #QI or dim X < co, and let 2 be a continuous weakly


independent preference relation on K. Then there exists a continuous linear functional &J on X such that for each x, y E K, &x)=4(y) implies x-y.

Proof. We assume, without loss of generality, that int K #a. For if int K = 0 and dim X < co we shall work in the afflne hull of K instead of X. Now if - has only one equivalence class the lemma is obvious. Let a, b E K such that aih. Since Z is continuous and K is connected (because it is convex), there is CE K such that a<c<b. Consider now the sets K, = {x~K/xtc} and K,={x~K(xic}. Then K, and K, are disjoint subsets of X with a non-empty interior. By Lemma 3.1, K, and K, are convex. Therefore by a standard separation theorem [see for example Theorem 8 in Dunford and Schwartz (1958, p. 417)], there exist LYE R and a non-zero continuous linear functional 4 on X such that 4(x) zcr for x E K 1 and 4(x)sa for XE K,. Let x1,x2 E K such that &x,)=4(x2). We will show that x1 -x2. By the continuity of 2, we may assume that x1,x2 lint K. Assume, on the contrary, that x1<x2. Then by the connectedness of K there is x,<d<x, such that not d-c. Consider the sets K,={x~Klx>d} and K, = {XE K ) xid}. Then K, and K, are disjoint convex subsets of X with a non-empty interior. Therefore there exist /?E R and a continuous linear functional II, on X such that t,+(x) zp if x E K, and Ii/(x) sfi if x E K,. Since

4(x1) = (b(x2) and +(x1) 2 fi <$(x2) (b ecause x2 E int K3), there exist p, qeX such that 4(p) <4(q) and t,b(p)>$(q). Assume now that c>d. Then K, c K, and K,c K,. Let x0 lint K such that 4(x,,) =u. Let E>O such that x0 + ~(p - q) E K. Then Cp(xo + e(p - q)) < c( and 4(x0 + E(q - p)) > CC Therefore x0 + .$q-p)EKI. Since K,cK3 and $(xo+4p-q))>Il/(xo+4q-p)), x0? ~(p-qq)~: K3. Denote y, =xo+&(q-p), y,=x,+~(p-q). Since y, E K, and y, E K,, y,>y,. Let YE K such that It/(y) c/3. For each 0s t 5 1 define f(t)=$(tyl +(l -t)y), g(t)=$(ty,+(l -t)y). Then f and g are strictly increasing and continuous in [0, l] and f(t) <g(t) for each 0 < t 5 1. Since g(1) >p, there is O< to< 1 such that g(t,) >fl and f(to) -CD. Therefore

to~2+(1-to)~~to~1+(l-to)y. As Y,>Y,, this contradicts the assumption that 2 is weakly independent. If d>c, we choose x0 lint K such that $(x0) = p and use the same argument as above.

Let 2 be a preference relation on a set X. Then 2 is countably dense if there exists a countable subset Z of X such that for each x, YEX with xty there is ZEZ which satisfies xiziy. It is well known that a countably dense preference relation on any set has a utility representation [see for example Proposition 5 in Debreu (1964) and Theorem 3.1 in Fishburn (1970)].

Proof of Theorem A. It is clear that if 2 has a utility representation u on K, of the form u = f o 4, where 4 is a continuous linear functional on X and f is monotonic and continuous on 4(K), then it is continuous and weakly independent. So assume that 2 is continuous and weakly independent on K.


By Lemma 3.2, there exists a continuous linear functional 4 on X such that

for each x,y~ K, &x)=@(y) implies x-y. We will show now that t is countably dense on K. Let I =4(K) and let ri, r2, r3,. . . be an enumeration of the rational numbers in I. For each natural number n let Z,EK such that &z,) =r”. Define Z= {zl,zz,. . .}. Then Z is a countable subset of K. Let x,y~ K such that x<y. We will show that there is Z,EZ such that x<z,<y. Since K is connected, there is z~int K such that x<z<y. Since 2 is continuous, there exists a neighbourhood V of z in X such that I/c K and for each VE r! x<u<y. Choose a rational number r,,~& V). Then there is PE V such that &z,)=&p). Therefore P-Z,, which implies that x<z,<y. Thus 2 is countably dense on K and hence it has a utility representation u on K. Since 2 is continuous on K and K is connected, by Theorem 3.5 of Fishburn (1970), 2 has a continuous utility representation on K. Therefore we may assume that u is continuous on K. Define f:Z+R by ~(~(x))=u(x) for each x E K. Then f is well defined. By Lemma 3.1 u is quasi-atIme on K (i.e., u is quasi-convex and quasi-concave on K). Therefore f is quasi-affine on I, and hence it is monotonic (a function which is defined on an interval of real numbers is quasi-affine iff it is monotonic). We will show that f is continuous on I. Indeed, since u is continuous on K and K is connected, u(K) is an interval. Thus, f is a monotonic function on I and its range is the interval u(K). Therefore fmust be continuous on I.

Let K and X be like in Theorem A. The following corollary of Theorem A implies that any continuous and independence preference relation on K has a von Neumann-Morgenstern utility which is the restriction of a continuous linear functional on X to K.

Corollary 3.4. Let K be a closed convex subset of a topological vector space X such that int K #@ or dim X < co, and let t: be a non-trivial preference relation on K (i.e., N has at least two equivalence classes). Then 2 is independent and continuous iff there is a continuous linear functional Cp on X which is a utility representation of 2 on K. Moreover, q5 is unique up to multiplication by positive numbers.

Proof The sufficiency part is clear. So assume that t: is continuous and independent. By Theorem A, 2 has a utility representation on K of the form u = f o 4, where f is monotonic and continuous, and I$ is a continuous linear functional on X. Without loss of generality f is non-decreasing (otherwise we can consider u’(x) = -f (( - 4(x)), x E K). Let x, YE K. We will show that xky iff 4(x)24(y). Since &x)24(y) implies x>y, it is sufficient to show that x-y implies 4(x)=+(y). Assume, on the contrary, that 4(x)<&y). Without loss of generally, there is ZE K such that z>x. Since 4(x) <b(y), there is 0 <t < 1 such that &tz + (1 - t)x) = 4(y). Therefore tz + (1 - t)x _ y. But this is


impossible because x-y and by the independence of 2, tz +( 1 - t)x tx + (1 - t)x =x. To prove the uniqueness of 4 up to multiplication by positive scalar, assume that $ is linear functional on X which also represents 2 on K. Let x0 lint K such that $(x0), 4(x0) #O. Then ~(x,,)~(x,,) >O. For if for example, $(x0) <O < 4(x,,), then we choose E > 0 such that (1 +&)x0 E K. Then $(( 1 +&)x0) < $(x0) and 4(x,, +&x0) > 4(x,), which is impossible. So assume, without loss of generality, that 4(x,,) and $(x,,) are positive. Let $1 = (l/+(x,))4, rc/1 =(l/lc/(xO))$. We will show that $1 =41. For otherwise, there is XEK such that $1(x)< 1 Cam, which is impossible because 41 and 11/1 represent 2 on K.

Using the special representations of linear functionals over vector spaces of measures, Theorem A and Corollary 3.4 enable us to give expected utility representations for preferences over measures (see Corollaries 3.5 and 3.7 below).

Let S be a set and Z and algebra of subsets of S. Denote by B(S,C) the Banach space of all bounded, real-valued, measurable functions on S with the supremum norm. Let FA be the Banach space of all bounded finitely additive measures on C with the variation norm. It is well known that FA is the dual of B(S, C) [see Theorem IV.5.1. in Dunford and Schwartz (1958)].

Corollary 3.5. Let K be a closed convex subset of the space FA endowed with the weak*-topology. Assume that int K #@ or ISI < 00, and let >- be a non- trivial preference relation on K. Then 2 is continuous and weakly independent iff there exist a function UE B(S, C) and a non-decreasing and continuous

function f on the interval I = {js u dp 1 p E K} such that for each p, II E FA,

(3.1)

Moreover, u is unique up to multiplication by positive numbers.

Proof. The existence of u and f such that (3.1) is satisfied is clear from Theorem A. We will show that u is unique up to multiplication by positive numbers. Assume that there is VE B(S, C) and a non-decreasing and continuous function g on {ssv dp 1 p E K} such that ptl iff g&v dp) zg(j, v dA) for each p, 2 E K. Since f is not constant on I = {Is u dp 1 p E K) (2 is not trivial), there is t E int I such that f is strictly increasing in some neighborhood V of t in I. Define 4: FA+R, $: FA-+R by +(p)=j,udp and $(p)=j,vdp for each ~1 E FA. Let p. E $- ‘(V) such that +(po) #O and $(po) #O. Then +(~o)$(~o) > 0. If for instance &(po) <O < $(po), we choose E>O such that

(l+Go~K1(l/l. Then f(4((l+4p0))<f(4(po)) while g(W+4pO))2


g($(pO)), which is impossible. Without loss of generality, 4(p0) and r+Qe) are positive. Let 4r =(1/4(p0))$ and tjl =( l/r&,))+. We will show that d1 = el. Assume not. Since $i(,~~)=Ic/~(~,,)= 1 (but $i #I,+~), there are pi, ,u~ ~$-i(k’)

such that &(P) < 1~ $i(~). Therefore f(@(~)) <f(&~o)), while &HP)) 1 g($(pe)), which is impossible. Let c= $&)/I&). Then c>O and 4 =c$. Thus Js u dp = Js cu dp for each p E FA. Therefore u = co.

Remark 3.6. Let S={s, ,..., s,} be a finite set and let As be the simplex of probability distributions on S. Assume that >- is continuous and weakly independent on As. Then by Corollary 3.5 there exist a real valued function u on S and a non-decreasing function f:conv u(S)+R such that for each p, q E As, p2q iff f(Cy= I p(s,)u(s,)) 2 f(c;= 1 q(si)u(si)). Thus the set of the best elements of 2 in As includes those for which the expectation of u with respect to them is maximal.

The following corollary shows that by strengthening weak independence to independence in Corollary 3.5 we can get rid of the function f to obtain an expected utility representation for the preference relation.

Corollary 3.7. Let K be a closed convex subset of the space FA endowed with the weak*-topology. Assume that int K #@ or ISI < co, and let 2 be a non- trivial preference relation on K. Then 2 is continuous and independent ijjf there exists a function u E B(S, C) such that for each p, 1 E FA,

Moreover, u is unique up to multiplication by positive numbers.

Proof By Corollary 3.4, there exists a continuous linear functional 4 on FA which is a utility representation of 2 on K. Now there is UE B(S, C) such that 4(p) =isudp for each ALE B(S, C). Thus (3.2) is satisfied. The uniqueness of u up to multiplication by positive numbers follows from the fact that $J is unique up to multiplication by positive numbers.

4. Preferences which are consistent under translations

In this section we study preference relations on convex sets that have some invariance properties under translations.

A preference relation 2 on a subset X of a vector space is consistent (with respect to translations) if there do not exist x,, x2, y,, y, in X such that x,+y,~X for i,jE{l,2), and x,+y,>x,+y,, x,+y,<x,+y,. It is easy to verify that a consistent and continuous preference relation on a closed convex set K which contains 0 is weakly affine. Indeed, assume that


x,y, a E K and x, ~>-a. We will show that )x+ +y2a. Assume not. Then +x+3x = x>+y + fx while +x + fy<y = +y + +y which contradicts the consistency of 2, by the continuity of 2, %x+(1-A)yta for each O<A<l. Therefore 2 is weakly convex. The proof that 2 is weakly concave is similar. Using the weak affinity of a continuous consistent preference relation, some modifications in the proof of Lemma 3.2 yield an analog of it to a continuous consistent preference relation (provided that OEK). Indeed,

let Kr, K,, K,, &, 6, $ and x1, x2, CI, /I as in the proof of Lemma 3.2. Let x,~int K such that @(x,-J #O and $(xO)#O. Define

As x0 E int K and +1(x0) = $r(xO) = 1 (but 41 #$i), there is yO~ K such that O<~$,(y,-J<l<tj,(y,). Assume first that 4(x0)tj(x0)>0. Let a>0 such that l+s<l(/,(y,) and there are yr,y,~int K with ~$~(y~)=a,--v, 11/1(y2)=B1- (~+a~). Take z1 =.syO, z2=(.s+fs2)x0. Then for sufficiently small E, y,, y,, zl, Z~EK and yi+zjEK, i,j~:(l,2}. NOW ~l(y,+z,)<a,, $l(yl+z2)>a,, $1(y2+z1)>/11, and $r(y2+z2)<f11, which contradicts the consistency of t. If ~$(x,,)ll/(x~) ~0, we will choose 0 <E < 5 such that there are y,, y, E int K which satisfy 41(yl)=~r -2s, $r(y2)=fi1 -2s and yi+ZjE K, i, j E { 1,2}, where z1 = 3&x0, z2 =&x0. Then 4,(y1+z1)>al, &i(yr+z~)<a,, I~/~(Yz+z,)> fll and $r(y,+z,) <fir. Since &X,)$(X,)-CO, this contradicts the consistency of 2.

Since the proof of Theorem A is based only on Lemmata 3.1 and 3.2, a version of this theorem is true for consistent preference relations. Thus we have

Theorem B. Let K be a closed convex set in a topological vector space X such that OE K, and int K # % or dim X -C cc. Let 2 be a preference relation on K. Then 2 is continuous and consistent if’ it has a utility representation of the form u = f o 4, where 4 is a continuous linear functional on X and f is a monotonic and continuous function on 4(K).

Remark 4.1. It is interesting to note that each independent preference relation on a convex subset K of a vector space is consistent (without assuming that OE K). Indeed, assume that 2 on K is independent. If > is not consistent there are xi, x2, y,, ~,EK such that xi+yj~K for each i,jE{1,2} and x,+y,>x,+y,, x,+y,>x,+y,. Denote a=x,+y,, b= x2 +y,, c=x2 +y,, d=x, +y2. Then by the independence of t, we have fa + $d > +b + &d and fc + +a> )d + +a, which is impossible because fb+$d=fc+$a.

A preference relation on a subset X of a vector space is translation


invariant if for each x,y,z~X such that x+z, y+zeX, x&-y implies x+z>y+z. It is completely translation inoariant if it satisfies xty iff x + z>y + z. Trockel (1987) has studied translation invariant preferences which are defined on Euclidean spaces and provides some applications in various economic fields like demand theory and social choice. In Aumann (1962) and Kannai (1963) partial orders on a topological vector space which possess the translation invariant property were studied in connection with the problem of the existence of linear utility which represents a partial order.

Theorem 4.2. Let K be a closed convex set in a topological vector space X such that OE K, and int K #Qr or dim X < co. Assume that 2 is a continuous and translation invariant preferencee relation on K. Then either there is a continuous linear functional on X which is a utility representation of 2 on K, or 2 has a utility representation of the form:

4-4 = 4(x) 2 @z 4(x) <c19 or 4-4 = 4(x) 4(x) > a XEK, (4.1) t(

4(x) 5 a’

where a E R, and C#I is a continuous linear functional on X. Moreover if 0 E int K, then there is a continuous linear functional on X which represents >- on K.

Proof: Since 2 is translation invariant on K, it is consistent. Therefore by Theorem B, > has a utility representation of the form u = f o c$, where C#J is a continuous linear functional on X and f is monotonic and continuous on l=+(K). We will show that there do not exist c,,c*~R such that c1 #c,,

If -‘(cl)1 > 1 and If - ‘(cz)( > 1 (h ere 1.1 denotes the cardinality of a set). Assume on the contrary, that there exist such c1 and c2. Without loss of generality, c1 < c2. Let xi, x2 E K such that ci = f (4 (x,)), c2 = f (4(x2)). Denote KI={x~K~u(x)=cl}, K2={ x E K 1 u(x) = c2}. Without loss of generality, assume, that f is non-decreasing on I (the proof for non-increasing f is similar). Then SUP,,~~ 4(x) <infxeK2 4(x). Let /?=supxeKl 4(x). We assume that there is XE K such that 4(x) >O. Otherwise, we shall do the same analysis with fl= infxsKz 4(x). Since (f -‘(cJ > 1, there is X~E K, such that #(x3) c/3. Thus, @(x3) <B-C &x2). Therefore there is x0 Eint K such that 4(x,,) =p. The continuity of u and the connectedness of K imply that x0 E K,. Let y,~int K such that &xJ<&(y,) </I. There is ZE K such that d(z)>O, x,+z, y,+z~ K and 4(y,+z)<j?<&x, +z). Now u(xo)=u(yo) and u(xo +z) > u(y, +z), which is impossible because 2 is translation invariant on K.

What we have shown above implies that either f is strictly increasing on I or there is exactly one real number c such that If -‘(c)l > 1. If f is strictly


increasing, then q5 is a utility representation of 2 on K. Assume that there is CER such that If-‘(c)l> 1. We claim that f(t)sc for each trzl or f(t)lc for each t E I. For otherwise let K’= {XE K 1 u(x) =c}. Then inf,,,. 4(x) and supxEK, 4(x) are attained in K’, and we can use the above arguments to get a contradiction. Therefore f is either constant (in this case the theorem is trivial) or 2 has utility representation of one of the forms in (4.1).

It remains to show that if O~int K then there is a continuous linear functional on X which represents >- on K. As OE int K, there is z E K such that 4(z) >O and fz E K. Therefore by using the same arguments as above (assuming that 2 is not trivial), it can be shown that If-‘(c)1 5 1 for each c E R. In order words f is strictly monotonic. Therefore in this case, 4 (or -4) represents 2 on K.

Corollary 4.3. If in Theorem 4.2, we assume that 2 is completely translation invariant (without the assumption that O~int K), then there is a continuous linear functional 4 on X which represents 2 on K. Moreover, 4 is unique up to multiplication by positive numbers.

Proof. The proof follows easily from Theorem 4.2 and it is omitted.

References

Aumann, R.J., 1962, Utility theory without the completeness axiom, Econometrica 30, 445462. Debreu, G., 1964, Continuity properties of Paretian utility, International Economic Review 5,

285-293. Dunford, N. and J. Schwartz, 1958, Linear operators, Part I (Wiley, Chichester). Fishburn, P.C., 1970, Utility theory for decision making (Wiley, Chichester). Fishburn, P.C., 1982, The foundations of expected utility (Reidel, Dordrecht). Kannai, Y., 1963, Existence of utility in an infinite dimensional partially ordered space, Israel

Journal of Mathematics 1, 229-234. Trockel, W., 1987, An invariance theorem for preference and some applications. Working paper

no. 157 (IMW, University of Bielefeld, Bielefeld).

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On preference relations which satisfy weak independence property