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Continuous representation of Von Neumann–Morgenstern preferences
Journal
of Mathematical
Economics
19 (1990) 323-340.
North-Holland
CONTINUOUS REPRESENTATION OF VON NEUMANN-MORGENSTERN PREFERENCES Mitsunobu University
MIYAKE*
ofTsukuba, Tsukuba, Ibaraki 305, Japan
Submitted
May 1988, accepted
May 1989
This paper considers a continuous representation of preference relations satisfying Grandmont’s (1972) Expected Utility Hypothesis, We equip the preferences with the topology of closed convergence, then we show the existence of a jointly continuous expected utility function and consider its uniqueness. Furthermore, we construct an embedding map of the preferences into the set of expected utility functions.
1. Introduction A continuous representation of a set of preferences is a continuous selection of a suitable utility function from the class of utility functions representing each preference. The continuity of the representation is considered in the sense that a convergence of the preferences implies a convergence of the selected utility functions. In an economic model, the continuous representation of a set of preferences or its variant, a jointly continuous utility function both on alternatives and preferences, is a useful tool when we consider continuity or stability properties of a solution of the model against its data (e.g. individual preferences or endowments). The jointly continuous utility function is introduced by Kannai (1970) to investigate continuity properties of the core of an economy and is discussed by Mas-Cole11 (1977, 1985), Back (1986) and others.’ Especially Milgrom and Weber (1981) consider continuous representation under uncertainty which can be applied to some classes of games and economic models.2 However, their expected utility functions are not derived from preference axioms. Thus, there is no general relation between convergence of the utility *I would like to thank Professors Yoshihiko Otani, Herbert Heyer and Toru Maruyama for their valuable comments and advice. Of course, any remaining errors are mine. ‘See the references of Back (1986). ‘The stability of solutions for cooperative [non-cooperative] games is studied in Jansen and Tijs (1983) [Van Damme (1983, 2.4)], respectively. and the continuity properties of economic equilibria are analyzed in Allen (1983). Chichilnisky (1985) investigates a continuous social welfare function. 03044068/90/$3,50
0
1990, Elsevier Science
Publishers
B.V. (North-Holland)
324
M. Miyake, Von Neumann-Morgenstern
preferences
functions and convergence of the underlying preferences which the utility functions represent, since the expected utility function is unique up to positive affine transformations. In this paper, we start from Grandmont’s (1972) axiomatic approach to a continuous von Neumann-Morgenstern preference which satisfies Paretiantype continuity as well as the Expected Utility Hypothesis. Then we equip the space of preferences with the closed convergence topology and consider their continuous representation by expected utilities. As a main result, we show, using a constructive method based on Kannai (1970), the existence of the jointly continuous (expected) utility function and its uniqueness property for the full class of von Neumann-Morgenstern preferences, which is a generalization of Grandmont’s Expected Utility Theorem. Furthermore, we show that the continuous utility function actually establishes an embedding (map) of a class of von Neumann-Morgenstern preferences into the set of expected utility functions. Thus the similarity of the von NeumannMorgenstern preferences with respect to the topology is identical to the similarity of their (suitably selected) expected utility functions. Hence we can translate the stability of a solution of a model in terms of (small perturbations of) utilities into the stability in terms of the preferences. The next section reviews Grandmont’s Expected Utility Theorem for a continuous von Neumann-Morgenstern preference. In section 3, we introduce the topology of closed convergence on the set of preferences and present the existence and uniqueness results of the jointly continuous expected utility function. Section 4 gives their extensions which admit the set of alternatives not to be locally compact and the preferences to be statedependent. The proofs are given in section 5 and the appendices. In our proof of the main result, we dispense with the continuous selection theorem which is crucial for Mas-Colell’s (1977) proof, since we follow Kannai’s constructive method, i.e., normalization of each utility function.
2. A von Neumann-Morgenstern
preference and the Expected Utility Theorem
The set of (pure) alternatives is a separable metric space X. The metric on X is denoted by 1,. Let @ be the a-field of all Bore1 subsets of X and let (P(X),,) be a pair of the set of (countably additive) probability measures on %? and the topology of weak convergence on P(X).3 A measure PEP(X) is called a mixed alternative and interpreted as a lottery in which the possible prize is XEX. The set P(X) can be regarded as a convex subset of the linear space of all real-valued functions on SY. A convex combination is interpreted
3For the topology
of weak convergence,
see Hildenbrand
(1974, D.I.).
M. Miyake, Van Neumann-Morgenstern
325
preferences
as a two-stage lottery.4 Identifying a pure alternative x EX as the one-point probability measure p such that p({x)) = 1, x belongs to P(X), i.e., Xc P(X). A preference 2 is a binary relation on P(X). The expression pkq means that p is preferred or indifferent to q. For any preference 2, the strict preference > and the indifference relation ‘v are defined by p>q iff not qkp; p-q iff pkq and qkp. A continuous von Neumann-Morgenstern preference 2 on (P(X), o) is a preference satisfying the following three axioms: Axiom 1.
Complete Preordering.
2 is a complete preordering on P(X).
Axiom 2. Continuity. {q E P(X): qkp) sets in (P(X),o) for all pE P(X). Axiom 3.
Substitutability.
and {qEP(X):pkq)
are closed sub-
For any p, q and r in P(X), pm q implies
As a corollary of Grandmont’s Expected Utility Theorem (1972, Theorem 3), we have the following theorem: Expected Utility Theorem. Suppose that X is a separable metric space and P(X) is endowed with the topology of weak convergence w. Let 2 be a continuous von Neumann-Morgenstern preference on (P(X), w). (A)
There exists a real-valued continuous function u on P(X) satisfying:
(4 u(p)Zu(q)op?zq for all p,q E P(X). (ii) The restriction u, of u on X is a bounded continuous function on X such that u(p) =sx u,(x) dp for all PEP(X). (B) Let F(k) be the set of real-valued continuous functions on P(X) satisfying the conditions (i)-(ii) and let u be a function in F(k). Then a realvalued continuous function v on P(X) belongs to F(k) if and only if there exist real numbers tl> 0, fl such that v(p) = CY* u(p) + f_3for all p E P(X).
Assertion (A) states that there exists a continuous expected utility function such that its expectation is taken in a general integral form and its restriction on X becomes a Paretian continuous utility function. Assertion (B) states that the utility function is unique up to a positive affine transformation. If X is finite, Axiom 2 is reduced to Herstein and Milnor’s (1953) Continuity Axiom (Axiom 2) and our Expected Utility Theorem is essentially equivalent to Herstein and Milnor’s theorem. 3. Continuous representation of von Neumann-Morgenstern In this section we analyze the topological ‘%ee Fishburn
properties
(1982, p. 13) and Lute and Raiffa (1957, Ch. 2).
preferences
of the set of von
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M. Miyake, Von Neumann-Morgenstern
preferences
Neumann-Morgenstern preferences and their derive continuous representation. Suppose that the set of pure alternatives X is a separable metric space as in section 2. In addition we assume that X is locally compact, which is a very general condition in economic analysis. Let W be the set of the continuous von Neumann-Morgenstern preferences on (P(X),cu) and let ~8 be the class of all closed subsets of P(X) x P(X). Identifying 2 EB by its graph G( 2) = {(p, q) E P(X) x P(X): pkq}, W is a subset of .&?’be Lemma A in appendix A. In order to introduce the topology of closed convergence z on k‘, we have to additionally assume that X is compact. Since X is compact if and only if P(X) is locally compact whenever X is locally compact by Heyer (1977, p. 26, Theorem 1.1.12), and since the closed convergence is topological in J%’ only when P(X) x P(X) is locally compact by Watson (1953), we cannot drop the compactness of X as long as we consider the topology on _c&‘.This assumption is very natural when we consider bidding games or bargaining games. Under this assumption, the (relative) topology of closed convergence on ,?# has the natural properties as Paretian preferences [Hildenbrand (1974, p. 96, Theorem l)]. Namely we have the following lemma: Lemma 1.5 (a) (2,~) is induced by the Hausdorff metric. (b) A sequence (2,) in 92 converges to 2 E.%? if and only if Li G(kJ= G(k) =Ls G(&), where Li G(kJ and Ls G(k,,) are the topological limes inferior and superior of {k”}, respectively. (c) Let kO be the null preference such that p-,,q for all p,qEP(X). kO is an isolated point in (2,~) if and only if X is a finite set.6 Proof.
Appendix A.
Next we consider the Kannai topology introduced in Kannai (1970) on the set of the non-null preferences B* E B - {kO> under the assumption that X is a compact metric space. To define and characterize the topology we need several definitions. Since X is a compact metric space, P(X) is a compact metric space by Hildenbrand [1974, p. 49, (30)]. Therefore there exists a countable basis II E {Bi} such that Bie II is a relatively compact open set for all i. Let T={(i,j):BinBj=4 and Bi>Bj for some 2 EB} where Bi is the closure of Bi and Bi>Bj means that p>q for all PE Bi,q~ Bj. For any t-(i,j)E7: let ~~-~~i,j,={~~E:Bi~Bj~. Note that UfeTgt=B* by 5For the mathematical concepts in Lemma 1, see Hildenbrand (1974, B. II.). 6This condition plays an important role in Chichilnisky’s (1985) social choice analysis. We derive it without the assumption for interpersonal utility comparison assumed in Chichilnisky (1985).
M. Miyake, Von Neumann-Morgenstern
preferences
327
Lemma A in appendix A. The Kannai topology is defined to be the topology on .4?* which has the class {,c!J~)~~~as a sub-basis. The relative topology of q on gLt is denoted by g, for all t E 7: Let {bi} be a countable subset of P(X) such that big Bi for all i, and let u, be the function on P(X) x S?Z for each t = (i, j) E T defined by
3UEf(k).
(1)
Though a value of u,(p, 2) depends on the selection of {bi} from P(X), it does not depend on the selection of u E F(k) by the Expected Utility Theorem (B). Therefore, o,(p, 2) is well-defined and then u,( . ,k) is the normalized utility function of 2 e.%Yl,such that a,(b,, 2) = 1 and v,(bj, 2) = - 1 for each t E T Then we have the following lemma: Lemma 2. (a) The Kannai topology n is the coarsest Hausdorff topology on 9?‘* for which the . set A-{(k,p,q)E9*xP(X)xP(X):p>q) is open in 9?* x P(X) x P(X). (b) For all t E 7: the relative topology n1 on 9t is induced by the following metric pt:
(c) n coincides with the topology of closed convergence 7: on .B*. Proof:
Appendix
A.
By Lemma 2, the topology of closed convergence r on SY* is characterized as the minimal Hausdorff topology such that A is open, and u,(p, 2) is already a jointly continuous utility function on P(X) x 95!( for all t E T For Paretian preferences the corresponding assertions to those in Lemma 2 are given in Kannai (1970, Theorems 3.1 and 3.2) and Hildenbrand (1974, p. 105, Problem 6). Although such assertions hold within the class of monotone Paretian preferences, in our case of von Neumann-Morgenstern preferences Lemma 2(a,c) hold for non-null preferences B* and Lemma 2(b) holds on each ,c#~. The main result of this section is the following theorem: Theorem 1. Suppose that X is a compact metric space and P(X) is endowed with the topology of weak convergence CO.Let &? be the set of continuous von
328
M. Miyake, Von Neumann-Morgenstern
Neumann-Morgenstern closed convergence z.
preferences
on (P(X),o)
preferences
endowed
with the topology
of
(A)
There exists a real-valued function U on P(X) x 9 such that: (i) U(p, 2) is jointly continuous on P(X) x 9’; (ii) U(p, k)g U(q, z)opzq for all p,qEP(X) and 2 E.@; (iii) U(p, 2) = lx Ux(x, 2) d p f or all (p, 2)~ P(X) x 6% where U, is the restriction of U on X x W. (B) Let 4Y be the set of real-valued functions on P(X) x W satisfying the conditions (i)giii) and let U be a function in a. Then a real-valued function V on P(X) x B belongs to 6& if and only if there exist continuous functions a(k) and /I( 2) from 93 to [ - CO,CD] such that: (iv) V(p, 2) E P(X) x B*: V(p, 2) = a( 2). U(p, 2:) + B( 2) and a( 2) > 0; (v)
~P~fYX):
=
V(P, 20)
a(kJ.U(p,
kO)+/?(kO)
if
X is afinite
set,
otherwise
C
where c = lim,, + coC~~(~~).U(P,~~)+B(~~)I for some PEP(X) and some convergent sequence (k,,} in W* such that lim,, + J12” = ko.
Assertion (A) states the existence of a jointly continuous utility function on P(X) x 9 such that U(. , 2) satisfies Grandmont’s Expected Utility Hypothesis, and Assertion (B) states the jointly continuous utility function is unique up to a positive affine transformation in a functional form. U(p, 2)
Corollary 1.1. Under the assumptions of Theorem I, suppose 9 to be the set of real-valued continuous functions in P(X) endowed with the topology of untform convergence $. Let U be a function in @ and let t? be a function from 92 to 9 such that 0( 2) = U( ,2). Then tI is a homeomorphism from (9*, z) to (t3(9?*), $) c (9, $), i.e., 0 is an embedding of (W*, z) into (F, II/). In addition, if X is a finite set, tI is an embedding of (9, z) into (9, II/). Corollary 1.2. Under the assumptions of Theorem 1, let 94?_,= (2 for any x, YEX and let v?(p) be a function on P(X) such that
v&4 = CW - WllCW
- u(y)1 for some u E U 2).
The function 8: .!JQ,+F defined by t3(2) =vt is an embedding (9, $) such that e( 2) E P( 2) for all 2 E%‘,...
Corollary
lB:x>y)
1.1 states that the jointly
continuous
of (.@._,,T) into
utility function can be
M. Miyake, Van Neumann-Morgenstern
preferences
329
recognized as a homeomorphism between the class of non-null preferences and the class of expected utility functions and the homeomorphism can be extended to the whole space 3 if X is a finite set. Specifically, Corollary 1.2 states that the function which assigns the normalized expected utility function a E r(k) such that u(x) = 1 and u(y) = 0 to each preference 2 in BxY is a homeomorphism between gxp and e(&?:,,).
4. Extensions In order to represent individual preferences in an economic mode1 under uncertainty, we have to consider the case that the set of pure alternatives X is the positive orthant of the n-dimensional Euclidean space E’!+ and the preferences may be state-dependent. If the set of pure alternatives X is ET, then the set of probability measures on X is not locally compact and the closed convergence of the preferences on the measures is not topological as stated in section 3. In this section we introduce a new topology for a class of preferences based on the closed convergence on compacta. Then we derive a continuous representation of the preferences by the topology. Assume that the set of pure alternatives X is E’!+. Let (9, P(X)) be the pair of the set of Bore1 subsets of X and the set of probability measures on :‘A endowed with the topology of weak convergence o. Let &&,,,, be the class of the von Neumann-Morgenstern preferences 2 on (P(X),o) in .B satisfying: Axiom
4.
Weak Monotonicity
in X.
x > ysx>y
for all x, Y E X.
To define a topology on gWrno we need some definitions. Let x = {K: K is a compact subset of X}, zT*={KEX:~X,~EKX>~} and P,={~EP: p(K’) =O> for all K ~31r. For any 2 E&,,,,, and K E %‘*, let GK( 2) = G(k)n(P,xP,) where G(k) is the graph of 2, and let dK(&,k2)= hK(GK(kJ,GK(k2)) for all k1,k2~.&,,,, and KEAC* where h, is the Hausdorff metric on the closed subsets in P, x P,. Note that P, x P, is compact since P, is compact by Hildenbrand [1974, p. 49, (30)]. Then define D(&K,E)={~*E&,,~: dK(k,k*)<&) for any ~E&Y,,,~~, KEA?‘* and E>O. Let T* be the topology on &,,, which has the class {D( 2, K,E): 2 E&Ywmo,K E%*, E>O} as a sub-basis. The topology r* has the following property: Lemma 3.
(B’,,,,
z*) is induced by the following
metric d,:
M. Miyake, Von Neumann-Morgenstern
330
preferences
where N is the set of positive integers and (Kt}tEn is a countable family compact sets in Z* such that: (cr)KiCKi+l for all iEN; (/-3)UieNKi=X. Proof
Appendix
of
B.
Then we have the following
theorem:
Theorem 2. Suppose that X = E”, and P(X) is endowed with the topology of weak convergence Q. Let &,, be the class of the von Neumann-Morgenstern preferences 2 ~9 on (P(X),w) satisfying Axiom 4 endowed with the topology z*, and let C(X) be the set of continuous functions on X endowed with the compact-open topology y. Fixing any two pure alternatives x,ye X with x >y define a real-valued function U(p, 2) on P(X) x .%?~mo by WP, 22) = CU(P) -
4~M4x) - u(y)1 for some u E F(k).
Then U(p, 2) is a jointly continuous expected utility function on P, x B)wmo for all K EX. Furthermore, a function 0 defined by 8( 2) = Ux( . ,k) is an embedding of (%&,,,, z*) into (C(X), y) where U, is the restriction of U( ., 2) on X.’ Next we introduce states of the world explicitly and allow the preferences to be state-dependent.8 Suppose that the set of pure alternatives is X= E’!+ and the set of states of the world is 52= E,. Let Y = X x Q = E",+ ’ and let PJr, be the set of Bore1 subsets of Y The measure p may be interpreted as the two-stage lottery which selects state SEQ and pure alternative XEX sequentially. Namely, p selects SE Q by the marginal probability of p on R at first, then p selects XEX by the conditional probability of p on S at the second stage. Let P(Y) be the set of the probability measures on (Y&) with the topology of weak convergence w and assume Y c P( Y) as in section 2. Let 4 wmo be the set of the von Neumann-Morgenstern preferences 2 on (P( Y), 13) satisfying: Axiom 4*. Intra-state SEQ, x>y*(x,s)>(y,s).
Weak Monotonicity
in X.
For any x, YGX and any
Then we can define a topology ? on &,,,, in the same manner as we did for T* in Theorem 2 modifying x and 3C* as ~6 = {K: K is a compact subset in Y} and $C* = {K: K is a compact subset in Y and x >y for some (x,s),(y, s) E Y}. As a corollary of Theorem 2 there exists a real-valued ‘The basic set of continuous utility functions % in Allen (1983, p. 70) may be recognized as a subset of 0(S,,,). ‘For the derivations of state-dependent utilities, see Dreze (1987, Ch. 1, 9.1). In our derivations, it is implicitly assumed that a person is able to express preferences among the lotteries, conditionally on any exogenously given probability measures on the states of the world.
function U(p, 2) on P(Y) x ,&,,, such that: (i) U(p, 2) is jointly continuous on PKx&,, for each K E~C; (ii) U(p, k)z U(q, k)epkq, and (iii) Therefore, it holds from U(p, 2) = jy U,(y, 2) dp V p, q E P, V 2 E ~‘,,,. Fubini’s Theorem that for any Bore1 probability measures p on X and p on Sz U*(P, CL,2) = U(P x P> 2) = f U,(Y,
2)
4p
x 4
Y
=! Specifically,
cjxUx(.u,ST2) dP1&
we have
U*(X,P, 2) = J U,(x-, s, 2) dlc. R Thus we derive a continuous Morgenstern preferences which world.
representation of continuously depend
the von Neumannon the states of the
5. Proofs Proof of Theorem I. (A) Since u,(p, 2) defined in (1) is the jointly continuous utility function on P(X) x BI by Lemma 2(b, c). Then transforming r,(p, 2) to u,(p, 2) E [0, l] for any t -(i, j) E T by u,(p, 2) = Cc& 2)
-n,(~M~,(2)
-n,(k)l,
n,( 2) = minPEp(xI v,(p, 2) where m,(2) = maxpeptXj u,(p, 2), and extending continuously u,(p, 2) on P(X) x 9 to be u,(p, 2) =0 on P(X) x 3”:: as in the proof in Mas-Cole11 (1977), we derive a jointly continuous expected utility function on P(X) x 2 by U(p, k)=‘&i,j,ET2m(i+jJ. uci, jj(p, 2). (B) The ‘if part’ of Assertion (B) is obvious. Thus we show the opposite direction. Let U and I/ be any functions in ‘2, and let c((2) and p( 2) be the real-valued functions on %?* such that:
42)
= CVPI, 2) -
V(P,,
~MWPI,
2) -
U(P,,
?)I,
(2)
(3) for some pl, p2~P(X) such that pI>p2. Note that the values of ~(2) do not depend on the selection of p1,p2, and that a(k), /?( 2) satisfy the condition (iv) [see Fishburn (1982, p. 20)]. We shall show ~((2) is continuous on 4!*. Let { 2,) be a convergent sequence in (&!*, z) such that lim,,, 2, = k0 E W*. Since kanE*, there are q1,q2cP(X) such that q1 >aq2. By Assertion (A), it holds that 3 n,,: Vnn>n,,: q1 >nq2. Then it holds by (2) that
332
M. Miyake, Von Neumann-Morgenstern
preferences
Therefore we have lim,,, c((2,) =CI( 2,) by Assertion (A), which implies SI(2) is continuous. Then /I( 2) is continuous on B* by (3). Extending a(k) and /I( 2) to %Ycontinuously, we have that the extended c((2) and p(k) satisfy (iv) and (v). If X is a finite set, the extension is completed by assigning suitable real numbers to c((ko), /?(ko) by Lemma l(c). Q.E.D. Proof of Corollary 1.1. It follows from Assertion (A) of Theorem 1 and Dugundji (1966, XII, 3.1) that 0(k) = U(. ,k) is one-to-one and continuous. Conversely, assume that {fnj b e a convergent sequence in (0(2*),$) such that lim,, o?f,=f: Let 2 = K’(f) and k=,= K’(f,) for all n. Then we have f,(p) = U(p, 2,) V n and f(p) = U(p, 2). By 2 ES%‘*,it holds by Lemma A in appendix A that 3 t=(i,j)~ T; 2 ~2~. Let s=minasdi f(p)-max,,,lf’(p)>O. There exists an integer ~zi such that max pEP(X~lfn(~)-f(~)(<~/4 for all n>fil. Then min,,B, f,(p) -maxPSBj f,(p) 2 s/2 >O for all n > n,. Therefore 2” E &!‘rfor all n>n,. Further, since the jointly continuous expected utility function on P(X) x W, is determined up to positive afhne transformations defined by Theorem l(B)(iv), and since P(X) is compact, it holds by Hildenbrand (1974, p. 30, Corollary) that
=
=
lim max \u,(P, 2,)-u,(p,
tn-t2
lim max IA(2,) - U(p,
?n+;t
= max
=
2,)
+ B( k=,)
PEP(X)
-A(k).U(P,
PEP(X)
Zz)I
PEP(X)
lim
Z)-B(k)1
IA( kJ.
U(p, 2”) + B( 2,)
kn+l=
mix (A(~).U(p,~t)+B(R)-A(~).U(p,~)-B(~)(=O
PEP(X)
for some real-valued continuous functions A(k) and B(k) on &. Thus we > = 2 by Lemma 2(b,c). If X is a finite set, we can construct a have lim n-a, N” homeomorphism based on the above by assigning a suitable constant function to the null preference ko, since k. is isolated by Lemma l(c). Q.E.D.
M. Miyake, Von Neumann-Morgenstern
333
prejbences
Proof of Corollary 1.2. It follows from Lemma 2 that (gx,,z) is induced by the sub-basis {,%?)tn.@xy}teTand that (P&n.%!,,, T) is metrized by pr for each t. Let p: be another metric on B,n W,, defined by replacing u,(p, kk) in the definition of pt with u:(p, kk) = v?,(p) = [u(p) - u(y)]/[u(x) -u(y)] (k = 1,2). Since u,(p, 2) is given by a positive afline transformation [Theorem l(B)(iv)] of z$(p, k), pt and p: are topologically equivalent. Therefore, it is easy to show that O(k)=+ is continuous. Further we can show that 8-i is continuous on O(%?,,,) as in the proof of Corollary 1.1 replacing g*, 2, and pt with gxp, gtn B,.. and p:, respectively. It follows from the Expected Utility Theorem that O(k) E r( 2) for all 2 E g_,. Q.E.D.
Proqf of Theorem 2. Let x, y be any two alternatives in X with .Y> y, and let K be a compact subset in jT. Suppose that {p,}, {k,,) be two sequences such that pn+p in (PK,m) and 2,-k in (&Ywrno,~*)as n+co. Since X=E’!+, there exists a family of compact sets {Ki)icN in x* satisfying (CZ)and (p). By 3 we have that lim,,,d,,(k”, k)=O k-2 in (.%,,,, r*) and Lemma which implies that lim,,, d,,(k,,,k)=O for all HEN. Since KcKj and {x,Y} c Kj for some Jo N, it holds from Corollary 1.2 that U( . , 2,) converges uniformly V( . ,k) on PKj and P,. By p,+p in (PKu), we have U(p,, kJ+U(p, 2). Therefore, it follows from the Expected Utility Theorem and Husain (1977, p. 256, Corollary 9) that 0 is one-to-one and continuous. The continuity of OK1 can be proved by Corollary 1.2, Lemma 3 and Husain (1977, p. 256, Corollary 9). Q.E.D. Appendix A Appendix A proves Lemmas 1 and 2 under the assumption compact metric space. We need the following lemma: Lemma A. For any 2 ES?‘* and any p, qE P(X) with p>q, in the basis Il such that p E Bi, q E Bj and &>Bj. Proof.
Assume
that p>q.
By the Expected
Utility
that
X is a
there exists Bi, Bj
Theorem,
it follows
that
p>C(3/4)P+(1/4)ql>C(1/4)p+(3/4)ql~q and that Bik[(3/4)P+(1/4)ql~ [( 1/4)p+(3/4)q]kBj for some Bi, Bje I7 such that PE Bi, q E Bj. Then we have B,>B,
with PE&
Proof of Hildenbrand
qE Bj.
Q.E.D.
Lemma 1. Assertions (1974, p. 19, Theorem
(a) and (b) are direct corollaries of 2). In the following we show Assertion
(c). First, assume that X = {xi, x2,. . ,x,} and that there exists a z-convergent sequence {k,} in g* such that lim n_m kn=kO. Let Xij={k~E*:xi>xj and ~~2x2~~ for all xgX} (1 SisH, 15 jsH, i#j). It is easy to show that W* = uijXij. Since there exists a finite number of Xi,, there exists a pair of
334
M. Miyake, Von Neumann-Morgenstern
preferences
the integers (i, j) and a subsequence (2,) of { kn} such that ~,,,EX~~ for all m. We assume (i, j) =( 1,2) without loss of generality. Let Qk= {qeP(X): 1 jx fkdxk-fX fk dq( <(l/3)) = {q E P(X): qk(xk) >2/3} (k= 1,2), where xk is the one-point probability measure and fk is the function such that fk(x) = 1 if x=x,; &(x)=0 otherwise for k=1,2. Then Q=Qi xQ2 (Q*=Q2xQ1) is an open neighborhood of (x1,x2) ((x2,x1)) in P(X) x P(X), respectively. Therefore, it follows from the Expected Utility Theorem and the definition of Qk and X,, that
41~~c41(X,)~XI +(1-41(x,)).x,l~[c(2/3).x,
+(1/3).x21
for all VEX,, and all (ql,q2) EQ. Then it holds that ‘not q2kql’ for all and all (q2,ql)EQ*, which implies Q*nG(R)=4 for all 2 EX,,. ?z EX,, Hence we have that Vn*: 3 n zti*; Q* n G( k,,) = C#Jwhich implies (x,, x1)$ Since (x,, x1) E G( k,,) = P(X) x P(X), this contradicts lim,,, G(&-,,). lim n.+3o2,= k0 (in 5). Then k,, is an isolated point if X is a finite set. Next we assume that X contains an infinite number of elements. Since X is a compact metric space, there exists an accumulation point x,, in X and convergent sequence {x,,}~~~ in X-(x,,} such that lim,,, x.=x0. Define a sequence of functions u, by u,(x) =(2(x,, x0) - A(x, x,))/~(x,, x,,) if 2(x,,, x0) > 1.(x,x,,); u,(x)=0 otherwise, where 1, is the metric on X. Let (k,} be a sequence in %?* defined by pk,,q iff lx u, dpzJ,u, dq. Then we show that lim n_+oo&= k0 (in 5). Let (p, q) be a pair of alternatives in P(X) x P(X) and let Q be an open neighborhood of (p,q). Since the set of measures with finite support is dense in P(X) by Hildenbrand [1974, p. 49, (29)], there exists (p*, q*) E Q such that p* and q* have finite support. Therefore, it follows from the definition of {knl that In,: Vnzn,; p*~~x~ and 3n,: Vnzn,; q*wnxl. Hence VnZmax(n,,n,); p*- ,,q* which implies Qn G( k,,) #& Then we have Li G( 2,) = P(X) x P(X) = G( k,J. On the other hand, P(X) x P(X) = Li G(k,,)cLs G(k,J by Klein and Tompson (1984, p. 26, Lemma 3.2.6), we have that Ls G( 2,) =P(X) x P(X) = G(R,). Hence we have lim n-rm > -II =& and k,, is not isolated. Q.E.D. Proof of Lemma 2(u). First we show that q is a Hausdorff > _ r, kz be any two preferences in R+ such that 2 1# k2. show that
3p,qEP(W; p>lq
and q>g.
assume ‘3p,qeP(X); pxlq without loss of generality. ‘3 P,~EW); ~>~q and q-2~’ true, then we have (A.l). Let us assume the latter. By kl, Since
k1 # kz,
we
may
topology. Let Then we will
(A.11 and q>g’ or If the former is kz~E*, we can
M. Miyuke. Von Neumann-Morgenstern preferences
335
assume additionally r>,s for some r, SE P(X). Let p(s) =E.s+( 1 -s).p q(c) =E. r+(l -E) .q for all EE(O, 1). By the Expected Utility Theorem, have that p(E)>Iq(E)
q(E)>2p(E)
for some EE(O, 1);
and we
for all EE(O, 1).
Then it holds that 3s*~(O, 1); p(~*)>~q(~*) and q(.z*)>,p(E*), which implies (A.l). It follows from (A.l) and Lemma A that there exist Bi, Bj, B,, B, in n such that: Bi> ,B,; B,> 2Bj; B, n Bj # 4; B, n B, # 4. Since Bi CTBj and B, n B, are open, there exists B,, B, such that B, c Bin Bj and B, c B,n B,, which implies 2 1E BR,, and 2 2 E &,,. By %?,,,,,n 8$,,,,= 4, q is Hausdorff. Next we show A is open in the product topology. For any (p,q, 2) E A, it holds by Lemma A that there exists (i, j) E T such that p E Bi E Z7, q E BjE n and Bi>Bj. Therefore 2 E~~i,j) and (p, q,k) E(Bi x Bj x ,~,i,j,C A. Since Bi x Bj x W,i, j, is open, A is open. Finally if x is a Hausdorff topology for which A is open, we can show v]cx as in Kannai (1970, the proof of Theorem 3.1). Thus we complete the proof of Lemma 2(a). Q.E.D. ProofofLemma 2(b). Let [, be the topology induced by the metric pt. We can prove vtc[, in almost the same manner as the proof of Theorem 3.2 in Kannai (1970) using the Expected Utility Theorem. To prove i, c ql, we need a modification of Kannai’s proof. Fix t=(i, j) E 7; and write it, q,, pt, & and u, as [, q, p, 9 and u for simplicity in the following proof. Since the class of open balls defined by p is a basis for i, it suffices to show that for any E > 0, k,EW,
In order to prove this, we use the following
Let n be an integer
such that
assertion:
n > max(cc, /?). Then be
it follows
that
1 > l/2+
a/2n > 0; 1 > l/2 + a/2n > 0. Let p*, q* E P(X) p*=(l/2+a/2n)bi+(l/2_cl/2n)bj;
4*=(1/2+P/2n)bi+(1/2-~/2n)bj,
and let K={q~P(X):q=(l/n)p+(l-(l/n))b*
where b*=(1/2)bi+(1/2)bj. u(P*, 2)
= cc/n;
for some PEP(X)},
(A.4)
Then we have by (1) that o(q*,
2)
=/3/n
for all 2 EP&
(A.5)
336
M. Miyake, Von Neumann-Morgenstern
preferences
and we have that K is compact since P(X) is compact and since the linear transformation on P(X) is continuous. Then it follows from (A.4) and (AS) that
={k&:q*>p>p*VpEK}. We can show that { 2 ~9: q*>p>p* V PE K} EV and (A.3) as in Kannai’s proof, since K is compact. Q.E.D.
that
(A.2) holds
by
Proof of Lemma 2(c). First we show r crj. Let (kn} be a r-convergent sequence in 99 such that lim,,, k,,= 2 EP. Then we show lim,,, k,,= 2 in I?. For any W EV such that 2 E w it suffices to show 3 N: V n > N; 2, E W Since 2 E W and since {G?t}t.T is a sub-basis of y, there exists a finite subset H= {(ii, h), . . . >(ih, j,)} of T such that 2 E n;=i ~)(ik,jk~C n! In the following we show that for some integer N*,
64.6) It holds from Lemma l(b) and a>bo(b,a) $G(k) that if a>b then there such that !I N: exists neighborhoods Q1 (Qz) of b ( a ) in P(X), respectively, Vn> N: V(x,y)~Qi x Qz; y>,x, which implies that there exists a neighborhood Q of (a, b) in P(X) x P(X) such that 3N:Vn>N:V(x,y)EQ;x>,y.
(A.7)
it holds that Bi,>Bj, for all k. For any (a, b)E& x 2 E there exists a neighborhood Q of (a, b) satisfying (A.7). Since Bi, x Bj, is compact, we can choose a finite number of the neighborhoods covering Bik x Bjk. Therefore, for each k we have 3 N,: V n> N,: V(X, Y) E Bi, x Bjk; which implies Bi,>,,Bjk and knnEtik,jkJ. Hence for N*= x>nY, max (N,, . . . , NJ we have (A.6). Next we show kn--+k in (99*, q) implies k=,+ 2 in (9?*, r). Since 2 ~9*, there exists p*,q* EP(X) such that p*>q*. By Lemma A, there exist in (W*, ye), it holds that 3 N: t=(i,j)ET such that GEE,. Since 2, -2 V n> N; ?,,E&?~, which implies p,(kn, k)+O by Lemma 2(b). Then we have that
Since B,,,
n:=l
Wcik,jk),
M. Miyake, Van Neumann-Morgenstern
t’J. kn) uniformly
331
preferences
to u,( . , 2).
converges
W)
First we show G(k) cLi G(k,,). For any (p,q)~ G(k), let p(s) =s.p* (1 -s).p; q(c)=E.q*+(l -c).q for all EE[O, 11. Then it holds that
forall EE Ki11;
u,(P(~,2) > q(q(d,k) P(E)-P and q(E)+q
as
+
(A.9) (A.lO)
s--+0.
By (A.8) and (A.9), we have V/E>O: 3 N: V n > N; a,(p(~), k,,) > u,(q(E), k,,), which implies p(tz)Z,,q(E) and (~(E),~(E))EG(~,). Therefore, it holds by (A.lO) that (p,q) ELMG(k,,) and then G(k))cLiG(k,J.
(A.1 1)
Second, for any (p,q)~Ls G(kJ there exist convergent sequences ip,,,], {q,,,j such that p,,,-+p and qm-+q satisfying that Vm: V N: 3 n> N; ~‘,(p,,,, 2,) 2 UJq,,,, k,,) by the definition of Ls G(k,). Since vl(pm, .) and t‘,(q,, .) are continuous on g holds that it r,(pm, k)=fim,-, Qrn7 k) 2 2) for all m. Since t)J ., k,) is continuous, it holds lim,, rl v,(q m, k,J =v,(i:, that u,(p, 2) 2 u,(q, 2) which implies pkq and (p,q) E G( 2). Hence we have
LsG(L,)cG(k).
(A.12)
It follows from (A.1 1) and (A.12) and Klein and Tompson (1984, p. 26, Lemma 3.2.6) that G( 2) c Li G( k,J c Ls G( 2,) c G( 2). which implies G(2)=LiG(&)=LsG(kJ and k,-+k in (.%‘*,t). Q.E.D.
Appendix B Appendix B proves the following lemma: Lemma 6>0
B.
For
Lemma
any K EAT*,
such that { 2 E&T,,,,,: U&z,
Proof:
K ~33
for
Select
For any some jEN.
(U(P)--(Y))/(u(x)--U(Y))
3 under
ka~9&+,mo and F>O
for
that
X= E”,
We need
there exists a real number
k)<@cWkz.,K,4.
we have
any
the condition
x, y
by the definition of {KiJicN that KC Kj in K such that x >y. Let V(p, 2) =
SOme
u~r(?z)
and
let
pt(2,,
k2)=
max,,,,)V(p,~~)--l/(p,~~)) for any ~Z;1,~.2~&,,(Z=K,Kj). Sincegw,,c G?*,,, it follows from Lemma l(a) and Corollary 1.2 that d, is topologically equivalent to pz (Z=K,Kj). Since (~~.~~,,:~~~(~~,~)0 by KC Kj, we have by Husain (1977, p.
M. Miyake, Von Neumann-Morgenstern
338
Proposition
preferences
17) that for any E>O there exists a real number
that {~~E~,o:~Kj(~o,~)<61}={~~~~,,:~K(~~,~)6,>0. k&z,, k)<& * (l/2)j. CdKj(& al/cl [l +dK,(~~,~)]<2j.61/(2j+2j.~61)~dK1(~~,~)<61. {k~Ew,o:
Then
6, > 0 such it holds that
+&,(kn 2)1<& =z-C4&m aI/
d,(~~,~)<6,}cD(~~,Kj,6,)cD(~~,K,&).
Thus
we
have
that
Q.E.D.
Proof of Lemma 3. First we show d, is a metric on .!J?,,, for {Ki}i,,c.X* satisfying (a) and (b). It can be shown that d, satisfies non-negativity, symmetry and the triangle inequality as the Frechet metric [Husain (1977, p. SS)]. Then we show the separateness. It is obvious that k*A kI implies d,(ki, k2)=0. Conversely we assume d,(k,, 2J =O. Let P, be the set of probability measures in P(X) with finite support. Since ki = kzon P,* for all icz N and since P, is a Herstein and Milnor’s (1953) mixture set, we can construct an expected utility function representing both 2 I and 2 2 on P, by Herstein and Milnor’s Expected Utility Theorem. Since P, is a dense subset of P(X) by Hildenbrand [1974, p. 49, (29)], we have by our Expected Utility Theorem in section 2 that 2 1 = 2 2 on P(X). Let r~ be the metric topology induced by d,. We show r* cy in the following. For each 2 E%&,~, jE N and E>O, it suffices to show for any 2 1 E D( 2, K,, E) there exists a real number 6 > 0 such that
Since D(k, K,, E) is an open ball with respect to the semimetric an open ball D(kl,Kj,&*) such that D(~l,Kj,~*)~D(~,Kj,~). have by Lemma B that for some 6 > 0,
dK,, there is Then we
Therefore we have z*c~. Next we show rl CT*. For each 2 1ES%&,,,and E>O, it suffices to show that d,( 2 I, 2)
Since d, is a metric, There exists 2 > E* > 0 such that
M. Miyake, Von Neumann-Morgenstern
preferences
Let m be an integer such that m > - [log(&*/2)/log2], and let q for i = 1,. . . , m. Then it holds that for any 2 E ny= 1 D( kz, K;, ci), d,.(k
kz)=
c
2-‘f&,(k
22)/(1
+d,,(k
339
=E* .2'-'/m
22))l
ieN
< i i-1
2-i.(c*.2i-l/,n)+
2-i
f i=m+
1
Then we have ~~zE~=~I(z2,Ki,Ci)C(~tE.‘A,,,:d,(~2,~)
we
have
References Allen, B., 1983, Neighboring information and distributions of agents’ characteristics under uncertainty, Journal of Mathematical Economics 12. 633101. Back, K., 1986, Concepts of similarity for utility functions, Journal of Mathematical Economics 15, 129-142. Chichilnisky, G.. 1985, Van Neumann-Morgenstern utihties and cardinal preferences, Mathematics of Operations Research 10. 633-641. D&e, J.H.. 1987. Essays on economic decisions under uncertainty (Cambridge University Press, Cambridge). Dugundji, J., 1966. Topology (Allyn and Bacon. Boston, MA). Fishburn, P.C., 1982, The foundation of expected utility (Reidel, Dordrecht). Grandmont, J.-M., 1972, Continuity properties of a van Neumann-Morgenstern utility, Journal of Economic Theory 4, 45557. Heyer, H., 1977, Probability measures on locally compact groups (Springer. Berlin). Hildenbrand, W., 1974, Core and equilibria of a large economy (Princeton University Press, Princeton, NJ). Hernstein, I.N. and J. Milnor, 1953. An axiomatic approach to measurable utility, Econometrica 21, 291-297. Husain, T.. 1977. Topology and maps (Plenum Press. New York). Jansen, M.J.M. and S.H. Tijs, 1983. Continuity of bargaining solutions, International Journal of Game Theory 12, 91-105. Kannai, Y., 1970, Continuity properties of the core of a market, Econometrica 38. 791-815. Klein, E. and A.C. Tompson, 1984, Theory of correspondences (Wiley, New York). Lute. R.D. and H. Raiffa, 1957, Games and decisions (Wiley. New York). Mas-Colell, A.. 1977, On the continuous representation of preorder& International Economic Review 18. 509-513.
340
M. Miyake,
Von Neumann-Morgenstern
preferences
Mas-Colell, A., 1985, The theory of general economic equilibrium (Cambridge University Press, Cambridge). Milgrom, P.R. and R.J. Weber, 1981, Topologies on information and strategies in games with incomplete information, in: 0. Moeschlin and D. Pallaschke eds., Game theory and mathematical economics (North-Holland, Amsterdam). Van Damme, E.E.C., 1983, Refinements of the Nash equilibrium concept, Lecture Notes in Economics and Mathematical Systems 219 (Springer, Berlin). Watson, P.D., 1953, On the limits of sequence of sets, Quarterly Journal of Mathematics 4, l-3.
of Mathematical
Economics
19 (1990) 323-340.
North-Holland
CONTINUOUS REPRESENTATION OF VON NEUMANN-MORGENSTERN PREFERENCES Mitsunobu University
MIYAKE*
ofTsukuba, Tsukuba, Ibaraki 305, Japan
Submitted
May 1988, accepted
May 1989
This paper considers a continuous representation of preference relations satisfying Grandmont’s (1972) Expected Utility Hypothesis, We equip the preferences with the topology of closed convergence, then we show the existence of a jointly continuous expected utility function and consider its uniqueness. Furthermore, we construct an embedding map of the preferences into the set of expected utility functions.
1. Introduction A continuous representation of a set of preferences is a continuous selection of a suitable utility function from the class of utility functions representing each preference. The continuity of the representation is considered in the sense that a convergence of the preferences implies a convergence of the selected utility functions. In an economic model, the continuous representation of a set of preferences or its variant, a jointly continuous utility function both on alternatives and preferences, is a useful tool when we consider continuity or stability properties of a solution of the model against its data (e.g. individual preferences or endowments). The jointly continuous utility function is introduced by Kannai (1970) to investigate continuity properties of the core of an economy and is discussed by Mas-Cole11 (1977, 1985), Back (1986) and others.’ Especially Milgrom and Weber (1981) consider continuous representation under uncertainty which can be applied to some classes of games and economic models.2 However, their expected utility functions are not derived from preference axioms. Thus, there is no general relation between convergence of the utility *I would like to thank Professors Yoshihiko Otani, Herbert Heyer and Toru Maruyama for their valuable comments and advice. Of course, any remaining errors are mine. ‘See the references of Back (1986). ‘The stability of solutions for cooperative [non-cooperative] games is studied in Jansen and Tijs (1983) [Van Damme (1983, 2.4)], respectively. and the continuity properties of economic equilibria are analyzed in Allen (1983). Chichilnisky (1985) investigates a continuous social welfare function. 03044068/90/$3,50
0
1990, Elsevier Science
Publishers
B.V. (North-Holland)
324
M. Miyake, Von Neumann-Morgenstern
preferences
functions and convergence of the underlying preferences which the utility functions represent, since the expected utility function is unique up to positive affine transformations. In this paper, we start from Grandmont’s (1972) axiomatic approach to a continuous von Neumann-Morgenstern preference which satisfies Paretiantype continuity as well as the Expected Utility Hypothesis. Then we equip the space of preferences with the closed convergence topology and consider their continuous representation by expected utilities. As a main result, we show, using a constructive method based on Kannai (1970), the existence of the jointly continuous (expected) utility function and its uniqueness property for the full class of von Neumann-Morgenstern preferences, which is a generalization of Grandmont’s Expected Utility Theorem. Furthermore, we show that the continuous utility function actually establishes an embedding (map) of a class of von Neumann-Morgenstern preferences into the set of expected utility functions. Thus the similarity of the von NeumannMorgenstern preferences with respect to the topology is identical to the similarity of their (suitably selected) expected utility functions. Hence we can translate the stability of a solution of a model in terms of (small perturbations of) utilities into the stability in terms of the preferences. The next section reviews Grandmont’s Expected Utility Theorem for a continuous von Neumann-Morgenstern preference. In section 3, we introduce the topology of closed convergence on the set of preferences and present the existence and uniqueness results of the jointly continuous expected utility function. Section 4 gives their extensions which admit the set of alternatives not to be locally compact and the preferences to be statedependent. The proofs are given in section 5 and the appendices. In our proof of the main result, we dispense with the continuous selection theorem which is crucial for Mas-Colell’s (1977) proof, since we follow Kannai’s constructive method, i.e., normalization of each utility function.
2. A von Neumann-Morgenstern
preference and the Expected Utility Theorem
The set of (pure) alternatives is a separable metric space X. The metric on X is denoted by 1,. Let @ be the a-field of all Bore1 subsets of X and let (P(X),,) be a pair of the set of (countably additive) probability measures on %? and the topology of weak convergence on P(X).3 A measure PEP(X) is called a mixed alternative and interpreted as a lottery in which the possible prize is XEX. The set P(X) can be regarded as a convex subset of the linear space of all real-valued functions on SY. A convex combination is interpreted
3For the topology
of weak convergence,
see Hildenbrand
(1974, D.I.).
M. Miyake, Van Neumann-Morgenstern
325
preferences
as a two-stage lottery.4 Identifying a pure alternative x EX as the one-point probability measure p such that p({x)) = 1, x belongs to P(X), i.e., Xc P(X). A preference 2 is a binary relation on P(X). The expression pkq means that p is preferred or indifferent to q. For any preference 2, the strict preference > and the indifference relation ‘v are defined by p>q iff not qkp; p-q iff pkq and qkp. A continuous von Neumann-Morgenstern preference 2 on (P(X), o) is a preference satisfying the following three axioms: Axiom 1.
Complete Preordering.
2 is a complete preordering on P(X).
Axiom 2. Continuity. {q E P(X): qkp) sets in (P(X),o) for all pE P(X). Axiom 3.
Substitutability.
and {qEP(X):pkq)
are closed sub-
For any p, q and r in P(X), pm q implies
As a corollary of Grandmont’s Expected Utility Theorem (1972, Theorem 3), we have the following theorem: Expected Utility Theorem. Suppose that X is a separable metric space and P(X) is endowed with the topology of weak convergence w. Let 2 be a continuous von Neumann-Morgenstern preference on (P(X), w). (A)
There exists a real-valued continuous function u on P(X) satisfying:
(4 u(p)Zu(q)op?zq for all p,q E P(X). (ii) The restriction u, of u on X is a bounded continuous function on X such that u(p) =sx u,(x) dp for all PEP(X). (B) Let F(k) be the set of real-valued continuous functions on P(X) satisfying the conditions (i)-(ii) and let u be a function in F(k). Then a realvalued continuous function v on P(X) belongs to F(k) if and only if there exist real numbers tl> 0, fl such that v(p) = CY* u(p) + f_3for all p E P(X).
Assertion (A) states that there exists a continuous expected utility function such that its expectation is taken in a general integral form and its restriction on X becomes a Paretian continuous utility function. Assertion (B) states that the utility function is unique up to a positive affine transformation. If X is finite, Axiom 2 is reduced to Herstein and Milnor’s (1953) Continuity Axiom (Axiom 2) and our Expected Utility Theorem is essentially equivalent to Herstein and Milnor’s theorem. 3. Continuous representation of von Neumann-Morgenstern In this section we analyze the topological ‘%ee Fishburn
properties
(1982, p. 13) and Lute and Raiffa (1957, Ch. 2).
preferences
of the set of von
326
M. Miyake, Von Neumann-Morgenstern
preferences
Neumann-Morgenstern preferences and their derive continuous representation. Suppose that the set of pure alternatives X is a separable metric space as in section 2. In addition we assume that X is locally compact, which is a very general condition in economic analysis. Let W be the set of the continuous von Neumann-Morgenstern preferences on (P(X),cu) and let ~8 be the class of all closed subsets of P(X) x P(X). Identifying 2 EB by its graph G( 2) = {(p, q) E P(X) x P(X): pkq}, W is a subset of .&?’be Lemma A in appendix A. In order to introduce the topology of closed convergence z on k‘, we have to additionally assume that X is compact. Since X is compact if and only if P(X) is locally compact whenever X is locally compact by Heyer (1977, p. 26, Theorem 1.1.12), and since the closed convergence is topological in J%’ only when P(X) x P(X) is locally compact by Watson (1953), we cannot drop the compactness of X as long as we consider the topology on _c&‘.This assumption is very natural when we consider bidding games or bargaining games. Under this assumption, the (relative) topology of closed convergence on ,?# has the natural properties as Paretian preferences [Hildenbrand (1974, p. 96, Theorem l)]. Namely we have the following lemma: Lemma 1.5 (a) (2,~) is induced by the Hausdorff metric. (b) A sequence (2,) in 92 converges to 2 E.%? if and only if Li G(kJ= G(k) =Ls G(&), where Li G(kJ and Ls G(k,,) are the topological limes inferior and superior of {k”}, respectively. (c) Let kO be the null preference such that p-,,q for all p,qEP(X). kO is an isolated point in (2,~) if and only if X is a finite set.6 Proof.
Appendix A.
Next we consider the Kannai topology introduced in Kannai (1970) on the set of the non-null preferences B* E B - {kO> under the assumption that X is a compact metric space. To define and characterize the topology we need several definitions. Since X is a compact metric space, P(X) is a compact metric space by Hildenbrand [1974, p. 49, (30)]. Therefore there exists a countable basis II E {Bi} such that Bie II is a relatively compact open set for all i. Let T={(i,j):BinBj=4 and Bi>Bj for some 2 EB} where Bi is the closure of Bi and Bi>Bj means that p>q for all PE Bi,q~ Bj. For any t-(i,j)E7: let ~~-~~i,j,={~~E:Bi~Bj~. Note that UfeTgt=B* by 5For the mathematical concepts in Lemma 1, see Hildenbrand (1974, B. II.). 6This condition plays an important role in Chichilnisky’s (1985) social choice analysis. We derive it without the assumption for interpersonal utility comparison assumed in Chichilnisky (1985).
M. Miyake, Von Neumann-Morgenstern
preferences
327
Lemma A in appendix A. The Kannai topology is defined to be the topology on .4?* which has the class {,c!J~)~~~as a sub-basis. The relative topology of q on gLt is denoted by g, for all t E 7: Let {bi} be a countable subset of P(X) such that big Bi for all i, and let u, be the function on P(X) x S?Z for each t = (i, j) E T defined by
3UEf(k).
(1)
Though a value of u,(p, 2) depends on the selection of {bi} from P(X), it does not depend on the selection of u E F(k) by the Expected Utility Theorem (B). Therefore, o,(p, 2) is well-defined and then u,( . ,k) is the normalized utility function of 2 e.%Yl,such that a,(b,, 2) = 1 and v,(bj, 2) = - 1 for each t E T Then we have the following lemma: Lemma 2. (a) The Kannai topology n is the coarsest Hausdorff topology on 9?‘* for which the . set A-{(k,p,q)E9*xP(X)xP(X):p>q) is open in 9?* x P(X) x P(X). (b) For all t E 7: the relative topology n1 on 9t is induced by the following metric pt:
(c) n coincides with the topology of closed convergence 7: on .B*. Proof:
Appendix
A.
By Lemma 2, the topology of closed convergence r on SY* is characterized as the minimal Hausdorff topology such that A is open, and u,(p, 2) is already a jointly continuous utility function on P(X) x 95!( for all t E T For Paretian preferences the corresponding assertions to those in Lemma 2 are given in Kannai (1970, Theorems 3.1 and 3.2) and Hildenbrand (1974, p. 105, Problem 6). Although such assertions hold within the class of monotone Paretian preferences, in our case of von Neumann-Morgenstern preferences Lemma 2(a,c) hold for non-null preferences B* and Lemma 2(b) holds on each ,c#~. The main result of this section is the following theorem: Theorem 1. Suppose that X is a compact metric space and P(X) is endowed with the topology of weak convergence CO.Let &? be the set of continuous von
328
M. Miyake, Von Neumann-Morgenstern
Neumann-Morgenstern closed convergence z.
preferences
on (P(X),o)
preferences
endowed
with the topology
of
(A)
There exists a real-valued function U on P(X) x 9 such that: (i) U(p, 2) is jointly continuous on P(X) x 9’; (ii) U(p, k)g U(q, z)opzq for all p,qEP(X) and 2 E.@; (iii) U(p, 2) = lx Ux(x, 2) d p f or all (p, 2)~ P(X) x 6% where U, is the restriction of U on X x W. (B) Let 4Y be the set of real-valued functions on P(X) x W satisfying the conditions (i)giii) and let U be a function in a. Then a real-valued function V on P(X) x B belongs to 6& if and only if there exist continuous functions a(k) and /I( 2) from 93 to [ - CO,CD] such that: (iv) V(p, 2) E P(X) x B*: V(p, 2) = a( 2). U(p, 2:) + B( 2) and a( 2) > 0; (v)
~P~fYX):
=
V(P, 20)
a(kJ.U(p,
kO)+/?(kO)
if
X is afinite
set,
otherwise
C
where c = lim,, + coC~~(~~).U(P,~~)+B(~~)I for some PEP(X) and some convergent sequence (k,,} in W* such that lim,, + J12” = ko.
Assertion (A) states the existence of a jointly continuous utility function on P(X) x 9 such that U(. , 2) satisfies Grandmont’s Expected Utility Hypothesis, and Assertion (B) states the jointly continuous utility function is unique up to a positive affine transformation in a functional form. U(p, 2)
Corollary 1.1. Under the assumptions of Theorem I, suppose 9 to be the set of real-valued continuous functions in P(X) endowed with the topology of untform convergence $. Let U be a function in @ and let t? be a function from 92 to 9 such that 0( 2) = U( ,2). Then tI is a homeomorphism from (9*, z) to (t3(9?*), $) c (9, $), i.e., 0 is an embedding of (W*, z) into (F, II/). In addition, if X is a finite set, tI is an embedding of (9, z) into (9, II/). Corollary 1.2. Under the assumptions of Theorem 1, let 94?_,= (2 for any x, YEX and let v?(p) be a function on P(X) such that
v&4 = CW - WllCW
- u(y)1 for some u E U 2).
The function 8: .!JQ,+F defined by t3(2) =vt is an embedding (9, $) such that e( 2) E P( 2) for all 2 E%‘,...
Corollary
lB:x>y)
1.1 states that the jointly
continuous
of (.@._,,T) into
utility function can be
M. Miyake, Van Neumann-Morgenstern
preferences
329
recognized as a homeomorphism between the class of non-null preferences and the class of expected utility functions and the homeomorphism can be extended to the whole space 3 if X is a finite set. Specifically, Corollary 1.2 states that the function which assigns the normalized expected utility function a E r(k) such that u(x) = 1 and u(y) = 0 to each preference 2 in BxY is a homeomorphism between gxp and e(&?:,,).
4. Extensions In order to represent individual preferences in an economic mode1 under uncertainty, we have to consider the case that the set of pure alternatives X is the positive orthant of the n-dimensional Euclidean space E’!+ and the preferences may be state-dependent. If the set of pure alternatives X is ET, then the set of probability measures on X is not locally compact and the closed convergence of the preferences on the measures is not topological as stated in section 3. In this section we introduce a new topology for a class of preferences based on the closed convergence on compacta. Then we derive a continuous representation of the preferences by the topology. Assume that the set of pure alternatives X is E’!+. Let (9, P(X)) be the pair of the set of Bore1 subsets of X and the set of probability measures on :‘A endowed with the topology of weak convergence o. Let &&,,,, be the class of the von Neumann-Morgenstern preferences 2 on (P(X),o) in .B satisfying: Axiom
4.
Weak Monotonicity
in X.
x > ysx>y
for all x, Y E X.
To define a topology on gWrno we need some definitions. Let x = {K: K is a compact subset of X}, zT*={KEX:~X,~EKX>~} and P,={~EP: p(K’) =O> for all K ~31r. For any 2 E&,,,,, and K E %‘*, let GK( 2) = G(k)n(P,xP,) where G(k) is the graph of 2, and let dK(&,k2)= hK(GK(kJ,GK(k2)) for all k1,k2~.&,,,, and KEAC* where h, is the Hausdorff metric on the closed subsets in P, x P,. Note that P, x P, is compact since P, is compact by Hildenbrand [1974, p. 49, (30)]. Then define D(&K,E)={~*E&,,~: dK(k,k*)<&) for any ~E&Y,,,~~, KEA?‘* and E>O. Let T* be the topology on &,,, which has the class {D( 2, K,E): 2 E&Ywmo,K E%*, E>O} as a sub-basis. The topology r* has the following property: Lemma 3.
(B’,,,,
z*) is induced by the following
metric d,:
M. Miyake, Von Neumann-Morgenstern
330
preferences
where N is the set of positive integers and (Kt}tEn is a countable family compact sets in Z* such that: (cr)KiCKi+l for all iEN; (/-3)UieNKi=X. Proof
Appendix
of
B.
Then we have the following
theorem:
Theorem 2. Suppose that X = E”, and P(X) is endowed with the topology of weak convergence Q. Let &,, be the class of the von Neumann-Morgenstern preferences 2 ~9 on (P(X),w) satisfying Axiom 4 endowed with the topology z*, and let C(X) be the set of continuous functions on X endowed with the compact-open topology y. Fixing any two pure alternatives x,ye X with x >y define a real-valued function U(p, 2) on P(X) x .%?~mo by WP, 22) = CU(P) -
4~M4x) - u(y)1 for some u E F(k).
Then U(p, 2) is a jointly continuous expected utility function on P, x B)wmo for all K EX. Furthermore, a function 0 defined by 8( 2) = Ux( . ,k) is an embedding of (%&,,,, z*) into (C(X), y) where U, is the restriction of U( ., 2) on X.’ Next we introduce states of the world explicitly and allow the preferences to be state-dependent.8 Suppose that the set of pure alternatives is X= E’!+ and the set of states of the world is 52= E,. Let Y = X x Q = E",+ ’ and let PJr, be the set of Bore1 subsets of Y The measure p may be interpreted as the two-stage lottery which selects state SEQ and pure alternative XEX sequentially. Namely, p selects SE Q by the marginal probability of p on R at first, then p selects XEX by the conditional probability of p on S at the second stage. Let P(Y) be the set of the probability measures on (Y&) with the topology of weak convergence w and assume Y c P( Y) as in section 2. Let 4 wmo be the set of the von Neumann-Morgenstern preferences 2 on (P( Y), 13) satisfying: Axiom 4*. Intra-state SEQ, x>y*(x,s)>(y,s).
Weak Monotonicity
in X.
For any x, YGX and any
Then we can define a topology ? on &,,,, in the same manner as we did for T* in Theorem 2 modifying x and 3C* as ~6 = {K: K is a compact subset in Y} and $C* = {K: K is a compact subset in Y and x >y for some (x,s),(y, s) E Y}. As a corollary of Theorem 2 there exists a real-valued ‘The basic set of continuous utility functions % in Allen (1983, p. 70) may be recognized as a subset of 0(S,,,). ‘For the derivations of state-dependent utilities, see Dreze (1987, Ch. 1, 9.1). In our derivations, it is implicitly assumed that a person is able to express preferences among the lotteries, conditionally on any exogenously given probability measures on the states of the world.
function U(p, 2) on P(Y) x ,&,,, such that: (i) U(p, 2) is jointly continuous on PKx&,, for each K E~C; (ii) U(p, k)z U(q, k)epkq, and (iii) Therefore, it holds from U(p, 2) = jy U,(y, 2) dp V p, q E P, V 2 E ~‘,,,. Fubini’s Theorem that for any Bore1 probability measures p on X and p on Sz U*(P, CL,2) = U(P x P> 2) = f U,(Y,
2)
4p
x 4
Y
=! Specifically,
cjxUx(.u,ST2) dP1&
we have
U*(X,P, 2) = J U,(x-, s, 2) dlc. R Thus we derive a continuous Morgenstern preferences which world.
representation of continuously depend
the von Neumannon the states of the
5. Proofs Proof of Theorem I. (A) Since u,(p, 2) defined in (1) is the jointly continuous utility function on P(X) x BI by Lemma 2(b, c). Then transforming r,(p, 2) to u,(p, 2) E [0, l] for any t -(i, j) E T by u,(p, 2) = Cc& 2)
-n,(~M~,(2)
-n,(k)l,
n,( 2) = minPEp(xI v,(p, 2) where m,(2) = maxpeptXj u,(p, 2), and extending continuously u,(p, 2) on P(X) x 9 to be u,(p, 2) =0 on P(X) x 3”:: as in the proof in Mas-Cole11 (1977), we derive a jointly continuous expected utility function on P(X) x 2 by U(p, k)=‘&i,j,ET2m(i+jJ. uci, jj(p, 2). (B) The ‘if part’ of Assertion (B) is obvious. Thus we show the opposite direction. Let U and I/ be any functions in ‘2, and let c((2) and p( 2) be the real-valued functions on %?* such that:
42)
= CVPI, 2) -
V(P,,
~MWPI,
2) -
U(P,,
?)I,
(2)
(3) for some pl, p2~P(X) such that pI>p2. Note that the values of ~(2) do not depend on the selection of p1,p2, and that a(k), /?( 2) satisfy the condition (iv) [see Fishburn (1982, p. 20)]. We shall show ~((2) is continuous on 4!*. Let { 2,) be a convergent sequence in (&!*, z) such that lim,,, 2, = k0 E W*. Since kanE*, there are q1,q2cP(X) such that q1 >aq2. By Assertion (A), it holds that 3 n,,: Vnn>n,,: q1 >nq2. Then it holds by (2) that
332
M. Miyake, Von Neumann-Morgenstern
preferences
Therefore we have lim,,, c((2,) =CI( 2,) by Assertion (A), which implies SI(2) is continuous. Then /I( 2) is continuous on B* by (3). Extending a(k) and /I( 2) to %Ycontinuously, we have that the extended c((2) and p(k) satisfy (iv) and (v). If X is a finite set, the extension is completed by assigning suitable real numbers to c((ko), /?(ko) by Lemma l(c). Q.E.D. Proof of Corollary 1.1. It follows from Assertion (A) of Theorem 1 and Dugundji (1966, XII, 3.1) that 0(k) = U(. ,k) is one-to-one and continuous. Conversely, assume that {fnj b e a convergent sequence in (0(2*),$) such that lim,, o?f,=f: Let 2 = K’(f) and k=,= K’(f,) for all n. Then we have f,(p) = U(p, 2,) V n and f(p) = U(p, 2). By 2 ES%‘*,it holds by Lemma A in appendix A that 3 t=(i,j)~ T; 2 ~2~. Let s=minasdi f(p)-max,,,lf’(p)>O. There exists an integer ~zi such that max pEP(X~lfn(~)-f(~)(<~/4 for all n>fil. Then min,,B, f,(p) -maxPSBj f,(p) 2 s/2 >O for all n > n,. Therefore 2” E &!‘rfor all n>n,. Further, since the jointly continuous expected utility function on P(X) x W, is determined up to positive afhne transformations defined by Theorem l(B)(iv), and since P(X) is compact, it holds by Hildenbrand (1974, p. 30, Corollary) that
=
=
lim max \u,(P, 2,)-u,(p,
tn-t2
lim max IA(2,) - U(p,
?n+;t
= max
=
2,)
+ B( k=,)
PEP(X)
-A(k).U(P,
PEP(X)
Zz)I
PEP(X)
lim
Z)-B(k)1
IA( kJ.
U(p, 2”) + B( 2,)
kn+l=
mix (A(~).U(p,~t)+B(R)-A(~).U(p,~)-B(~)(=O
PEP(X)
for some real-valued continuous functions A(k) and B(k) on &. Thus we > = 2 by Lemma 2(b,c). If X is a finite set, we can construct a have lim n-a, N” homeomorphism based on the above by assigning a suitable constant function to the null preference ko, since k. is isolated by Lemma l(c). Q.E.D.
M. Miyake, Von Neumann-Morgenstern
333
prejbences
Proof of Corollary 1.2. It follows from Lemma 2 that (gx,,z) is induced by the sub-basis {,%?)tn.@xy}teTand that (P&n.%!,,, T) is metrized by pr for each t. Let p: be another metric on B,n W,, defined by replacing u,(p, kk) in the definition of pt with u:(p, kk) = v?,(p) = [u(p) - u(y)]/[u(x) -u(y)] (k = 1,2). Since u,(p, 2) is given by a positive afline transformation [Theorem l(B)(iv)] of z$(p, k), pt and p: are topologically equivalent. Therefore, it is easy to show that O(k)=+ is continuous. Further we can show that 8-i is continuous on O(%?,,,) as in the proof of Corollary 1.1 replacing g*, 2, and pt with gxp, gtn B,.. and p:, respectively. It follows from the Expected Utility Theorem that O(k) E r( 2) for all 2 E g_,. Q.E.D.
Proqf of Theorem 2. Let x, y be any two alternatives in X with .Y> y, and let K be a compact subset in jT. Suppose that {p,}, {k,,) be two sequences such that pn+p in (PK,m) and 2,-k in (&Ywrno,~*)as n+co. Since X=E’!+, there exists a family of compact sets {Ki)icN in x* satisfying (CZ)and (p). By 3 we have that lim,,,d,,(k”, k)=O k-2 in (.%,,,, r*) and Lemma which implies that lim,,, d,,(k,,,k)=O for all HEN. Since KcKj and {x,Y} c Kj for some Jo N, it holds from Corollary 1.2 that U( . , 2,) converges uniformly V( . ,k) on PKj and P,. By p,+p in (PKu), we have U(p,, kJ+U(p, 2). Therefore, it follows from the Expected Utility Theorem and Husain (1977, p. 256, Corollary 9) that 0 is one-to-one and continuous. The continuity of OK1 can be proved by Corollary 1.2, Lemma 3 and Husain (1977, p. 256, Corollary 9). Q.E.D. Appendix A Appendix A proves Lemmas 1 and 2 under the assumption compact metric space. We need the following lemma: Lemma A. For any 2 ES?‘* and any p, qE P(X) with p>q, in the basis Il such that p E Bi, q E Bj and &>Bj. Proof.
Assume
that p>q.
By the Expected
Utility
that
X is a
there exists Bi, Bj
Theorem,
it follows
that
p>C(3/4)P+(1/4)ql>C(1/4)p+(3/4)ql~q and that Bik[(3/4)P+(1/4)ql~ [( 1/4)p+(3/4)q]kBj for some Bi, Bje I7 such that PE Bi, q E Bj. Then we have B,>B,
with PE&
Proof of Hildenbrand
qE Bj.
Q.E.D.
Lemma 1. Assertions (1974, p. 19, Theorem
(a) and (b) are direct corollaries of 2). In the following we show Assertion
(c). First, assume that X = {xi, x2,. . ,x,} and that there exists a z-convergent sequence {k,} in g* such that lim n_m kn=kO. Let Xij={k~E*:xi>xj and ~~2x2~~ for all xgX} (1 SisH, 15 jsH, i#j). It is easy to show that W* = uijXij. Since there exists a finite number of Xi,, there exists a pair of
334
M. Miyake, Von Neumann-Morgenstern
preferences
the integers (i, j) and a subsequence (2,) of { kn} such that ~,,,EX~~ for all m. We assume (i, j) =( 1,2) without loss of generality. Let Qk= {qeP(X): 1 jx fkdxk-fX fk dq( <(l/3)) = {q E P(X): qk(xk) >2/3} (k= 1,2), where xk is the one-point probability measure and fk is the function such that fk(x) = 1 if x=x,; &(x)=0 otherwise for k=1,2. Then Q=Qi xQ2 (Q*=Q2xQ1) is an open neighborhood of (x1,x2) ((x2,x1)) in P(X) x P(X), respectively. Therefore, it follows from the Expected Utility Theorem and the definition of Qk and X,, that
41~~c41(X,)~XI +(1-41(x,)).x,l~[c(2/3).x,
+(1/3).x21
for all VEX,, and all (ql,q2) EQ. Then it holds that ‘not q2kql’ for all and all (q2,ql)EQ*, which implies Q*nG(R)=4 for all 2 EX,,. ?z EX,, Hence we have that Vn*: 3 n zti*; Q* n G( k,,) = C#Jwhich implies (x,, x1)$ Since (x,, x1) E G( k,,) = P(X) x P(X), this contradicts lim,,, G(&-,,). lim n.+3o2,= k0 (in 5). Then k,, is an isolated point if X is a finite set. Next we assume that X contains an infinite number of elements. Since X is a compact metric space, there exists an accumulation point x,, in X and convergent sequence {x,,}~~~ in X-(x,,} such that lim,,, x.=x0. Define a sequence of functions u, by u,(x) =(2(x,, x0) - A(x, x,))/~(x,, x,,) if 2(x,,, x0) > 1.(x,x,,); u,(x)=0 otherwise, where 1, is the metric on X. Let (k,} be a sequence in %?* defined by pk,,q iff lx u, dpzJ,u, dq. Then we show that lim n_+oo&= k0 (in 5). Let (p, q) be a pair of alternatives in P(X) x P(X) and let Q be an open neighborhood of (p,q). Since the set of measures with finite support is dense in P(X) by Hildenbrand [1974, p. 49, (29)], there exists (p*, q*) E Q such that p* and q* have finite support. Therefore, it follows from the definition of {knl that In,: Vnzn,; p*~~x~ and 3n,: Vnzn,; q*wnxl. Hence VnZmax(n,,n,); p*- ,,q* which implies Qn G( k,,) #& Then we have Li G( 2,) = P(X) x P(X) = G( k,J. On the other hand, P(X) x P(X) = Li G(k,,)cLs G(k,J by Klein and Tompson (1984, p. 26, Lemma 3.2.6), we have that Ls G( 2,) =P(X) x P(X) = G(R,). Hence we have lim n-rm > -II =& and k,, is not isolated. Q.E.D. Proof of Lemma 2(u). First we show that q is a Hausdorff > _ r, kz be any two preferences in R+ such that 2 1# k2. show that
3p,qEP(W; p>lq
and q>g.
assume ‘3p,qeP(X); pxlq without loss of generality. ‘3 P,~EW); ~>~q and q-2~’ true, then we have (A.l). Let us assume the latter. By kl, Since
k1 # kz,
we
may
topology. Let Then we will
(A.11 and q>g’ or If the former is kz~E*, we can
M. Miyuke. Von Neumann-Morgenstern preferences
335
assume additionally r>,s for some r, SE P(X). Let p(s) =E.s+( 1 -s).p q(c) =E. r+(l -E) .q for all EE(O, 1). By the Expected Utility Theorem, have that p(E)>Iq(E)
q(E)>2p(E)
for some EE(O, 1);
and we
for all EE(O, 1).
Then it holds that 3s*~(O, 1); p(~*)>~q(~*) and q(.z*)>,p(E*), which implies (A.l). It follows from (A.l) and Lemma A that there exist Bi, Bj, B,, B, in n such that: Bi> ,B,; B,> 2Bj; B, n Bj # 4; B, n B, # 4. Since Bi CTBj and B, n B, are open, there exists B,, B, such that B, c Bin Bj and B, c B,n B,, which implies 2 1E BR,, and 2 2 E &,,. By %?,,,,,n 8$,,,,= 4, q is Hausdorff. Next we show A is open in the product topology. For any (p,q, 2) E A, it holds by Lemma A that there exists (i, j) E T such that p E Bi E Z7, q E BjE n and Bi>Bj. Therefore 2 E~~i,j) and (p, q,k) E(Bi x Bj x ,~,i,j,C A. Since Bi x Bj x W,i, j, is open, A is open. Finally if x is a Hausdorff topology for which A is open, we can show v]cx as in Kannai (1970, the proof of Theorem 3.1). Thus we complete the proof of Lemma 2(a). Q.E.D. ProofofLemma 2(b). Let [, be the topology induced by the metric pt. We can prove vtc[, in almost the same manner as the proof of Theorem 3.2 in Kannai (1970) using the Expected Utility Theorem. To prove i, c ql, we need a modification of Kannai’s proof. Fix t=(i, j) E 7; and write it, q,, pt, & and u, as [, q, p, 9 and u for simplicity in the following proof. Since the class of open balls defined by p is a basis for i, it suffices to show that for any E > 0, k,EW,
In order to prove this, we use the following
Let n be an integer
such that
assertion:
n > max(cc, /?). Then be
it follows
that
1 > l/2+
a/2n > 0; 1 > l/2 + a/2n > 0. Let p*, q* E P(X) p*=(l/2+a/2n)bi+(l/2_cl/2n)bj;
4*=(1/2+P/2n)bi+(1/2-~/2n)bj,
and let K={q~P(X):q=(l/n)p+(l-(l/n))b*
where b*=(1/2)bi+(1/2)bj. u(P*, 2)
= cc/n;
for some PEP(X)},
(A.4)
Then we have by (1) that o(q*,
2)
=/3/n
for all 2 EP&
(A.5)
336
M. Miyake, Von Neumann-Morgenstern
preferences
and we have that K is compact since P(X) is compact and since the linear transformation on P(X) is continuous. Then it follows from (A.4) and (AS) that
={k&:q*>p>p*VpEK}. We can show that { 2 ~9: q*>p>p* V PE K} EV and (A.3) as in Kannai’s proof, since K is compact. Q.E.D.
that
(A.2) holds
by
Proof of Lemma 2(c). First we show r crj. Let (kn} be a r-convergent sequence in 99 such that lim,,, k,,= 2 EP. Then we show lim,,, k,,= 2 in I?. For any W EV such that 2 E w it suffices to show 3 N: V n > N; 2, E W Since 2 E W and since {G?t}t.T is a sub-basis of y, there exists a finite subset H= {(ii, h), . . . >(ih, j,)} of T such that 2 E n;=i ~)(ik,jk~C n! In the following we show that for some integer N*,
64.6) It holds from Lemma l(b) and a>bo(b,a) $G(k) that if a>b then there such that !I N: exists neighborhoods Q1 (Qz) of b ( a ) in P(X), respectively, Vn> N: V(x,y)~Qi x Qz; y>,x, which implies that there exists a neighborhood Q of (a, b) in P(X) x P(X) such that 3N:Vn>N:V(x,y)EQ;x>,y.
(A.7)
it holds that Bi,>Bj, for all k. For any (a, b)E& x 2 E there exists a neighborhood Q of (a, b) satisfying (A.7). Since Bi, x Bj, is compact, we can choose a finite number of the neighborhoods covering Bik x Bjk. Therefore, for each k we have 3 N,: V n> N,: V(X, Y) E Bi, x Bjk; which implies Bi,>,,Bjk and knnEtik,jkJ. Hence for N*= x>nY, max (N,, . . . , NJ we have (A.6). Next we show kn--+k in (99*, q) implies k=,+ 2 in (9?*, r). Since 2 ~9*, there exists p*,q* EP(X) such that p*>q*. By Lemma A, there exist in (W*, ye), it holds that 3 N: t=(i,j)ET such that GEE,. Since 2, -2 V n> N; ?,,E&?~, which implies p,(kn, k)+O by Lemma 2(b). Then we have that
Since B,,,
n:=l
Wcik,jk),
M. Miyake, Van Neumann-Morgenstern
t’J. kn) uniformly
331
preferences
to u,( . , 2).
converges
W)
First we show G(k) cLi G(k,,). For any (p,q)~ G(k), let p(s) =s.p* (1 -s).p; q(c)=E.q*+(l -c).q for all EE[O, 11. Then it holds that
forall EE Ki11;
u,(P(~,2) > q(q(d,k) P(E)-P and q(E)+q
as
+
(A.9) (A.lO)
s--+0.
By (A.8) and (A.9), we have V/E>O: 3 N: V n > N; a,(p(~), k,,) > u,(q(E), k,,), which implies p(tz)Z,,q(E) and (~(E),~(E))EG(~,). Therefore, it holds by (A.lO) that (p,q) ELMG(k,,) and then G(k))cLiG(k,J.
(A.1 1)
Second, for any (p,q)~Ls G(kJ there exist convergent sequences ip,,,], {q,,,j such that p,,,-+p and qm-+q satisfying that Vm: V N: 3 n> N; ~‘,(p,,,, 2,) 2 UJq,,,, k,,) by the definition of Ls G(k,). Since vl(pm, .) and t‘,(q,, .) are continuous on g holds that it r,(pm, k)=fim,-, Qrn7 k) 2 2) for all m. Since t)J ., k,) is continuous, it holds lim,, rl v,(q m, k,J =v,(i:, that u,(p, 2) 2 u,(q, 2) which implies pkq and (p,q) E G( 2). Hence we have
LsG(L,)cG(k).
(A.12)
It follows from (A.1 1) and (A.12) and Klein and Tompson (1984, p. 26, Lemma 3.2.6) that G( 2) c Li G( k,J c Ls G( 2,) c G( 2). which implies G(2)=LiG(&)=LsG(kJ and k,-+k in (.%‘*,t). Q.E.D.
Appendix B Appendix B proves the following lemma: Lemma 6>0
B.
For
Lemma
any K EAT*,
such that { 2 E&T,,,,,: U&z,
Proof:
K ~33
for
Select
For any some jEN.
(U(P)--(Y))/(u(x)--U(Y))
3 under
ka~9&+,mo and F>O
for
that
X= E”,
We need
there exists a real number
k)<@cWkz.,K,4.
we have
any
the condition
x, y
by the definition of {KiJicN that KC Kj in K such that x >y. Let V(p, 2) =
SOme
u~r(?z)
and
let
pt(2,,
k2)=
max,,,,)V(p,~~)--l/(p,~~)) for any ~Z;1,~.2~&,,(Z=K,Kj). Sincegw,,c G?*,,, it follows from Lemma l(a) and Corollary 1.2 that d, is topologically equivalent to pz (Z=K,Kj). Since (~~.~~,,:~~~(~~,~)
M. Miyake, Von Neumann-Morgenstern
338
Proposition
preferences
17) that for any E>O there exists a real number
that {~~E~,o:~Kj(~o,~)<61}={~~~~,,:~K(~~,~)
Then
6, > 0 such it holds that
+&,(kn 2)1<& =z-C4&m aI/
d,(~~,~)<6,}cD(~~,Kj,6,)cD(~~,K,&).
Thus
we
have
that
Q.E.D.
Proof of Lemma 3. First we show d, is a metric on .!J?,,, for {Ki}i,,c.X* satisfying (a) and (b). It can be shown that d, satisfies non-negativity, symmetry and the triangle inequality as the Frechet metric [Husain (1977, p. SS)]. Then we show the separateness. It is obvious that k*A kI implies d,(ki, k2)=0. Conversely we assume d,(k,, 2J =O. Let P, be the set of probability measures in P(X) with finite support. Since ki = kzon P,* for all icz N and since P, is a Herstein and Milnor’s (1953) mixture set, we can construct an expected utility function representing both 2 I and 2 2 on P, by Herstein and Milnor’s Expected Utility Theorem. Since P, is a dense subset of P(X) by Hildenbrand [1974, p. 49, (29)], we have by our Expected Utility Theorem in section 2 that 2 1 = 2 2 on P(X). Let r~ be the metric topology induced by d,. We show r* cy in the following. For each 2 E%&,~, jE N and E>O, it suffices to show for any 2 1 E D( 2, K,, E) there exists a real number 6 > 0 such that
Since D(k, K,, E) is an open ball with respect to the semimetric an open ball D(kl,Kj,&*) such that D(~l,Kj,~*)~D(~,Kj,~). have by Lemma B that for some 6 > 0,
dK,, there is Then we
Therefore we have z*c~. Next we show rl CT*. For each 2 1ES%&,,,and E>O, it suffices to show that d,( 2 I, 2)
Since d, is a metric, There exists 2 > E* > 0 such that
M. Miyake, Von Neumann-Morgenstern
preferences
Let m be an integer such that m > - [log(&*/2)/log2], and let q for i = 1,. . . , m. Then it holds that for any 2 E ny= 1 D( kz, K;, ci), d,.(k
kz)=
c
2-‘f&,(k
22)/(1
+d,,(k
339
=E* .2'-'/m
22))l
ieN
< i i-1
2-i.(c*.2i-l/,n)+
2-i
f i=m+
1
Then we have ~~zE~=~I(z2,Ki,Ci)C(~tE.‘A,,,:d,(~2,~)
we
have
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