A utility representation theorem with weaker continuity condition Tomoki Inoue∗† September 10, 2009

Abstract We prove that a mixture continuous preference relation has a utility representation if its domain is a convex subset of a finite dimensional vector space. Our condition on the domain of a preference relation is stronger than Eilenberg [Eilenberg, S., 1941. Ordered topological spaces. American Journal of Mathematics 63, 39-45] and Debreu [Debreu, G., 1959. Theory of Value. John Wiley & Sons, New York; Debreu, G., 1964. Continuity properties of Paretian utility. International Economic Review 5, 285-293], but our condition on the continuity of a preference relation is strictly weaker than the usual continuity assumed by them. JEL classification: C60; D11; D81 ∗

Institute of Mathematical Economics, Bielefeld University, P.O. Box 100131, 33501 Bielefeld, Ger-

many; [email protected]. † I would like to express my gratitude to Takuya Masuzawa and Shin-ichi Suda for helpful discussions. I also would like to thank Chiaki Hara, Atsushi Kajii, Tadashi Sekiguchi, and Walter Trockel for their valuable comments, and three anonymous referees for their comments and suggestions which improved the exposition of this paper. A part of the results of this paper was presented at the Third Asian Workshop on General Equilibrium Theory (GETA2006) held at Academia Sinica. I acknowledge a Grant-in-Aid for Research Fellow of the Japan Society for the Promotion of Science (JSPS) and financial support from the German Research Foundation (DFG).

1

Keywords: Mixture continuity; Utility representation

1

Introduction

In conjunction with the independence axiom, Herstein and Milnor [7] proved that every mixture continuous preference relation has an expected utility representation. The mixture continuity is a weaker notion than the usual continuity. It requires that a preference relation is continuous in the parameter space, and it is independent of the topology on the domain of a preference relation. As far as only utility representation is concerned, the independence axiom is dispensable. We prove that every mixture continuous preference relation on a convex subset of a finite dimensional vector space has a utility representation. Eilenberg [3] (see also Debreu [1] [2]) proved that every continuous preference relation on a connected, separable topological space has a continuous utility representation. Since any convex subset of a finite dimensional vector space is connected and separable with respect to the Euclidean topology, our condition on the domain of a preference relation is stronger than Eilenberg’s theorem. On the other hand, as Young and Young’s [12] example showed or as our example (Example 1) shows, the mixture continuity is strictly weaker than the usual continuity assumed in Eilenberg’s theorem. Since a mixture continuous preference relation may not be continuous, it may not have a maximal element in a compact set. This fact limits the application of the mixture continuity. For example, the mixture continuity is not sufficient for consumers’ demands to be nonempty. To resolve this problem, an additional property of a preference relation is needed. Inoue [9] proved that if a mixture continuous preference relation is convex or weakly monotone, it recovers the upper semi-continuity and, therefore, it has a maximal element in a compact set. The proof of our utility representation can be decomposed to two steps. In the first step, we prove that every mixture continuous preference relation is countably bounded, i.e., there exists a countable subset of vectors such that any vector can be preordered between some vectors in the countable set (Proposition 1). If we restrict a preference 2

relation onto a line segment, then the mixture continuity is equivalent to the usual continuity. Since any line segment is connected and separable, in the second step, by applying Eilenberg’s theorem repeatedly to the countable set obtained in the first step, we can obtain a utility representation. In a more general space, the countable boundedness is a necessary and sufficient condition for a mixture continuous or continuous preference relation to have a utility representation. Fishburn [5] proved that a convex, mixture continuous preference relation on a convex subset of a (possibly infinite dimensional) vector space has a utility representation if and only if it is countably bounded. Monteiro [11] proved that a continuous preference relation on a path connected topological space has a continuous utility representation if and only if it is countably bounded. In the case of a finite dimensional vector space, our proposition clarifies that the countable boundedness follows from the mixture continuity. The mixture continuity is defined independently of the topology on the domain of a preference relation. In a finite dimensional vector space, the Euclidean topology is the only Hausdorff linear topology. Hence, the mixture continuity is of special interest when the domain of a preference relation is infinite dimensional (see the introduction of Herstein and Milnor [7] and the notes of Chapter 4 of Debreu [1]). However, our utility representation theorem cannot be extended to an infinite dimensional case. Actually, a mixture continuous preference relation on an infinite dimensional vector space may not be countably bounded. Est´evez Toranzo and Herv´es Beloso [4] proved that in any nonseparable metric space, there always exists a continuous preference relation that is not countably bounded. Since every continuous preference relation is mixture continuous, in any non-separable, metrizable topological vector space, there exists a mixture continuous preference relation that is not countably bounded. Even in a separable, metrizable topological vector space, there can exist a mixture continuous preference relation that is not countably bounded. We give such an example (Example 3). This fact clarifies the difference between the mixture continuity and the usual continuity, because every continuous preference relation on a separable topological space is always countably bounded. Given a mixture continuous preference relation, if the set of its discontinuity points is 3

small, first, by Eilenberg’s theorem, we can obtain a continuous utility function on the set of continuity points, and second we can extend it to the whole domain of the preference relation. We discuss whether this method is valid for any mixture continuous preference relation in Section 4. This paper is organized as follows. In Section 2, we give the definition of the mixture continuity and give an example of a mixture continuous preference relation that is not continuous. In Section 3, we prove the utility representation theorem of a mixture continuous preference relation. In Section 4, we discuss the relationship between our utility representation theorem and Eilenberg’s theorem. In Section 5, we exemplify that our utility representation theorem cannot be extended to an infinite dimensional vector space.

2

Mixture continuous preference relations

Let X be a nonempty convex subset of the L-dimensional vector space RL which is equipped with the Euclidean topology.1 A preference relation % on X is a reflexive, transitive, and complete binary relation on X. Given a preference relation %, we define binary relations  and ∼ on X as follows: x  y if and only if not y % x; x ∼ y if and only if x % y and y % x. A utility function representing a preference relation % or a utility representation of % is a real-valued function u on X such that x % y if and only if u(x) ≥ u(y). In their expected utility representation, Herstein and Milnor [7] used the mixture continuity,2 a weaker notion than the usual continuity. It requires that a preference 1

As we see below, the mixture continuity is defined independently of the topology on X. In order

to discuss the relationship between the mixture continuity and the usual continuity, we equip X with the Euclidean topology. Note that any Hausdorff linear topology on a finite dimensional vector space is equivalent to the Euclidean topology. 2

This terminology is commonly used in decision theory in which X = {x ∈ RL +|

PL

i=1

x(i) = 1} is

a set of lotteries. For lottery x = (x(1) , . . . , x(L) ) ∈ X, x(i) represents the probability of consequence i occurring, and for lotteries x, y ∈ X and for t ∈ [0, 1], (1 − t)x + ty represents the mixed lottery of lotteries x and y with the respective probabilities 1 − t and t.

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relation is continuous in the parameter space. For x, y ∈ X, let I(x, y) = {t ∈ R | (1 − t)x + ty ∈ X}. Since X is convex, I(x, y) is an interval containing [0, 1]. Definition 1. A preference relation % on X is mixture continuous if for every x, y, z ∈ X, the sets {t ∈ I(x, y) | (1 − t)x + ty % z} and {t ∈ I(x, y) | z % (1 − t)x + ty} are closed in I(x, y) with respect to the Euclidean topology on R. Remark 1. For x, y ∈ X, let X(x, y) be the straight line in X which passes through x and y, i.e., X(x, y) = {(1 − t)x + ty | t ∈ R} ∩ X = {(1 − t)x + ty | t ∈ I(x, y)}. A preference relation % on X is mixture continuous if and only if for every x, y, z ∈ X, the sets {w ∈ X(x, y) | w % z} and {w ∈ X(x, y) | z % w} are closed in X(x, y). Note that point z may not lie on X(x, y). Thus, the mixture continuity of % is not equal to the continuity of % |X(x,y) for every x, y ∈ X, where % |X(x,y) is the restriction of % onto X(x, y) × X(x, y). Actually, the lexicographic ordering on R2 is not mixture continuous but its restriction onto any straight line is continuous. Remark 2. If a preference relation % on X is continuous, i.e., for every x ∈ X, the sets {y ∈ X | y % x} and {y ∈ X | x % y} are closed in X, then % is mixture continuous. The inverse of Remark 2 is not true. Actually, the binary relation generated from Young and Young’s [12] function is mixture continuous but is not continuous. We will give a different example from Young and Young’s one. Like Young and Young’s example, our example is generated from a real-valued function continuous with respect to every straight line. Definition 2. A real-valued function u : X → R is continuous with respect to every straight line if for every α ∈ R and every x, y ∈ X, the sets {t ∈ I(x, y) | u((1 − t)x + ty) ≥ α} and {t ∈ I(x, y) | α ≥ u((1 − t)x + ty)} are closed in I(x, y). Note that u : X → R is continuous with respect to every straight line if and only if for every x, y ∈ X, the restriction u|X(x,y) of u onto X(x, y) is continuous. Note also that even if u is a utility representation of a continuous preference relation, u may not be continuous. For example, the usual ordering ≥ on R is continuous and any 5

increasing function on R is a utility function representing ≥, but increasing function may be discontinuous at some points. Therefore, the continuity of a preference relation and the continuity of its utility representation are not equivalent. The following remark gives the relationship of continuities between a preference relation and its utility representation. Given a real-valued function u : X → R, a preference relation %u on X is defined by x %u y if and only if u(x) ≥ u(y). It is clear that u is a utility representation of %u . Remark 3.

(1) If u : X → R is continuous with respect to every straight line (resp.

continuous at x ∈ X), then %u is mixture continuous (resp. continuous at x).3 (2) Let % be a mixture continuous preference relation (resp. a preference relation continuous at x ∈ X) and let u be its utility representation. If u(X) is an interval, u is continuous with respect to every straight line (resp. continuous at x). Now, we are ready to give an example which illustrates that the mixture continuity is strictly weaker than the continuity. Example 1. A real-valued function u on R2 is defined by  2   2x y if (x, y) 6= (0, 0), x4 + y 2 u(x, y) =   0 if (x, y) = (0, 0). Clearly, u is continuous on R2 \ {(0, 0)}. At (0, 0), u is not continuous, because u(0, 0) = 0 and u(x, x2 ) = 1 for any x 6= 0. Since u is continuous on R2 \ {(0, 0)}, it is continuous on any straight line which does not pass through (0, 0). In addition, it can be easily shown that u is continuous on any straight line passing through (0, 0). Thus, u is continuous with respect to every straight line. 3

A preference relation % on X is continuous at x ∈ X if (i) for every w ∈ X with w  x, there exists

an open subset U of RL such that x ∈ U and w  x0 for every x0 ∈ X ∩ U and (ii) for every y ∈ X with x  y, there exists an open subset V of RL such that x ∈ V and x0  y for every x0 ∈ X ∩ V . Note that a preference relation % on X is continuous at every x ∈ X if and only if it is continuous, i.e., for every x ∈ X, the sets {y ∈ X | y % x} and {y ∈ X | x % y} are both closed in X.

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We now prove that u(R2 ) is an interval. Since u is continuous on R2 \ {(0, 0)} and R2 \{(0, 0)} is connected, u(R2 \{(0, 0)}) is an interval. From u(0, 0) = 0 ∈ u(R2 \{(0, 0)}), it follows that u(R2 ) = u(R2 \ {(0, 0)}). Therefore, u(R2 ) is an interval. From Remark 3, %u is mixture continuous but is not continuous at (0, 0). Because of the lack of continuity, a mixture continuous preference relation may not have a maximal element in a compact set. If a mixture continuous preference relation is convex or weakly monotone, however, it recovers the upper semi-continuity and, therefore, it has a maximal element in a compact set (see Inoue [9]). Following Young and Young [12], we can construct a mixture continuous preference relation whose discontinuity points make a dense subset of R2 . Example 2. Let Q2 = {(a1 , b1 ), (a2 , b2 ), . . .}, where Q is the set of rational numbers. For every natural number n, define un : R2 → R by un (x, y) = u(x − an , y − bn ), where u P −n un (x, y) is well-defined, is the function in Example 1. The function U (x, y) = ∞ n=1 2 because max(x,y)∈R2 |u(x, y)| = 1. Since u is continuous with respect to every straight line and u is discontinuous only at (0, 0), the function U is continuous with respect to every straight line and is not continuous at any (a, b) ∈ Q2 . Since U (R2 ) is an interval, from Remark 3, %U is mixture continuous and is not continuous at any (a, b) ∈ Q2 . Young and Young [12] constructed a function continuous with respect to every straight line in R2 such that the set of its discontinuity points is an uncountable dense subset of R2 . This function generates a mixture continuous preference relation whose discontinuity points make an uncountable dense subset of R2 . In Section 4, we will discuss the smallness of the set of discontinuity points of a mixture continuous preference relation.

3

Representation by a utility function

We prove that the mixture continuity is sufficient for the utility representation. Theorem. Let X be a nonempty convex subset of RL . If a preference relation % on X is mixture continuous, then there exists a real-valued function u : X → R such that (i) 7

a % b if and only if u(a) ≥ u(b), (ii) u(X) is an interval, and (iii) u is continuous with respect to every straight line. Before giving a proof, we compare this theorem with related works in the literature. Eilenberg [3] (see also Debreu [1] [2]) proved that every continuous preference relation on a connected, separable topological space can be represented by a continuous utility function. Monteiro [11] proved that a continuous preference relation % on a path connected topological space X has a continuous utility representation if and only if it is countably bounded, i.e, there exists a countable subset Y of X such that for every x ∈ X, there exist y, z ∈ Y with y % x % z. The domain of a preference relation in Eilenberg’s theorem and in Monteiro’s theorem may not be a vector space and even if it is a vector space, it may not be finite dimensional. Thus, our condition on the domain of a preference relation is stronger than their conditions. (Recall that a convex subset of RL is separable, connected, and path connected with respect to the Euclidean topology.) On the other hand, as we saw in the previous section, the mixture continuity is strictly weaker than the usual continuity. Fishburn [5] proved that a convex, mixture continuous preference relation on a (possibly infinite dimensional) vector space has a utility representation if and only if it is countably bounded. In our framework of a finite dimensional vector space, as we show in the next proposition, the countably boundedness follows from the mixture continuity of a preference relation, although it is a necessary and sufficient condition for a utility representation in Fishburn’s or in Monteiro’s framework. Proposition 1. Let X be a nonempty convex subset of RL . If a preference relation % on X is mixture continuous, then there exists a countable subset Y of X such that for every x ∈ X, there exist y, z ∈ Y with y % x % z. Proof. We only prove that there exists an upward countable subset Y of X such that for every x ∈ X, there exists a y ∈ Y with y % x. By a similar manner, we can prove the existence of a downward countable subset of X. We prove by induction on the dimension of the affine hull aff(X) of X. Let k = dim aff(X). Note that k ≤ L. Under 8

an appropriate affine transformation, aff(X) can be identified with Rk .4 Thus, we may assume that X is a subset of Rk . When k = 0, the proposition is clear. When k = 1, X is an interval. Therefore, X can be represented as a countable union of closed intervals, S say, X = ∞ n=1 [an , bn ]. Since % is mixture continuous, it is continuous on every [an , bn ] and, therefore, for every n, there exists a maximal element yn ∈ [an , bn ] for % on [an , bn ]. Let Y = {y1 , y2 , . . .}. Then, Y satisfies the required property. Suppose that the proposition is true for k ≤ l but not true for k = l + 1. Then, we have: (a) for every countable subset Y of X, there exist x1 and x2 in X such that for every y ∈ Y , x1  x2  y. Let pr1 : Rl+1 → R be the projection into the first coordinate, i.e., pr1 (x(1) , . . . , x(l+1) ) = x(1) . Since dim aff(X) = l + 1, pr1 (X) is a non-degenerate interval. Therefore, pr1 (X) ∩ Q is a countably infinite set, where Q is the set of rational numbers. Hence, we may write pr1 (X) ∩ Q = {q1 , q2 , . . .}. Since for every n, the set X ∩ ({qn } × Rl ) is a convex set with at most dimension l, by the induction hypothesis, we have: (b) for every n, there exists a countable subset Yn of X ∩ ({qn } × Rl ) such that for every x ∈ X ∩ ({qn } × Rl ), there exists a y ∈ Yn with y % x. Let Y =

S∞

n=1

Yn . Then, Y is countable and, therefore, from (a), it follows that:

(c) there exist x∗1 and x∗2 in X such that for every y ∈ Y , x∗1  x∗2  y. ∗(1)

Since pr1 (X) is non-degenerate, there exists a w ∈ X such that w(1) 6= x1 . From the mixture continuity of %, we have: (d) there exists a t0 ∈ [0, 1[ such that for every t ∈ [t0 , 1], (1 − t)w + tx∗1  x∗2 . ∗(1)

∗(1)

Since t0 < 1 and w(1) 6= x1 , there exists a t∗ ∈ [t0 , 1] such that (1 − t∗ )w(1) + t∗ x1 Therefore, for some n∗ , (1 − t∗ )w(1) + 4

∗(1) t∗ x1

∈ Q.

= qn∗ . By (b), there exists a y ∗ ∈ Y

Note that an affine transformation maps straight lines to straight lines. Thus, the mixture continuity

of a preference relation is not affected by an affine transformation.

9

such that y ∗ % (1 − t∗ )w + t∗ x∗1 . On the other hand, from (c) and (d), it follows that (1 − t∗ )w + t∗ x∗1  x∗2  y ∗ , which is a contradiction. This completes the proof of Proposition 1. This proposition cannot be extended to the case of an infinite dimensional vector space as we will discuss in Section 5. Once we know that a preference relation is countably bounded, we can prove the Theorem by applying Eilenberg’s [3] theorem repeatedly. The formal proof is as follows: Proof. From Proposition 1, there exist two countable sets {y1 , y2 , . . .} and {z1 , z2 , . . .} such that for every x ∈ X, there exists n with yn % x % zn . Since [y1 , z1 ] = {(1 − t)y1 + tz1 | 0 ≤ t ≤ 1} is connected and separable, and % is continuous on [y1 , z1 ], from Eilenberg’s [3] theorem (see also Debreu [1] [2]), there exists a continuous utility function u on [y1 , z1 ]. Note that u([y1 , z1 ]) is a bounded interval, because u is continuous on [y1 , z1 ], and [y1 , z1 ] is connected and compact. Let y10 (resp. z10 ) be a maximal (resp. minimal) element on [y1 , z1 ]. Since for every x ∈ X with y10 % x % z10 , there exists a wx ∈ [y1 , z1 ] with x ∼ wx , we can extend u to the set {x ∈ X | y10 % x % z10 } by defining u(x) = u(wx ). If yn  y10 for some n, u has not been defined on a subinterval [yn , v[= {(1 − t)yn + tv | 0 ≤ t < 1} of [yn , y10 ] such that u(v) has already been defined. Note that v is the closest point in [yn , y10 ] from yn with the property v ∼ y10 . Again, from Eilenberg’s theorem, there exists a continuous function un on [yn , v] with un (v) = u(v) = u(y10 ). Note that un ([yn , v]) is a bounded closed interval. Let yn0 be a maximal element on [yn , v]. Since for every x ∈ X with yn0 % x % y10 , there exists a wx ∈ [yn , v] with x ∼ wx , we can define u(x) = un (wx ). Thus, we have extended u to the set {x ∈ X | yn0 % x % z10 }. Note that, by construction of u, u({x ∈ X | yn0 % x % z10 }) is a bounded closed interval. By repeating this argument, we can extend u to the whole space X, because % is countably bounded. By construction, u is a utility representation of % and u(X) is an interval. Therefore, from Remark 3, u is continuous with respect to every straight line. This completes the proof of the Theorem.

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4

Relationship with Eilenberg’s (1941) theorem

Given a mixture continuous preference relation, if the set of its discontinuity points is small, first, by Eilenberg’s theorem, we can obtain a utility function on the set of continuity points, and second we can extend the utility function to the whole domain of the preference relation. In this section, we discuss the validity of this method for any mixture continuous preference relation. Let % be a mixture continuous preference relation on a convex subset X of RL as in the Theorem. Also, let D = {x ∈ X | % is not continuous at x}. Note that the set X \ D of continuity points is separable. Thus, if (1) X \ D is connected, then, by Eilenberg’s theorem, there exists a continuous utility function u on X \ D. Since u is continuous on a connected set X \ D, u(X \ D) is an interval. We may assume that u(X \ D) is bounded.5 For a discontinuity point x ∈ D, if there exist a and b in X \ D with a % x % b, then there must exist a wx ∈ X \ D with x ∼ wx , because % is continuous on a connected set X \ D. Thus, we can extend u to such discontinuity point x by defining u(x) = u(wx ). Any remaining discontinuity point x ∈ D satisfies either x  a for every a ∈ X \ D or a  x for every a ∈ X \D. For a discontinuity point x ∈ D with x  a for every a ∈ X \D (resp. a  x for every a ∈ X \ D), define u(x) = sup u(X \ D) (resp. u(x) = inf u(X \ D)). Then, we have obtained a real-valued function u on the whole domain X. If (2) for every x ∈ D, there exists a y ∈ X and a sequence (xn )n in (X \ D) ∩ [x, y] such that (xn )n converges to x, then we can show that the resulting function u : X → R is actually a utility representation of %.6 Hence, once we know that a mixture continuous preference relation satisfies 5

If u(X \ D) is not bounded, we have only to replace u by Tan−1 ◦ u, where Tan−1 : R →] − π/2, π/2[

is the principal value of the arc tangent. 6 For the proof of this, see Inoue [8, Proposition 4].

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properties (1) and (2), we can obtain its utility representation by direct application of Eilenberg’s theorem. In addition, we can show that: Proposition 2. Every mixture continuous preference relation on a convex subset of a Euclidean space satisfies properties (1) and (2). For the proof of this proposition, see Inoue [8, Propositions 2 and 3]. From this proposition, for any mixture continuous preference relation, its utility representation can be obtained by direct application of Eilenberg’s theorem. It should be noticed that this fact does not imply that our utility representation theorem is dispensable, because we rely on our utility representation theorem when we prove Proposition 2. Actually, in order to prove Proposition 2, we have to characterize the set of discontinuity points of a mixture continuous preference relation. The proof of this characterization without using our utility representation theorem will be complicated even if possible, because we cannot use the properties of a real-valued function: the set of discontinuity points of any real-valued function is a Fσ -set; and a continuous function on a compact set is uniformly continuous. By using these properties, Kershner [10] characterized the set of discontinuity points of a unicontinuous function which is weaker than a function continuous with respect to every straight line. Inoue’s [8] proof of Proposition 2 relies on Kershner’s characterization.

5

Infinite dimensional case

The mixture continuity is not sufficient for the countable boundedness of a preference relation if its domain is a subset of an infinite dimensional vector space. Indeed, Fishburn [6] gave an example of a preference relation on a convex subset of l1 ([0, ω1 )), where ω1 is the first uncountable ordinal, such that it is mixture continuous and convex, but it is not countably bounded. Monteiro [11] extended Fishburn’s example. He constructed a preference relation on a convex subset of l1 ([0, ω1 )) such that it is continuous, convex, and monotone, but it is not countably bounded. In both examples by Fishburn and Monteiro, 12

the domain of preference relations is non-separable with respect to l1 -norm. Est´evez Toranzo and Herv´es Beloso [4] obtained a more general negative result. They proved that in any non-separable metric space, there always exists a continuous preference relation that is not countably bounded. Since every continuous preference relation is mixture continuous (see Remark 2), by Est´evez Toranzo and Herv´es Beloso’s theorem, in any non-separable, metrizable topological vector space,7 there exists a mixture continuous preference relation that is not countably bounded. Even in a separable, metrizable topological vector space, there can exist a mixture continuous preference relation that is not countably bounded. The next example illustrates this point. This fact clarifies the difference between the mixture continuity and the continuity, because every continuous preference relation on a separable topological space is countably bounded. Example 3. Let l∞ be the space of all bounded real sequences, i.e., l∞ = {x = (xn )n | xn ∈ R for every n ∈ N and sup |xn | < ∞}. n

We equip two norms on l∞ . For x = (xn )n ∈ l∞ , let kxk∞ = sup |xn | and kxk = n

∞ X

2−n |xn |.

n=1

Note that (l∞ , k · k∞ ) is a non-separable Banach space. It can be easily shown that (l∞ , k · k) is a separable normed space.8 For every k ∈ N, let Dk = {x = (xn )n ∈ l∞ | xn ∈ Q if 1 ≤ n ≤ k, xn = 0 if n ≥ k + 1}. Then,

S∞

n=1

Dk is a countable dense subset of (l∞ , k · k).

By Est´evez Toranzo and Herv´es Beloso’s [4] theorem, there exists an k · k∞ -continuous preference relation on l∞ that is not countably bounded. This preference relation is clearly mixture continuous. Since the mixture continuity is independent of the topology on the 7 8

Note that any non-separable Hausdorff topological vector space is infinite dimensional. The space (l∞ , k·k) is not complete, because the sequence (xk )k = ((xkn )n )k with xkn = 3n /2n if n ≤ k

and xkn = 0 if n ≥ k + 1 is a Cauchy sequence in (l∞ , k · k) but (xk )k does not converge in (l∞ , k · k).

13

domain l∞ , this is a mixture continuous preference relation on a separable normed space (l∞ , k · k) and it is not countably bounded.

References [1] Debreu, G., 1959. Theory of Value. John Wiley & Sons, New York. [2] Debreu, G., 1964. Continuity properties of Paretian utility. International Economic Review 5, 285-293. [3] Eilenberg, S., 1941. Ordered topological spaces. American Journal of Mathematics 63, 39-45. [4] Est´evez Toranzo, M., Herv´es Beloso, C., 1995. On the existence of continuous preference orderings without utility representations. Journal of Mathematical Economics 24, 305-309. [5] Fishburn, P.C., 1983a. Transitive measurable utility. Journal of Economic Theory 31, 293-317. [6] Fishburn, P.C., 1983b. Utility functions on ordered convex sets. Journal of Mathematical Economics 12, 221-232. [7] Herstein, I.N., Milnor, J., 1953. An axiomatic approach to measurable utility. Econometrica 21, 291-297. [8] Inoue, T., 2008a. A utility representation theorem with weaker continuity condition. IMW Working Paper 401, Institute of Mathematical Economics, Bielefeld University; available at http://www.imw.uni-bielefeld.de/papers/files/imw-wp-401.pdf [9] Inoue, T., 2008b. Linearly continuous preferences and equilibrium analysis in a finite dimensional commodity space. Institute of Mathematical Economics, Bielefeld University, unpublished manuscript.

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[10] Kershner, R., 1943. The continuity of functions of many variables. Transactions of the American Mathematical Society 53, 83-100. [11] Monteiro, P.K., 1987. Some results on the existence of utility functions on path connected spaces. Journal of Mathematical Economics 16, 147-156. [12] Young, W.H., Young, G.C., 1910. Discontinuous functions continuous with respect to every straight line. Quarterly Journal of Pure and Applied Mathematics 41, 87-93.

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A utility representation theorem with weaker continuity ...

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On Multi$Utility Representation of Equitable ...
class of functions with domain A and range in B. Let us recall the standard .... To check anonymity of ^/ , let x ( X and xπ be a profile with the utilities of the ith and.

Mechanism Design with Weaker Incentive Compatibility Constraints1
Jun 13, 2005 - grateful to my advisors Jeff Ely and Michael Whinston. I also thank Paul Beaudry and two anonymous referees for helpful comments. 2Department of Economics, The University of British Columbia, #997-1873 East Mall, Vancouver,. BC, V6T 1Z

Representation of transferable utility games by coalition ...
∗School of Business Administration, Faculty of Urban Liberal Arts, Tokyo Metropolitan ... production economy, the class of TU games generated by coalition .... The first economy E1(v) is essentially the same as the induced coalition production.

Mechanism Design with Weaker Incentive Compatibility Constraints1
Jun 13, 2005 - Game Theory. The MIT Press, Cambridge/London. [4] Green, J. R., Laffont, J.J., 1986. Partially verifiable information and mechanism design.

Weaker Condition.pdf
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A minmax theorem for concave-convex mappings with ...
Sion [4] or see Sorin [5] and the first chapter of Mertens-Sorin-Zamir [2] for a .... (5). Then X and Y are both finite dimensional but unbounded, f is concave-.

A Folk Theorem with Private Strategies
Mar 31, 2011 - The main contribution here is to apply the techniques from that .... For any player i, let ϕi(ai,bi) be the probability of failure conditional on a ...

A Folk Theorem for Stochastic Games with Private ...
Page 1 ... Keywords: Stochastic games, private monitoring, folk theorem ... belief-free approach to prove the folk theorem in repeated prisoners' dilemmas.

k-NN Aggregation with a Stacked Email Representation
Number of Emails. Dean-C. 0.205. Lucci-P. 753. Watson-K. 0.214. Bass-E. 754. Heard-M. 0.270 ..... Marc A. Smith, Jeff Ubois, and Benjamin M. Gross. Forward ...

Harsanyi's Aggregation Theorem with Incomplete Preferences
... Investissements d'Ave- nir Program (ANR-10-LABX-93). .... Bi-utilitarianism aggregates two utility functions ui and vi for each individual i = 1, … , I, the former ...

EXPONENTIAL UTILITY WITH NON-NEGATIVE ...
Utility maximization constitutes a primary field of research in financial math- ematics, since it offers a well-posed methodology for studying decision making under uncertainty. The most ... (2008), Henderson (2009), and Frei et al. (2011)), incomple

Oates' Decentralization Theorem with Imperfect ...
Nov 26, 2013 - In our model, agents are heterogeneous so that their result does ...... Wildasin, D. E. (2006), “Global Competition for Mobile Resources: Impli-.

Robust Utility Maximization with Unbounded Random ...
pirical Analysis in Social Sciences (G-COE Hi-Stat)” of Hitotsubashi University is greatly ... Graduate School of Economics, The University of Tokyo ...... Tech. Rep. 12, Dept. Matematica per le Decisioni,. University of Florence. 15. Goll, T., and

Harsanyi's Aggregation Theorem with Incomplete Preferences
rem to the case of incomplete preferences at the individual and social level. Individuals and society .... Say that the preference profile ( ≿ i) i=0. I satisfies Pareto ...

eBook Download Business Continuity Management System: A ...
Book Synopsis. A Business Continuity. Management System (BCMS) is a management framework that creates controls to address risks and measure an.

A STRUCTURE THEOREM FOR RATIONALIZABILITY ...
under which rti (ai) is a best reply for ti and margΘXT−i. (πti,rti (ai)) = κti . Define a mapping φti,rti (ai),m : Θ* → Θ* between the payoff functions by setting. (A.5).