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This note shows that if the space of events is sufficiently rich and the subjective probability function of each individual is non-atomic, then there is a σ-algebra of events over which everyone will have the same probability function, and moreover, the range of this common probability is the entire unit interval.

This note uses the Theorem of the Alternative to prove new results on the implementability of general, asymmetric auctions, and to provide simpler proofs of known results for symmetric auctions. The tradeoff is that type spaces are taken to be finite.

We present a new simple and short proof of the existence of equilibria in the classical single-sector growth model.

In this survey problems from the consideration of infinite dimensional commodity spaces in the Arrow-Debreu-McKenzie model are discussed. It has become clear in the last couple of decades that in order to address real policy questions, economic models that are both stochastic and dynamic have to be employed. These models lead naturally to infinite dimensional commodity spaces.

Machina has proposed a definition of dynamic consistency which admits non-expected utility functionals. We show that even under this new definition a dynamically consistent preference relation that is differentiable becomes arbitrarily close to an expected utility preference after the realization of a low probability event.

An oddsmaker who does not update odds in accordance with conditional probability can be subjected to a sure loss. Such a losing situation is called a Dutch book. This is often presented as an argument for setting odds in accordance with conditional probability. The authors point out that if the strategic aspects of oddsmaking are taken into account (choice of odds is a strategic choice in games against bettors), then it is possible that, in the equilibrium of the game, posted odds need not be updated 'correctly.' This calls into question the role of such arguments in the foundations of decision theory. Copyright 1994 by Royal Economic Society.

This paper shows that a choice function is either consistent with the expected utility hypothesis or is ex ante dominated for some prior.

This note shows that if the space of events is sufficiently rich and the subjective probability function of each individual is non-atomic, then there is a sigma-algebra of events over which everyone will have the same probability function, and moreover, the range of these probabilities is the whole [0, 1] segment.

A well-known Dutch book, due to de Finetti, shows how violations of the additivity law of probability theory (Pr(A ∪ B) = Pr(A)+ Pr(B)-Pr(A ∩ B)) open the door to a sequence of bets leading to a sure loss. In this paper we show that in a market environment, when bookies act strategically, it may well be optimal for them to post incoherent (nonadditive) odds. This is true even when the bookie's own preferences are expected utility, provided she plays against at least two nonexpected utility bettors.

In Chapter 4 we introduced the concept of σ-algebra of sets to capture the properties of events in probability theory. We used the traditional terminology of referring to sets belonging to such a σ-algebra as measurable sets. While we have good pedagogical reasons for introducing the material in this order, it is not obvious what a σ-algebra has to do with measurement of anything. In this chapter we hope to remedy this omission. Historically, mathematicians were interested in generalizing the notions of length, area, and volume. The most useful generalization of these concepts is the notion of a measure. In its abstract form a measure is a set function with additivity properties that reflect the properties of length, area, and volume. A set function is an extended real function defined on a collection of subsets of an underlying measurable space. (We also impose the restriction that a measure assumes at most one of the values ∞ and —∞.) In this chapter we consider set functions that have some of the properties ascribed to area. The main property is additivity. The area of two regions that do not overlap is the sum of their areas. A charge is any nonnegative set function that is additive in this sense. A measure is a charge that is countably additive. That is, the area of a sequence of disjoint regions is the infinite series of their areas. A probability measure is a measure that assigns measure one to the entire space. Charges and measures are intimately entwined with integration, which we take up in Chapter 11. But here we study them in their own right.

One way to think of functional analysis is as the branch of mathematics that studies the extent to which the properties possessed by finite dimensional spaces generalize to infinite dimensional spaces. In the finite dimensional case there is only one natural linear topology. In that topology every linear functional is continuous, convex functions are continuous (at least on the interior of their domains), the convex hull of a compact set is compact, and nonempty disjoint closed convex sets can always be separated by hyperplanes. On an infinite dimensional vector space, there is generally more than one interesting topology, and the topological dual, the set of continuous linear functionals, depends on the topology. In infinite dimensional spaces convex functions are not always continuous, the convex hull of a compact set need not be compact, and nonempty disjoint closed convex sets cannot generally be separated by a hyperplane. However, with the right topology and perhaps some additional assumptions, each of these results has an appropriate infinite dimensional version.

Unless otherwise indicated, in this chapter X is a metrizable topological space, and P (X) (or simply P) is the set of all probability measures on the Borel sets B of X. As usual, C b (X) denotes the Banach lattice of all bounded continuous real functions on X. The reason we focus on probability measures is that every finite measure is the difference of measures each of which is a nonnegative multiple of a probability measure. That is, the probability measures span the space of all signed measures of bounded variation.

We discuss the two fundamental theorems of welfare economics in the context of the Arrow-Debreu-McKenzie model with an infinite dimensional commodity space. As an application, we prove the existence of competitive equilibrium in the standard single agent growth model.

In this chapter we present some useful odds and ends that should be a part of everyone’s mathematical tool kit, but which don’t conveniently fit anywhere else. Our presentation is informal and we do not prove many of our claims. We also use this chapter to standardize some terminology and notation. In particular, Section 1.3 introduces a number of kinds of binary relations.

Chapter 8 dealt with measures and charges defined on abstract semirings or σ-algebras of sets. In applications there is often a natural topological or metric structure on the underlying measure space. By combining topological and set theoretic notions it is possible to develop a richer and more useful theory. Some of these connections between measure theory and topology are discussed in this chapter.

Among the most important and simplest normed and Banach spaces are the sequence spaces—vector subspaces of the vector space ℝℕ of all real sequences. The sequence spaces can be thought of as the “building blocks” of Banach spaces and Banach lattices. Whether they are embedded in a Banach space or a Banach lattice reflect the topological and order structure of the space. In this chapter, we introduce the classical sequence spaces, φ, c 0, l 1, l ∞, and l p (0 < p < ∞). We isolate each one of these sequence spaces and investigate its basic properties.

In this chapter, we introduce the classical L p -spaces and study their basic properties. For a measure space (X, Σ, μ) and 0 < p

Over the last fifty years advanced mathematical tools have become an integral part in the development of modern economic theory. Economists continue to invoke sophisticated mathematical techniques and ideas in order to understand complex economic and social problems. In the last ten years the theory of Riesz spaces (vector lattices) has been successfully applied to economic theory. By now it is understood relatively well that the lattice structure of Riesz spaces can be employed to capture and interpret several economic notions. On April 16-20, 1990, a small conference on Riesz Spaces, Positive Opera­ tors, and their Applications to Economics took place at the California Institute of Technology. The purpose of the conference was to bring mathematicians special­ ized in Riesz Spaces and economists specialized in General Equilibrium together to exchange ideas and advance the interdisciplinary cooperation between math­ ematicians and economists. This volume is a collection of papers that represent the talks and discussions of the participants at the week-long conference. We take this opportunity to thank all the participants of the conference, especially those whose articles are contained in this volume. We also greatly ap­ preciate the financial support provided by the California Institute of Technology. In particular, we express our sincerest thanks to David Grether, John Ledyard, and David Wales for their support. Finally, we would like to thank Susan Davis, Victoria Mason, and Marge D'Elia who handled the delicate logistics for the smooth running of the conference.

Depending on the school of thought, expected utility theory states that choices among lotteries either should be made or actually made by maximizing the expected value of a real valued function of the outcomes—a utility function. This article provides a look at some of the functional analytic results used in expected utility theory. I concentrate on applications to the theory of stochastic dominance relations and the revealed preference approach to expected utility. Few of these results are deep, given the underlying tools, but many of them are not widely known, and their combination is novel. In particular, the revealed preference results of Border [4] are extended to higher degree stochastic dominance relations.

There are several solutions to the Nash bargaining problem in the literature. Since various authors have expressed preferences for one solution over another, we find it useful to study preferences over solutions in their own right. We identify a set of appealing axioms on such preferences that lead to unanimity in the choice of solution, which turns out to be the solution of Nash.

AN AUCTION IS A MECHANISM for allocating a single indivisible object to one of several competing bidders. The winner is the bidder who is awarded the object. The rules of the auction specify two functions. The first is the probability with which a bidder wins, as a function of everyone's bids. The second is the payment each bidder makes to the seller, as a function of all the bids and whether or not he wins. For instance, a first-price auction awards the object to the highest bidder with probability one (providing there are no tie bids), the winner pays his bid, and the losers pay nothing. The bidders in an auction differ significantly. These differences are captured by the bidder's type. A type may be the bidder's personal valuation of the object for sale, his degree of risk aversion, or perhaps his information about the object. (Maskin and Riley (1984) discuss a number of different economically meaningful examples of bidder types.) From the viewpoint of the seller and the other bidders, each bidder's type is a random variable. In this analysis we confine attention to auctions in which the types are independently and identically distributed according to a known probability distribution. The Revelation Principle asserts that every auction is strategically equivalent to an auction in which bidders bid by announcing their type and no bidder has any incentive to lie. Such an auction is called an incentive compatible direct auction. We will confine our attention to the probability functions for direct auctions, and let the incentive compatibility conditions restrict the payment functions. Each bidder can compute the probability that he wins, conditional on his own type, by averaging over the types of the other bidders. The function relating a bidder's type to his probability of winning is the reduced form of the auction. The literature on "optimal" auctions usually addresses the problem of maximizing expected revenue for the seller. For this purpose, all the relevant information about the probability function of an auction is contained in its reduced form. It is the reduced form that determines each bidder's behavior and hence the seller's expected revenue. In a symmetric auction each bidder's reduced form is identical, so that expected revenue is a functional defined on reduced forms, which are functions of one variable, namely, types. This makes the seller's problem somewhat tractable. To design an auction, a seller must be able to recognize a reduced form and recover the underlying auction. Reduced forms satisfy an intuitive feasibility condition. Given a set of types, the

It is shown for the case of private goods economies that every social welfare function satisfying a weak nonimposition condition and the independence of irrelevant alternatives axiom is of one of the following forms. It is either null, or the class of decisive coalitions is an ultrafilter, or the class of anti-decisive coalitions is an ultrafilter. In the case of a private goods economy with finitely many traders, the latter conditions imply the existence of either a dictator or anti-dictator. By requiring the Pareto principle as well, it is easily seen that the social welfare function must be dictatorial.

If individuals and society both obey the expected utility hypothesis and social alternatives are uncertain, then the social utility must be a linear combination of the individual utilities, provided the society is indifferent when all its members are. This result was first proven by Harsanyi [4] who made implicit assumptions in the proof not actually needed for the result (see [5]). This note presents a straightforward proof of Harsanyi's theorem based on a separating hyperplane argument.

This paper examines the implications for social welfare functions of restricting the domain of individual preferences to type-one preferences. Type-one preferences assume that each person has a most preferred alternative in a euclidean space and that alternatives are ranked according to their euclidean distance from this point. The result is that if we impose Arrow's conditions of collective rationality, IIA, and the Pareto principle on the social welfare function, then it must be dictatorial. This result may not seem surprising, but it stands in marked contrast to the problem considered by Gibbard and Satterthwaite of finding a social-choice function. With unrestricted domain, under the Gibbard-Satterthwaite hypotheses, choices must be dictatorial. With type-one preferences this result has been previously shown not to be true. This finding identifies a significant difference between the Arrow and Gibbard-Satterthwaite problems.