Tom Armstrong

The core-walras equivalence

The equivalence of cores and competitive equilibrium sets in the very general framework of Boolean rings and algebras is proved. The results include most previous results as special cases. In particular, it is shown that Theorems 1 and 2 of M. K. Richter (J. Econom. Theory 3 (1971), 323–334) remain true without the usual assumptions of δ-algebras for coalitions and δ-additive measures for allocations. This permits economies with countably many agents, rather than requiring continuumly many.

Discovering patterns in real-valued time series

This paper describes an algorithm for discovering variable length patterns in real-valued time series. In contrast to most existing pattern discovery algorithms, ours does not first discretize the data, runs in linear time, and requires constant memory. These properties are obtained by sampling the data stream rather than processing all of the data. Empirical results show that the algorithm performs well on both synthetic and real data when compared to an exhaustive algorithm.

Existence of nonatomic core-walras allocations☆

Abstract We prove the existence of Walras allocations, where allocations are understood as measures on a nonatomic Boolean algebra. The algebra need not be a σ-algebra, as usually assumed, and the measures need not be countably additive and need not have densities with respect to the endowment allocation. Except for a strengthening of the continuity hypothesis, this is the framework we used earlier to prove the equivalence of core and Walras allocations. Our proof rests on two propositions of special interest in their own right: one asserts extensibility of coalition preferences, and the other characterizes coalition preferences.

Conglomerability of probability measures on Boolean algebras

For the case where B is a Boolean algebra of events and P is a probability (finitely additive) deFinetti (1972) considered the question of conglomerability of P and found that in many circumstances this natural notion was equivalent to countable additivity of P. Schervish, Seidenfeld, and Kadane (1984) pursued these investigations on the connection between countable additivity and conglomerability in greater detail for the case where B is a σ-algebra. Hill and Lane (1985) and Zame (1988) give alternative proofs. This article is an extension (for the most part) of Schervish, Seidenfeld, and Kadanes work to the case where B is an arbitrary Boolean algebra. The more restrictive notion of positive conglomerability for a class of algebras, including the countable algebras, σ-complete algebras, and inifinite product algebras is treated completely. This class is described by the requirement that a {0, 1}-valued measure be countably additive if every countable family of negligible sets is contained within a negligible set (i.e., corresponds to a P-point of of the Stone space). In general positive conglomerability fails to be equivalent to countable additivity though the degree of failure is minor. Building on techniques of Hill, Lane, and Zame, we obtain partial results on conglomerability for non-σ-complete algebras.

Hierarchical Arrow Social Welfare Functions

In Armstrong [1985] I examined precisely dictatorial social welfare functions. These are those social welfare functions o with associated dictator k so that, for a profile / of preferences, the social preference a(f) agrees precisely with the preference /(/c) of the dictator rather than merely extending that preference. The characterization is that of relational monotonicity for g so that if a preference profile g was obtained by extending the preferences of individuals v in a society V from that of another profile /, (so that as subsets oflxl, where X is the alternative space, the weak order f(v) is contained in the weak order g(v) for all veV) then a(f) must also be extended by a{g). As mentioned in that paper Sarbadhikari had brought to my attention the fact that Arrow Social Welfare Functions (ASWF) other than the precisely dictatorial ones exist. The simplest example is where there is a secondary dictator whose preferences between alternatives are followed in case the primary dictator is indifferent between these alternatives. Indeed one could allow tertiary dictators or quaternary dictators etc. That is, one can conceive of a dictatorial pecking order, finite in extent for finite V, but possibly infinite in extent for infinite V. The purpose of this paper is to characterize which Arrow Social Welfare Functions have such a dictatorial hierarchy. This characterization is that the social welfare function satisfy, in addition, a certain Pareto condition which we call comonotonicity for its intimate connection with current non-linear expected utility literature. Precisely dictatorial ASWFS may be considered conservative or minimally decisive in expressing preference between as few alternatives as possible. We examine here maximally decisive ASWF and characterize them as having maximal hierarchies. At the conclusion we make some remarks as to the rationalizability of ASWFs by weighted majority rule.

Remarks Related to Finitely Additive Exchange Economies

Inspired mainly by the desire to work with perfectly competitive exchange economies with only countably many agents Brown, Pallaschke, Klein, Weiss and Armstrong-Richter have examined economies given by non-atomic finitely additive (rather than countably additive) measures. In Section 2 a fairly standard measure theoretic model of perfect competition is presented. In Section 3 it is seen based on Skala in part, that assumptions of fewer than \( C = {2^{\aleph {}_0}}\) traders forces one to drop countable additivity (subject to one’s axioms of set theory). In Section 4 another reason to drop the assumption of countable additivity is examined. This is the consideration of the limit economies of a non-tight perfectly competitive sequence of finite economies. In Section 7 a rather “constructive” model of coalition formation is given based on work of Klein leading naturally only to algebras of coalitions rather than σ-algebras. In this context the assumption of countable additivity is often unnatural or unverifiable.

Notes on Jordan fields based on an article by Maharam

Abstract Results of Maharam on Jordan fields associated with a finitely additive measure are extended to take into consideration the natural topology induced by an algebra of sets namely that induced by the Stone space. Based on this the question of when the Jordan field is a σ-algebra and the question of the completeness of associated L 1 spaces are answered.

Bayes' rule and hidden variables

We show that a quantum system admits hidden variables if and only if there is a rich set of states which satisfy a Bayesian rule. The result is proved using a relationship between Bayesian type states and dispersion-free states. Various examples are presented. In particular, it is shown that for classical systems every state is Bayesian and for traditional Hilbert space quantum systems no state is Bayesian.