Dynamic programming and mathematical economics.

"Prepared for United States Air Force Project Rand."

A note on the multiple partners assignment game

In the assignment game of Shapley and Shubik [Shapley, L.S., Shubik, M., 1972. The assignment game. I. The core, International journal of Game Theory 1, 11-130] agents are allowed to form one partnership at most. That paper proves that, in the context of firms and workers, given two stable payoffs for the firms there is a stable payoff which gives each firm the larger of the two amounts and also one which gives each of them the smaller amount. Analogous result applies to the workers. Sotomayor [Sotomayor, M., 1992. The multiple partners game. In: Majumdar, M. (Ed.), Dynamics and Equilibrium: Essays in Honor to D. Gale. Mcmillian, pp. 322-336] extends this analysis to the case where both types of agents may form more than one partnership and an agent`s payoff is multi-dimensional. Instead, this note concentrates in the total payoff of the agents. It is then proved the rather unexpected result that again the maximum of any pair of stable payoffs for the firms is stable but the minimum need not be, even if we restrict the multiplicity of partnerships to one of the sides. (C) 2009 Elsevier B.V. All rights reserved.

Existence of a semicontinuous or continuous utility function: a unified approach and an elementary proof

In this paper, we present a new unified approach and an elementary proof of a very general theorem on the existence of a semicontinuous or continuous utility function representing a preference relation. A simple and interesting new proof of the famous Debreu Gap Lemma is given. In addition, we prove a new Gap Lemma for the rational numbers and derive some consequences. We also prove a theorem which characterizes the existence of upper semicontinuous utility functions on a preordered topological space which need not be second countable. This is a generalization of the classical theorem of Rader which only gives sufficient conditions for the existence of an upper semicontinuous utility function for second countable topological spaces. (C) 2002 Elsevier Science B.V. All rights reserved.

The non-existence of a utility function and the structure of non-representable preference relations

In this paper we investigate the structure of non-representable preference relations. While there is a vast literature on different kinds of preference relations that can be represented by a real-valued utility function, very little is known or understood about preference relations that cannot be represented by a real-valued utility function. There has been no systematic analysis of the non-representation problem. In this paper we give a complete description of non-representable preference relations which are total preorders or chains. We introduce and study the properties of four classes of non-representable chains: long chains, planar chains, Aronszajn-like chains and Souslin chains. In the main theorem of the paper we prove that a chain is non-representable if and only it is a long chain, a planar chain, an Aronszajn-like chain or a Souslin chain. (C) 2002 Published by Elsevier Science B.V.

Lexicographic decomposition of chains and the concept of a planar chain

In an earlier paper [Journal of Mathematical Economics, 37 (2002) 17-38], we proved that if a preference relation on a commodity space is non-representable by a real-valued function then that chain is necessarily a long chain, a planar chain, an Aronszajn-like chain or a Souslin chain. In this paper, we study the class of planar chains, the simplest example of which is the Debreu chain (R-2, <(l)). (C) 2002 Elsevier Science B.V. All rights reserved.

Value and prices in Russian economic thought (1890-1920)

The subject "Value and prices in Russian economic thought (1890--1920)" should evoke several names and debates in the reader's mind. For a long time, Western scholars have been aware that the Russian economists Tugan-Baranovsky and Bortkiewicz were active participants to the Marxian transformation problem, that the mathematical models of Dmitriev prefigured forthcoming neoricardian based models, and that many Russian economists were either supporting the Marxian labour theory of value or being revisionists. Moreover, these ideas were preparing the ground for Soviet planning. Russian scholars additionally knew that this period was the time of introduction of marginalism in Russia, and that, during this period, economists were active in thinking the relation of ethics with economic theory. All these issues are well covered in the existing literature. But there is a big gap that this dissertation intends to fill. The existing literature handles these pieces separately, although they are part of a single, more general, history. All these issues (the labour theory of value, marginalism, the Marxian transformation problem, planning, ethics, mathematical economics) were part of what this dissertation calls here "The Russian synthesis". The Russian synthesis (in the singular) designates here all the attempts at synthesis between classical political economy and marginalism, between labour theory of value and marginal utility, and between value and prices that occurred in Russian economic thought between 1890 and 1920, and that embraces the whole set of issues evoked above. This dissertation has the ambition of being the first comprehensive history of that Russian synthesis. In this, this contribution is unique. It has always surprised the author of the present dissertation that such a book has not yet been written. Several good reasons, both in terms of scarce availability of sources and of ideological restrictions, may accounted for a reasonable delay of several decades. But it is now urgent to remedy the situation before the protagonists of the Russian synthesis are definitely classified under the wrong labels in the pantheon of economic thought. To accomplish this task, it has seldom be sufficient to gather together the various existing studies on aspects of this story. It as been necessary to return to the primary sources in the Russian language. The most important part of the primary literature has never been translated, and in the last years only some of them have been republished in Russian. Therefore, most translations from the Russian have been made by the author of the present dissertation. The secondary literature has been surveyed in the languages that are familiar (Russian, English and French) or almost familiar (German) to the present author, and which are hopefully the most pertinent to the present investigation. Besides, and in order to increase the acquaintance with the text, which was the objective of all this, some archival sources were used. The analysis consists of careful chronological studies of the authors' writings and their evolution in their historical and intellectual context. As a consequence, the dissertation brings new authors to the foreground - Shaposhnikov and Yurovsky - who were traditionally confined to the substitutes' bench, because they only superficially touched the domains quoted above. In the Russian synthesis however, they played an important part of the story. As a side effect, some authors that used to play in the foreground - Dmitriev and Bortkiewicz - are relegated to the background, but are not forgotten. Besides, the dissertation refreshes the views on authors already known, such as Ziber and, especially, Tugan-Baranovsky. The ultimate objective of this dissertation is to change the opinion that one could have on "value and prices in Russian economic thought", by setting the Russian synthesis at the centre of the debates.

Max-convex decompositions for cooperative TU games

We show that any cooperative TU game is the maximum of a finite collection of convex games. This max-convex decomposition can be refined by using convex games with non-negative dividends for all coalitions of at least two players. As a consequence of the above results we show that the class of modular games is a set of generators of the distributive lattice of all cooperative TU games. Finally, we characterize zero-monotonic games using a strong max-convex decomposition