An elementary proof of the Knaster-Kuratowski-Mazurkiewicz-Shapley Theorem

Summary This note provides an elementary short proof of the Knaster-Kuratowski-Mazurkiewicz-Shapley (K-K-M-S) Theorem based on Brouwer's fixed point theorem. The usefulness of the K-K-M-S Theorem lies in the fact that it can be applied to prove directly Scarf's (1967) Theorem, i.e. any balanced game has a non-empty core. We also show that the K-K-M-S Theorem and the Gale-Nikaido-Debreu Theorem can be proved by the same arguments.We wish to thank Roko Aliprantis for useful comments.     

On Kakutani's fixed point theorem,the K-K-M-S theorem and the core of a balanced game

Summary We provide elementary proofs of Scarf's theorem on the non-emptiness of the core and of the K-K-M-S thoerem, based on Kakutani's fixed point theorem. We also show how these proofs can be modified to apply a coincidence theorem of Fan instead of Kakutani's fixed point theorem, for some additional simplicity.The results presented here were first reported in Shapley (1987) and Vohra (1987). A version of our proof of Theorem 1 has also been presented in a recent book by C.D. Aliprantis, D.J. Brown and O. Burkinshaw,Existence and Optimality of Competitive Equilibria (1989) Springer-Verlag. We are grateful to Ky Fan, Wanda Gorgol, Tatsuro Ichiishi and Ali Khan for comments on earlier drafts. Vohra's research has been supported in part by NSF grant SES-8605630.     

A simple proof of K-K-M-S theorem

Summary We give a simple proof of the K-K-M-S theorem based on the Kakutani fixed point theorem, the separation theorem for convex sets and the Berge maximum theorem.The author is grateful to Gerard Debreu for his advice and to Tatsuro Ichiishi, Michael Todd, Rajiv Vohra, Nicholas Yannelis, Lin Zhou and two anonymous referees for their comments. The original version of this note was written in the fall 1992 while the author was visiting the Department of Economics, University of California at Berkeley.     

On equilibrium existence in payoff secure games

We propose a single framework for studying the existence of approximate and exact pure strategy equilibria in payoff secure games. Central to the framework is the notion of a multivalued mapping with the local intersection property. By means of the Fan-Browder collective fixed point theorem, we first show an approximate equilibrium existence theorem that covers a number of known games. Then a short proof of Reny’s (Econometrica 67:1029–1056, 1999) equilibrium existence theorem is provided for payoff secure games with metrizable strategy spaces. We also give a simple proof of Reny’s theorem in its general form for metric games in an appendix for the sake of completeness.     

A generalization of Condorcet’s Jury Theorem to weighted voting games with many small voters

We extend Condorcet’s Jury Theorem (Essai sur l’application de l’analyse à la probabilité des décisions rendues à la pluralité des voix. De l’imprimerie royale, 1785) to weighted voting games with voters of two kinds: a fixed (possibly empty) set of ‘major’ voters with fixed weights, and an ever-increasing number of ‘minor’ voters, whose total weight is also fixed, but where each individual’s weight becomes negligible. As our main result, we obtain the limiting probability that the jury will arrive at the correct decision as a function of the competence of the few major players. As in Condorcet’s result the quota q = 1/2 is found to play a prominent role. I wish to thank Maurice Koster, Moshé Machover, Guillermo Owen and two anonymous referees for helpful comments.     

Universality of Nash components

We show that Nash equilibrium components are universal for the collection of connected polyhedral sets. More precisely for every polyhedral set we construct a so-called binary game—a game where all players have two pure strategies and a common utility function with values either zero or one—whose success set (the set of strategy profiles where the maximal payoff of one is indeed achieved) is homeomorphic to the given polyhedral set. Since compact semi-algebraic sets can be triangulated, a similar result follows for the collection of connected compact semi-algebraic sets.We discuss implications of our results for the strategic stability of success sets, and use the results to construct a Nash component with index k for any fixed integer k.     

Mixed-strategy equilibria in the Nash Demand Game

In the Nash Demand Game, each of the two players announces the share he demands of an amount of money that may be split between them. If the demands can be satisfied, they are; otherwise, neither player receives any money. This game has many pure-strategy equilibria. This paper characterizes mixed-strategy equilibria. The condition critical for an equilibrium is that players’ sets of possible demands be balanced. Two sets of demands are balanced if each demand in one set can be matched with a demand in the other set such that they sum to one. For Nash’s original game, a complete characterization is given of the equilibria in which both players’ expected payoffs are strictly positive. The findings are applied to the private provision of a discrete public good.     

Optimal balanced growth in a general multi-sector endogenous growth model with constant returns

I will study a multi-sector endogenous growth model with general constant returns to scale technologies and demonstrate the existence, uniqueness and the saddle-path stability of the balanced growth equilibrium. I will first demonstrate the existence of a balanced growth equilibrium, by showing that the balanced growth rate associated with the balanced growth equilibrium is solely determined by solving a Frobenius root problem of the price equations derived from the Euler equations and the property of the nonsubstitution theorem. Then I will show the saddle-path stability of the balanced growth equilibrium without any capital intensity conditions, which is a generalized property proved in the two-sector endogenous growth models by de Guevara et al. (J Econ Dyn Control 21, 115–143, 1997), Bond et al. (J Econ Theory 68, 149–173 1996) and Mino (Int Eco Rev 37, 227–251 1996). The theorem clearly implies that the balanced growth equilibrium has a transition path in the neighborhood of the balanced growth equilibrium. The paper was presented at the conferences “Irregular Growth: Beyond Balanced Growth” held on June 19–21, 2003 in Paris and “Economic Growth and Distribution: On the Nature and Causes of the Wealth of Nations” held on June 16–18, 2004 in Lucca, Italy. From the discussion with Alain Venditti at CNRS-GREQAM, Gerhard Sorger at University of Vienna and the conference participants, I have been benefited much by writing this paper. Especially Alain Venditte had given me a chance to take a look at his unpublished paper titled ” Indeterminacy and the Role of Factor Substitutability” jointly written with Kazuo Nishimura at Kyoto University and published in Macroeconomic Dynamics, Vol. 8. The author also would like to thank an anonymous referee for useful suggestions.     

Nash and Walras equilibrium via Brouwer

Summary. The existence of Nash and Walras equilibrium is proved via Brouwer's Fixed Point Theorem, without recourse to Kakutani's Fixed Point Theorem for correspondences. The domain of the Walras fixed point map is confined to the price simplex, even when there is production and weakly quasi-convex preferences. The key idea is to replace optimization with “satisficing improvement,” i.e., to replace the Maximum Principle with the “Satisficing Principle.” Received: July 9, 2001; revised version: February 25, 2002 RID="*" ID="*" I wish to thank Ken Arrow, Don Brown, and Andreu Mas-Colell for helpful comments. I first thought about using Brouwer's theorem without Kakutani's extension when I heard Herb Scarf's lectures on mathematical economics as an undergraduate in 1974, and then again when I read Tim Kehoe's 1980 Ph.D dissertation under Herb Scarf, but I did not resolve my confusion until I had to discuss Kehoe's presentation at the celebration for Herb Scarf's 65th birthday in September, 1995. RID="*" ID="*"Correspondence to: C. D. Aliprantis     

On the rationalizability of observed consumers’ choices when preferences depend on budget sets and (potentially) on anything else

We prove that defining consumers’ preferences over budget sets is both necessary and sufficient to make every fully informative and finite set of observed consumption choices rationalizable by a collection of preferences which are transitive, complete, and monotone with respect to own consumption. Our finding has two important theoretical consequences. First, assuming that preferences depend on budget sets is illegitimate under the scientific commitments of revealed preference theory. Second, as long as consumers’ preferences are not defined over budget sets, we can assume that preferences depend on observable objects other than own consumption without compromising the logical possibility to reject the model against observation. We however point out that, despite this logical possibility, in practice it can be almost impossible to reject a model where preferences are defined over objects that depend on budget sets. As an example of this we show that if preferences are defined over consumption choices of other individuals then rationalization fails only in cases of negligible practical interest.     

Over- and under-investment according to different benchmarks

In a two-stage oligopoly, with investment in the first stage and quantity or price competition in the second stage, there is a “Common Wisdom” Theorem which states that we find over-investment if the goods are substitutes and competition is in strategic substitutes, or if goods are complements and competition is in strategic complements, and that we find under-investment if we have complements and strategic substitutes or substitutes and strategic complements. The existing literature, however, lacks a proof of this theorem and, in particular, it lacks a systematic comparison of the different benchmarks for over- and under-investment. A “naive” benchmark is the cost efficient investment with respect to the subgame perfect (closed loop) equilibrium quantities. Alternative benchmarks (which are more often proposed) are the open loop equilibrium investment or the welfare maximizing investment. The chosen benchmark is critical because the Common Wisdom Theorem applies (under certain conventional conditions) only for the naive benchmark. The other two benchmarks give rise to subcases.     

Pivotal voters: A new proof of arrow's theorem

This paper presents a new proof of Arrow's ‘General Possibility Theorem’, focusing on the way how ‘social’ preferences change in response to changes in the preferences of individuals, under given social welfare functions.     

Noncooperative foundations of the nucleolus in majority games

This paper studies coalition formation, payoff division and expected payoffs in a “divide the dollar by majority rule” game with random proposers. A power index is called self-confirming if it can be obtained as an equilibrium of the game using the index itself as probability vector. Unlike the Shapley value and other commonly used power indices, the nucleolus has this property. The proof uses a weak version of Kohlberg's [SIAM J. Appl. Math. 20 (1971) 62] balancedness result reinterpreting the balancing weights as probabilities in a mixed strategy equilibrium.     

Debreu's Gap Theorem

Summary Debreu's fundamental result on the existence of utility functions that are continuous with respect to the order topology requires his Gap Theorem. A new proof of this result is given.     

A necessary and sufficient condition for non-emptiness of the core of a non-transferable utility game

It is well-known that a transferable utility game has a non-empty core if and only if it is balanced. In the class of non-transferable utility games balancedness or the more general π-balancedness due to Billera (SIAM J. Appl. Math. 18 (1970) 567) is a sufficient, but not a necessary condition for the core to be non-empty. This paper gives a natural extension of the π-balancedness condition that is both necessary and sufficient for non-emptiness of the core.     

Excess demand functions, equilibrium prices, and existence of equilibrium

Summary. For continuous aggregate excess demand functions of economies, the existing literature (e.g. Sonnenschein (1972, 1973), Mantel (1974), Debreu (1974), Mas-Colell (1977), etc.) achieves a complete characterization only when the functions are defined on special subsets of positive prices. In this paper, we allow the functions to be defined on a larger class of price sets, (allowing, for example the closed unit simplex, including its boundary). Besides characterizing excess demands for a larger class of economies, our extension provides a useful tool for proving other results. It allows us to characterize the equilibrium price set for a larger class of economies. It also permits extending Uzawa's observation (1962), by showing that Brouwer's Fixed-Point Theorem is implied by the Arrow-Debreu Equilibrium Existence Theorem (1954, Theorem I). Received: October 18, 1995; revised version June 28, 1996     

Supporting weakly Pareto optimal allocations in infinite dimensional nonconvex economies

In this paper, we prove a new version of the Second Welfare Theorem for economies with a finite number of agents and an infinite number of commodities, when the preference correspondences are not convex-valued and/or when the total production set is not convex. For this kind of nonconvex economies, a recent result, obtained by one of the authors, introduces conditions which, when applied to the convex case, give for Banach commodity spaces the well-known result of decentralization by continuous prices of Pareto-optimal allocations under an interiority condition. In this paper, in order to prove a different version of the Second Welfare Theorem, we reinforce the conditions on the commodity space, assumed here to be a Banach lattice, and introduce a nonconvex version of the properness assumptions on preferences and the total production set. Applied to the convex case, our result becomes the usual Second Welfare Theorem when properness assumptions replace the interiority condition. The proof uses a Hahn-Banach Theorem generalization by Borwein and Jofré (in Joper Res Appl Math 48:169–180, 1997) which allows to separate nonconvex sets in general Banach spacesThis work was partially supported by Nucleo Complex Engineering System. The successive versions of the paper were partly prepared during visits of Alejandro Jofré to CERMSEM and of Monique Florenzano and Pascal Gourdel to the Centro de Modelamiento Matematico. The hospitality of both institutions and the support of the french Coopération régionale Cone Sud are gratefully aknowledged. The authors thank Ali Khan for stimulating exchange of ideas and literature, Roko Aliprantis, Jean-Marc Bonnisseau, Alain Chateauneuf, Roger Guesnerie, Filipe Martins Da Rocha, Moncef Meddeb, B. Mordukovich, Lionel Thibault and Rabee Tourky for valuable discussions     

Existence of perfect equilibria: a direct proof

We provide a direct proof of the existence of perfect equilibria in finite normal form games and extensive games with perfect recall. It is done by constructing a correspondence whose fixed points are precisely the perfect equilibria of a given finite game. Existence of a fixed point is secured by a generalization of Kakutani theorem, which is proved in this paper. This work offers a new approach to perfect equilibria, which would hopefully facilitate further study on this topic. We also hope our direct proof would be the first step toward building an algorithm to find the set of all perfect equilibria of a strategic game.     

Equilibria in infinite dimensional commodity spaces revisited

Summary. The aim of the paper is to provide a new proof of the Mas-Colell–Richard existence of equilibrium result when preferences are non-transitive and incomplete. Our proof generalizes the main ideas of the Negishi approach to the case of unordered preferences. Received: January 10, 1996; revised version: November 23, 1999     

The nature of Coasean property

The Coase Theorem is widely regarded as pointing to the importance of positive transaction costs for the analysis of economic institutions. Various interpretations of the Coase Theorem regard transaction costs as some set of impediments to contracting, or more broadly, as the costs of providing institutional solutions to conflicts over resource use. The abstract nature of the Coasean hypothetical tends to promote an abstract notion of property as a thin entitlement: a right in a designated person to take certain actions or derive value from a set of resource attributes. On this view, property is like a collection of tiny contracts. The property rights furnished by actual property law are much more coarse grained than this, and property is correspondingly “incomplete” for transaction costs reasons. Property and contract are substitutes in some situations, but they often are not interchangeable—because of Coasean transaction costs.