Gross substitutability of point-to-set correspondences

Polterovich, Victor; Spivak, V.A.

doi:10.1016/0304-4068(83)90032-0

Cited by 26 publications

(15 citation statements)

References 10 publications

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“…We could state Assumption 8 for correspondences, following Polterovich and Spivak (1983), but we use it in conjunction with Assumption 6, hence for functions, and therefore, state it for functions.…”

Section: Assumptionmentioning

confidence: 99%

Continua of underemployment equilibria reflecting coordination failures, also at Walrasian prices

Citanna¹,

Crès²,

Drèze³

et al. 2001

Journal of Mathematical Economics

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In this paper, the existence of unemployment is partly explained as being the result of coordination failures. It is shown that as a result of self-fulfilling pessimistic expectations, even at Walrasian prices, a continuum of equilibria results, among which an equilibrium with approximately no trade and a Walrasian equilibrium. These coordination failures also arise at other price systems, but then unemployment is the result of both a wrong price system and coordination failures. Some properties of the set of equilibria are analyzed. Generically, there exists a continuum of non-indifferent equilibrium allocations. Under a condition implied by gross substitutability, there exists a continuum of equilibrium allocations in the neighborhood of a competitive allocation, when prices are Walrasian. For a specialized economy, a dynamic illustration is offered.

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Section: Assumptionmentioning

confidence: 99%

Continua of underemployment equilibria reflecting coordination failures, also at Walrasian prices

Citanna¹,

Crès²,

Drèze³

et al. 2001

Journal of Mathematical Economics

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“…Polterovich and Spivak [45] extended the characterization of the Separation Lemma to scenarios where the demand is a set-valued function of the prices, which includes in particular the exchange model with linear utilities. This extension says that for any equilibrium priceπ, and non-equilibrium price π, and any vector z ∈ R n that is chosen from the set of aggregate excess demands of the market at π, we haveπ · z > 0.…”

Section: Lemma 3 [Separation Lemma]mentioning

confidence: 99%

“…Cutting plane methods that exploit characterizations like that of the Separation Lemma and its extension in [45] have been studied (see the paper by Primak [47] and the references therein). Primak [46] (see also [47]) has also shown that the Polterovich-Spivak characterization holds when traders have linear utilities and production sets are limited to the positive orthant R n + .…”

Section: Lemma 3 [Separation Lemma]mentioning

confidence: 99%

The computation of market equilibria

2004

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The market equilibrium problem has a long and distinguished history. In 1874, Walras published the famous "Elements of Pure Economics", where he describes a model for the state of an economic system in terms of demand and supply, and expresses the supply equals demand equilibrium conditions [62]. In 1936, Wald gave the first proof of the existence of an equilibrium for the Walrasian system, albeit under severe restrictions [61]. In 1954, Nobel laureates Arrow and Debreu proved the existence of an equilibrium under milder assumptions [3]. This existence result, along with the two fundamental theorems of welfare are the pillars of modern equilibrium theory. The First Fundamental Theorem of Welfare showed the Pareto optimality of allocations at equilibrium prices, thereby formally expressing Adam Smith's "invisible hand" property of markets and providing important social justification for the theory of equilibrium.

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“…The technical lynchpin for the algorithm is a strong separation lemma that allows the use of the ellipsoid method. This lemma stregthens the lemma from [77] in a way that is analagous to how [23] strengthens Lemma 1.2.1.…”

Section: Thesis Contributions To the Computation Of Market Equilibriamentioning

confidence: 86%

“…Following Polterovich and Spivak [77], we define gross substitutability (GS) correspondences. Let π 1 and π 2 be price vectors for a market with n goods.…”

Section: Gross Substitutability Correspondencesmentioning

confidence: 99%

Algorithmic game theory and the computation of market equilibria

McCune¹

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It is demonstrated that for certain markets where traders have constant elasticity of substitution utility (CES) functions, the existence of a price equilibrium can be determined in polynomial time. It is also shown that for a certain range of elasticity of substitution where the CES market does not satisfy gross subsitutability that price equilibira can be computed in polynomial time. It is also shown that for markets satisfying gross substitutability, equilibria can be computed in polynomial time even if the excess demand is a correspondence. On the experimental side, equilibrium computation algorithms from computer science without running time guarantees are shown to be competitive with software packages used in applied microeconomics.Simulations also lend support to the Nash equilibrium solution concept by showing that agents employing heuristics in a restricted form of Texas Holdem converge to an approximate equilibrium. Monte Carlo simulations also indicate the long run preponderance of skill over chance in Holdem tournaments. Abstract Approved: Thesis Supervisor To my wife Jennifer, my mother Patricia, my father Daniel, and my sister Julie for their many years of love and support ii I am especially grateful to my advisor, Dr. Kasturi Varadajan. Having come to Iowa without having taken a course in computer science, it was Kasturi who first introduced me to the study of algorithms through his excellent course and sparked my first real interest in theoretical computer science. Kasturi was always generous with both time and ideas. This was critical as time after time, many days of confusion would be cleared upon his blackboard. This work would not have been possible iv without his kind advice, erudition, encouragement and, not least, his patience. And for that, I will remain grateful. v ABSTRACT It is demonstrated that for certain markets where traders have constant elasticity of substitution utility (CES) functions, the existence of a price equilibrium can be determined in polynomial time. It is also shown that for a certain range of elasticity of substitution where the CES market does not satisfy gross subsitutability that price equilibira can be computed in polynomial time. It is also shown that for markets satisfying gross substitutability, equilibria can be computed in polynomial time even if the excess demand is a correspondence. On the experimental side, equilibrium computation algorithms from computer science without running time guarantees areshown to be competitive with software packages used in applied microeconomics.Simulations also lend support to the Nash equilibrium solution concept by showing that agents employing heuristics in a restricted form of Texas Holdem converge to an approximate equilibrium. Monte Carlo simulations also indicate the long run preponderance of skill over chance in Holdem tournaments.vi Proof: Both σ and σ 0 are in B, so it must be the case that for each j, |σ j − σ 0j | < 1.Therefore, we only need to bound each component of ▽g(σ 0 ) −ᾱ by ǫ ′ 4n . One can see the formula...

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Gross substitutability of point-to-set correspondences

Cited by 26 publications

References 10 publications

Continua of underemployment equilibria reflecting coordination failures, also at Walrasian prices

Continua of underemployment equilibria reflecting coordination failures, also at Walrasian prices

The computation of market equilibria

Algorithmic game theory and the computation of market equilibria

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