The Jacobian matrix and global univalence of mappings

Gale, David; Nikaidō, Hukukane

doi:10.1007/bf01360282

Cited by 581 publications

(385 citation statements)

References 6 publications

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“…Proof of this result can be found in Gale and Nikaido (1965) and some further developments that deal with boundary equilibria are found in Rosen (1965). Notice that the univalent mapping argument is somewhat weaker than the contraction mapping argument.…”

Section: Methods 3 Univalent Mapping Argumentmentioning

confidence: 91%

Game Theory in Supply Chain Analysis

Cachon

Netessine

2004

International Series in Operations Research &Amp; Management Science

306

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Game theory has become an essential tool in the analysis of supply chains with multiple agents, often with conflicting objectives. This chapter surveys the applications of game theory to supply chain analysis and outlines game-theoretic concepts that have potential for future application. We discuss both non-cooperative and cooperative game theory in static and dynamic settings. Careful attention is given to techniques for demonstrating the existence and uniqueness of equilibrium in non-cooperative games. A newsvendor game is employed throughout to demonstrate the application of various tools. 1

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Section: Methods 3 Univalent Mapping Argumentmentioning

confidence: 91%

Game Theory in Supply Chain Analysis

Cachon

Netessine

2004

International Series in Operations Research &Amp; Management Science

306

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“…If this mapping is single-dimensional R 1 → R 1 , then it is quite clear that the condition sufficient for the mapping to be one to one is quasiconcavity of π i . Similarly, for the R n → R n mapping to be one to one, we require quasiconcavity of the mapping, which translates into quasidefiniteness of the Hessian: Proof of this result can be found in Gale and Nikaido [40] and some further developments that deal with boundary equilibria are found in Rosen [80]. Notice that the univalent mapping argument is somewhat weaker than the contraction mapping argument.…”

Section: Methods 3 Univalent Mapping Argumentmentioning

confidence: 97%

Game Theory in Supply Chain Analysis

Cachon¹,

Netessine²

2006

Models, Methods, and Applications for Innovative Decision Making

166

168

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Game theory has become an essential tool in the analysis of supply chains with multiple agents, often with conflicting objectives. This chapter surveys the applications of game theory to supply chain analysis and outlines game-theoretic concepts that have potential for future application. We discuss both noncooperative and cooperative game theory in static and dynamic settings. Careful attention is given to techniques for demonstrating the existence and uniqueness of equilibrium in noncooperative games. A newsvendor game is employed throughout to demonstrate the application of various tools.

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“…Two of the best known papers providing a generic set of sufficient conditions for uniqueness in games of the form analysed here are Gale andNikaido (1965) andRosen (1965).…”

Section: A Univalent Mapping Theoremsmentioning

confidence: 99%

A New Existence and Uniqueness Theorem for Continuous Games

Hogan

2011

The B.E. Journal of Theoretical Economics

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Abstract:This paper derives a general sufficient condition for existence and uniqueness in continuous games using a variant of the contraction mapping theorem applied to mapping from a subset of the real line on to itself. We first prove this contraction mapping variant, and then show how the existence of a unique equilibrium in the general game can be shown by proving the existence of a unique equilibrium in an iterative sequence of games involving such R mappings. Finally, we show how a general condition for this to occur is that a matrix derived from the Jacobean matrix of best-response functions be have positive leading principal minors, and how this condition generalises some existing uniqueness theorems for particular games.-to-R

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The Jacobian matrix and global univalence of mappings

Cited by 581 publications

References 6 publications

Game Theory in Supply Chain Analysis

Game Theory in Supply Chain Analysis

Game Theory in Supply Chain Analysis

A New Existence and Uniqueness Theorem for Continuous Games

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