Pure and randomized equilibria in the stochastic von Neumann-Gale model

Igor V. Evstigneev, Klaus Reiner Schenk-Hoppé

Research output: Contribution to journalArticlepeer-review


The paper examines the problem of the existence of equilibrium for the stochastic analogue of the von Neumann-Gale model of economic growth. The mathematical framework of the model is a theory of set-valued random dynamical systems defined by positive stochastic operators with certain properties of convexity and homogeneity. Existence theorems for equilibria in such systems may be regarded as generalizations of the Perron-Frobenius theorem on eigenvalues and eigenvectors of positive matrices. The known results of this kind are obtained under rather restrictive assumptions. We show that these assumptions can be substantially relaxed if one allows for randomization. The main result of the paper is an existence theorem for randomized equilibria. Some special cases (models defined by positive matrices) are considered in which the existence of pure equilibria can be established. © 2007 Elsevier B.V. All rights reserved.
Original languageEnglish
Pages (from-to)871-887
Number of pages16
JournalJournal of Mathematical Economics
Issue number7-8
Publication statusPublished - Sept 2007


  • Pure and mixed strategies
  • Randomization
  • Randomized von Neumann path
  • Stochastic von Neumann-Gale model
  • von Neumann equilibrium


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