On Gradient Like Properties of Population Games, Learning Models and Self Reinforced Processes View Full Text


Ontology type: schema:Chapter      Open Access: True


Chapter Info

DATE

2015

AUTHORS

Michel Benaim

ABSTRACT

We consider ordinary differential equations on the unit simplex of \(\mathbb{R}^{n}\) that naturally occur in population games, models of learning and self reinforced random processes. Generalizing and relying on an idea introduced in Dupuis and Fisher (On the construction of Lyapunov functions for nonlinear Markov processes via relative entropy, 2011), we provide conditions ensuring that these dynamics are gradient like and satisfy a suitable “angle condition”. This is used to prove that omega limit sets and chain transitive sets (under certain smoothness assumptions) consist of equilibria; and that, in the real analytic case, every trajectory converges toward an equilibrium. In the reversible case, the dynamics are shown to be C 1 close to a gradient vector field. Properties of equilibria -with a special emphasis on potential games—and structural stability questions are also considered. More... »

PAGES

117-152

References to SciGraph publications

  • 1992-03. Vertex-reinforced random walk in PROBABILITY THEORY AND RELATED FIELDS
  • 1982. Geometric Theory of Dynamical Systems, An Introduction in NONE
  • 1999. Dynamics of stochastic approximation algorithms in SÉMINAIRE DE PROBABILITÉS XXXIII
  • 1996-01. Asymptotic pseudotrajectories and chain recurrent flows, with applications in JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS
  • Book

    TITLE

    Dynamics, Games and Science

    ISBN

    978-3-319-16117-4
    978-3-319-16118-1

    Author Affiliations

    Identifiers

    URI

    http://scigraph.springernature.com/pub.10.1007/978-3-319-16118-1_8

    DOI

    http://dx.doi.org/10.1007/978-3-319-16118-1_8

    DIMENSIONS

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