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2015 | OriginalPaper | Chapter

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

Author : Michel Benaim

Published in: Dynamics, Games and Science

Publisher: Springer International Publishing

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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.

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previous chapter Periodic Homogenization of Deterministic Control Problems via Limit Occupational Measures
next chapter Wave Interaction with Floating Bodies in a Stratified Multilayered Fluid
Appendix
Available only for authorised users
Footnotes
1
By this we mean that L is the restriction to Δ of a C 1 map defined in a neighborhood of Δ in \(aff(\varDelta ) =\{ x \in \mathbb{R}^{n}\,:\sum _{i}x_{1} = 1\}\).
 
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Metadata
Title
On Gradient Like Properties of Population Games, Learning Models and Self Reinforced Processes
Author
Michel Benaim
Copyright Year
2015
Book
Dynamics, Games and Science

Print ISBN: 978-3-319-16117-4

Electronic ISBN: 978-3-319-16118-1

Copyright Year: 2015

DOI
https://doi.org/10.1007/978-3-319-16118-1_8

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