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Evolutionary Games in Natural, Social, and Virtual Worlds$
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Daniel Friedman and Barry Sinervo

Print publication date: 2016

Print ISBN-13: 9780199981151

Published to Oxford Scholarship Online: August 2016

DOI: 10.1093/acprof:oso/9780199981151.001.0001

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(p.102) 4 Equilibrium
Evolutionary Games in Natural, Social, and Virtual Worlds

Daniel Friedman

Barry Sinervo

Daniel Friedman

Barry Sinervo

Oxford University Press

Chapter 4 connects the dynamic steady state concepts analyzed in previous chapters to the classic static concepts of Nash equilibrium (NE) and evolutionarily stable state (ESS). It shows that NE is a necessary condition for a steady state to be dynamically stable, and that ESS is sufficient under replicator dynamics, but is not always sufficient for other monotone dynamics. It applies equilibrium ideas to established models of Fisherian runaway sexual selection, and also introduces the concept of an adjustment matrix, which modifies the payoff matrix via the Hadamard product (element-by-element matrix multiplication). A technical appendix explains three main techniques of stability analysis: eigenvalues of (projected) Jacobian matrices, Lyapunov functions, and Liouville’s theorem.

Keywords:   equilibrium, dynamic steady state, Nash equilibrium, ESS, runaway sexual selection, Hadamard product, eigenvalues, Lyapunov functions, Liouville’s theorem

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