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Games in Game Theory Read chapter Read first chapter

2015 | OriginalPaper | Chapter

10. Evolutionary Stability\(^*\)

Author : Takako Fujiwara-Greve

Published in: Non-Cooperative Game Theory

Publisher: Springer Japan

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Abstract

It is possible to talk about the stability of a strategy distribution without relying on an individual player’s choice. This is evolutionary stability. In this chapter, we give some “static” stability notions based on the payoff comparison and one dynamic stability concept based on the replicator dynamic, which is most widely used.

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previous chapter Equilibrium Selection
Footnotes
1
Before Maynard-Smith and Price [11], Hamilton [7] and others already formulated a mathematical model of natural selection.
 
2
This assumption is plausible in many realistic cases, such as animal encounters. If both players need to agree to end the partnership, like in a divorce process, then the same set of outcomes as that of an infinitely repeated game arises, because players can force each other to stay in the game forever.
 
3
Other possible information that a player may base her/his actions upon is her/his own past histories (including actions by past opponents, e.g., Kandori [8] and Ellison [3]) and the time count since (s)he entered the game. For simplicity, we only consider partnership-independent strategies, ignoring these kinds of information.
 
4
To be precise, this definition does not specify actions in off-path information sets, because they do not affect the NSS analysis.
 
5
Again, this specification is not complete for off-path information sets. Any strategy with this on-path action plan works.
 
6
To be precise, a sufficient condition is needed. See Fujiwara-Greve and Okuno-Fujiwara [5].
 
7
Rigorously speaking, we must prove that any mutant strategy cannot thrive when the population consists of \(\alpha ^*\) \(c_0\)-players and \((1-\alpha ^*)\) \(c_1\)-players. This is done in [5].
 
8
The exogenous factors of birth rates and death rates can be omitted without affecting the resulting dynamic equation.
 
9
Consider the vector \(\mathbf {x}\) as a row vector and \(\mathbf {x}^T\) as its transpose, i.e., a column vector.
 
10
A state \(x^*\) is Lyapunov stable if, for any neighborhood V of \(x^*\), there exists a neighborhood W of \(x^*\) such that, starting from any \(x(0)\in W\), \(x(t) \in V\) for any \(t \geqq 0\).
 
11
A state \(x^*\) is asymptotically stable if it is Lyapunov stable and there exists a neighborhood V such that for any \(x(0)\in V\), \(\lim _{t\rightarrow \infty } x(t) = x^*\).
 
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Weibull J (1995) Evolutionary game theory. MIT Press, CambridgeMATH
Metadata
Title
Evolutionary Stability
Author
Takako Fujiwara-Greve
Copyright Year
2015
Publisher
Springer Japan
Book
Non-Cooperative Game Theory

Print ISBN: 978-4-431-55644-2

Electronic ISBN: 978-4-431-55645-9

Copyright Year: 2015

DOI
https://doi.org/10.1007/978-4-431-55645-9_10