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2017 | OriginalPaper | Chapter

Replicator Equations as Limits of Evolutionary Games on Complete Graphs

Author : Petr Stehlík

Published in: Advances in Difference Equations and Discrete Dynamical Systems

Publisher: Springer Singapore

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Abstract

In this paper we discuss connections between the evolutionary games on graphs and replicator equations. On the traditional examples of social dilemma games we introduce the basic ideas of replicator dynamics and the mathematical concepts behind evolutionary games on graphs. We show that the stability regions of evolutionary games on complete graphs with the sequential and synchronous updating with deterministic imitation dynamics converge to the stability regions of replicator equations. Finally, we show that by a finer choice of a time scale and a stochastic imitation dynamic update rule not only the stability regions but also the trajectories of evolutionary games on graphs converge to those of replicator equations.

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previous chapter A Dynamical Trichotomy for Structured Populations Experiencing Positive Density-Dependence in Stochastic Environments
next chapter Connection Between Continuous and Discrete Delay and Halanay type Inequalities
Footnotes
1
In some cases one should rather speak about agent-based models, since the graph structure varies and agents are allowed to interact in a very complex fashion [24, 29], see also [10].
 
2
Another motivation for our study had been the small size of cooperation macroeconomic networks, see e.g. [8, 14, 15]. Note that this is in contrast to the focus on large, often scale-free, networks in the physical and biological applications [27].
 
3
We denote by \(\mathbb {T}^2_\ge \) the set
$$\begin{aligned} \mathbb {T}^2_\ge := \{(t,s)\in \mathbb {T}^2 : t \ge s\}. \end{aligned}$$
 
4
We say that a state \(x\in \{0,1\}^V\) is a coexistence equilibrium (coexistence fixed point) of the evolutionary game on a graph \((G, p, u, \mathscr {T}, \varphi )\) if (a) it is a fixed point, i.e., \(\varphi (t+1,t,x)=x\) for all \(t\in \mathbb {T}\), and (b) it is a coexistence state, i.e., \(0<\sum _{i\in V} x_i < |V|\), see [6] for more details.
 
5
The reason why Theorem 6 is not stated in this way directly is the fact that it could be applied also in cases when the problem (23) does not have a unique solution.
 
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Metadata
Title
Replicator Equations as Limits of Evolutionary Games on Complete Graphs
Author
Petr Stehlík
Copyright Year
2017
Publisher
Springer Singapore
Book
Advances in Difference Equations and Discrete Dynamical Systems

Print ISBN: 978-981-10-6408-1

Electronic ISBN: 978-981-10-6409-8

Copyright Year: 2017

DOI
https://doi.org/10.1007/978-981-10-6409-8_4

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