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Human–Environment–Social System and Evolutionary Game Theory Read chapter Read first chapter

2015 | OriginalPaper | Chapter

2. Fundamental Theory for Evolutionary Games

Author : Jun Tanimoto

Published in: Fundamentals of Evolutionary Game Theory and its Applications

Publisher: Springer Japan

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Abstract

In this chapter, we take a look at the appropriate treatment of linear dynamical systems, which you may be familiar with if you have taken some standard engineering undergraduate classes. The discussion is then extended to non-linear systems and their general dynamic properties. In this discussion, we introduce the 2-player and 2-strategy (2 × 2) game, which is the most important archetype among evolutionary games. Multi-player and 2-strategy games are also introduced. In the latter parts of this chapter, we define the dilemma strength, which is useful for the universal comparison of the various reciprocity mechanisms supported by different models.

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Footnotes
1
Concerning detail of this discussion, you should consult with Tanimoto (2014).
 
2
This argument derives from the fact that the time evolution of the error between the numerical solution and the explicit solution obeys the original equation.
 
3
The “linear” quality of a system is truly beneficial in engineering. No sharp fluctuations develop over time; therefore, future behavior is easily extrapolated from currently available information.
 
4
To precisely know about GID & RAD and D g and D r , you should consult with Tanimoto and Sagara (2007a).
 
5
This is referred to as strategy adaptation.
 
6
This is referred to as evolution.
 
7
The proportion of cooperative members at the start of a series of games is 0.5.
 
8
To precisely know about R-reciprocity and ST-Reciprocity, you should consult with Tanimoto and Sagara (2007b).
 
9
Precisely speaking, we should call it a 1st order free-rider, because in the models considering the punishment mechanism, which many previous studies have investigated, there are 1st order free-riders, meaning simple defectors, as well as 2nd order free-riders, implying cooperators who are not punishing other defectors.
 
10
But there have been several indications that “tragedy of commons”, or say a multi-players Chicken game is insufficient to model with general environmental problems on the ground that the model does not consider any dynamics of the environment. Even though a grass field or fishing field temporarily becomes exhaustive, it can gradually recover according to dynamics the environment has. More intimately, you can consult with Tanimoto (2005).
 
11
Strictly speaking PD satisfying D g  = D r .
 
12
This situation accords with common sense. If a game is played against the same partner each time rather than against an unknown one, both individuals should accept the cooperation option to avoid strategies leading merely to short term profit. If both individuals take the defection option P, neither will benefit long-term. Our daily behavior follows the former pattern.
 
13
Many of these dynamics can be verified by simulation. Games are repeated between multiple agents in a simulated society; this approach is known as multi-agent simulation.
 
14
There are so many literatures concerning this point. Because of space limitation, we cite only five of those; Wynne-Edwards (1962), Williams (1996), Wilson (1975), Maynard Smith (1976), Slatkin and Wade (1978).
 
15
Because of space limitation, we can cite here only five of those; Hassell et al. (1994), Ebel and Bornholdts (2002), Santos and Pacheco (2005), Santos et al. (2006), Yamauchi et al. (2010).
 
16
Also, multi-player Chicken game is sometimes used as a template for the discussion on environmental problems like Hardin’s tragedy of commons (Hardin 1968) as mentioned before.
 
17
It is worthwhile to note that an individual agent in the assumed model is exposed to direct reciprocity situation because the number of games played with a same opponent is presumed infinite. But, viewing each of 32 strategies (not viewing each of agents), we can say this society is well-mixed because we applied replicator dynamics to solve the equilibrium distribution of strategies.
 
18
That is just part of the Chicken area in Fig. 2.34 (a), because, in Fig. 2.10, we assumed R = 1 & P = 1, and \( -1\le {D}_g\le 1 \) & \( -1\le {D}_r\le 1 \).
 
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Metadata
Title
Fundamental Theory for Evolutionary Games
Author
Jun Tanimoto
Copyright Year
2015
Publisher
Springer Japan
Book
Fundamentals of Evolutionary Game Theory and its Applications

Print ISBN: 978-4-431-54961-1

Electronic ISBN: 978-4-431-54962-8

Copyright Year: 2015

DOI
https://doi.org/10.1007/978-4-431-54962-8_2