📝 Original Info
- Title: Prisoners Dilemma in One-Dimensional Cellular Automata: Visualization of Evolutionary Patterns
- ArXiv ID: 0708.3520
- Date: 2007-09-18
- Authors: Researchers from original ArXiv paper
📝 Abstract
The spatial Prisoner's Dilemma is a prototype model to show the emergence of cooperation in very competitive environments. It considers players, at site of lattices, that can either cooperate or defect when playing the Prisoner's Dilemma with other z players. This model presents a rich phase diagram. Here we consider players in cells of one-dimensional cellular automata. Each player interacts with other z players. This geometry allows us to vary, in a simple manner, the number of neighbors ranging from one up to the lattice size, including self-interaction. This approach has multiple advantages. It is simple to implement numerically and we are able to retrieve all the previous results found in the previously considered lattices, with a faster convergence to stationary values. More remarkable, it permits us to keep track of the spatio-temporal evolution of each player of the automaton. Giving rise to interesting patterns. These patterns allow the interpretation of cooperation/defection clusters as particles, which can be absorbed and collided among themselves. The presented approach represents a new paradigm to study the emergence and maintenance of cooperation in the spatial Prisoner's Dilemma.
💡 Deep Analysis
Deep Dive into Prisoners Dilemma in One-Dimensional Cellular Automata: Visualization of Evolutionary Patterns.
The spatial Prisoner’s Dilemma is a prototype model to show the emergence of cooperation in very competitive environments. It considers players, at site of lattices, that can either cooperate or defect when playing the Prisoner’s Dilemma with other z players. This model presents a rich phase diagram. Here we consider players in cells of one-dimensional cellular automata. Each player interacts with other z players. This geometry allows us to vary, in a simple manner, the number of neighbors ranging from one up to the lattice size, including self-interaction. This approach has multiple advantages. It is simple to implement numerically and we are able to retrieve all the previous results found in the previously considered lattices, with a faster convergence to stationary values. More remarkable, it permits us to keep track of the spatio-temporal evolution of each player of the automaton. Giving rise to interesting patterns. These patterns allow the interpretation of cooperation/defection
📄 Full Content
arXiv:0708.3520v2 [physics.comp-ph] 18 Sep 2007
October 24, 2018
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International Journal of Modern Physics C
c⃝World Scientific Publishing Company
PRISONER’S DILEMMA IN ONE-DIMENSIONAL CELLULAR
AUTOMATA: VISUALIZATION OF EVOLUTIONARY PATTERNS
MARCELO ALVES PEREIRA†, ALEXANDRE SOUTO MARTINEZ‡
Departamento de F´ısica e Matem´atica
Faculdade de Filosofia Ciˆencias e Letras de Ribeir˜ao Preto
Universidade de S˜ao Paulo
Av. Bandeirantes, 3900, 14040-901
Ribeir˜ao Preto, SP, Brazil
†marceloapereira@usp.br
‡asmartinez@usp.br
AQUINO LAURI ESP´INDOLA
Departamento de F´ısica e Matem´atica
Faculdade de Filosofia Ciˆencias e Letras de Ribeir˜ao Preto
Departamento de Medicina Social
Faculdade de Medicina de Ribeir˜ao Preto
Universidade de S˜ao Paulo
Av. Bandeirantes, 3900, 14049-900
Ribeir˜ao Preto, SP, Brazil
aquinoespindola@usp.br
Received Day August 2007
Revised Day Month 2007
The spatial Prisoner’s Dilemma is a prototype model to show the emergence of cooper-
ation in very competitive environments. It considers players, at site of lattices, that can
either cooperate or defect when playing the Prisoner’s Dilemma with other z players. This
model presents a rich phase diagram. Here we consider players in cells of one-dimensional
cellular automata. Each player interacts with other z players. This geometry allows us
to vary, in a simple manner, the number of neighbors ranging from one up to the lat-
tice size, including self-interaction. This approach has multiple advantages. It is simple
to implement numerically and we are able to retrieve all the previous results found in
the previously considered lattices, with a faster convergence to stationary values. More
remarkable, it permits us to keep track of the spatio-temporal evolution of each player
of the automaton. Giving rise to interesting patterns. These patterns allow the interpre-
tation of cooperation/defection clusters as particles, which can be absorbed and collided
among themselves. The presented approach represents a new paradigm to study the
emergence and maintenance of cooperation in the spatial Prisoner’s Dilemma.
Keywords: Prisoner’s Dilemma; Game Theory; Evolutionary dynamics; Computational
modeling; Sociophysics.
PACS Nos.: 02.50.Le, 07.05.Tp
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1. Introduction
Games amuse mankind since emerging of the first civilizations. Besides amusement,
they are a branch of Mathematics known as Game Theory, consolidated by John
von Neumann.1 The main purpose of such theory is the determination of strategies
to get the maximum returns in situations where multiple rational players have the
same aim. The most prominent game, due to the cooperation emergence among
competitive rational players, is the Prisoner’s Dilemma (PD), originally framed by
Merrill Flood and Melvin Dresher in 1950.2 The game formalization with prison
sentence payoffs and the name “Prisoner’s Dilemma” is due to Albert W. Tucker,
when he wanted to make the ideas of Flood and Dresher more accessible to an
audience at Stanford University in the same year.3
There are several problems that can be modeled using the PD game. In real
life, there are many situations of conflict, i.e., one person trying to reach his/her
personal aim is incompatible to collective aims. Because PD is a conflict situation
between two players, it is possible to make an analogy between the game and real
life, e.g., in Politics (Sociophysics)4, Economics (Econophysics)5,6, and Biology.7
In the classical version of PD, two players can either cooperate or defect. Under
mutual cooperation players get a payoffR (reward), otherwise if they are defectors,
the payoffis P (punishment). When a player cooperates and the other defects, they
get S (sucker) and T (temptation), respectively. These payoffvalues must satisfy
the inequalities: T > R > P > S and T + S < 2R. The former relationship assures
the existence of the dilemma, whereas the latter prevents that a couple of players,
that alternate between cooperation and defection, get the same payoffor the payoff
be higher than a cooperative couple payoff.
In a single round game, the best choice for a rational player is to defect. Defec-
tion assures the largest payoffindependently of the other player’s decision (Nash
equilibrium). In successive rounds, one has the Iterated Prisoner Dilemma (IPD).
In this case, the best choice is not necessarily the defection, it is convenient only
to retaliate a previous non-cooperative play. The IPD game became popular with
the computer tournament proposed by Axelrod8,9. This tournament intended to
compare the strategies. It turned out, that a simple strategy, with only one time
step memory, called tit-for-tat (TFT), was by far the most stable one.
Nowak and May10 have shown the emergence of cooperation between players
with strategies with no memory, in the presence of spatial structure. This version of
the game is known as Spatial Prisoner’s Dilemma (SPD). This is a simple, purely
dete
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Reference
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