Evolutionary Kaleidoscopes in the Prisoner's Dilemma

For deterministic update rules (synchronous lattice update, best player in neighborhood reproduces) and symmetrical initial configurations this can lead to fascinating spatio-temporal patterns. Such evolutionary kaleidoscopes are certainly only of limited scientific interest but they do have quite some entertainment value.

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  [math]\displaystyle{ t=0 }[/math]

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  [math]\displaystyle{ t=50 }[/math]

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  [math]\displaystyle{ t=100 }[/math]

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  [math]\displaystyle{ t=150 }[/math]

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  [math]\displaystyle{ t=200 }[/math]

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  [math]\displaystyle{ t=250 }[/math]

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  [math]\displaystyle{ t=300 }[/math]

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  [math]\displaystyle{ t=350 }[/math]

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  [math]\displaystyle{ t=400 }[/math]

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  [math]\displaystyle{ t=450 }[/math]

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  [math]\displaystyle{ t=500 }[/math]

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  [math]\displaystyle{ t=550 }[/math]

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  [math]\displaystyle{ t=600 }[/math]

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  [math]\displaystyle{ t=650 }[/math]

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  [math]\displaystyle{ t=700 }[/math]

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  [math]\displaystyle{ t=750 }[/math]

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  [math]\displaystyle{ t=800 }[/math]

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  [math]\displaystyle{ t=850 }[/math]

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  [math]\displaystyle{ t=900 }[/math]

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  [math]\displaystyle{ t=950 }[/math]

EvoLudoLab

Color code:	Cooperators	Defectors
	New cooperator	New defector

Payoffs:	Low High

Note: The gradient of the payoff scale is augmented by pale shades of the strategy colours to mark payoffs that are achieved in homogeneous populations of the corresponding type.

Spatial Prisoner's Dilemma: evolutionary kaleidoscopes

For symmetrical initial configurations and determinsitic update rules fascinating spatio-temporal patterns resembling evolutionary kaleidoscopes can emerge in spatially structured populations where individuals are arranged on a lattice and interact only within a limited local neighborhood. Different values for the paramters [math]\displaystyle{ R, S, T }[/math] and [math]\displaystyle{ P }[/math] give rise to different types of kaleidoscopes. Here we set [math]\displaystyle{ R = 1, P = 0, S = 0.0 }[/math] and[math]\displaystyle{ T = 1.4 }[/math] and individuals interact with their four nearest neighbors on a square [math]\displaystyle{ 101\times 101 }[/math] lattice with periodic boundary conditions.

Patient people will find that the evolving patterns come to a sudden an unexpected end after more than 200'000 generations (MC steps) when the system relaxes into a cyclic state with short period. The time until an absorbing or cyclic state with short period is reached sensitively depends on the lattice size. Interestingly there is no simple relation between system size and relaxation time.

Data views

Strategies - Structure	Snapshot of the spatial arrangement of strategies.
Strategies - Mean	Time evolution of the strategy frequencies.
Fitness - Structure	Snapshot of the spatial distribution of payoffs.
Fitness - Mean	Time evolution of average population payoff bounded by the minimum and maximum individual payoff.
Fitness - Histogram	Snapshot of payoff distribution in population.
Structure - Degree	Degree distribution in structured populations.
Statistics - Fixation probability	Statistics of fixation probabilities.
Statistics - Fixation time	Statistics of fixation and absorption times.
Console log	Message log from engine.

Game parameters

The list below describes only the few parameters related to the Prisoner's Dilemma, Snowdrift and Hawk-Dove games. Follow the link for a complete list and detailed descriptions of the user interface and further parameters such as spatial arrangements or update rules on the player and population level.

--paymatrix <a00,a01;a10,a11>

2x2 payoff matrix. Type \(A\) has index 0 and type \(B\) index 1.

--reward <a11>

the reward for mutual cooperation. The payoff of type \(A\) against its own type (see --paymatrix).

--temptation <a10>

the temptation to defect. The payoff of type \(B\) against type \(A\) (see --paymatrix).

--sucker <a01>

the sucker's payoff of an exploited cooperator. The payoff of type \(A\) against type \(B\) (see --paymatrix).

--punishment <a00>

the punishment for mutual defection. The payoff of type \(B\) against its own type (see --paymatrix).

--init <a,b>

initial frequencies of type \(A\) and \(B\), respectively. Frequencies that do not add up to 100% are scaled accordingly.

--inittype <type>

type of initial configuration:

frequency

random distribution with given frequency

uniform

uniform random distribution

monomorphic

monomorphic initialization

mutant

single mutant in homogeneous population of another type. Mutant and resident types are determined by the types with the lowest and highest frequency, respectively (see option --init).

stripes

stripes of traits

kaleidoscopes

(optional) configurations that produce evolutionary kaleidoscopes for deterministic updates (players and population). Not available for all types of games.