# 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.

## EvoLudoLab

Along the bottom of the applet there are several buttons to control the execution and the speed of the simulations - for details see the *EvoLudo* GUI documentation. Of particular importance are the parameters button and the data views pop-up list along the top. The former opens a panel that allows to set and change various parameters concerning the game as well as the population structure, while the latter displays the simulation data in different ways.

Color code: | Cooperators | Defectors |
---|---|---|

New cooperator | New defector |

Payoff code: | Low | High |
---|

*Note:* The shades of grey of the payoff scale are augmented by blueish and reddish shades indicating payoffs for mutual cooperation and defection, respectively.

## 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>R, S, T</math> and <math>P</math> give rise to different types of kaleidoscopes. Here we set <math>R = 1, P = 0, S = 0.0</math> and<math>T = 1.4</math> and individuals interact with their four nearest neighbors on a square <math>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 | |

Snapshot of the spatial arrangement of strategies. | |

Time evolution of the strategy frequencies. | |

Snapshot of the spatial distribution of payoffs. | |

Time evolution of average population payoff bounded by the minimum and maximum individual payoff. | |

Snapshot of payoff distribution in population. | |

Degree distribution in structured populations. | |

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.

- Reward
- reward for mutual cooperation.
- Temptation
- temptation to defect, i.e. payoff the defector gets when matched with a cooperator. Without loss of generality two out of the four traditional payoff values \(R, S, T\) and \(P\) can be fixed and set conveniently to \(R = 1\) and \(P = 0\). This means mutual cooperation pays \(1\) and mutual defection zero. For example for the prisoner's dilemma \(T > R > P > S\) must hold, i.e. \(T > 1\) and \(S < 0\).
- Sucker
- sucker's payoff which denotes the payoff the cooperator gets when matched with a defector.
- Punishment
- punishment for mutual defection.
- Init Coop, init defect
- initial fractions of cooperators and defectors. If they do not add up to 100%, the values will be scaled accordingly. Setting the fraction of cooperators to 100% and of defectors to zero, then the lattice is initialized with a symmetrical configuration suitable for observing evolutionary kaleidoscopes.