# Evolutionary Kaleidoscopes in the Snowdrift Game

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 Snowdrift or Hawk-Dove game: evolutionary kaleidoscopes

In spatially structured lattice populations with symmetrical initial configurations and deterministic updates fascinating spatio-temporal patterns emerge for suitable parameters <math>R, S, T</math> and <math>P</math> which resemble changing persian carpets or evolutionary kaleidoscopes. The initial symmetry is preserved by the determinsitic updates, i.e. synchronized lattice updates and only the individual with the highest payoff reproduces. Variation of the parameters leads to different types of kaleidoscopes. Here the cost-to-benefit ratio is set to <math>r = 0.6</math>, i.e. <math>R = 1, P = 0, S = 0.4</math> and <math>T = 1.6</math> and individuals interact with their four nearest neighbors on a square <math>101\times 101</math>lattice with periodic boundary conditions.

## 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.