# EvoLudoLab: Rock-Paper-Scissors - ODE with Mutations

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: | Rock | Scissors | Paper |
---|---|---|---|

New rock | New scissors | New paper |

Payoff code: | Low | High |
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## Replicator-mutator dynamics

The interior fixed point \(\hat x\) is an unstable focus. Near \(\hat x\) all trajectories spiral away but the boundary of the simplex \(S_3\) is repelling because of mutations, which gives rise to a stable limit cycle. For larger mutation rates the interior fixed point becomes stable again even for \(s<1\).

The parameters are \(s = 0.2\), \(\mu=0.005\) using numerical integration of the replicator-mutator equation.

## Data views | |

Snapshot of the spatial arrangement of strategies. | |

Snapshot of the spatial arrangement of strategies in 3D. | |

Time evolution of the strategy frequencies. | |

Trajectories of strategy frequencies shown in the simplex \(S_3\). Double clicks in the interior of \(S_3\) set the initial frequencies of strategies. | |

Snapshot of the spatial distribution of payoffs. | |

Snapshot of the spatial distribution of payoffs in 3D. | |

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

Payoff distribution of each strategy in population. | |

Degree distribution in structured populations. | |

Displays messages, warnings and errors reported by the simulation engine plus information on the applet/application. |

## Game parameters

The list below describes only the parameters related to the rock-scissors-paper game and the population dynamics. Follow the link for a complete list and descriptions of all other parameters such as spatial arrangements or update rules on the player and population level.

- Payoffs
- general \(3\times 3\) payoff matrix for the three strategic types \(R, S, P\). The rock-paper-scissors game requires cyclic dominance between the three types.
- Init Rock, Scissors, Paper
- initial frequencies of rock, paper, and scissors types. If they do not add up to 100%, the values will be scaled accordingly.

*Tip:* Use the context menu to switch between forward and backward integration in time.