# EvoLudoLab: Rock-Paper-Scissors - SDE 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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## Stochastic dynamics with mutations - Langevin equation

The interior fixed point \(\hat x\) of the replicator dynamics is an unstable focus. Even without stochasticity all trajectories spiral away from \(\hat x\) toward the boundary of the simplex \(S_3\). However, due to mutations, the boundary is repelling, which results in a stochastic analog of a stable limit cycle. For larger mutation rates the interior fixed point becomes stable again even for \(s<1\).

The parameters are \(s = 0.2\), \(N=1000\), \(\mu=0.005\) using numerical integration of the stochastic differential equation based on Euler's method with \(dt=0.01\).

## 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.