# EvoLudoLab: Rock-Paper-Scissors - SDE

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 - Langevin equation

The interior fixed point \(\hat x\) is a stable focus of the replicator dynamics. Demographic stochasticity arises from the finite population size of \(N=100\). In the absence of mutations, the boundaries remain absorbing and even though the interior fixed point is attracting, stochastic fluctuations nevertheless eventually drive the population to the absorbing boundaries.

The parameters are \(s = 1.4\) using numerical integration (\(dt=0.01\)) of the Langevin equation derived from a continuums approximation of the stochastic dynamics in finite populations. The starting point is close to the attractor \(\hat x=(1/2, 1/3, 1/6)\).

## 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.