# EvoLudoLab: Continuous Snowdrift Game - Branching (sqrt)

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 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: | Maximum | Minimum | Mean |
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Strategy code: | Defect | Cooperate |
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Payoff code: | Low | High |
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*Note:* The shades of grey of the payoff scale are augmented by blueish and reddish shades indicating payoffs for mutual cooperation and defection, respectively.

## Continuous Snowdrift game: Attractor & Repellor

For more complicated payoff functions several singular strategies \(x^*\) may be found. In this example we use \(B(x) = b_1 \sqrt{x+y}\) and \(C(x) = c_1 \ln(c_2 x+1)\). For the parameters indicated below this results in a repellor near \(x_1^*\approx 3.9\) together with a branching point near \(x^*_2 \approx 0.7\). Starting with \(x_0 < 3.9\) drives the population towards lower investments until the branching point is reached. At \(x^*_2\) two branches emerge and diverge until the upper branch reaches the boundary of the trait range. Note that for the dimorphic population the repellor near \(x_1^*\approx 3.9\) no longer exists. The trait range in the above simulation is \([0,5]\).

The parameters are set to \(b_1 = 1, c_1 = 1, c_2 = 0.6\) with players imitating better strategies proportional to the payoff difference and an initial traits/investment of \(2.8 \pm 0.02\) in a population of \(5'000\) individuals. Mutations occur with a probability of 1% and the standard deviation of the Gaussian distributed mutations is \(0.01\).

## Data views | |

Snapshot of the spatial arrangement of strategies. | |

Time evolution of the strategy frequencies. | |

Snapshot of strategy distribution in population | |

Time evolution of the strategy distribution | |

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 continuous snowdrift game. Follow the link for a complete list and descriptions of all other parameters e.g. referring to update mechanisms of players and the population.

- Benefit/Cost Functions
- A variety of different combinations of cost and benefit functions can be selected.
- Benefit \(b_0,\ b_1\)
- Two parameters for the benefit function. Note that not all functions require both.
- Cost \(c_0,\ c_1\)
- Two parameters for the cost function. Note that not all functions require both.
- Mean invest
- Mean trait value of initial population.
- Sdev invest
- Standard deviation of initial population. If set to negative values, the population will be initialized with uniform distributed traits.