# 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
 Strategy code: Defect Cooperate
 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.

## Continuous Snowdrift game: Attractor & Repellor

For more complicated payoff functions several singular strategies $$x^*$$ may be found. In this example we use $$B(x) = b_0 \sqrt{x+y}$$ and $$C(x) = c_0 \ln(c_1 x+1)$$. 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 $$b_0 = 1, c_0 = 1, c_1 = 0.6$$ with players imitating better strategies proportional to the payoff difference and an initial traits/investment of $$3.2 \pm 0.02$$ in a population of $$2'000$$ individuals. Mutations occur with a probability of 1% and the standard deviation of the Gaussian distributed mutations is $$0.01$$.

### Data views

Strategies - Structure

Snapshot of the spatial arrangement of strategies.

Strategies - Mean

Time evolution of the strategy frequencies.

Strategies - Histogram

Snapshot of strategy distribution in population

Strategies - Distribution

Time evolution of the strategy distribution

Fitness - Structure

Snapshot of the spatial distribution of payoffs.

Fitness - Mean

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

Fitness - Histogram

Snapshot of payoff distribution in population.

Structure - Degree

Degree distribution in structured populations.

Console log

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.