EvoLudoLab: Continuous Snowdrift Game - Repellor

Color code:	Maximum	Minimum	Mean

Investments:	Minimum Maximum

Payoffs & Densities:	Low High

Continuous Snowdrift game: Repellor

In this scenario, selection and mutation drives the population away from the singular strategy \(x^* = 0.6\), i.e. \(x^*\) is a repellor. The final state of the population now depends on the initial configuration of the population. If the initial strategy was \(x_0 < x^*\) then the investments decrease over time and defectors reign. However, if \(x_0 > x^*\) holds the population evolves towards a cooperative state with maximal investments. Also note that if \(x_0\) lies close to \(x^*\) then few mutants may diffuse to the other side of \(x^*\) and then again two branches evolve. But in contrast to evolutionary branching, this process is not generic as it requires a particular preparation of the initial configuration.

The parameters are set to \(b_2 = -0.5, b_1 = 3.4, c_2 = -1.5, c_1 = 4\) with players imitating better strategies proportional to the payoff difference and an initial traits/investment of \(0.5 \pm 0.05\) in a population of 5'000 individuals. Mutations occur with a probability of 1% and the standard deviation of the Gaussian distributed mutations is \(0.01\). Note that in this case it may or may not happen that a high investing branch evolves, depending on whether early mutants managed to have investment levels higher than \(x^* = 0.6\).

Data views

Strategies - Structure	Snapshot of the spatial arrangement of strategies.
Strategies - Structure 3D	3D view of 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 - Structure 3D	3D view of 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.

--benefitfcn <f1[,f2[...]]>

benefit function for each trait:

0: \(B(x,y)=b_0\ y\)

benefits linear in opponents investment \(y\).

1: \(B(x,y)=b_0\ y+b_1\ y^2\)

benefits quadratic in opponents investment \(y\).

2: \(B(x,y)=b_0 \sqrt{y}\)

\(\sqrt{\ }\)-saturating benefits for opponents investment \(y\)

3: \(B(x,y)=b_0 \ln(b_1\ y+1)\)

\(\ln\)-saturating benefits for opponents investment \(y\)

4: \(B(x,y)=b_0 (1-\exp(-b_1\ y))\)

\(\exp\)-saturating benefits for opponents investment \(y\)

10: \(B(x,y)=b_0 (x+y)\)

benefits linear in joint investments \(x+y\).

11: \(B(x,y)=b_0 (x+y)+b_1\ (x+y)^2\)

benefits quadratic in joint investments \(x+y\) (default).

12: \(B(x,y)=b_0 \sqrt{x+y}\)

\(\sqrt{\ }\)-saturating benefits for joint investments \(x+y\)

13: \(B(x,y)=b_0 \ln(b_1\ (x+y)+1)\)

\(\ln\)-saturating benefits for joint investments \(x+y\)

14: \(B(x,y)=b_0 (1-\exp(-b_1\ (x+y)))\)

\(\exp\)-saturating benefits for joint investments \(x+y\)

20: \(B(x,y)=b_0 x+b_1\ y+b_2\ x\ y\)

benefits linear in investments \(x\) and \(y\) as well as cross term \(x\,y\).

30: \(B(x,y)=b_0 x\)

benefits linear in own investments \(x\).

31: \(B(x,y)=b_0 x+b_1\ x^2\)

benefits quadratic in own investments \(x\).

32: \(B(x,y)=b_0 x+b_1\ x^2+b_2\ x^3\)

benefits cubic in own investments \(x\).

--benefitparams <b0>[,<b1>[...[;<b'0>[,<b'1>[...]]]]]

parameters \(b_i\) for benefit function of each trait.

--costfcn <f1[,f2[...]]>

cost function for each trait:

0: \(C(x,y)=c_0\ x\)

costs linear in own investment \(x\).

1: \(C(x,y)=c_0\ x+c_1\ x^2\)

costs quadratic in own investment \(x\) (default).

2: \(C(x,y)=c_0 \sqrt{x}\)

\(\sqrt{\ }\)-saturating costs for own investment \(x\)

3: \(C(x,y)=c_0 \ln(c_1\ x+1)\)

\(\ln\)-saturating costs for own investment \(x\)

4: \(C(x,y)=c_0 (1-\exp(-c_1\ x))\)

\(\exp\)-saturating costs for own investment \(x\)

10: \(C(x,y)=c_0 (x+y)\)

costs linear in joint investments \(x+y\).

11: \(C(x,y)=c_0 (x+y)+c_1\ (x+y)^2\)

costs quadratic in joint investments \(x+y\).

12: \(C(x,y)=c_0 (x+y)+c_1\ (x+y)^2+c_2\ (x+y)^3\)

costs cubic in joint investments \(x+y\).

13: \(C(x,y)=c_0 (x+y)+c_1\ (x+y)^2+c_2\ (x+y)^3+c_3\ (x+y)^4\)

costs quartic in joint investments \(x+y\).

20: \(C(x,y)=c_0 x+c_1\ y+c_2\ x\ y\)

costs linear in investments \(x\) and \(y\) as well as cross term \(x\,y\).

--costparams <c0>[,<c1>[...[;<c'0>[,<c'1>[...]]]]]

parameters \(c_i\) for cost function of each trait.

--init <m[,s]>

Initial configuration with mean trait m and standard deviation s (or mutant trait, see --inittype).

--inittype <t>

type of initial configuration:

uniform

uniform trait distribution.

mono

monomorphic trait distribution for mean trait (see --init <m[,s]>).

gaussian

Gaussian trait distribution with mean m and standard deviation s (see --init <m,s>).

delta

mutant with trait s in monomorphic population with trait m (see --init <m,s>).