EvoLudoLab: Continuous Snowdrift Game - Branching (exp)

Color code:	Maximum	Minimum	Mean

Investments:	Minimum Maximum

Payoffs & Densities:	Low High

Continuous Snowdrift game: Saturating investments

In all examples so far, if higher investments were advantageous (at least in one branch) then the investments would continue to increase until the upper boundary of the trait range is reached. This must not be the case. In this last example we choose \(B(x+y) = b_1[1-\exp(-b_2 x)]\) and \(C(x) = c_1 \ln(c_2 x+1)\). For the parameters below, we again observe a branching point near \(x^*_1\approx 0.7\) accompanied by a repellor near \(x^*_2\approx 0.2\). Starting with a population \(x_0 > 0.2\), selection and mutations drive the population towards the branching point but now the emerging upper branch grows only to trait values of around \(2.2\). Obviously, when starting with \(x_0 < 0.2\) branching can not occur and investment levels stay close to zero. Again note that the dimorphic population no longer has a repellor near \(x^*_2\approx 0.2\) and therefore the lower branch evolves straight to minimal investments.

The parameters are set to \(b_1 = 5, b_2 = 1, c_1 = 1, c_2 = 10\) 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 \(0.02\).

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>).