EvoLudoLab: 2x2 Game - Bistability

Color code:	Cooperators	Defectors
	New cooperator	New defector

Payoffs:	Low High

Note: The gradient of the payoff scale is augmented by pale shades of the strategy colours to mark payoffs that are achieved in homogeneous populations of the corresponding type.

Bi-stability

Depending on the initial configuration, i.e. the initial fraction of type $A$ players, the population either evolves towards a homogenous state with all $A$ or all $B$ . Both states are stable and hence the name bi-stability. This is an instance of a coordination game, such as the Staghunt Game.

Whenever type $A$ players exceed the threshold $x_{3}=(P-S)/(R-S-T+P)$ they thrive and type $B$ players. In the above simulations a finite population of size 10'000 starts close to the threshold $x_{3}$ . The small amount of noise introduced by considering finite populations triggers whether the population evolves towards cooperation or defection. This can be verified by restarting the simulations a few times. Note that the population can linger around the unstable equilibrium point $x_{3}$ for quite a while before converging to the homogenous state of either all cooperators or all defectors.

The parameters are set to $R=1,P=0,T=0.9$ and $S=-0.6$ and players imitating better strategies proportional to the payoff difference. According to the above formula, the initial fraction of cooperators was set to 85.4%.

Data views

Strategies - Structure	Snapshot of the spatial arrangement of strategies.
Strategies - Mean	Time evolution of the strategy frequencies.
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.
Statistics - Fixation probability	Statistics of fixation probabilities.
Statistics - Fixation time	Statistics of fixation and absorption times.
Stationary distribution	Statistics of the stationary distribution of the numbers of each strategic type. Note, only available for non-zero mutation rates.
Console log	Message log from engine.

Module parameters

The list below describes only the few parameters related to the Prisoner's Dilemma, Snowdrift and Hawk-Dove games. Follow the link for a complete list and detailed descriptions of the user interface and further parameters such as spatial arrangements or update rules on the player and population level.

--paymatrix <a00,a01;a10,a11>

2x2 payoff matrix. Type $A$ has index 0 and type $B$ index 1.

--inittype <type>

type of initial configuration:

frequency <f0>,<f1>...: random distribution with given trait frequencies, f0, f1,.... Note, only available for frequency based modules and models.
density <d0>,<d1>...: initial trait densities <d1,...,dn>. Note, only available for density based modules and models.
uniform: uniform random distribution, equal frequencies of all traits.
monomorphic <t>[,<v>]: monomorphic initialization with trait t. Note, for modules with variable population densities, the optional parameter v indicates the initial frequency of vacant sites. If omitted the monomorphic trait is initialized at its (estimated) carrying capacity.
mutant <m>,<r>[,<v>]: single mutant with trait m in homogeneous resident population of type r. The mutant is placed in a location selected uniformly at random (mutants arising through cosmic rays). Note, for modules with variable population densities, the optional parameter v indicates the initial frequency of vacant sites. If omitted the resident trait is initialized at its (estimated) carrying capacity.
temperature <m>,<r>[,<v>]: single mutant with trait m in homogeneous resident population of type r. The mutant is placed in a location selected proportional to the in-degree of nodes (temperature initialization, mutants arising through errors in reproduction). Note, for modules with variable population densities, the optional parameter v indicates the initial frequency of vacant sites. If omitted the resident trait is initialized at its (estimated) carrying capacity.
stripes: stripes of traits. Note, only available for 2D lattices.
kaleidoscopes: configurations that produce evolutionary kaleidoscopes for deterministic updates (players and population). Note, only available for some modules.

Note, for modules that admit multiple species, the initialization types for each species can be specified as an array separated by ;. With more species than initialization types, they are assigned in a cyclical manner.