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 [math]\displaystyle{ A }[/math] players, the population either evolves towards a homogenous state with all [math]\displaystyle{ A }[/math] or all [math]\displaystyle{ B }[/math]. 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 [math]\displaystyle{ A }[/math] players exceed the threshold [math]\displaystyle{ x_3 = (P-S)/(R-S-T+P) }[/math] they thrive and type [math]\displaystyle{ B }[/math] players. In the above simulations a finite population of size 10'000 starts close to the threshold [math]\displaystyle{ x_3 }[/math]. 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 [math]\displaystyle{ x_3 }[/math] for quite a while before converging to the homogenous state of either all cooperators or all defectors.

The parameters are set to [math]\displaystyle{ R = 1, P = 0, T = 0.9 }[/math] and [math]\displaystyle{ S = -0.6 }[/math] 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.
Console log	Message log from engine.

Game 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.

--reward <a11>

the reward for mutual cooperation. The payoff of type \(A\) against its own type (see --paymatrix).

--temptation <a10>

the temptation to defect. The payoff of type \(B\) against type \(A\) (see --paymatrix).

--sucker <a01>

the sucker's payoff of an exploited cooperator. The payoff of type \(A\) against type \(B\) (see --paymatrix).

--punishment <a00>

the punishment for mutual defection. The payoff of type \(B\) against its own type (see --paymatrix).

--init <a,b>

initial frequencies of type \(A\) and \(B\), respectively. Frequencies that do not add up to 100% are scaled accordingly.

--inittype <type>

type of initial configuration:

frequency

random distribution with given frequency

uniform

uniform random distribution

monomorphic

monomorphic initialization

mutant

single mutant in homogeneous population of another type. Mutant and resident types are determined by the types with the lowest and highest frequency, respectively (see option --init).

stripes

stripes of traits

kaleidoscopes

(optional) configurations that produce evolutionary kaleidoscopes for deterministic updates (players and population). Not available for all types of games.