EvoLudoLab: 2x2 Game - Bistability: Difference between revisions
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{{EvoLudoLab:2x2| | {{EvoLudoLab:2x2| | ||
options="--run --delay 200 --view | options="--game 2x2 --run --delay 200 --view Strategies_-_Mean --reportfreq 0.5 --popsize 10000 --popupdate async --playerupdate imitate --geometry M --intertype a --numinter 1 --references a --init 14.6,85.4 --inittype frequencies --mutation 0.0 --basefit 1.0 --selection 1.0 --reward 1.0 --punishment 0.0 --temptation 0.9 --sucker -0.6"| | ||
title=Bi-stability| | title=Bi-stability| | ||
doc=Depending on the initial configuration, i.e. the initial fraction of type <math>A</math> players, the population either evolves towards a homogenous state with all <math>A</math> or all <math>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. | doc=Depending on the initial configuration, i.e. the initial fraction of type <math>A</math> players, the population either evolves towards a homogenous state with all <math>A</math> or all <math>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. |
Latest revision as of 15:53, 13 October 2023
Color code: | Cooperators | Defectors |
---|---|---|
New cooperator | New defector |
Payoffs: | Low High
|
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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
Snapshot of the spatial arrangement of strategies. | |
Time evolution of the strategy frequencies. | |
Snapshot of the spatial distribution of payoffs. | |
Time evolution of average population payoff bounded by the minimum and maximum individual payoff. | |
Snapshot of payoff distribution in population. | |
Degree distribution in structured populations. | |
Statistics of fixation probabilities. | |
Statistics of fixation and absorption times. | |
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.