# EvoLudoLab: 2x2 Game - Bistability

Along the bottom of the applet there are several buttons to control the execution and the speed of the simulations - for details see the EvoLudo GUI documentation. Of particular importance are the parameters button and the data views pop-up list along the top. The former opens a panel that allows to set and change various parameters concerning the game as well as the population structure, while the latter displays the simulation data in different ways.

Color code: Cooperators Defectors New cooperator New defector
 Payoff code: Low High

Note: The shades of grey of the payoff scale are augmented by blueish and reddish shades indicating payoffs for mutual cooperation and defection, respectively.

## Bi-stability

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

The parameters are set to [itex]R = 1, P = 0, T = 0.9[/itex] and [itex]S = -0.6[/itex] 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.
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.

Reward
reward for mutual cooperation.
Temptation
temptation to defect, i.e. payoff the defector gets when matched with a cooperator. Without loss of generality two out of the four traditional payoff values \(R, S, T\) and \(P\) can be fixed and set conveniently to \(R = 1\) and \(P = 0\). This means mutual cooperation pays \(1\) and mutual defection zero. For example for the prisoner's dilemma \(T > R > P > S\) must hold, i.e. \(T > 1\) and \(S < 0\).
Sucker
sucker's payoff which denotes the payoff the cooperator gets when matched with a defector.
Punishment
punishment for mutual defection.
Init Coop, init defect
initial fractions of cooperators and defectors. If they do not add up to 100%, the values will be scaled accordingly. Setting the fraction of cooperators to 100% and of defectors to zero, then the lattice is initialized with a symmetrical configuration suitable for observing evolutionary kaleidoscopes.