# 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 |
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*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 <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.

Whenever type <math>A</math> players exceed the threshold <math>x_3 = (P-S)/(R-S-T+P)</math> they thrive and type <math>B</math> players. In the above simulations a finite population of size 10'000 starts close to the threshold <math>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>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>R = 1, P = 0, T = 0.9</math> and <math>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. | |

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.