by Christoph Hauert, Version 1.0, April 2004.
In well-mixed populations the equilibrium fractions of cooperators and defectors are easily calculated using the replicator equation:
where xi denotes the frequency, Pi the payoff (fitness) of strategy i and P the average population payoff. The replicator equation simply states that the success of a strategy depends on its relative performance in the population. Therefore, strategies with a higher than average payoff will spread. In a population with a fraction of x cooperators and y = 1 - x defectors the replicator equation reduces to
where Pc and Pd denote the average payoffs of cooperators and defectors, respectively. The above equation has three equilibria: two trivial ones with x1 = 0 and x2 = 1 as well as a non-trivial one for Pc = Pd which leads to
x3 = (S - P) / (T + S - R - P).
The stability of all three equilibrium points is easily obtained by checking dx/dt near the equilibium points: x1 = 0 is stable if P > S, x2 = 1 stable if R > T and x3 is stable if S > P and T > R, i.e. whenever both x1 and x3 are unstable.
Instead of referring to the fraction of cooperators, x may equally refer to the propensity of cooperation in a continuous strategy space. This equivalent interpretation leaves the above calculations and conclusions unaffected but it does affect the individual based simulations. In particular, we need to introduce mutations. The replicator equation then determines the fate of a rare mutant y when competing against the resident x. If the mutation is favorable the mutant will spread and usually displace the resident.
Along the bottom of the VirtualLab are several buttons to control the execution and the speed of the simulations. Of particular importance are the Param button and the data views pop-up list on 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.
|New cooperator||New defector|
Note: The yellow and green colors are very useful to get an intuition of the activitiy in the system. The shades of grey of the payoff scale are augmented by blueish and reddish shades, which indicate the payoffs for mutual cooperation and defection, respectively.
|Params||Pop up panel to set various parameters.|
|Views||Pop up list of different data presentations.|
|Slider||Idle time between updates. On the right your CPU clock determines the update speed while on the left updates are made roughly once per second.|
|Mouse||Mouse clicks on the graphics panels generally start, resume or stop the simulations.|
|Structure - Strategy||Snapshot of the spatial arrangement of strategies. Mouse clicks cyclically change the strategy of the respective site for the preparation of custom initial configurations.|
|Mean frequency||Time evolution of the strategy frequencies.|
|Structure - Fitness||Snapshot of the spatial distribution of payoffs.|
|Mean Fitness||Time evolution of the mean payoff of each strategy together with the average population payoff.|
|Histogram - Fitness||Histogram of payoffs for each strategy.|
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 all other parameters such as spatial arrangements or update rules on the player and population level.
- reward for mutual cooperation.
- temptation to defect, i.e. payoff the defector gets when matched with a cooperator.
- sucker's payoff which denotes the payoff the cooperator gets when matched with a defector.
- 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.