2×2 games: well-mixed populations

In well-mixed populations the equilibrium fractions of cooperators and defectors are easily calculated using the replicator equation. If x denotes the fraction of cooperators (and 1-x the fraction of defectors) then their evolutionary fate is given by

dx/dt = x (1 - x)(P_c - P_d)

where P_c and P_d denote the average payoffs of cooperators and defectors, respectively. The replicator equation basically states that the more successful strategy, i.e. the one with the higher payoff will increase in abundance. The above equation has three equilibria: two trivial ones with x₁ = 0 and x₂ = 1 as well as a non-trivial equilibrium for P_c = P_d which leads to

x + (1 - x) S = x T
x₃ = S/(T + S - 1)

with the payoff for mutual cooperation set to R = 1 and mutual defection set to P = 0. The replicator equation allows to shift and normalize the payoffs without affecting the dynamics because the performance of cooperators and defectors only depends on the relative payoffs, i.e. on payoff differences. Note that x₃ does not necessarily lie in the interval (0,1). This leads to the four basic evolutionary scenarios discussed below.

Different scenarios

All of the following examples and suggestions are meant as inspirations for further experimenting with the virtual lab. If your browser has JavaScript enabled, the following links open a new window containing a running lab that has all necessary parameters set as appropriate.

Legend

Time evolution of cooperators and defectors in well-mixed populations where individuals interact according different 2×2 games.

Color code:	Cooperators	Defectors

Cooperation

For T < 1 and S > 0 the only stable equilibrium is x₂ = 1 (x₁ = 0 is unstable and x₃ does not exist, i.e. not in (0,1)). Thus, regardless of the initial configuration of the population cooperators will increase and eventually reach fixation. Cooperation dominates defection. In biology, this situation relates to by-product mutualism.

Defection

Conversely, for T > 1 and S < 0 the only stable equilibrium is x₁ = 0 (x₂ = 1 is unstable and x₃ does not exist), i.e. cooperators eventually go extinct irrespective of the initial configuration of the population. Defection dominates cooperation. In biology, this situation is represented by the famous Prisoner's Dilemma (for further details see the tutorial on cooperation in structured populations).

Co-existence

For T > 1 and S > 0 both trivial equilibria are unstable and the only stable equilibrium is x₃. In that case, cooperators and defectors co-exist. This situation corresponds to the Snowdrift game, Chicken or Hawk-Dove game used to model cooperation and competition in biology (for further details see the tutorial on cooperation in structured populations).

Bi-stability

Finally, for T < 1 and S < 0 both trivial equilibria are stable with an unstable equilibrium x₃ in the interior. Depending on the initial configuration of the population either cooperators or defectors will increase and reach fixation. The position of x₃ determines the basin of attraction of the evolutionary end states of all cooperation or all defection. If the initial fraction of cooperators exceeds x₃ then cooperators thrive but vanish otherwise.

VirtualLab

The small applet below illustrates the different components. Along the bottom there 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. Clicking on the examples below opens a new window with a larger applet and all parameters preset accordingly.

Color code:	Cooperators	Defectors
	New cooperator	New defector

Payoff code:	Low								High

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.

Java applet on 2×2 games in well-mixed populations. Sorry, but you are missing the fun part!

	Controls
Params	Pop up panel to set various parameters.
Views	Pop up list of different data presentations.
Reset	Reset simulation
Run	Start/resume simulation
Next	Next generation
Pause	Interrupt simulation
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.

	Data views
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.

Game parameters

The list below describes only the few parameters relevant for specifying the 2×2 game. Follow the link for a complete list and descriptions of all other parameters e.g. referring to update mechanisms of players and the population.

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.

2×2 games:

Well-mixed populations

by Christoph Hauert, Version 2.1, June 2003.