Structured populations:
Pure strategies
by Christoph Hauert, Version 1.0, April 2004.
 Location:
 VirtualLabs
 » Cooperation
 » Structured populations  pure strategies
The equilibrium frequencies of cooperators and defectors in structured populations can differ considerably from the results for wellmixed populations with random encounters. To model spatial structures the individuals are arranged on a lattice and interact only within a limited local neighborhood. Various choices for the lattice geometry are possible: square lattice with neighborhood size N = 4 or N = 8, triangular with N = 3 or hexagonal with N = 6. Whenever a site x is updated, the present occupant and all its neighbors compete for recolonizing the focal site x with their offspring. We assume that a neighbor y succeeds in reproducing with a probability proportional to the difference in payoffs P_{x}  P_{y} provided it is positive and with probability zero otherwise. In wellmixed populations, this assumption leads to the replicator equation. A spatial analogue of the replicator dynamics is then to set the competitive success of neighbor y proportional to
where a is a normalization constant such that 0 < w_{y} < 1 holds (a = T  S in the Prisoner's Dilemma and a = T  P in the Snowdrift or HawkDove game). With probability p_{x} = (1  w_{y1})(1  w_{y2}) … (1  w_{yN}) all neighbors y_{i} fail to reproduce and the focal individual succeeds in placing its own offspring in site x. Otherwise, with probability 1  p_{x}, one neighbor takes over the focal site. The relative probability for success of a neighbor y is w_{y}/w where w = w_{1} + w_{2} + … + w_{N}.
Updating of the lattice can be synchronous referring to populations with discrete nonoverlapping generations or asynchronous for populations with overlapping generations in continuous time. For synchronous updates, all individuals interact and accumulate payoffs and then everbody attempts to reproduce with successrates relative to each individuals payoff within its neighborhood. For asynchronous updates only a single randomly selected focal site gets updated at a time: first the payoffs of the selected individual and all its neighbors are determined and then they compete to repopulate the focal site as outlined above.
Dynamical regimes
The following examples illustrate and highlight different relevant or otherwise interesting scenarios but at the same time they are meant as suggestions and starting points for further exploring and experimenting with the dynamics of the system. 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 spatially structured populations where individuals interact only within a limited local neighborhood and engage in prisoner's dilemma and snowdrift games.
 

Evolutionary kaleidoscopes:
 
Evolutionary kaleidoscopes:
 
Space promotes cooperation:
 
Space inhibits cooperation:
 
Mechanisms:
 
Mechanisms:

The key result that spatial structure promotes cooperation in the Prisoner's Dilemma but fails to do so under the apparently relaxed conditions of the Snowdrift or HawkDove game is very robust. In particular, the qualitative results remain unaffected by changes in the updating of the lattice (synchronous versus asynchronous) or the lattice geometry (square, triangular or hexagonal lattices) leading to different neighborhood sizes. This can be easily veryfied using the VirtualLabs and changing e.g. the lattice geometry on the Population tab of the parameter panel.
VirtualLab
Along the bottom of the VirtualLab applet are several buttons arranged to control the execution and the speed of the simulations. Of particular importance are the Param button and the data views popup 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.
Color code:  Cooperators  Defectors 

New cooperator  New defector 
Payoff code:  Low  High 

Note: The yellow and green strategy 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 indicating payoffs for mutual cooperation and defection, respectively.
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 related to the Prisoner's Dilemma, Snowdrift and HawkDove 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:
 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.