Volunteering in Public Goods games:
by Christoph Hauert, Version 2.2, September 2005.
In structured populations, the individuals occupy the nodes of a lattice, graph or network. The performance or fitness of all individuals is determined through interactions within their neighborhood. The evolutionary dynamics can be modeled in different ways but in any case, the strategy of an individual is updated by probabilistically comparing its fitness with the fitnesses of its neighbors. This updating can be done in synchrony, e.g. reflecting an annual reproductive cycle, or in asynchrony which approximates a continuous time system. In addition, there are various meaningful ways to implement the probabilistic comparison and the derivation of the fitness.
In well-mixed populations, the replicator dynamics predicts periodic oscillations driven by the cyclic dominance of cooperators, defectors and loners. However, all trajecories are structurally unstable such that noise eventually drives the system to the boundary of the simplex S3 and thus eliminating two strategies resulting in a homogenous state (usually all loners). In particular, this also happens in the case of finite population sizes. Thus, the long-term maintenance of cooperation requires stabilization. This can be achieved in different ways, e.g. through modifications of the dynamics or by considering structured populations.
In contrast to well-mixed populations, cooperators can survive in structured populations already in compulsory public goods games by forming compact clusters. Voluntary participation and the loner option largely extends the range where cooperators persist because loners provide additional protection to cooperators by reducing exploitation. This results in four different dynamical regimes.
All collections provide more detailed information on different aspects of the dynamics of voluntary public goods games in structured populations including many preconfigured simulations illustrating and highlighting particular scenarios.
The buttons along the bottom 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. Use the above collections to run simulations illustrating particular scenarios with all parameters preset accordingly.
|New cooperators||New defectors||New loners|
Note: The pale strategy colors are very useful to get an intuition of the activitiy in spatially structured systems. The shades of grey of the payoff scale are augmented by blueish and reddish shades, which indicate 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.|
|Simplex S3||Frequencies plotted in the simplex S3. Mouse clicks set the initial frequencies of strategies.|
|Phase Plane 2D||Frequencies plotted in the phase plane spanned by the frequency of participants (x + y = 1 - z) and the relative frequency of cooperators (f = x / (x + y)). Mouse clicks set the initial frequencies of strategies or stop the simulations.|
|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 voluntary public goods 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.
- multiplication factor r of public good.
- cost of cooperation c (investment into common pool).
- payoff for loners. Typically this value is positive but smaller than r - 1 such that groups of cooperators are better of but loners are better off than groups of defectors.
- Lone cooperator, lone defector:
- payoffs for cooperators and defectors which are forced to act as loners because they could not find interaction partners. Usually this will be the same as the loner's payoff.
- Init Coop, init defect, init loner:
- initial fractions of cooperators, defectors and loners. If they do not add up to 100%, the values will be scaled accordingly. Setting the fraction of cooperators to 100% (and the others to zero) results in a symmetrical initial configuration suitable for observing evolutionary kaleidoscopes.