Evolutionary Games and Population Dynamics:
Public goods games
by Christoph Hauert, Version 1.0, October 2006.
In a population of varying density, an attempt at gathering N individuals that engage in a public goods interaction might not always be successful at low population densities and instead of a group of size N, only S≤N individuals participate. If S = 0 or 1 no interaction occurs. This leads to a natural feedback between population density and game theoretical interactions. The dynamics of cooperators and defectors in public goods interactions is determined by their respective payoffs obtained in randomly formed groups of S individuals. Independent of whether the focal individual is a cooperator or a defector, it receives the same expected payoff from its S - 1 co-players. Hence, the sole determinant of success is the return of the individual's own investment c, which is (r/S - 1) c. For 1 < r < S defectors are always better off as required by the traditional formulation of the public goods game. However, for r > S the social dilemma is relaxed and cooperation dominates. Nevertheless, defectors outperform cooperators in any group consisting of both types (this represents an instance of Simpson's paradox). Also note that this is a fleeting state since thriving cooperators increases the average population payoff and hence the population density which in turn leads to larger interaction groups and puts defectors back into control.
The negative feedback between population density and interaction group size hinges on the fact that the group size can become smaller than r. For pairwise prisoner's dilemma interactions this is not the case: because S cannot vary (and is always equal to N = 2), either r < S always holds (in which case the population goes extinct) or r > S always holds (in which case defectors disappear but cooperators persist). The dynamic feed back cannot operate in either case.
This tutorial complements scientific articles co-authored with Miranda Holmes, Michael Doebeli and Joe Yuichiro Wakano and provides interactive Java applets to visualize and explore the systems' dynamic for parameter settings of your choice.
The following panels illustrate the rich dynamics of this system. The phase space is spanned by the population density x + y (or 1 - z) and the relative fraction of cooperators f = x / (x + y). The left boundary (z = 1) is attracting and consists of a line of stable fixed points (filled circles), which represent states where the population cannot maintain itself and disappears. Conversely, the right boundary, which denotes the maximal population density (z = 0), is repelling. In absence of cooperators (bottom boundary, f = 0), population densities decrease and eventually vanish. Finally, in absence of defectors (top boundary, f = 1), there are two saddle points (open circles) except for the last scenario where one is a stable node (filled circle). In addition, there may be an interior fixed point Q present.
The following list illustrates the different dynamical scenarios. A click on the image to the left, opens a new window with an interactive Java applet that allows to explore the dynamics by numerical integration of the differential equations. The initial condition is set with a mouse-click in the phase plane. Hit 'Run' for forward and backward integration of the differential equations for the given initial condition. The numerical integration can be interrupted by another mouse-click on the phase plane.
Trajectory of the interior fixed point Q for increasing multiplication factors r. Along this trajectory, the system undergoes a series of various types of bifurcations:
- No Q - extinction
- Q unstable node - extinction
- Q unstable focus - extinction
- Q stable focus - co-existence
- Q stable node - co-existence
- No Q - cooperation
For larger group sizes N fascinating and much more complex Hopf bifurcations and dynamical scenarios are possible, which includes multiple, stable and unstable limit cycles. However, also note that r values for which these fascinating bifurcations occur is restricted to a tiny interval. Thus, despite their appeal from a dynamical systems' perspective, the limit cycles might be of only limited relevance for biological applications.
In a population with a fraction x cooperators, y defectors and z = 1 - x - y available space, then the average payoff to cooperators fC and defectors fD is given by:
fC = fD - F(z)
Note that this ivation assumes that the benefits of the public good is contingent on social interactions, i.e. a single participant in the public goods interaction cannot increase its capital. For a detailed derivation of the formulas please consult the scientific articles in the reference section.
The 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.
|New cooperators||New defectors||New empty|
Note: The shades of grey of the payoff scale are augmented by blueish and reddish shades indicating 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 start, resume or stop the simulations.|
|Structure - Strategy||Snapshot of the spatial arrangement of strategies.|
|Mean frequency||Time evolution of the strategy frequencies.|
|Simplex S3||Frequencies plotted in the simplex S3. Mouse clicks set the initial frequencies of strategies or stops the simulations.|
|Phase Plane 2D||Frequencies plotted in the phase plane spanned by the population density (x + y = 1 - z) and the relative frequency of cooperators (f = x / (x + y)). Mouse clicks set the initial frequencies of strategies, stop the simulations or switch to backward integration.|
|Structure - Fitness||Snapshot of the spatial distribution of payoffs.|
|Mean Fitness||Time evolution of average population payoff bounded by the minimum and maximum individual payoff.|
|Histogram - Fitness||Snapshot of payoff distribution in population.|
The list below describes only the parameters related to the public goods game and the population dynamics. 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).
- Lone cooperator's payoff:
- payoff for a cooperator if no one else joins the public goods interaction.
- Lone defector's payoff:
- payoff for a defector if no one else joins the public goods interaction.
- Base birthrate:
- baseline reproductive rate of all individuals. The effective birthrate is affected by the individual's performance in the public goods game and additionally depends on the availability of empty space.
- constant death rate of all individuals.
- Init Coop, init defect, init empty:
- initial densities of cooperators, defectors and empty space. If they do not add up to 100%, the values will be scaled accordingly.