Volunteering in Public Goods games:
Well-mixed populations
by Christoph Hauert, Version 2.0, March 2003.
- Location:
- VirtualLabs
- » Volunteering
- » Well-mixed populations
In well-mixed populations a group of players is randomly drawn and offered to participate in a public goods game. Cooperators and defectors will accept (with different intentions though) and loners refuse to participate and prefer to rely on some small but fixed income. The performance of the three strategies in these games determines whether they will spread in the population. The dynamics and the long term behavior of the system heavily depends on the mechanism of transmitting strategies from parents to offspring or of the imitation behavior. For example, if the reproduction rate of each individual is proportional to its relative performance in the population (i.e. the payoff achieved in the public goods interactions relative to the average population payoff) where players replicate asexually transmitting their strategy to their offspring results in the replicator dynamics. Similarly, if players imitate the strategy of a randomly chosen model member of the population with a probability proportional the difference in their payoffs (provided it is positive) again leads to the replicator dynamics. If players have perfect knowledge about the composition of the population and choose the best strategy based on that knowledge gives the best-reply dynamics.
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.
Fixed point Q in simplex S3
The position of the interior fixed point Q depends on the values of the multiplication factor r, the interaction group size N and the loner's payoff σ. The figure on the left indicates the relocation of Q when increasing each of the three paramters independently. Most interestingly, increasing r at first benefits cooperators by increasing their equilibrium fraction but further increases tend to lower the fraction of cooperators while mainly benefiting defectors. Conversely, increasing the loner's payoff σ mostly benefits cooperators.
The stability but not the location of Q depends on the dynamics of the system. For the replicator dynamics Q is neutrally stable and surrounded closed periodic orbits, which lead to everlasting oscillations of the three strategies. For the best-reply dynamics, Q is stable (but not globally) and for imitate-the-better it can be stable, or unstable depending on the parameter values and thus may be a source, sink or center.
Legend | Time evolution of cooperators, defectors and loners in well-mixed populations where individuals engage in voluntary public goods interactions.
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Simplex S3 representations & fixed pointsEach point in the interior of the simplex S3 spanned by the three strategic types (cooperators, defectors and loners) indicates a particular composition of the population. The corners of S3 denote homogenous populations of all cooperators, defectors or loners, respectively. They are all saddle points because of the cyclic dominance of the three strategies. For a sufficiently high multiplication factor r of the public good (r > 2) an additional interior fixed point Q appears. | ||||
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Homoclinic orbitsFor low multiplication factors on the public good, only homoclinic orbits are observed issuing from and returning to homogenous states of all loners. The boundary of S3 is a heteroclinic cycle resulting from the rock-scissors-paper type of dominance between the three strategies cooperators, defectors and loners. | ||||
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Periodic orbitsSufficiently high multiplication factors lead to an interior fixed point Q. It can be shown analytically that Q is a center, i.e. neutrally stable and surrounded by closed orbits. This results in periodic oscillations of the three strategies - at least in infinite populations. For the replicator dynamics the difference between infinte and finite populations is particularly pronounced. The structurally unstable center Q in the interior of the simplex becomes unstable in finite populations. All trajectories spiral outwards and eventually reach the boundary of the simplex and end in a homogenous state of the population. | ||||
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Stable cooperationFor very high multiplication factors (i.e. larger or equal to the sampled group size) cooperation becomes stable. In the case of equality, the boundary ec to ed consists of fixed points. The upper half ec to Q is stable (Nash equilibria, black dots) and lower half Q to ed unstable (open circles). Through random shocks and the occasional introduction of the loner strategy, the state will eventually get close to ec. For still larger factors the dynamics drives the system into that homogenous state of all cooperators. The reason for this is that the social dilemma is relaxed for sufficiently high multiplication factors. This happens exactly when the return of each dollar invested in the public good becomes equal to or larger than unity. Note, however, that even though at that point any investment is actually profitable, defectors nevertheless outperform cooperators in any mixed group. | ||||
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Imitate better playersIn the replicator dynamics, individuals imitate strategies of randomly sampled members of the population with a probability proportional to the difference in payoffs (provided the difference is positive). Assuming that individuals always imitate better performing sampled members, the dynamics changes. Depending on the parameter values Q is a source, sink or center. On the left, the structurally unstable case with closed orbits and Q a center is shown. The dashed lines indicate isoclines where cooperators and defectors, defectors and loners, as well as loners and cooperators perform equally well. In finite populations Q tends to become unstable too, but often a weakly attracting periodic orbit emerges, which allows for persistent co-existence of all three strategies. | ||||
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Best-reply dynamicsIf all individuals have perfect information about the composition of the population, then each individual that reassesses its strategy will obviously choose the best-reply to the current composition. This leads to the so-called best-reply dynamics. In that case the interior fixed point Q is stable and most trajectories spiral towards this mixed equilibrium. Note that if initially the population contained a large fraction of defectors, the dynamics drives the population into a homogenous state with all loners. However, the occasional appearance of cooperators will initiate trajectories that again converge to Q. The dashed lines again indicate isoclines as introduced above. Finite population sizes introduce some noise but do not lead to any significant changes in the dynamics. |
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 | Loners |
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| New cooperators | New defectors | New loners |
| Payoff code: | Low | High |
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Note: The pale 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 the payoffs for mutual cooperation and defection, respectively.
| Controls | |
| Params | Pop up panel to set various parameters. |
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| 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. |
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| 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. |
Game parameters
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.
- Interest:
- multiplication factor r of public good.
- Cost:
- cost of cooperation c (investment into common pool).
- Loner:
- 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.