Synergy and Discounting of Cooperation in Social Dilemmas:
by Christoph Hauert, Version 1.0, March 2006.
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 local neighborhood. As in well-mixed populations, the severity of the social dilemma arising in these local interations is determined by the cost-to-benefit ratio of cooperation c/b, the discount/synergy factor of accumulated benefits w, and the group size N. Darwinian selection again requires that individuals with a higher fitness (payoff) have an increased propensity to proliferate and transmit their strategy either in terms of reproduction or through imitation by other individuals. However, and in contrast to well-mixed populations, both interaction and competition now occur locally. This results in local pattern formation which, for example, enables cooperators to form compact clusters and thereby reduces exploitation by defectors. The dynamics of such spatially structured populations can become very complex and thus no longer allows for a full analytical treatment. This is nicely exemplified by the fact that in the traditional public goods game the extinction of cooperators when increasing c/b exhibits a critical phase transition. Instead, the dynamics can be studied by means of individual based simulations and the results can be compared to the analytic results for well-mixed populations in order to point out the detailed effects of spatial structuring with limited local interactions.
All collections provide more detailed information on different aspects of the dynamics of synergy and discounting of cooperation in social dilemmas when individuals interact in structured populations and include many preconfigured simulations illustrating and highlighting particular scenarios.
Whenever the updating of all individuals in the population is done in synchrony and the updating of the individuals' strategies follows deterministic rules, fascinating spatio-temporal patterns can emerge ressembling evolutionary kaleidoscopes or dynamic persian carpets - enjoy!
Structured populations and limited local interactions often affect the equilibrium proportions of cooperators and defectors. Generally, spatial structure promotes cooperation in prisoner's dilemma type interactions and increases the basin of attraction of the cooperative state in bi-stable coordination games. However, in snowdrift type interactions where cooperators and defectors co-exist in well-mixed populations, spatial structure can both enhance as well as inhibit or even eliminate cooperation.
The details of the update rules determining the propagation of strategies has substantial effects on the equilibria. For an update rule insipred by the replicator dynamics, i.e. based on payoff differences, the phase plane to the left indicates changes to the equilibrium frequencies of cooperators as compared to well-mixed populations. For parameters in blue shaded areas cooperators are promoted whereas in red shaded areas they are reduced. The saturation of the colors indicates the strength of the effect.
The Moran process can be either interpreted in terms of a birth-death or a death-birth process. In well-mixed populations the sequence of events has essentially no effect but this is not true for structured populations. In the birth-death scenario, a focal individual is randomly selected for reproduction with a probability proportional to its fitness. Then, one of its neighbors is randomly selected to be eliminated and is replaced by clonal offspring of the focal individual. As compared to the above updating based on payoff differences, the birth-death process favors cooperation and the parameter range where spatial structure inhibits cooperation is largely reduced.
In the death-birth scenario, first a random individual is selected and eliminated. Then the neighbors of the vacant site compete to repopulate the site with their clonal offspring. Each neighbor succeeds with a probability proportional to its fitness. This updating rule is the strongest promoter of cooperation and reduces detrimental effects of spatial structure to a tiny area in parameter space.
The individual based approach outlined for well-mixed populations can be easily adapted to spatially structured populations by two simple changes: (i) to determine each individuals fitness, the N -1 other members of the interaction group are randomly selected among the individuals' neighbors. Naturally, the neighborhood size puts an upper bound on the size of the interaction group, e.g. N ≤ 9 for the Moore neighborhood. (ii) the model individual is a randomly chosen neighbor of the focal individual. Thus, evolutionary dynamics leads to probabilistic comparisons of fitnesses within a local neighborhood. Following the case of well-mixed populations the probabilistic updating is based either on the replicator dynamics or the Moran process.
In spatially structured populations the replicator type updating remains unchanged. However, for the Moran process it is important to note that traditional formulation of the Moran process assumes well-mixed populations and represents a birth-death process. However, a death-birth process is equally plausible, where first a randomly selected individual is eliminated and only then another individual is randomly chosen for reproduction with a probability proportional to its fitness. It is quite obvious that the two processes become identical in the limit of infinite (well-mixed) populations. Interestingly, however, considerable differences occur between birth-death and death-birth processes in structured populations. In fact, it turns out that the death-birth process favors cooperation as compared to the birth-death process. The intuitive reason is that in the birth-death process a cooperator-defector pair competing for reproduction are necessarily neighbors - which means that the cooperator has directly contributed to the fitness of its defecting competitor and thus decreased its own chances to produce offspring. Conversely, in the death-birth process, cooperators and defectors compete to recolonize a vacant site and hence they are usually not direct neighbors. Thus, spatial separation largely prevents cooperators from nourishing their opponents.
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|
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.|
|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 synergy and discounting in social dilemmas. Follow the link for a complete list and descriptions of all other parameters e.g. referring to update mechanisms of players and the population.
- Total benefit b created by each cooperator and divided among all N members of the group (including the cooperator itself).
- cost of cooperation c (investment into common pool).
- synergy or discounting factor w determining the increase in value for accumulated cooperative benefits.
- 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 the others to zero) results in a symmetrical initial configuration suitable for observing evolutionary kaleidoscopes.