Volunteering in Public Goods games:
Dynamical regimes
by Christoph Hauert, Version 1.0, September 2005.
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The evolutionary dynamics in structured populations can be conveniently modeled as follows: first, the payoff or fitness of a randomly drawn focal individual is determined by a single interaction within its neighborhood. If the interaction group size N is smaller than the neighborhood size (including the focal individual), then N-1 random neighbors are selected for the interaction (plus the focal individual). Second, one of the focal individual's neighbors is chosen at random and neighbor's fitness is similarly determined from a single interaction within its respective neighborhood. Finally, a probabilistic comparison of the two payoffs determines which individual's offspring replaces the focal individual. For the following illustration, a particularly simple comparison is used where the neighbor's offspring replaces the focal individual with a probability proportional to the payoff difference provided that the neighbor performed better than the focal individual and with probability zero otherwise (for other updating mechanisms see below). This represents an individual based analogy of the replicator dynamics and actually recovers it in the limit of infinite population size and increasing neighborhood sizes. Note that this update procedure covers unstructured populations, too, which would correspond to networks where every individual is linked to every other member of the population (fully connected graph).
Dynamical regimes
Frequency of cooperators (blue), defectors (red) and loners (yellow) in optional public goods interactions as a function of the multiplication factor of the common good. Players are arranged on a lattice with Moore neighborhood (eight neighbors) and interact in randomly formed groups of size N = 5. For small multiplication factors r < σ + 1 = 2 loners dominate. The reason is simple: in this case even in a group of cooperators the payoffs do not exceed the loner's income. For σ + 1 < r < rL = 3.01 all three strategies co-exist in dynamical equilibrium. Above the threshold rL loners vanish and cooperators thrive through cluster formation alone and eventually eliminate defectors (r > rD = 3.68). Thus, the dynamics reverts the voluntary public goods game back to a compulsory interaction.
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Optional public goods games in lattice populationsThe dynamics of cooperators (blue), defectors (red) and loners (yellow) arranged on a lattice and engaging in optional public goods interactions with their neighbors depends on the multiplication factor r of the common good. Four qualitatively different dynamical regimes are observed: for low r only loners survive; for higher r all three strategies co-exist; for still higher r loners disappear but cooperators and defectors still co-exist; finally, for very high r defectors also disappear and cooperators dominate. |
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Scenarios
The snapshots below illustrate typical lattice configurations for each one of the four dynamical regimes outlined above. A click on the snaphsot opens another window with a running applet that has all parameters set accordingly in order to reproduce the respective scenario. Various parameters can be altered to investigate the systems response.
Legend | Time evolution of cooperators, defectors and loners in well-mixed populations where individuals engage in voluntary public goods interactions.
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Loners dominateFor r < σ + 1 = 2 loners dominate because they outperform even groups of cooperators. Irrespective of the initial configuration, the system quickly approaches a state of a sea of loners with few isolated cooperators and defectors interspersed. Note that isolated cooperators and defectors act as loners because they continuously fail to find interaction partners for the public goods game. The snapshot on the left is taken after about 50 generations for r = 1.98. Strictly speaking, such r violate the conditions for the loners payoff. | ||||||||
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Co-existenceAll three strategies co-exist in a dynamical equilibrium. Note that in the compulsory public goods game, i.e. in absence of loners, cooperators would be doomed and defection reign for such r. With optional participation, clusters of cooperators enjoy additional protection against exploitation by adjacent loners. Thus, clusters of cooperators expand into loner territory while being dimished by defectors on the other side. The cyclic dominance produces travelling waves propagating across the lattice. The snapshot on the left is taken after about 1000 generations for r = 2.5. | ||||||||
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Return to compulsory interactionsFor sufficiently high r, the advantage arising from cluster formation is sufficient to ensure the survival of cooperators and the loners go extinct. Interestingly, this corresponds to a voluntary return to compulsory interactions. Another tutorial is dedicated to the dynamics of such compulsory public goods games. The extinction of loners exhibits a critical phase transition in the universality class of directed percolation. The snapshot on the left is taken after about 1000 generations for r = 3.3. | ||||||||
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Cooperators persevereFor still higher multiplication factors r, cooperators finally manage to take over the population. Note that this occurs well below the threshold in well-mixed populations (r=N). When starting with equal initial proportions, the loners vanish quickly but then it takes a long time and patience until the last defectors is eliminated. The extinction of defectors ressembles a branching and annihilating random walk and shows the characteristics of another critical phase transition. The snapshot on the left is taken after about 5000 generations for r = 3.7 and it would take quite a few more generations to get rid of the last defector (typically 20'000 - 30'000 generations). |
Representative snapshots of the optional public goods games on a square lattice with synchronous lattice updates. A click on each snapshot starts a running applet in a new window with all parameters set accordingly. These figures refer to the article Hauert, Ch., De Monte, S., Hofbauer, J. & Sigmund, K. (2002) Volunteering as Red Queen Mechanism for Cooperation in Public Goods Games, Science 296, 1129-1132.
Legend | Time evolution of cooperators, defectors and loners in well-mixed populations where individuals engage in voluntary public goods interactions.
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(a) Best takes over, low multiplication factorDeterministic update rule where every player adopts the strategy of the neighbor (including itself) that performed best. For low multiplication rates (r = 2.2) all three strategies co-exist in a dynamical equilibrium with traveling waves sweeping across the lattice. Note that cooperators tend to die out within the first few generations due to the finite (and rather small) lattice size. To get the depicted patterns, you might have to Reset the board several times or choose a larger lattice size. | ||||||||
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(b) Best takes over, high multiplication factorSame update rule as in (a) but for a higher multiplication factor of r = 3.8. The clustering advantage of cooperators is now strong enough such that they can thrive on their own and without the assistance and protection by loners. Consequentially the loners quickly go extinct. | ||||||||
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(c) Imitation/replication, low multiplication factorStochastic update rule prescribing that 80% of all players adopt more successful neighboring strategies, with a probability proportional to the payoff difference. In this case, all participants reassess their strategy before interacting and the fitness of each individual is given by the average payoff achieved since the last strategy switch. The multiplication factor is set to r = 2.2 as in (a). As in (a) all three strategies co-exist in dynamical equilibrium with traveling waves sweeping across the lattice driven by the rock-scissors-paper type cyclic dominance of the three strategies. | ||||||||
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(d) Imitation/replication, high multiplication factorUpdate rule as in (c) but for the higher multiplication factor of r = 3.8. In absence of loners, cooperators would now go extinct due to the randomness. In a typical configuration, clusters of cooperators are surrounded by defectors and the latter again surrounded by loners. Every now and then cooperators manage to break through the defectors clutch and invade domains of loners. |
- Hauert, Ch., De Monte, S., Hofbauer, J. & Sigmund, K. (2002) Volunteering as Red Queen Mechanism for Cooperation in Public Goods Games, Science 296, 1129-1132 (click on title to access the article online).
- Szabó, G. & Hauert, Ch. (2002) Phase transitions and volunteering in spatial public goods games, Phys. Rev. Lett. 89 118101.
- Hauert, Ch. & Szabó, G. (2005) Game theory and physics, Am. J. Phys. 73, 405-414.
- Hauert, Ch. (2006) Cooperation, Collectives Formation and Specialization, Advances in Complex Systems 9, 315-335.