
This is the supplementary webpage to the publication on The Outbreak of Cooperation in PNAS.
Description of cover figure: Illustration of selforganized pattern formation in migration games. The snapshot, taken after 200 time steps, shows the outcome of the spatial interaction of mobile individuals, who strategically interact with their four nearest neighbors and copy the strategy of their most successful neighbor. The simulation assumes that individuals earn no payoff, if they meet individuals who use the same strategy, while their outcome is positive, if interacting with individuals using another strategy. Strategy 1 is represented by blue disks, strategy 2 by red disks. A change to strategy 2 in the last time step is indicated in green, a change to strategy 2 by yellow. White spaces are empty. Individuals perform successdriven migration, i.e. they move to empty locations with a higher expected payoff. The maximum migration distance in one time step was assumed to be 5. Starting with uniformly distributed strategies, one observes an agglomeration of individuals, and individuals with different strategies mix. This is to be distinguished from the migratory dynamics in other spatial games, which lead to segregation patterns, for example, as studied by Nobel prize winner Thomas Schelling. For animated videos see /migrationgames. 
Spontaneous outbreak of prevalent cooperation in the spatial prisoner's dilemma with random relocations and strategy mutations. The prisoner's dilemma describes social interactions, in which it is risky to cooperate and tempting to defect (i.e. to cheat or freeride). The simulations are for 49x49grids (red = defector, blue = cooperator, white = empty site, green = defector who became a cooperator, yellow = cooperator who turned into a defector in the last iteration). The simulation starts with the initial configuration of a circular cluster of defectors (red) at time t=0. In each time step (iteration), the strategies and locations of all individuals have been updated in a random sequential order. The video shows one snapshot every time step. In Phase I, the cluster of defector splits up, and defectors disperse over the space due to random relocations. Nevertheless, small defective clusters are formed, as the payoff for mutual defection is higher than when defectors do not have any neighbors. Cooperators are generated randomly at a very small rate due to strategy mutations (green), but usually turn into defectors quickly (yellow). The video is cut, because the features of the spatiotemporal patterns do not change over more than 20,000 iterations. Phase II is displayed more slowly to highlight the sudden outbreak of cooperation: Around the time t=25,510, a small, but overcritical cluster of cooperators appears in the upper right corner (green and blue). This happens by random coincidence of strategy mutations, which creates cooperators in neighboring locations by chance. The overcritical cluster of cooperators (blue) does not only allow cooperators to survive; neighboring defectors also start to imitate them due to their greater payoff (green). The evolution in Phase III is shown again at the previous movie speed: Once a large enough cooperative cluster has appeared, cooperation is "exported" to other locations by random and successdriven migration, and it spreads quickly among individuals almost everywhere. For more videos see /migrationgames
Related Publications:
[1] D. Helbing (1992) A mathematical model for behavioral changes by pair interactions. Pages 330348 in: G. Haag, U. Mueller, and K. G. Troitzsch (eds.) Economic Evolution and Demographic Change. Formal Models in Social Sciences (Springer, Berlin).
[2] D. Helbing (1993) Stochastic and Boltzmannlike models for behavioral changes, and their relation to game theory. Physica A 193, 241258.
[3] D. Helbing (1996) A stochastic behavioral model and a `microscopic' foundation of evolutionary game theory. Theory and Decision 40, 149179.
[4] D. Helbing (1998) Microscopic foundation of stochastic game dynamical equations. Pages 211224 in: W. Leinfellner and E. Köhler (eds.) Game Theory, Experience, Rationality (Kluwer Academic, Dordrecht).
[5] D. Helbing and T. Vicsek (1999) Optimal selforganization. New Journal of Physics 1, 13.113.17.
[6] D. Helbing and T. Platkowski (2000) Selforganization in space and induced by fluctuations. International Journal of Chaos Theory and Applications 5(4), 4762.
[7] D. Helbing and T. Platkowski (2002) Drift or fluctuationinduced ordering and selforganisation in driven manyparticle systems. Europhysics Letters (EPL) 60, 227233.
[8] D. Helbing (2009) Pattern formation, social forces, and diffusion instability in games with successdriven motion European Physical Journal B 67, 345–356.
[9] W. Yu and D. Helbing (2009) Game theoretical interactions of moving agents. Forthcoming.
[10] D. Helbing, M. Schönhof, H.U. Stark, and J. A. Holyst (2005) How individuals learn to take turns: Emergence of alternating cooperation in a congestion game and the prisoner's dilemma. Advances in Complex Systems 8, 87116.
[11] D. Helbing, M. Schönhof, and D. Kern (2002) Volatile decision dynamics: Experiments, stochastic description, intermittency control, and traffic optimization. New Journal of Physics 4, 33.133.16.
[12] D. Helbing (2004) Dynamic decision behavior and optimal guidance through information services: Models and experiments. Pages 4795 in: M. Schreckenberg and R. Selten (eds.) Human Behaviour and Traffic Networks (Springer, Berlin).
[13] D. Helbing (1994) A mathematical model for the behavior of individuals in a social field. Journal of Mathematical Sociology 19 (3), 189219.
[14] D. Helbing (1993) Boltzmannlike and BoltzmannFokkerPlanck equations as a foundation of behavioral models. Physica A 196, 546573.
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