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Volunteering in Public Goods games:

Kaleidoscopes

by Christoph Hauert, Version 2.1, May 2003.

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For symmetrical initial configurations and deterministic updating of the system, the initial symmetry is preserved and the evolving lattice resembles a dynamically changing persian carpet or an evolutionary kaleidoscope driven by the cyclic dominance of the three strategies. Deterministic updating requires synchronous updating of all players in the population (this corresponds to e.g. an annual reproductive cycle as opposed to asynchronous updating, which approximates continuous time) as well as deterministic updating of all players' strategies. One particularly simple rule is to adopt the strategy of the best performing neighbor - and in case there is a tie, keep your own strategy. Note that a tie can also occur between two better performing but different strategies. If this happens the focal player will adopt the dominant strategy according to the cyclic dominance of cooperators, defector and loners. Otherwise, this 'degeneracy' can be resolved by choosing slightly different parameters.

Similar evolutionary kaleidoscopes can be found in 2×2 games with the prisoner's dilemma and snowdrift game as prominent representatives. Interestingly, the traditional formulation of the public goods game, i.e. in the absence of loners, no such dynamic patterns are observed. However, the synergy and discounting framework readily produce appealing dynamical patterns for both public goods type as well as snowdrift type interactions. Another three-strategy game that produces nice patterns is the rock-scissors-paper game.

Clicking either on one of the pictures below or the corresponding link to the right will open a new window with a running applet illustrating the respective scenario. You can use this as a starting point to study effects of variations of the parameters.

Legend	Time evolution of cooperators, defectors and loners in structured populations where individuals engage in voluntary public goods interactions. Color code: Cooperators Defectors Loners   New cooperators New defectors New loners
	Moore neighborhood The Moore neighborhood limits interactions of an individual to its eight neighbors reachable by a chess-kings-move. The rectangular lattice is initialized with a single 3×3 cluster of loners in the center surrounded by one layer of defectors in a sea of cooperators. The snapshot on the left was taken after 300 generations.
	von Neumann neighborhood The von Neumann neighborhood is smaller than the Moore neighborhood and each player interacts only with its four nearest neighbors to the north, east, south and west. This neighborhood puts a stronger emphasis on the lattice structure because spreading velocities are twice as fast in the lattice orientation than in diagonal directions. The symmetrical initialization is the same as above. The snapshot on the left was taken after 300 generations.
	Hexagonal lattices On hexagonal or honeycomb lattices each player has six neighbors. The lattice is initialized with a single honeycomb cluster of seven loners in the center surrounded by one layer of defectors in a sea of cooperators. The snapshot on the left was taken after 300 generations.
	Triangular lattices No parameter combinations and initial configurations have been found so far that produce kaleidoscopes on triangular lattices where each player interacts with three neighbors, i.e. the group size can be up to N = 4. - If you do find suitable parameters and configurations, please let me know!