In order to study effects arising from the geometrical arrangement of the players on different lattices, the following examples consider the situation where individuals interact in groups of constant size N = 4. The interaction partners are randomly selected among all neighbors. When strategies are updated, individuals compete with a single randomly selected neighbor. All examples use identical game parameters with r = 3.4, c = 1 and individuals imitate the strategy of a competing neighbor with a probability proportional to the payoff difference z inspired by condensed matter physics: 1/(1 + exp[-z/T]) with a temperature or noise term T = 0.1.

In this setting two adverse effects compete and affect cooperation: on one hand, the random sampling of the interaction partners among all neighbors introduces some sort of noise which is generally in favor of defectors but on the other hand higher degrees of local clustering would appear to be generally in favor of cooperators. Also recall that in absence of population structures, i.e. in well-mixed populations, cooperators are invariably driven to extinction.



Time evolution of the frequency of cooperators in well-mixed populations where individuals engage in different kinds of interactions.

Color code:CooperatorsDefectors
 New cooperatorNew defector

Square lattice - Moore neighborhood

Each individual interacts with three randomly chosen individuals out of eight neighbors (those reachable by a chess kings move) resulting in groups of N = 4. The system equilibrates at around 28% cooperators. The relatively large neighborhood size acts similar to a source of noise which generally acts against cooperators. This effect is apparently stronger than clustering advantages resulting from the small minimal loop size of three (two neighbors have four common neighbors).


Hexagonal lattice - six neighbors

Individuals are arranged on a honeycomb or hexagonal lattice and interact with a random sample of three out of six neighbors. The system equilibrates at much higher frequencies of cooperators around 65%. This is approximately the same abundance as for square lattices with von Neumann neighborhood (four neighbors). Thus, this suggests that the hexagonal arrangement of players favors cooperation. One intuitive reason for this result seems to be the small loop size of three, i.e. any two neighboring individuals have two common neighbors.


Square lattice - von Neumann neighborhood

Individuals are again arranged on a square lattice but have only four neighbors to the north, east, south, and west. As before, groups are formed by randomly selecting three out of these four neighbors. The equilibrium frequency of cooperators is roughly 65% cooperators. Somewhat surprisingly this is roughly identical to the hexagonal lattice. In analogy to the reasoning above, it is worth noting that the minimal loop size of four is larger (instead of three) and neighboring individuals have no other neighbors in common.


Triangular lattice - three neighbors

Individuals are arranged on a triangular lattice with three neighbors each. In each public goods game they interact with all their neighbors (N = 4 as before). This results in a high equilibrium fraction of around 78% cooperators. The minimal loop size of six is rather big but on the other hand there is less noise in this system because each individual interacts with all its neighbors instead of a randomly selected sample.