Phase transitions and symmetry breaking of cooperation on lattices

Abstract

The donation game is an instance of a social dilemma with a single parameter given by the cost-to-benefit ratio of cooperation, r. In spatial settings limited local interactions and clustering are capable of supporting cooperation by reducing exploitation from defectors. Traditionally the interaction and competition neighbourhoods are identical. Here we discuss intriguing differences in the dynamics that arise when separating the neighbourhoods. On the square lattice disjoint interaction and competition neighbourhoods are easily realized by considering nearest neighbour interactions and second nearest neighbour competition. Incidentally, the number of first and second neighbours is the same. More importantly, this separates the population into two competing sub-populations, with interactions solely between sub-populations but competition within sub-populations. For negative cost-to-benefit ratios, r, the donation game turns into a harmony game and defection becomes an act of spite. In the traditional setup the extinction of cooperators under harsh conditions, large r, and that of spiteful defectors, r < 0, exhibits critical phase transitions with characteristics of directed percolation. In contrast, with two sub-populations spiteful behaviour cannot persist, while the extinction of cooperators exhibits the same characteristics. Most intriguingly, however, for smaller r spontaneous symmetry breaking in the levels of cooperation between the two sub-populations is observed. The symmetry breaking resembles the sub-lattice ordering occurring in the anti-ferromagnetic Ising model. Within the twofold degenerated phases, decreasing the cost-to-benefit ratio induces extremely large fluctuations (bursts) in the frequencies of cooperation. These bursts eventually drive the system into one of the absorbing states: occasionally homogeneous defection in both sub-lattices but usually only in one and homogeneous cooperation in the other, achieving perfect asymmetry. PACS numbers89.65.-s, 89.75.Fb, 87.23.Kg