Phase diagrams for the spatial public goods game with pool-punishment
  The efficiency of institutionalized punishment is studied by evaluating the stationary states in the spatial public goods game comprising unconditional defectors, cooperators, and cooperating pool-punishers as the three competing strategies. Fine and cost of pool-punishment are considered as the two main parameters determining the stationary distributions of strategies on the square lattice. Each player collects its payoff from five five-person public goods games, and the evolution of strategies is subsequently governed by imitation based on pairwise comparisons at a low level of noise. The impact of pool-punishment on the evolution of cooperation in structured populations is significantly different from that reported previously for peer-punishment. Representative phase diagrams reveal remarkably rich behavior, depending also on the value of the synergy factor that characterizes the efficiency of investments payed into the common pool. Besides traditional single and two-strategy stationary states, a rock-paper-scissors type cyclic dominance can emerge in strikingly different ways. 
Attila Szolnoki, Gyorgy Szabo, Matjaz Perc
physics.soc-ph cond-mat.stat-mech physics.bio-ph q-bio.PE