Ordering in spatial evolutionary games for pairwise collective strategy updates

Evolutionary $2 \times 2$ games are studied with players located on a square lattice. During the evolution the randomly chosen neighboring players try to maximize their collective income by adopting a random strategy pair with a probability dependent on the difference of their summed payoffs between the final and initial state assuming quenched strategies in their neighborhood. In the case of the anti-coordination game this system behaves alike an anti-ferromagnetic kinetic Ising model. Within a wide region of social dilemmas this dynamical rule supports the formation of similar spatial arrangement of the cooperators and defectors ensuring the optimum total payoff if the temptation to choose defection exceeds a threshold value dependent on the sucker's payoff. The comparison of the results with those achieved for pairwise imitation and myopic strategy updates has indicated the relevant advantage of pairwise collective strategy update in the maintenance of cooperation.