  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.
