Abstract Evolutionary potential games provide an analogy for equilibrium statistical physics under suitable conditions that is broken if cyclic components are added. Using Monte Carlo simulations we study the effect of weak cyclic perturbations on the equilibrium in a system of players located on a square lattice while the dynamics is controlled by a logit rule. The pair interaction is composed of a symmetric three-strategy coordination game of unit strength, a weak self-dependent term, and a cyclic (rock–paper–scissors) component. The self-dependent component favors the first (rock) strategy which dominates the system behavior at low noises if the strength of the cyclic component is below a threshold value. In the opposite case the predator of the first strategy (paper) rules the behavior at low noises while the first strategy can occasionally form growing domains (bursts) diminished after the appearance of the third one. The variation of noise modifies the invasion velocities as well as the rate of nucleation processes and yields drastic changes in the strategy frequencies, fluctuations, average payoffs, and spatio-temporal patterns.