  We study the rock-paper-scissors game in structured populations, where the
invasion rates determine individual payoffs that govern the process of strategy
change. The traditional version of the game is recovered if the payoffs for
each potential invasion stem from a single pairwise interaction. However, the
transformation of invasion rates to payoffs also allows the usage of larger
interaction ranges. In addition to the traditional pairwise interaction, we
therefore consider simultaneous interactions with all nearest neighbors, as
well as with all nearest and next-nearest neighbors, thus effectively going
from single pair to group interactions in games of cyclic dominance. We show
that differences in the interaction range affect not only the stationary
fractions of strategies, but also their relations of dominance. The transition
from pairwise to group interactions can thus decelerate and even revert the
direction of the invasion between the competing strategies. Like in
evolutionary social dilemmas, in games of cyclic dominance too the indirect
multipoint interactions that are due to group interactions hence play a pivotal
role. Our results indicate that, in addition to the invasion rates, the
interaction range is at least as important for the maintenance of biodiversity
among cyclically competing strategies.
