  We consider evolutionary games in which the agent selected for update
compares their payoff to q neighbours, rather than a single neighbour as in
standard evolutionary game theory. Through studying fixed point stability and
fixation times for 2x2 games with all-to-all interactions, we find that the
flow changes significantly as a function of q. Further, we investigate the
effects of changing the underlying topology from an all-to-all interacting
system to an uncorrelated graph via the pair approximation. We also develop the
framework for studying games with more than two strategies, such as the
rock-paper-scissors game where we show that changing q leads to the emergence
of new types of flow.
