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.

This work was supported by the the Agencia Estatal de Investigaci´on and Fondo Europeo de Desar- rollo Regional (FEDER, UE) under project APASOS (PID2021-122256NB-C21, PID2021-122256NB- C22), the Mar´ıa de Maeztu programme for Units of Excellence, CEX2021-001164-M funded by MCIN/AEI/10.13039/501100011033. We also acknowledge a studentship by the Engineering and Physical Sciences Research Council (EPSRC, UK), reference EP/T517823/1.

With funding from the Spanish government through the "Severo Ochoa Centre of Excellence" accreditation (CEX2021-001164-M).

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