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We present a simple game model where agents with different memory lengthscompete for finite resources. We show by simulation and analytically that aninstability exists at a critical memory length, and as a result, differentmemory lengths can compete and co-exist in a dynamical equilibrium. Ouranalytical formulation makes a connection to statistical urn models, and weshow that temperature is mirrored by the agent's memory. Our analysis is easilygeneralisable to many other game models with implications that we brieflydiscuss.

Does good memory help you win games?