Delayed response in the Hawk Dove game

We consider a group of agents playing the Hawk-Dove game. These agents have a finite memory of past interactions which they use to optimize their play. By both analytical and numerical approaches, we show that an instability occurs at a critical memory length, and we provide its characterization. We show also that when the game is stable, having a long memory is beneficial but that instability, which may be produced by excessively long memory, hands the advantage to those with shorter memories.