The optimal strategy against Fictitious Play in infinitely repeated games

With the rapid development of Artificial Intelligence(AI), the game between human and machine/AI becomes more and more common and important. In this paper we will investigate such infinitely repeated games. We will study the optimal play of the human given the machine/AI adopting the Fictitious Play(FP) for all different kinds of
$2 \times 2$
games under the assumption of a payoff parameter being rational. We give the optimal strategy of human which is proven to lead the system into a cycle after finite times. The results show that FP algorithm can be exploited easily and heavily. Analysis in this paper might shed some light on the result for more general and complicated situations.