Due to several economical and technological factors, the integration of distributed microgrids into power networks is growing fast. In this paper, the impacts of false data injection attacks on power generation of distributed microgrids are considered. It is shown that the optimal power generation of distributed microgrids depends on the sensor measurements when economical factors as well as voltage angle regulation at buses are taken into account. Consequently, the attacker is able to change the optimal behavior of microgrids by spoofing the sensor measurements. The effects of the false data injection attacks on the behavior of microgrids in the power network are studied in a game theoretic framework. The interaction between the attacker and the system defender in a power network with multiple areas is considered as a zero-sum Markov game. The game solution specifies the optimal proportion of the times that the attacker and defender should attack and defend different measurements, respectively. The effectiveness of the proposed approach and properties of the game are studied through simulation studies.