Phasor Measurement Units (PMUs) are used in the measurement, control and protection of power grids. However, deploying PMUs at every bus in a power system is prohibitively expensive, necessitating their optimal placement to ensure the system observability with minimum cost. This partial PMU placement renders the system vulnerable to False Data Injection Attacks (FDIAs). This paper proposes a zero-sum game-based approach to strategically place an additional PMU (after the initial optimal PMU deployment for full observability) to bolster robustness against FDIAs by introducing redundancy in attack-susceptible areas. To compute the Nash equilibrium (NE) solution, we leverage a reinforcement learning algorithm that mitigates the need for complete knowledge of the opponent's actions. The proposed PMU deployment algorithm increases the detection rate of FDIA by 36% compared to benchmark approaches.