Title: Can We Find Nash Equilibria at a Linear Rate in Markov Games?. Abstract: We study decentralized learning in two-player zero-sum discounted Markov games where the goal is to design a policy optimization algorithm for either agent satisfying two properties. First, the player does not need to know the policy of the opponent to update its policy. Second, when both players adopt the algorithm, their joint policy converges to a Nash equilibrium of the game. To this end, we construct a meta-algorithm, dubbed as $\texttt{Homotopy-PO}$, which provably finds a Nash equilibrium at a global linear rate. In particular, $\texttt{Homotopy-PO}$ interweaves two base algorithms $\texttt{Local-Fast}$ and $\texttt{Global-Slow}$ via homotopy continuation. $\texttt{Local-Fast}$ is an algorithm that enjoys local linear convergence while $\texttt{Global-Slow}$ is an algorithm that converges globally but at a slower sublinear rate. By switching between these two base algorithms, $\texttt{Global-Slow}$ essentially serves as a ``guide'' which identifies a benign neighborhood where $\texttt{Local-Fast}$ enjoys fast convergence. However, since the exact size of such a neighborhood is unknown, we apply a doubling trick to switch between these two base algorithms. The switching scheme is delicately designed so that the aggregated performance of the algorithm is driven by $\texttt{Local-Fast}$. Furthermore, we prove that $\texttt{Local-Fast}$ and $\texttt{Global-Slow}$ can both be instantiated by variants of optimistic gradient descent/ascent (OGDA) method, which is of independent interest.