Online reviews provide product evaluations for customers to make decisions. Unfortunately, the evaluations can be manipulated using fake reviews ("spams") by professional spammers, who have learned increasingly insidious and powerful spamming strategies by adapting to the deployed detectors. Spamming strategies are hard to capture, as they can be varying quickly along time, different across spammers and target products, and more critically, remained unknown in most cases. Furthermore, most existing detectors focus on detection accuracy, which is not well-aligned with the goal of maintaining the trustworthiness of product evaluations. To address the challenges, we formulate a minimax game where the spammers and spam detectors compete with each other on their practical goals that are not solely based on detection accuracy. Nash equilibria of the game lead to stable detectors that are agnostic to any mixed detection strategies. However, the game has no closed-form solution and is not differentiable to admit the typical gradient-based algorithms. We turn the game into two dependent Markov Decision Processes (MDPs) to allow efficient stochastic optimization based on multi-armed bandit and policy gradient. We experiment on three large review datasets using various state-of-the-art spamming and detection strategies and show that the optimization algorithm can reliably find an equilibrial detector that can robustly and effectively prevent spammers with any mixed spamming strategies from attaining their practical goal. Our code is available at https://github.com/YingtongDou/Nash-Detect.