In MANET, the cooperative authentication mechanism requires the cooperation of the neighbor nodes and significantly enhances the authentication probability. However, it exposes location privacy of neighbor nodes and is costly. How to balance the authentication and location privacy is a key issue. In this paper, we use game theory to analyze the behavior of neighbor nodes in cooperative authentication and gain the optimal strategy. Every node seeks to obtain most reward at least location privacy loss and cost. We first build the static game with complete information and obtain two pure-strategy and one mixed-strategy Nash equilibria. These equilibria can be used efficiently to balance authentication and location privacy. Then, we build the static game with incomplete information and obtain the Bayesian Nash equilibria.