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It is critical to maintain significant convergence and diversity in many-objective optimization problems (MaOPs) for the performance of many-objective evolutionary algorithms (MaOEAs). However, some issues pose serious challenges to MaOEAs, such as the intensification of the conflict between convergence and diversity, and the lack of Pareto selection pressure. To address the above issues, we propose a cooperative-competitive two-stage game mechanism assisted many-objective evolutionary algorithms (MaOEA-GM). The algorithm is divided into two stages, such as competition and cooperative. In competitive game stage, a strategy pool is constructed, including angle penalty distance strategy and favorable convergence strategy. In addition, a new game utility function is designed to balance convergence and diversity. This method promotes the selection of genetically superior parents for inheritance, so that the population can quickly approach the true Pareto front. In cooperative game stage, individuals choose their preferred environmental selection mechanism by voting. The final scheme is determined using the criterion of minority obedience to the majority, thereby increasing the algorithm Pareto selection pressure. Experimental results demonstrate that compared with five advanced MaOEAs, The MaOEA-GM algorithm has not only preponderance in convergence and diversity indicators, but also higher competitiveness in solving MaOPs. [ABSTRACT FROM AUTHOR]