Carbon trading is an effective tool to reduce emissions. And the implementation of the policy requires the cooperation of all participants in the carbon market, then it will cause a contradiction between social benefits and individual benefits. To break the dilemma faced by the carbon market, it is necessary to fully consider the behavioral strategies of all participants. However, no known literature discusses the effect of quota allocation under the banking mechanism on the behavior strategies of carbon market participants from the limited rational perspective. This paper analyzes the effect of the banking mechanism on each party's strategy in the carbon market. Results show that, under the regulation of the carbon trading management center, the promotion of carbon trading policy is separated into four periods, respectively, policy trial, policy promotion, policy adjustment, and policy stability. The evolutionary stability equilibrium represents the optimal equilibrium point of the system. Adopting a banking mechanism do not affect the outcome of evolutionary stability equilibrium, but has an effect on the rate convergence to the evolutionary stability equilibrium. The findings can be utilized as a guide for accelerating the achievement of the carbon-neutral goal, perfecting the carbon market mechanism, and creating targeted emission-reduction incentive policies.