Analysis of digital financial innovation behavior based on fractional-order evolutionary game

This paper delves into the government’s regulatory issues regarding the innovation of digital finance companies. Under static and dynamic reward–punishment mechanisms, fractional-order evolutionary game models are established between the government and digital finance enterprises. Considering the timeliness of rewards and punishments, time-delayed fractional-order models are further established to refine the classification of government regulatory actions. Applying stability theory, the consistencies, differences, and advantages of evolutionary stability between fractional-order and integer-order replicated dynamical systems are compared in depth, and stability and Hopf bifurcation criteria are derived. Theoretical analysis and numerical simulation results demonstrate that under the static reward–punishment mechanism, there are fundamental differences in the stability between the fractional-order and integer-order models; in the dynamic situation, both show consistency and disparity, and the fractional-order model exhibits more complex dynamic behaviors. These findings can effectively tackle the evolutionary stability issue of internal equilibrium points that integer-order models fail to address, and provide the government with valuable theoretical foundation and model selection space to conduct precise supervision of the innovative behaviors of digital finance companies.