Mathematical dynamic integration model based on game theory and optimization theory

As technology advances, the research on mathematical dynamic integration models has yielded a potent instrument for the analysis and optimization of complex systems, thereby fostering progress across multiple domains. To solve the issues of traditional dynamic integration models, such as the lack of solid mathematical theoretical support, imbalanced weight allocation among multiple participants, and insufficient dynamic adaptability, this study integrates the Shapley value method from cooperative game theory with Monte Carlo sampling techniques to construct an information evaluation model. The calculation of the marginal contributions of participants in different alliances enables the quantification of the value of multi-source information, thus resolving the issue of imbalanced weight allocation. Furthermore, a dynamic optimization model is built by integrating Nash equilibrium and Markov decision processes, utilizing state transition probabilities to capture temporal changes in strategies and achieve stable strategy optimization. Finally, a mathematical model that includes integration is designed. Experiments that compare the two models show that the average time it takes to make a decision goes up by only 0.2 seconds when the number of samples is between 25 and 150, and the amount of memory used goes up by only 1.2GB. In the InfiMM-WebMath-40B dataset, the model's standard deviation increases from 0.020 in 25 samples to 0.030 in 150 samples, an increase of only 0.010. In the MV-MATH dataset, the standard deviation increases by only 0.011. Furthermore, the proposed model outperforms the comparative models in all metrics, reducing convergence time by 14.7%, training time by 15.3%, energy consumption by 16.3%, and iteration count by 21.5%, while maintaining a low central processing unit utilization rate. Therefore, the proposed mathematical dynamic integration model can more accurately handle uncertain information in complex systems, optimize system decisions, reveal the inherent laws governing dynamic changes in systems, and provide stronger theoretical support and methodological guidance for solving practical problems.