UC Santa Barbara

Stochastic differential games and those with delay play a crucial role in modeling complex, real-world phenomena. The ability to find Nash equilibria in these games enhances the predictive capabilities of scientists and professionals across various fields and informs optimal decision-making processes. These problems can be computationally demanding to solve, especially in the case of delayed dynamics and interaction among a large number of agents.

This dissertation begins with an overview of stochastic differential games and existing machine learning methodologies designed to find their Nash equilibria. We then extend these existing methodologies to the challenging case of stochastic delay differential games with a new algorithm for finding their closed-loop Nash equilibria. To evaluate the effectiveness of our proposed algorithm, we test it on problems with known solutions. In particular, we introduce new financial models based on competing portfolio managers taking into consideration delayed tax-effects. We derive analytical Nash equilibrium solutions for these newly introduced stochastic delay differential games, serving as additional benchmarks to assess the performance of our proposed machine learning approach.

Finally, building on the existing machine learning methodologies for stochastic differential games, we introduce a new modified algorithm that we use to solve the Nash equilibrium problem for a game-theoretic, stochastic SEIR (Susceptible-Exposed-Infectious-Recovered) model applied to the COVID-19 pandemic. Solving this proposed model demonstrates the effects of differing policies on the spread of disease over different regions and how these policies affect each other, illustrating the practical effectiveness of the proposed numerical approach.