CAREER: Stochastic Games on Large Graphs in the Mean Field Regime and Beyond

Diverse areas of science rely on mathematical models of large-scale systems of interacting and competing agents. The agents may be investors, drivers, or viruses, and the large systems may be financial markets, highways, or epidemics, for example. The most widely used framework for modeling competition between many interacting agents, known as 'mean field game theory,' is fundamentally limited in scope to symmetric models, in which each agent interacts equally with each of the others. Many important phenomena, on the other hand, are governed by networks---such as those formed by social ties or financial obligations – that determine which agents interact with each other. Network structures have important implications that the standard theory cannot capture. This project tackles this limitation by developing new mathematical frameworks to handle large-scale models of competition with heterogeneous interactions governed by networks. Both graduate and undergraduate students are involved in the research activities, and local K-12 outreach efforts are designed to foster an interest in applied mathematics and continued education.

This project builds a rigorous foundation for a new era of network-based mean field game theory and applications. The first goal is to identify the range of network structures for which the standard mean field approximation is valid, in order to substantially extend the scope of the mean field paradigm. For those network structures for which the standard mean field approximation fails, the second goal is to identify and analyze tractable alternatives. Certain important features of mean field approximations, such as asymptotic independence, fail in sparse (as opposed to dense) networks, and the third goal of the research is to identify precise sparsity thresholds for this and other phenomena. This project adapts and extends recent methodologies of graph limit theories and large deviations, which have a strong track record of solving network-based problems in statistical physics. New mathematical challenges arise in the game-theoretic setting due to the long-range dependencies induced by competitive equilibrium, and new techniques must be developed to address these challenges.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.