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

不同的科学领域依赖于相互作用和竞争主体的大规模系统的数学模型。代理人可能是投资者、司机或病毒，而大系统可能是金融市场、高速公路或流行病。被称为“平均场博弈论”的最广泛使用的框架是对许多相互作用的代理之间的竞争进行建模，它的范围基本上局限于对称模型，在对称模型中，每个代理与其他代理之间的相互作用是平等的。另一方面，许多重要的现象是由网络控制的——比如由社会关系或经济义务形成的网络——这些网络决定了哪些主体相互作用。网络结构具有标准理论无法捕捉的重要含义。本项目通过开发新的数学框架来解决这一限制，以处理由网络控制的异构相互作用的大规模竞争模型。研究生和本科生都参与了研究活动，当地K-12的推广工作旨在培养对应用数学和继续教育的兴趣。本课题为基于网络的平均场博弈论及其应用的新时代奠定了坚实的基础。第一个目标是确定标准平均场近似有效的网络结构的范围，以便大大扩展平均场范式的范围。对于那些标准平均场近似失败的网络结构，第二个目标是识别和分析可处理的替代方案。平均场近似的某些重要特征，如渐近独立性，在稀疏（相对于密集）网络中失效，研究的第三个目标是为这种现象和其他现象确定精确的稀疏性阈值。该项目适应并扩展了图极限理论和大偏差的最新方法，这些方法在解决统计物理中基于网络的问题方面有很强的记录。由于竞争均衡引起的长期依赖，博弈论中出现了新的数学挑战，必须开发新的技术来解决这些挑战。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.