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

科学的不同领域依赖于相互作用和相互竞争的大系统的数学模型。例如，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。最广泛使用的建模许多相互作用的代理之间的竞争的框架，被称为“平均场博弈理论”，从根本上限制在对称模型的范围内，在对称模型中，每个代理彼此平等地交互。另一方面，许多重要的现象是由网络管理的-例如那些由社会关系或财务义务形成的网络--这些网络决定了哪些代理人相互作用。网络结构具有标准理论无法捕捉到的重要含义。该项目通过开发新的数学框架来处理大规模竞争模型，以及由网络管理的异质交互，从而解决了这一限制。研究生和本科生都参与了研究活动，当地的K-12推广活动旨在培养人们对应用数学和继续教育的兴趣。该项目为基于网络的平均场博弈理论和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对其有效的网络结构的范围，以便实质性地扩展平均场范例的范围。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容易处理的替代方案。平均场近似的某些重要特征，如渐近独立性，在稀疏(而不是密集)网络中是失败的，研究的第三个目标是确定这种现象和其他现象的精确稀疏阈值。这个项目采用和扩展了最近的图形极限理论和大偏差的方法，这些方法在解决统计物理中基于网络的问题方面有着很好的记录。由于竞争均衡引起的长期依赖，博弈论环境中出现了新的数学挑战，必须开发新的技术来应对这些挑战。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.