Collaborative Research: Nash Equilibrium Problems under Uncertainty

纳什均衡模型为研究竞争环境中决策者之间的相互依赖关系提供了数学基础。这些模型提供了许多多智能体社会技术系统，其中包括通信，电力和交通网络的操作的见解。这些社会技术系统不仅必须适应技术限制（例如，电力流动的物理定律），但也必须承认几个决策者的目标，所有这些决策者都在不确定性下做出选择，并与他们的竞争对手竞争。作为一个具体的例子，考虑现代电力网络：一些发电是由大型机组进行的，而分布式发电和存储可能由个人运营，输电和配电资产由系统运营商和公用事业公司运营。此外，可再生能源（高度不稳定）的日益普及，燃料价格的不确定性，新技术和能源政策，为公司和/或个人的经营创造了一个具有挑战性的环境。该项目结合了系统约束的需要，以及在决策者之间缺乏合作的情况下风险和不确定性的影响，将解决一些最具挑战性的问题，即不确定性下的NE模型的结构，算法，可扩展性和解释。该项目更广泛的影响包括电力和通信网络市场的研究，以及研究生的培训和教育。目前的理论和算法方案在适应风险度量、分层和基于资源的决策以及耦合策略集的能力方面相对有限。出于这些缺陷，本研究将利用一个广泛的框架，可以容纳私人和耦合的约束，允许线性，二次和凸模型采取追索权，并捕捉风险规避通过合并的复合和偏差为基础的风险措施。本研究的目标包括以下内容：（i）分析：通过分析基于风险的变分和拟变分不等式问题，验证所得平衡点的存在性和唯一性;（ii）算法：设计，分析（收敛性，复杂性）和统计性质的估计算法依赖于联合抽样和近似方案;和（iii）应用：在竞争和不确定性下的电力市场和其他规划领域的问题将在两个阶段和层次的设置考虑。

Nash Equilibrium (NE) models provide the mathematical basis to study the interdependence between decision-makers in a competitive environment. These models provide insights into the operations of many multi-agent socio-technical systems, which include communications, electric power, and transportation networks. These socio-technical systems must not only accommodate technical constraints (e.g., physical laws of electrical power flow), but must also acknowledge the goals of several decision-makers, all of whom make choices under uncertainty and compete with their rivals. As a concrete example, consider the modern electricity network: some generation is carried out by large units, whereas distributed generation and storage may be operated by individuals, and transmission and distribution assets are operated by system operators and utilities. In addition, the growing penetration of renewable energy (highly volatile), the uncertainty about fuel prices, new technology, and energy policy, create a challenging setting in which the firms and/or individuals operate. This project, which combines the need to incorporate system constraints, as well as the effects of risk and uncertainty in the absence of cooperation across decision-makers, will address some of the most challenging questions regarding the structure, algorithms, scalability, and interpretations for NE models under uncertainty. The broader impact of the project includes the study of markets in power and communication networks, and the training and education of graduate students.Current theories and algorithmic schemes are relatively restrictive in their ability to accommodate risk measures, hierarchical and recourse-based decision-making, and coupled strategy sets. Motivated by these lacunae, this research will utilize a broad framework that can accommodate both private and coupled constraints, allows for linear, quadratic and convex models for taking recourse, and captures risk-aversion through the incorporation of composite and deviation-based risk measures. The goals of this research include the following: (i) Analysis: Verifiable statements for the existence and uniqueness of the resulting equilibria through the analysis of the resulting risk-based variational and quasi-variational inequality problems; (ii) Algorithms: Design, analysis (convergence, complexity) and statistical properties of estimators derived on algorithms reliant on joint sampling and approximation schemes; and (iii) Applications: Problems in electricity markets and other areas of planning under competition and uncertainty will be considered in two-stage and hierarchical settings.