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.
