Research Projects

Large-scale interacting rational entities are ubiquitous in many disciplines of science and engineering, such as smart grids, financial engineering, and the spread of epidemics. One usually resorts to game-theoretic models to analyze such systems, and Nash equilibrium (NE) is a well-accepted solution concept that models their outcome. This research proposal aims to analyze NE of many-player games and use these insights to design incentive schemes for distributed electricity production in smart grids. Since analyzing large-player games is challenging, one usually considers a corresponding mean field game (MFG) and the notion of a mean field game equilibrium (MFGE) to approximate NE of many-player games. This approximation procedure (referred to as the convergence problem) has been made rigorous in many models of games with finite horizon costs. However, the convergence problem is poorly understood in several situations, such as (1) Markovian closed-loop controls, (2) infinite-horizon discounted cost models, or (3) when one has multiple MFGEs (referred to as the equilibrium selection problem). The first goal of this project is to analyze the above questions on the convergence problem in MFGs. While MFGs consider symmetrically interacting agents, many applications motivate us to consider games on sparse random graphs that model local interactions among the agents. This project aims to develop a general theory of games on sparse graphs, propose a notion of equilibrium in the spirit of MFGs, analyze the convergence of NE of games on suitable families of large random graphs to this equilibrium, and understand the equilibrium selection problem in the non-uniqueness regime. This project also aims to apply these results to the context of smart grids. There has been a recent interest in using solar panels for electricity generation in households and possibly supplying them to the grid. One can model this as a game among the households by assigning suitable cost functions and modeling their interactions (caused by the electricity price). A central planner could potentially incentivize the agents through prizes or subsidies to achieve a certain level of electricity production from renewable sources. This opens up the question of designing suitable incentive schemes to reach a desired level of distributed electricity production from renewable sources. We aim to address this question by applying our theoretical results (especially on the equilibrium selection problem). In conclusion, this project aims to understand fundamental theoretical questions about large-scale multi-agent systems using the framework of mean field games and games on sparse random graphs, as well as to apply these insights to aid the design of incentive schemes and prizes for distributed energy production in electricity networks.