TA5 - Théorie des jeux / Game theory 3

This paper establishes a data-driven solution for infinite horizon linear quadratic Gaussian Mean Field Games with network-coupled heterogeneous agent populations where the dynamics of the agents are unknown. The solution technique relies on Integral Reinforcement Learning and Kleinman’s iteration for solving algebraic Riccati equations (ARE). The resulting algorithm uses trajectory data to generate network-coupled MFG strategies for agents and does not require parameters of agents’ dynamics. Under technical conditions on the persistency of excitation and on the existence of unique stabilizing solution to the corresponding AREs, the learned network-coupled MFG strategies are shown to converge to their true values.

We study valid inequalities for binary bilevel integer programs whose lower level is a shortest-path problem, arising, e.g., from a decision diagram reformulation. Using disjunctive programming, we characterize all classes of inequalities for the convex hull of bilevel-feasible solutions and reveal a connection between these inequalities and flows on the follower’s graph.

Keywords: Bilevel Programming; Integer Programming; Cutting plane.

In this work, we address the optimal energy trading (OET) problem in distribution grids managed by an energy grid operator (EGO) and a large number of prosumer households. We first introduce a clustering architecture that partitions the grid into residential prosumer clusters (RPCs), each coordinated by an aggregator responsible for internal energy exchanges and external interactions with neighbouring clusters and the EGO. To optimize energy exchanges within and across clusters, we develop a novel decentralized control framework based on major-minor graphon mean field game theory~(MM-GMFG). This framework models the OET as a decentralized dynamic game, deriving optimal control strategies that minimize each prosumer’s individual energy cost (minor agents) and optimal strategies for the EGO (major agent). Household (minor agent) dynamics and cost functions are influenced by: (i) local aggregate effects within their RPC and global interactions across interconnected cluster representing the minor agents local and global effect, and (ii) the major agent decisions (EGO strategies). EGO dynamics and cost function are influenced by the aggregate effect of all the minor agents. Optimal control strategies, interpreted as Nash Equilibria for the formulated game, are found via an application of MM-GMFG. Numerical experiments conducted on a dense energy network with 100 clusters, each containing 200 uniform households, validate the effectiveness of the proposed method in achieving scalable and cost-efficient energy coordination.