Dynamic game theory (DGT) provides a mathematical framework for analyzing multi-agent decision processes that evolve over time. DGT has found successful applications in engineering, management science, and economics, where dynamic multi-agent decision problems naturally arise. Some representative engineering applications include cyber-physical systems, smart grids, communications and networking. A significant portion of existing DGT models in the literature is formulated in an unconstrained setting. However, real-world multi-agent decision situations involve practical aspects like saturation constraints, bandwidth limitations, production capacity and budget constraints. These considerations translate into equality and inequality constraints when specifying the dynamic game model. Consequently, the decisions available to each player become intricately linked to the choices made by other players at every stage of the game. As a result, players' decision sets are interdependent, often referred to as being coupled. In the literature on static games, which deal with decision problems where players act only once, generalized Nash equilibrium has been proposed as a solution concept (as an extension of Nash equilibrium), when the decision sets of the players are coupled. The related literature in the dynamic setting still remains limited.

In the first part of the seminar, we analyse a class of linear quadratic difference games with inequality constraints In particular, we assume that players' decision sets at each stage are coupled, and characterized by affine-inequality constraints involving state and control variables. We show that the necessary conditions for the existence of generalized open-loop Nash equilibria (GOLNE) result in two coupled discrete-time linear complementarity systems.The sufficient conditions are then derived through an equivalence between the solutions of these systems and the convexity of players' objective functions. With additional assumptions, we show that GOLNE strategies can be obtained by solving a large-scale linear complementarity problem.

The second part of the seminar focuses on mean-field-type difference games (MFTDGs). MFTDGs are a specific class of stochastic dynamic games that allow for the inclusion of not just the state and control terms, but also their distributions in the objective functionals and state dynamics. We focus on a discrete-time setting with finite-horizon, scalar linear-state dynamics, and quadratic objectives. Our approach incorporates coupled affine-inequality constraints on the mean values of state and control variables. Using the direct method, also known as completion of squares, we establish a connection between the existence of a solution for these MFTDGs and the existence of a multiplier process satisfying implicit complementarity conditions.