Title: Linear quadratic dynamic games with constraints
Speaker: Dr. Puduru Viswanatha Reddy, GERAD, HEC Montreal
Time: 930am, 4/11/15, Wednesday
Venue: Room 205, ME Building
Abstract: Dynamic game theory, which deals with decision making processes involving multiple persons occurring over time, has been successfully used to study a large variety of problems in many fields, such as engineering, economics and management science. A significant share of these studies selected a linear-quadratic dynamic game (LQDG) structure to model the situation at hand. The popularity of LQDGs is essentially due to two reasons. First, while being tractable, LQDGs capture interactions between the control and state variables of the different players and allow for a non-constant return to scale, two features that occur quite naturally in many applications. The linear dynamics, which are a priori restrictive, are still considered an acceptable approximation of a possibly non-linear specification. Second, LQDGs yield closed-form expressions for equilibrium strategies, and importantly, theorems characterizing the existence and uniqueness of equilibria are available. In brief, the conceptual and methodological grounds of LQDGs are very much established and provide an attractive ready-for-use framework for many applications. In this work, we consider LQDGs where the players face equality and inequality constraints that jointly involve the state and control variables. Although such constraints are inherently present in many applications, there are no general results available on the existence and uniqueness of equilibria in constrained linear-quadratic dynamic games (Con-LQDGs). To achieve this objective, we consider linear dynamics and linear constraints in the dynamic game model. Then, we make two simplifying assumptions. First, we suppose that the players have two types of control variables, namely, (i) variables that (directly) affect the dynamics but do not enter the constraints, and (ii) variables that enter the constraints jointly with the state variables but do not appear in the dynamics. Second, we assume that the objective functions are separable in the two types of control variables, that is, that there are no cross-terms between them. We characterize the existence of Nash equilibria under constrained open-loop and constrained feedback information structures. In the open-loop case, we show that the existence of constrained Nash equilibrium is closely related to the solvability of a parametric two-point boundary value problem. In the feedback case, we show that the constrained feedback Nash equilibrium can be obtained from a feedback Nash equilibrium associated with an unconstrained parametric linear-quadratic game. Further, with few additional assumptions, we show that the Nash equilibria, under both the informational settings, have a geometric interpretation as intersection of two parametric curves. Finally, for this class of games we show that the constrained Nash equilibria can be obtained as solutions of a single large scale linear complementarity problem.
Based on the work P. V. Reddy and G. Zaccour. Open-loop Nash Equilibria in a Class of Linear-Quadratic Difference Games with Constraints. IEEE Transactions on Automatic Control, 60(9), 2559 - 2564, 2015 P. V. Reddy and G. Zaccour. Feedback Nash Equilibria in Linear-Quadratic Difference Games with Constraints. (submitted)
Short Bio: Dr. Puduru Viswanadha Reddy is currently a post-doc at GERAD (Group for Research in Decision Analysis), HEC Montreal, Canada. He holds a Ph.D in Operations Research from Tilburg University, The Netherlands, M.S. from the University of Maryland, M.Sc (Engg) from the Indian Institute of Science and B.Tech from Sri Venkateswara University. His research interests are in the areas of game theory and control theory with applications in engineering, management science and economics.