Min LQG games and collective discrete choice problems

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We introduce a class of finite horizon linear quadratic Gaussian games involving distinct potential finite destination states, interpreted as discrete choices under social pressure. The model provides stylized interpretations of opinion swings in elections, the dynamics of discrete societal choices, as well as a framework for achieving communication constrained group decision making in micro‐robotic based exploration. Two distinct cases are considered: 1. The zero noise or deterministic case where agents are initially randomly distributed over their range space; 2. The fully stochastic case. Under mild technical conditions, the existence of epsilon ‐Nash equilibria is established in both cases although these equilibria may in general be multiple. The corresponding agent control strategies are of a decentralized nature and are characterized in each case by the fixed points of a specific finite dimensional operator. Individual agent destination choices are fixed at the outset in case 1, while by contrast, their probability distribution evolves randomly along trajectories in case 2, with a deterministic limit for the complete population as the latter grows to infinity. This is joint work with Rabih Salhab and Jérôme Le Ny.

Prof. Roland Malhamé, École Polytechnique de Montréal

Dr. Roland Malhamé received the Bachelor’s, Master’s and Ph.D. degrees in Electrical Engineering from the American University of Beirut, the University of Houston, and the Georgia Institute of Technology in 1976, 1978 and 1983 respectively. After single year stays at University of Quebec, and CAE Electronics Ltd (Montreal), he joined in 1985 École Polytechnique de Montréal, where he is Professor of Electrical Engineering. In 1994, 2004, 2012, 2018 he was on sabbatical leave respectively with LSS CNRS (France), École Centrale de Paris,University of Rome Tor Vergata, and National Technical University of Athens. His interest in statistical mechanics inspired approaches to the analysis and control of large-scale systems has led him to contributions in the area of aggregate electric load modeling, and to the early developments of the theory of mean field games. His current research interests are in collective decentralized decision making schemes, and the development of mean field based control algorithms in the area of smart grids. From June 2005 to June 2011, he headed GERAD, the Group for Research on Decision Analysis. He is past Asoociate Editor of IEEE Transactions on Automatic Control, and International Transactions on Operations Research. He was elected Fellow of IEEE in 2022