Game theory : models, numerical methods and applications

Game theory is the theory of "strategic thinking". Developed for military purposes and defense, in the past it has also been used as an alternative and complementary approach to deal with robustness in the presence of worst-case uncertainties or disturbances in many areas such as economics, engineering, computer science, just to name a few. However, game theory is recently gaining ground in systems and control engineering, mostly in engineered systems involving humans, where there is a trend to use game theoretic tools to design protocols that will provide incentives for people to cooperate. For instance, scientists tend to use game theoretic tools to design optimal traffic flows, or predicting or avoiding blackouts in power networks or congestion in cyber-physical networked controlled systems. Incentives to cooperate are also crucial in dynamic resource allocation, multi-agent systems and social models (including social and economic networks). This paper assembles the material of two graduate courses given at the Department of Engineering Science of the University of Oxford in June-July 2013 and at the Department of Electrical and Electronic Engineering of Imperial College, in October-December 2013. The paper covers the foundations of the theory of noncooperative and cooperative games, both static and dynamic. It also highlights new trends in cooperative differential games, learning, approachability (games with vector payoffs) and mean-field games (large number of homogeneous players). The course emphasizes theoretical foundations, mathematical tools, modeling, and equilibrium notions in different environments.