I am looking for a good introductory level textbook (or lecture notes) on mean field games that would be suitable for a graduate course. Ideally, it would include some brief words about optimal control and dynamic programming. Thanks!
The lecture notes on mean-field games of Lions, from a course at the Collège de France, have been typed out by Pierre Cardaliaguet. They address both optimal control and dynamic programming. I would think it will be hard to beat these for an authoritative exposition. For a summary of open problems you could look here.
Here's my go to links:
PDE flavor notes by Ryzhik: https://math.stanford.edu/~ryzhik/STANFORD/MEAN-FIELD-GAMES/notes-mean-field.pdf
Probability flavor notes by Lacker: http://www.columbia.edu/~dl3133/MFGSpring2018.pdf
Background material on mean-field interacting processes by Golse: https://arxiv.org/abs/1301.5494
1). There is a two volume textbook "Probabilistic Theory of Mean Field Games with Applications " by Carmona and Delarue
2). A very comprehensive textbook on MFG is being written by Tamer Basar, but I don't know if it is ready yet.