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I was reading this neat piece by Fudenberg http://fudenberg.fas.harvard.edu/predictive%20game%20theory.pdf on Predictive Game Theory. He argues correctly that most traditional work in Game Theory is not suited for prediction in real world set-ups.

Do you know any work that has advanced the theory or empirics of predictive game theory?

I am aware of some recent work of partial identification for games but I am more interested in learning in games and off-equilibrium dynamics.

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Actually, Section 3 of the paper you linked is probably one of the most comprehensive lists you could find of the movers and the shakers with regard to predictive game theory. I know that many of them have both published and working papers addressing these issues. Fudenberg even addresses those specific two topics. A few selections from the paper:

Ignacio Esponda, Philippe Jehiel, and David K. Levine are leaders in studying adaptive processes in extensive form games, and the sorts of non-Nash equilibrium outcomes that can persist even when players have a lot of experience with the game.

Michel Benaïm, Josef Hofbauer, William Sandholm, and Sylvain Sorin are making important advances in the mathematics of dynamical systems and applying them to non-equilibrium dynamics.

Chaim Fershtman and Ariel Pakes are developing estimation methods for field data that allow for incorrect off-path beliefs.

Jeff Shamma is a pioneer in bringing techniques from the feedback-control literature to the study of learning in games.

I would recommend going through these authors' websites and looking for papers that fit.

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