# CMU Randomized Algorithms

Randomized Algorithms, Carnegie Mellon: Spring 2011

## Lecture #14: Game Theory

Today we discussed some key concepts in game theory and connections between (some of) them and results in online learning.

We began by discussing 2-player zero-sum games. The Minimax Theorem states that these games have a well-defined value V such that (a) there exists a mixed strategy p for the row-player that guarantees the row player makes at least V (in expectation) no matter what column the column-player chooses, and (b) there exists a mixed strategy q for the column-player that guarantees the row player makes at most V (in expectation) no matter what row the row-player chooses. We then saw how we could use the results we proved about the Randomized Weighted Majority algorithm to prove this theorem.

We next discussed general-sum games and the notion of Nash equilibria. In a zero-sum game, a Nash equilibrium requires the two players to both be playing minimax optimal strategies, but in general there could be Nash equilibria of multiple quality-levels for each player. We then proved the existence of Nash equilibria. However, unlike in the case of zero-sum games, the proof gives no idea of how to find a Nash equilibrium or even to approach one. In fact, doing so in a 2-player nxn game is known to be PPAD-hard.

Finally we discussed the notion of correlated equilibria and the connection of these to swap-regret, a generalization of the kind of regret notion we discussed last time. In particular, any set of algorithms with swap-regret sublinear in T, when played against each other, will have an empirical distribution of play that approaches (the set of) correlated equilibria.  See lecture notes.