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Randomized Algorithms, Carnegie Mellon: Spring 2011

February 24, 2011

Posted by on 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.

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