This is a chapter from Jim Ratliff's Graduate-Level Game-Theory Course. See outline for the entire course. I no longer maintain, update, or correct these notes. However, I would appreciate hearing from people who download these notes and find them useful. Also I may eventually post problem sets and their solutions. Let me know if you'd like to be notified of such changes. (Please email me.)
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We'll now see explicitly how to find the set of (mixed-strategy) Nash equilibria for general two-player games where each player has a strategy space containing two actions (i.e. a "2x2 matrix game").
We first compute the best-response correspondence for a player. We partition the possibilites into three cases: The player is completely indifferent; she has a dominant strategy; or, most interestingly, she plays strategically (i.e., based upon her beliefs about her opponent's play).
The Nash equilibria of the game are the strategy profiles in the intersection of the two players' best-response correspondences. We use this fact and the possible forms of players' best-response correspondences to explore the possible sets of Nash equilibria in these games
We then apply this technique to two particular games. The first game is a typical and straightforwardly solved example; the second is nongeneric in the sense that it has an infinite number of equilibria. For each game we will compute the graph of each player's best-response correspondence and identify the set of Nash equilibria by finding the intersection of these two graphs.
§3.1: Nash Equilibrium
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§4.1: Introduction to Extensive-Form Games