# lecture41 - Lecture IV Nash Equilibrium Markus M Mbius o...

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Lecture IV: Nash Equilibrium Markus M. M¨obius February 19, 2004 Readings: Gibbons, sections 1.1.C and 1.2.B Osborne, sections 2.6-2.8 and sections 3.1 and 3.2 Iterated dominance is an attractive solution concept because it only as- sumes that all players are rational and that it is common knowledge that every player is rational (although this might be too strong an assumption as our experiments showed). It is essentially a constructive concept - the idea is to restrict my assumptions about the strategy choices of other players by eliminating strategies one by one. For a large class of games iterated deletion of strictly dominated strategies significantly reduces the strategy set. However, only a small class of games are solvable in this way (such as Counot competition with linear demand curve). Today we introduce the most important concept for solving games: Nash equilibrium. We will later show that all finite games have at least one Nash equilibrium, and that the set of Nash equilibria is a subset of the strategy pro- files which survive iterated deletion. In that sense, Nash equilibrium makes stronger predictions than iterated deletion would but it is not excessively strong in the sense that it does not rule out any equilibrium play for some games. Definition 1 A strategy profile s * is a pure strategy Nash equilibrium of G if and only if u i ( s * i , s * - i ) u i ( s i , s * - i ) for all players i and all s i S i . 1

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Definition 2 A pure strategy NE is strict if u i ( s * i , s * - i ) > u i ( s i , s * - i ) A Nash equilibrium captures the idea of equilibrium. Both players know what strategy the other player is going to choose, and no player has an incentive to deviate from equilibrium play because her strategy is a best response to her belief about the other player’s strategy.
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