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Title: Nash.pdf Created Date: 12/11/2001 4:05:15 PM

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A subgame-perfect Nash equilibrium is a Nash equilibrium because the entire game is also a subgame. The converse is not true. There can be a Nash Equilibrium that is not subgame-perfect. For example, the above game has the following equilibrium: Player 1 plays in the beginning, and they would have played ( ) in the proper subgame, as

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Pure Strategy Nash Equilibrium A strategy vector s = (s 1;:::;s k) is a pure strategy Nash Equilibrium (pure Nash) if c i (s) c i(s0;s i) for all i, and for all s0 i 2S i. Intuitively, no player is able to decrease their cost through unilateral action (choosing another of their strategies while everybody else remains the same).

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profile is a mixed‐strategy Nash equilibriumif and only if playing is a best response to ? . That is: Ü Ü ? Ü Ü Ü ? Üfor each Ü Ü • Fact #1 about mixed‐strategy Nash Equilibrium:A mixed strategy is Üis a best response to ? Üonly if Üassigns positive probability exclusively to

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Nash Equilibrium: Theory A very wide range of situations may be modeled as strategic games. For exam-ple, the players may be rms, the actions prices, and the preferences a reection of the rms’ prots. Or the players may be candidates for political ofce, the actions

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Nash equilibrium for a strategic game is a pro le of actions such that each action is a best response to the other actions. Let B i(a i) ˆA i be the set of player i’s best response actions against a i 2A i. Here is the formal de nition of Nash equilibrium. Nash Equilibrium a = (a 1

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