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Lecture 2 Nash Equilibrium 2010

Lecture 2 Nash Equilibrium 2010 - ECON 4109H LECTURE 2 NASH...

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ECON 4109H LECTURE 2 NASH EQUILIBRIUM September 13, 2010 ECON 4109H LECTURE 2
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Strategic Games Definition A strategic game (with ordinal preferences) consists of: a set of players for each player, a set of actions for each player, preferences over the set of action profiles. ECON 4109H LECTURE 2
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What Is an Action Profile? We may assume that the set of players is { 1, 2, . . . , n } We can denote the set of actions for player i by A i . An action for player i is then a i . An action profile is a list ( a 1 , a 2 , . . . , a n ) of actions, one for each player. Players care about the entire action profile, not just a particular action. ECON 4109H LECTURE 2
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Example: Meeting in New York Suppose that there are two players, Ann and Bob, who must meet in New York, but cannot communicate ahead of time (no cell phones). They can meet at the Empire State Building (E) or Grand Central Station (G). Thus they each have two actions: A Ann = A Bob = { E , G } . The action profiles are ( E , E ) , ( E , G ) , ( G , E ) , ( G , G ) . Suppose that the players don’t care where they meet, but just that they meet. So they both prefer ( E , E ) and ( G , G ) to both ( E , G ) and ( G , E ) . This shows why players may care about action profiles, not just their own actions, or just the other person’s actions. ECON 4109H LECTURE 2
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Simultaneous Moves In previous example, assumed no cell phones. In general in a strategic form game, it is assumed that players make their choices independently and simultaneously. Later we will study extensive form games, which allow us to model back and forth interaction over time. ECON 4109H LECTURE 2
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Simultaneous Moves Simultaneous moves are more general than they appear. For example, suppose Ann moves first and may chooses a location. Only after she arrives may she use a cell phone to tell Bob where she is. Then Bob chooses his location. Ann moves first, and Bob moves second, but can we think of this as a simultaneous moves game? ECON 4109H LECTURE 2
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Simultaneous Moves Suppose that Bob chooses a plan . Let X Y mean: “If Ann tells me she is at X , I will go to Y .” Then assuming that when Ann calls she must tell the truth, Ann still has 2 actions A Ann = { E , G } , where E means: “Go to the Empire State Building and Call Bob to tell him I am there,” and G is similar. Bob now has 4 actions: A Bob = 1. E E , G E 2. E E , G G 3. E G , G E 4. E G , G G Bob’s choice of plan is simultaneous with and independent of Ann’s choice of location.
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