26_ContinuousGames_ToPost

26_ContinuousGames_ToPost - ECON 410 Continuous Games...

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Unformatted text preview: ECON 410 Continuous Games “’Over’? Did you say ‘over’? Nothing is over until we decide it is!” – Bluto Blutarsky 2 Class 26 - Continuous Games 3 Class 26 - Continuous Games 4 Class 26 - Continuous Games 5 Class 26 - Continuous Games Normal-Form Game: 1. A set of players 2. Moves each player can make 3. Payoffs the players might receive under each combination of moves Cardinality of action space if infinite If there are an infinite number of actions, then there must be a corresponding infinite number of payoffs. 6 Class 26 - Continuous Games Problem: 1. Cardinality of action space is infinite: Define the action space as an interval (or nicely defined set) instead of listing every possibility. Example: “Player 1 has actions {1, 2, 3, 4, 5, 6, 7, 8, 9,10, 11…} Player 2 has actions {1, 2, 3, 4, 5, 6, 7, 8, 9,10, 11…} Player 3 has actions {1, 2, 3, 4, 5, 6, 7, 8, 9,10, 11…}” can be written } 3 , 2 , 1 { , ∈ ∈ i N a i 7 Class 26 - Continuous Games Problem: 2. If there are an infinite number of actions, then there must be a corresponding infinite number of payoffs: Use a payoff function, which maps every possible strategy set to a payoff. Example: “If P1 plays 1 and P2 plays 1 P1’s payoff=2 P2’s payoff=1 If P1 plays 1 and P2 plays 2 P1’s payoff=5 P2’s payoff=1 If P1 plays 2 and P2 plays 2 P1’s payoff=6 P2’s payoff=2” can be written 1 2 2 2 1 ; 1 a a a = Π + = Π 8 1, -1-1, 1-1, 1 1, -1 Player 2 Tails Head s Tails Heads Player 1 (1- ) (1- ) Class 26 - Continuous Games 9 Class 26 - Continuous Games Normal Form Game 1. Set of players : {Player 1, Player 2} 2. Strategy s pace of each player: Player 1: [0,1], where is the probability of playing Heads Player 2: [0,1], where is the probability of playing Heads 3. Payoff each player receives given any s trategy profile: Player 1: 1=[(1)() + (-1)(1-)] + (1- ) [(-1)() + (1)(1-)] Player 2: 2= [(1)() + (-1)(1- )] + (1- ) [(-1)() + (1)(1- )] 10 Best Response Graph A Best Response Graph is a graph that shows a player’s best response to every possible strategy her opponent can play. Class 26 - Continuous Games 11 BR 1 BR 2 Class 26 - Continuous Games 12 Class 26 - Continuous Games Steps to Solve for the NE of a 2-Player Continuous Game 1. For Player 1, derive the BR Graph (or Function) by finding Player 1’s BR for each possible strategy of Player 2. 2. For Player 2, derive the BR Graph (or Function) by finding Player 2’s BR for each possible strategy of Player 1. 3. To find the Nash, find the set of strategies that simultaneously lies on each player's BR Graph. This is just the graphical interpretation of “find the set of strategies that simultaneously solves the BR functions”. 13 Class 26 - Continuous Games q1 q2 BR 1 BR 2 * 1 2 0.5 5 q q = - + * 2 1 0.5 5 q q = - + 2 2 * * 0.5( ) 5 .5 5 q q = +- +- 2 2 * * 0.25 2.5 5 q q =- + 2 * 0.75 2.5 q = * 2 3.33 q = * 1 (0.5) 3.3 ( ) 5 3 q = - + * 1 3.33 q = 14 Class 26 - Continuous Games Group-Clicker Question (P): A firm with market power faces an inverse...
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This note was uploaded on 01/05/2012 for the course ECON 410 taught by Professor Codrin during the Fall '07 term at UNC.

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26_ContinuousGames_ToPost - ECON 410 Continuous Games...

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