ECS1050.03.post

ECS1050.03.post - UNIT II: The Basic Theory 6/30 Zero-sum...

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UNIT II: The Basic Theory Zero-sum Games Nonzero-sum Games Nash Equilibrium: Properties and Problems Bargaining Games Bargaining and Negotiation Review Midterm 7/16 6/30
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Zero-sum Games The Essentials of a Game Extensive Game Matrix Game Dominant Strategies Prudent Strategies Solving the Zero-sum Game The Minimax Theorem
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The Essentials of a Game 1. Players : We require at least 2 players (Players choose actions and receive payoffs.) 2. Actions : Player i chooses from a finite set of actions, S = {s 1 ,s 2 ,…. .,s n }. Player j chooses from a finite set of actions T = {t 1 ,t 2 ,……,t m }. 3. Payoffs : We define P i (s,t) as the payoff to player i, if Player i chooses s and player j chooses t. We require that P i (s,t) + P j (s,t) = 0 for all combinations of s and t. 4. Information : What players know (believe) when choosing actions. ZERO-SUM
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The Essentials of a Game 4. Information : What players know (believe) when choosing actions. Perfect Information : Players know their own payoffs other player(s) payoffs the history of the game, including other(s) current action* *Actions are sequential (e.g., chess, tic-tac-toe). Common Knowledge
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Extensive Game Player 1 chooses a = {1, 2 or 3} Player 2 b = {1 or 2} Player 1 c = {1, 2 or 3} Payoffs = a 2 + b 2 + c 2 if /4 leaves remainder of 0 or 1. -(a 2 + b 2 + c 2 ) if /4 leaves remainder of 2 or 3. Player1’s decision nodes -3 -6 -11 -6 9 -14 -6 9 -14 9 12 17 -11 -14 -19 -14 17 -22 GAME 1. “Square the Diagonal” (Rapoport: 48-9) Player 2’s decision nodes 1 3 2 1 2 1 2 3
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Extensive Game How should the game be played? Solution : a set of “advisable” strategies, one for each player. Strategy : a complete plan of action for every possible decision node of the game, including nodes that could only be reached by a mistake at an earlier node . Player1‘s advisable Strategy in red -3 -6 -11 -6 9 -14 -6 9 -14 9 12 17 -11 -14 -19 -14 17 -22 GAME 1. 1 3 2 1 2 1 2 3 Start at the final decision nodes (in red) Backwards-induction
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Extensive Game How should the game be played? Solution : a set of “advisable” strategies, one for each player. Strategy : a complete plan of action for every possible decision node of the game, including nodes that could only be reached by a mistake at an earlier node . Player1‘s advisable Strategy in red -3 -6 -11 -6 9 -14 -6 9 -14 9 12 17 -11 -14 -19 -14 17 -22 GAME 1. Player2’s advisable strategy in green 1 3 2 1 2 1 2 3 Player1’s advisable strategy in red
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Extensive Game How should the game be played? If both player’s choose their advisable (prudent) strategies, Player1 will start with 2, Player2 will choose 1, then Player1 will choose 2.
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ECS1050.03.post - UNIT II: The Basic Theory 6/30 Zero-sum...

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