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Lecture12-new

Lecture12-new - 6.254 Game Theory with Engineering...

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6.254 : Game Theory with Engineering Applications Lecture 12: Extensive Form Games Asu Ozdaglar MIT March 16, 2010 1
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Game Theory: Lecture 12 Introduction Outline Extensive Form Games with Perfect Information Backward Induction and Subgame Perfect Nash Equilibrium One-stage Deviation Principle Applications Reading: Fudenberg and Tirole, Chapter 3 (skim through Sections 3.4 and 3.6), and Sections 4.1-4.2. 2
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Game Theory: Lecture 12 Extensive Form Games Extensive Form Games We have studied strategic form games which are used to model one-shot games in which each player chooses his action once and for all simultaneously. In this lecture, we will study extensive form games which model multi-agent sequential decision making. Our focus will be on multi-stage games with observed actions where: All previous actions are observed, i.e., each player is perfectly informed of all previous events. Some players may move simultaneously at some stage k . Extensive form games can be conveniently represented by game trees . Additional component of the model, histories (i.e., sequences of action profiles). 3
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Game Theory: Lecture 12 Extensive Form Games Example 1 – Entry Deterrence Game: Entrant In Out A F Incumbent (2,1) (0,0) (1,2) There are two players. Player 1, the entrant, can choose to enter the market or stay out. Player 2, the incumbent, after observing the action of the entrant, chooses to accommodate him or fight with him. The payoffs for each of the action profiles (or histories) are given by the pair ( x , y ) at the leaves of the game tree: x denotes the payoff of player 1 (the entrant) and y denotes the payoff of player 2 (the incumbent). 4
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Game Theory: Lecture 12 Extensive Form Games Example 2 – Investment in Duopoly Player 1 Invest Not Invest Player 2 Cournot Game I c 1 = 0 c 2 = 2 Cournot Game II c 1 = 2 c 2 = 2 There are two players in the market. Player 1 can choose to invest or not invest. After player 1 chooses his action, both players engage in a Cournot competition. If player 1 invests, then they will engage in a Cournot game with c 1 = 0 and c 2 = 2. Otherwise, they will engage in a Cournot game with c 1 = c 2 = 2. We can also assume that there is a fixed cost of f for player 1 to invest. 5
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Game Theory: Lecture 12 Extensive Form Games Extensive Form Game Model A set of players, I = { 1, . . . , I } . Histories : A set H of sequences which can be finite or infinite. h 0 = initial history a 0 = ( a 0 1 , . . . , a 0 I ) stage 0 action profile h 1 = a 0 history after stage 0 . . . . . . h k + 1 = ( a 0 , a 1 , . . . , a k ) history after stage k If the game has a finite number ( K + 1) of stages, then it is a finite horizon game. Let H k = { h k } be the set of all possible stage k histories. Then H K + 1 is the set of all possible terminal histories , and H = K + 1 k = 0 H k is the set of all possible histories. 6
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Game Theory: Lecture 12 Extensive Form Games Extensive Form Game Model Pure strategies for player i is defined as a contingency plan for every possible history h k .
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