Chapter13b

Extensive presentation game tree normal form

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Extensive Presentation Game Tree. Normal Form Presentation Payoff Matrix. Player 2 (U,U) (U,D) (D,U) (D,D) Player 1 R (3,6) (3,6) (0,2) (0,2) L (0,2) (6,3) (0,2) (6,3)

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Equilibrium of a Game Sub-Game Perfect Nash Equilibrium Sub-game Perfect Nash equilibrium A player’s best response to a given strategy played by another player. Q: How do I find a sub-game perfect Nash equilibrium? A: Take game tree and use method called backward induction Remember, that the SPNE is a set of strategies, not an outcome or a sequence of actions The SPNE: (Player 1, Player 2) = (L;U,D) What are the other two NE? They are non-credible threats.
Equilibrium of a Game Sub Game Perfect Nash Equilibrium. Extensive Presentation Game Tree. Normal Form Presentation Payoff Matrix. Player 2 (U,U) (U,D) (D,U) (D,D) Player 1 R (3,6) (3,6) (0,2) (0,2) L (0,2) (6,3) (0,2) (6,3) SPNE

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Game Theory What can we conclude? Always use normal form/pay-off matrix to solve a simultaneous game. First identify if there exists dominant strategy for any of the players. If there is, the equilibrium must contain such dominant strategy. If no dominant strategy exists, solve the game using the very definition of Nash Equilibrium: my best response given your action. Always use the game tree to solve a sequential game using backward induction. The equilibrium is a sub-game perfect nash equilibrium. Eliminate all the non-credible threats.
Duopoly - Revisited Consider the following game: The players: Firm 1 and Firm 2 with same cost functions. Timing of a game: Simulteneous Information Availability: Perfect. Possible strategies: Firm1 (P B ,P M ) and Firm 2(P B ,P M ) The payoffs associated with each combination of strategies (table below) Repetition: Non - repeated. The decision rule: Max own payoff. Nash Equilibrium: (Player 1, Player 2) = (P B ;P B ) with payoff (1,1) Non-cooperate the lowest welfare outcome. How to induce cooperation?

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Duopoly - Revisited Penalty: Suppose there is a enforceable penalty of 5 charged to the firm who does not comply with the agreement (always play P M ) and is given to the one that suffers from that noncompliance.
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