ECS1050.04.post

# ECS1050.04.post - UNIT II: The Basic Theory Zero-sum Games...

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UNIT II: The Basic Theory Zero-sum Games Nonzero-sum Games Nash Equilibrium: Properties and Problems Bargaining Games Review Midterm 7/16 7/2

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Nonzero-sum Games Examples: Bargaining Duopoly International Trade
Nonzero-sum Games The Essentials of a Game Eliminating Dominated Strategies Best Response Nash Equilibrium Duopoly: An Application Solving the Game Existence of Nash Equilibrium Properties and Problems See : Gibbons, Game Theory for Applied Economists (1992): 1-51.

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Solving the 2x2 Game T 1 T 2 3. Prisoner’s Dilemma 4. Stag Hunt 5. Chicken 6. Battle of the Sexes S 1 S 2 x 1 ,x 2 w 1 ,w 2 z 1 ,z 2 y 1 ,y 2
T 1 T 2 S 1 S 2 3,3 0,5 5 ,0 1 ,1 3. Prisoner’s Dilemma NE = {(S 2 ,T 2 )} Solving the Game

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T 1 T 2 S 1 S 2 5 ,5 0,3 3,0 1 ,1 4. Stag Hunt (also, Assurance Game) NE = {(S 1 ,T 1 ), (S 2 ,T 2 )} Solving the Game
T 1 T 2 S 1 S 2 3,3 1 ,5 5 ,1 0,0 5. Chicken (also Hawk/Dove) NE = {(S 1 ,T 2 ), (S 2 ,T 1 )} Solving the Game

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O F O F 5 ,3 0,0 0,0 3 ,5 6. Battle of the Sexes NE = {(O,O), (F,F)} Solving the Game Find the mixed strategy Nash Equilibrium
Solving the Game O F O F 5,3 0,0 0,0 3,5 Let (p,1-p) = prob 1 (O, F ) (q,1-q) = prob 2 (O, F ) Then EP 1 (O) = 5q EP 1 (F) = 3-3q q* = 3/8 EP 2 (O) = 3p EP 1 (O) = 5-5p p* = 5/8 NE = {(1, 1); (0, 0); (5/8, 3/8)} ); (0, 0); (5/8, 3/8 Game 6. p (1-p) q (1-q)

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Solving the Game O F O F 5,3 0,0 0,0 3,5 Let (p,1-p) = prob 1 (O, F ) (q,1-q) = prob 2 (O, F ) Then EP 1 (O|q) = 5q EP 1 (F|q) = 3-3q q* = 3/8 EP 2 (O|p) = 3p EP 2 (F) = 5-5p p* = 5/8 NE = {(1, 1); (0, 0); (5/8, 3/8) } ); (0, 0); (5/8, 3/8 Game 6. Equalizers
q OPERA 1 FIGHT 0 0 1 p Find the Best Response Functions for Pat p*(q); and Chris q*(p) Game 6. Solving the Game

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q OPERA 1 3/8 FIGHT 0 0 5/8 1 p Game 6. q*(p) if p<5/8, then Player 2’s best response is q* = 0 (FIGHT) if p>5/8 q* = 1 (OPERA) Solving the Game
q OPERA 1 3/8 FIGHT 0 0 5/8 1 p Game 6. q*(p) p*(q) NE = {(1, 1); (0, 0); (5/8, 3/8)} Solving the Game (p, q); (p, q)

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The Battle of the Sexes FIGHT OPERA 5, 3 0, 0 0, 0 3, 5 FIGHT OPERA efficiency equity bargaining power or skill P 1 P 2 NE = {(1, 1); (0, 0); (5/8, 3/8)} Game 6. (0,0) (1,1) (5/8, 3/8) Solving the Game
Existence of Nash Equilibrium Prisoner’s Dilemma Button-Button Battle of the Sexes GAME 3. GAME 2. GAME 6. 0 1 0 1 0 1 p q 1 0 There can be (i) a single pure-strategy NE; (ii) a single mixed-strategy NE; or (iii) two pure-strategy NEs plus a single mixed-strategy NE (for x=z; y=w). (i) (ii) (iii)

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Properties SADDLEPOINT v. NASH EQUILIBRIUM (Zero-sum) (Nonzero-sum) EXISTENCE: Does a solution always exist for the class of games? YES YES STABILITY: Is it self-enforcing? YES YES UNIQUENESS: Does it identify an unambiguous course of action? YES NO EFFICIENCY: Is it at least as good as any other outcome for all players? --- (YES) NOT ALWAYS SECURITY: Does it ensure a minimum payoff? YES NO
Properties [I]f game theory is to provide a unique solution to a game- theoretic problem then the solution must be a Nash equilibrium in the following sense. Suppose that game theory makes a unique prediction about the strategy each player will choose. In order for this prediction to be correct, it is necessary that each player be willing to choose the strategy predicted by the theory. Thus each player’s

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## This note was uploaded on 03/21/2010 for the course ECON 1050 taught by Professor Neugeboren during the Summer '09 term at Harvard.

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ECS1050.04.post - UNIT II: The Basic Theory Zero-sum Games...

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