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Unformatted text preview: Game Theory: MixedStrategy Nash Equilibrium Todd Sarver Northwestern University Econ 3102 Fall 2011 Todd Sarver (Northwestern University) Game Theory Econ 3102 Fall 2011 1 / 31 Outline 1 Games without PureStrategy NE 2 Mixed Strategies Definition Expected Utility MixedStrategy Nash Equilibrium 3 Tricks for Finding MSNE 4 Combining IESDS with MSNE 5 Existence of MSNE Todd Sarver (Northwestern University) Game Theory Econ 3102 Fall 2011 2 / 31 Not All Games Have a PureStrategy Nash Equilibrium Example (Penalty Kick) Suppose the match comes down to a penalty kick. . . Kicker L R Goalkeeper L 1 , 1 1 , 1 R 1 , 1 1 , 1 Todd Sarver (Northwestern University) Game Theory Econ 3102 Fall 2011 3 / 31 Not All Games Have a PureStrategy Nash Equilibrium Example (Penalty Kick) Suppose the match comes down to a penalty kick. . . Kicker L R Goalkeeper L 1 , 1 1 , 1 R 1 , 1 1 , 1 There is no PureStrategy Nash Equilibrium in this game. (Notice this game is strategically equivalent to Matching Pennies.) Todd Sarver (Northwestern University) Game Theory Econ 3102 Fall 2011 3 / 31 Not All Games Have a PureStrategy Nash Equilibrium Example (Penalty Kick) Suppose the match comes down to a penalty kick. . . Kicker L R Goalkeeper L 1 , 1 1 , 1 R 1 , 1 1 , 1 There is no PureStrategy Nash Equilibrium in this game. (Notice this game is strategically equivalent to Matching Pennies.) What predictions can we make in games like this one? Should we consider a different solution concept than NE? Todd Sarver (Northwestern University) Game Theory Econ 3102 Fall 2011 3 / 31 Not All Games Have a PureStrategy Nash Equilibrium Example (Penalty Kick) Suppose the match comes down to a penalty kick. . . Kicker L R Goalkeeper L 1 , 1 1 , 1 R 1 , 1 1 , 1 There is no PureStrategy Nash Equilibrium in this game. (Notice this game is strategically equivalent to Matching Pennies.) What predictions can we make in games like this one? Should we consider a different solution concept than NE? Or, instead, should we consider allowing for more strategies? Todd Sarver (Northwestern University) Game Theory Econ 3102 Fall 2011 3 / 31 Example (Penalty Kick) Kicker L R Goalkeeper L 1 , 1 1 , 1 R 1 , 1 1 , 1 Lets think about what really happens in this game to get an idea of how to proceed. Todd Sarver (Northwestern University) Game Theory Econ 3102 Fall 2011 4 / 31 Example (Penalty Kick) Kicker L R Goalkeeper L 1 , 1 1 , 1 R 1 , 1 1 , 1 Lets think about what really happens in this game to get an idea of how to proceed. If a kicker always kicks in one direction, he becomes predictable and the goalkeeper will be able to block his kick. Todd Sarver (Northwestern University) Game Theory Econ 3102 Fall 2011 4 / 31 Example (Penalty Kick) Kicker L R Goalkeeper L 1 , 1 1 , 1 R 1 , 1 1 , 1 Lets think about what really happens in this game to get an idea of how to proceed....
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 Spring '08
 SARVER
 Microeconomics, Game Theory

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