Chapter-7 - Mixed Strategies In baseball, what type of...

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Mixed Strategies In baseball, what type of pitch does the pitcher throw: fastball, splitter, curveball, slider? What type of pitch does the batter prepare for. • In penalty kicks of soccer, where should the kick direct the ball: left, middle, or right? Where should the goalkeeper dive?
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Police Patrol and Drug Trade A drug dealer can either work his trade at a street corner and a park. • A police officer can either patrol on the street corner or at the park. Drug dealer Police officer 80, 20 0, 100 10, 90 60, 40 Street corner Park Street corner Park
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Police Patrol and Drug Trade No (pure) strategy Nash equilibrium in this game. • Players may randomize, or play mixed strategies . -The police patrols at street corner with probability p, and patrols at park with probability 1-p. -The drug dealer chooses street corner with probability d, and chooses park with probability 1-d. -The randomizations are independent.
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Police Patrol and Drug Trade pd × Table: Outcomes in the Police Patrol and Drug Trade game 40 60 park park 90 10 streets park 100 0 park streets 20 80 streets streets Dealer’s payoff Officer’s payoff Probability Dealer’s choice Officer’s choice ( ) 1 × ( ) 1 × ( ) ( ) 11 p d ×−
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Police Patrol and Drug Trade • A strategy pair can be represented by • Given the expected payoff to the police officer: The expected payoff to the drug dealer: ( ) ,, pd ( ) p d ( ) ( ) ( ) ( ) ( ) ,8 0 1 0 1 1 0 1 1 6 0 60 60 50 130 PO Vp d p d p d p d p d p d = × ×+ × −× + =− + () ( ) ( ) ( ) ( ) , 20 1 100 1 90 1 1 40 40 60 50 130 DD dp d p d p d p d p d × +×− × +− × × ×− × =+ +
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Police Patrol and Drug Trade • Suppose the drug dealer works at street corner with probability d.
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Chapter-7 - Mixed Strategies In baseball, what type of...

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