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Unformatted text preview: ECON 4109H LECTURE 9 Mixed Strategy Equilibrium October 25, 2010 ECON 4109H LECTURE 9 Matching Pennies H T H 1, 1 1, 1 T 1, 1 1, 1 players match player 2 prefers to switch choice. players dont match player 1 prefers to switch choice. So there is no Nash equilibrium. Strategic games with ordinal preferences may not have any Nash equilibria. ECON 4109H LECTURE 9 Matching Pennies H T H 1, 1 1, 1 T 1, 1 1, 1 What if player 2 randomizes? 2 chooses heads with probability p , tails with probability ( 1 p ) . How does player 2 weigh the uncertain prospects that result from choosing heads? Either choice for player 1 results in an uncertain prospect. ECON 4109H LECTURE 9 Matching Pennies H T H 1, 1 1, 1 T 1, 1 1, 1 2 chooses H with probability p and T with probability 1 p . For 1, heads results in: match with probability p mismatch with probability 1 p . Tails results in mismatch with probability p match with probability 1 p . ECON 4109H LECTURE 9 Matching Pennies H T H 1, 1 1, 1 T 1, 1 1, 1 Assume 1 wants to maximize the probability of a match Let Prob ( Match  H ) = prob. of match if 1 chooses heads. Prob ( Match  T ) = prob. of match if 1 chooses tails. ECON 4109H LECTURE 9 Matching Pennies H T H 1, 1 1, 1 T 1, 1 1, 1 1s problem : max { Prob ( Match  X ) : X { H , T }} Equivalent to: max { [ 1 Prob ( Match  X )]+[( 1 ) Prob ( Mismatch  X )] : X { H , T }} ECON 4109H LECTURE 9 Matching Pennies [ 1 Prob ( Match  H )] + [( 1 ) Prob ( Mismatch  H )] > [ 1 Prob ( Match  T )] + [( 1 ) Prob ( Mismatch  T )] [ 1 Prob ( Match  H )] + [( 1 ) ( 1 Prob ( Match  H ))] > [ 1 Prob ( Match  T )] + [( 1 ) ( 1 Prob ( Match  T ))] [ 2 Prob ( Match  H )]  1 > [ 2 Prob ( Match  T )]  1 Prob ( Match  H ) > Prob ( Match  T ) ECON 4109H LECTURE 9 Matching Pennies with Randomization Players 2 players. Actions Each player chooses a probability p of playing heads. Preferences Preferences are given by the following utility functions: U 1 ( p 1 , p 2 ) = [ 1 p 1 p 2 ] + [( 1 ) p 1 ( 1 p 2 )] +[( 1 ) ( 1 p 1 ) p 2 ] + [ 1 ( 1 p 1 )( 1 p 2 )] U 2 ( p 1 , p 2 ) = [( 1 ) p 1 p 2 ] + [ 1 p 1 ( 1 p 2 )] +[ 1 ( 1 p 1 ) p 2 ] + [( 1 ) ( 1 p 1 )( 1 p 2 )] Player 1 wants to maximize the probability of matching. Player 2 wants to minimize the probability of matching. ECON 4109H LECTURE 9 Matching Pennies with Randomization p 1 = 1 corresponds to 1 playing heads. p 1 = 0 corresponds to 1 playing tails. U 1 ( p 1 , p 2 ) = [ 1 p 1 p 2 ] + [( 1 ) p 1 ( 1 p 2 )] +[( 1 ) ( 1 p 1 ) p 2 ] + [ 1 ( 1 p 1 )( 1 p 2 )] U 1 ( 1, p 2 ) = [ 1 p 2 ] + [( 1 ) ( 1 p 2 )] U 1 ( 1, 1 ) = 1 U 1 ( 1, 0 ) =  1 ECON 4109H LECTURE 9 Matching Pennies with Randomization Theorem The unique equilibrium in matching pennies with randomization is p * 1 = p * 2 = 1 / 2 ....
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 Fall '09
 Notsure

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