Lecture 9_Mixed Strategy Equilibrium2010

Lecture 9_Mixed Strategy Equilibrium2010 - ECON 4109H...

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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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Lecture 9_Mixed Strategy Equilibrium2010 - ECON 4109H...

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