This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 don’t 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 1’s 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 ....
View
Full
Document
This note was uploaded on 01/19/2012 for the course ECON 4109H taught by Professor Notsure during the Fall '09 term at Minnesota.
 Fall '09
 Notsure

Click to edit the document details