This preview shows pages 1–10. Sign up to view the full content.
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 3 NASH EQUILIBRIUM CONTINUED September 23, 2010 ECON 4109H LECTURE 3 Guess 2/3 of the Average Each student submits a real number between 0 and 100. All students whose numbers are closest to 2/3 of the average evenly split $1. All other students get $0. Assume that there are at least 2 students. ECON 4109H LECTURE 3 Guess 2/3 of the Average Theorem There is a unique Nash equilibrium in the Guess 2/3 of the Average game. In that equilibrium, all students submit 0. ECON 4109H LECTURE 3 Guess 2/3 of the Average First argue: all students submitting 0 is a Nash equilibrium. If all submit 0, all receive payment of $1 / n where n = # of students. Hold fixed everyones behavior except i . Suppose that i submits r > 0. Then 2/3 of the new average is 2 / 3 ( r / n ) . But: 2 3 r n < r n 6 r 2 , (last inequality follows from the fact that there are at least 2 students.) So 0 is closer to 2/3 of the average than r . So i is not a winner and makes $0 instead of $1 / n . So he is worse off. So everyone submitting 0 is a Nash equilibrium. ECON 4109H LECTURE 3 Theorem There is a unique Nash equilibrium in the Guess 2/3 of the Average game. In that equilibrium, all students submit 0. ECON 4109H LECTURE 3 Guess 2/3 of the Average Next argue nothing except everyone bidding 0 is a Nash equilibrium. In order to do this, I must prove a lemma. ECON 4109H LECTURE 3 Guess 2/3 of the Average Lemma Fix a strategy profile a = ( a 1 , a 2 , . . . , a i , . . . , a n ) . If, instead of bidding a j , player j bids a j := 2 3 n 2 i I \ j a i , j will be bidding exactly 2/3 of the average and will be one of the winners. ECON 4109H LECTURE 3 Guess 2/3 of the Average 2 3 1 n ( X i I \ j a i + a j ) = 2 3 1 n ( X i I \ j a i + 2 3 n 2 X i I \ j a i ) = 2 3 1 n ( 1 + 2 3 n 2 ) X i I \ j a i = 2 3 1 n ( 3 n 2 3 n 2 + 2 3 n 2 ) X i I \ j a i = 2 3 1 n 3 n 3 n 2 X i I \ j a i = 2 3 n 2 X i I \ j a i = a j ECON 4109H LECTURE 3 Guess 2/3 of the Average Lemma Fix a strategy profile a = ( a 1 , a 2 , . . . , a i , . . . , a n ) . If, instead of bidding a j , player j bids a j := 2 3 n 2 i I \ j a i , he will be one of the winners....
View
Full
Document
 Fall '09
 Notsure

Click to edit the document details