Lecture 3 Nash Equilibrium Continued 2010

# Lecture 3 Nash Equilibrium Continued 2010 - ECON 4109H...

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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 everyone’s 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....
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Lecture 3 Nash Equilibrium Continued 2010 - ECON 4109H...

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