Game Theory
Solution to Tutorial 1
2011/2012 Semester I MA4264 Tutor: Xiang Sun∗ August 24, 2011 Game Theory
1
Review
• “Static” means one-shot, or simultaneous-move; “Complete information” means that the payoff functions are common knowledge. • Normal-form representation: G = {S1 , . . . , Sn ; u1 , . . . , un }, where n is finite. • si is strictly dominated by si , if ui (si , s−i ) < ui (si , s−i ), ∀s−i ∈ S−i .
• Rational players do not play strictly dominated strategies, since they are always not optimal no matter what strategies others would choose. • Iterated elimination of strictly dominated strategies. This process is orderindependent. • Given other players’ strategies s−i ∈ S−i , Player i’s best response, denoted by Ri (s−i ), is the set of maximizers of maxsi ∈Si ui (si , s−i ), i.e., Ri (s−i ) = si ∈ Si : ui (si , s−i ) = max ui (si , s−i ) ⊂ Si . si ∈Si
We call Ri the best-response correspondence for player i. • Given s−i , the best response Ri (s−i ) is a set. • In the n-player normal-form game G = {S1 , . . . , Sn ; u1 , . . . , un }, the strategy profile (s∗ , . . . , s∗ ) is a pure-strategy Nash equilibrium if 1 n s∗ ∈ Ri (s∗ ), i −i equivalently, ui (s∗ , s∗ ) = max ui (si , s∗ ), −i i −i si ∈Si
∀i = 1, . . . , n,
∀i = 1, . . . , n.
• {Nash equilibrium(a)} ⊂ {Outcomes of IESDS}.
∗
Email: xiangsun@nus.edu.sg; Mobile: 9169 7677; Office: S17-06-14.
1
MA4264 Game Theory
2/10
Solution to Tutorial 1
2
Tutorial
Exercise 1. In the following normal-form games, what strategies survive iterated elimination of strictly dominated strategies? What are the pure-strategy Nash equilibria? T M B L 2, 0 3, 4 1, 3 C 1, 1 1, 2 0, 2 R 4, 2 2, 3 3, 0 U M D L 1, 3 −2, 0 0, 1 R −2, 0 1, 3 0, 1
Solution. 1. In the left game, for Player 1, B is strictly dominated by T and will be eliminated. Then the bi-matrix becomes to the reduced bi-matrix G1 . In the bi-matrix G1 , for Player 2, C is strictly dominated by R and the bimatrix G1
2011/2012 Semester I MA4264 Tutor: Xiang Sun∗ August 24, 2011 Game Theory
1
Review
• “Static” means one-shot, or simultaneous-move; “Complete information” means that the payoff functions are common knowledge. • Normal-form representation: G = {S1 , . . . , Sn ; u1 , . . . , un }, where n is finite. • si is strictly dominated by si , if ui (si , s−i ) < ui (si , s−i ), ∀s−i ∈ S−i .
• Rational players do not play strictly dominated strategies, since they are always not optimal no matter what strategies others would choose. • Iterated elimination of strictly dominated strategies. This process is orderindependent. • Given other players’ strategies s−i ∈ S−i , Player i’s best response, denoted by Ri (s−i ), is the set of maximizers of maxsi ∈Si ui (si , s−i ), i.e., Ri (s−i ) = si ∈ Si : ui (si , s−i ) = max ui (si , s−i ) ⊂ Si . si ∈Si
We call Ri the best-response correspondence for player i. • Given s−i , the best response Ri (s−i ) is a set. • In the n-player normal-form game G = {S1 , . . . , Sn ; u1 , . . . , un }, the strategy profile (s∗ , . . . , s∗ ) is a pure-strategy Nash equilibrium if 1 n s∗ ∈ Ri (s∗ ), i −i equivalently, ui (s∗ , s∗ ) = max ui (si , s∗ ), −i i −i si ∈Si
∀i = 1, . . . , n,
∀i = 1, . . . , n.
• {Nash equilibrium(a)} ⊂ {Outcomes of IESDS}.
∗
Email: xiangsun@nus.edu.sg; Mobile: 9169 7677; Office: S17-06-14.
1
MA4264 Game Theory
2/10
Solution to Tutorial 1
2
Tutorial
Exercise 1. In the following normal-form games, what strategies survive iterated elimination of strictly dominated strategies? What are the pure-strategy Nash equilibria? T M B L 2, 0 3, 4 1, 3 C 1, 1 1, 2 0, 2 R 4, 2 2, 3 3, 0 U M D L 1, 3 −2, 0 0, 1 R −2, 0 1, 3 0, 1
Solution. 1. In the left game, for Player 1, B is strictly dominated by T and will be eliminated. Then the bi-matrix becomes to the reduced bi-matrix G1 . In the bi-matrix G1 , for Player 2, C is strictly dominated by R and the bimatrix G1