# Game Theory

2011/2012 Semester I MA4264 Tutor: Xiang Sun∗ August 24, 2011 Game Theory

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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.

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MA4264 Game Theory

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Solution to Tutorial 1

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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 becomes to the reduced bi-matrix G2 . In the bi-matrix G2 , for Players 1 and 2, no strategies is strictly dominated. Hence the strategies T , M , L and R will survive iterated elimination of strictly dominated strategies. L 2, 0 3, 4 C 1, 1 1, 2 G1 R 4, 2 2, 3 L 2, 0 3, 4 R 4, 2 2, 3 G2 U M D L 1, 3 −2, 0 0, 1 H R −2, 0 1, 3 0, 1

T M

T M

In the bi-matrix G2 , we will obtain that the Nash equilibria are (M, L) and (T, R) (red pairs in the bi-matrix). 2. In the right game, it is easy to see that no strategy is strictly dominated. Hence all strategies will survive iterated elimination of strictly dominated strategies. From the bi-matrix H, we will obtain that the Nash equilibria are (U, L) and (M, R) (red pairs in the bi-matrix). Exercise 2. An old lady is looking for help crossing the street. Only one person is needed to help her; more are okay but no better than one. You and I are the two people in the vicinity who can help, each has to choose simultaneously whether to do so. Each of us will get pleasure worth of 3 from her success (no matter who helps her). But each one who goes to help will bear a cost of 1, this being the value of our time taken up in helping. Set this up as a game. Write the payoﬀ table, and ﬁnd all pure-strategy Nash equilibria. Solution. • There are two players: You (Player 1) and I (Player 2);

• For each player, he/she has 2 strategies: “Help” and “Not Help”.

MA4264 Game Theory

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Solution to Tutorial 1

Player 1

Help Not help

Player 2 Help Not help 2, 2 2, 3 3, 2 0, 0 K

• Since there are 2 players, and 2 pure strategies for each player, the payoﬀ function can be represented by a bi-matrix K: From the bi-matrix K, we will ﬁnd the Nash equilibria are (Help, Not help) and (Not help, Help) (red pairs in the bi-matrix). Exercise 3. There are three computer companies, each of which can choose to make large (L) or small (S) computers. The choice of company 1 is denoted by S1 or L1 , and similarly, the choices of...

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