Game Theory HW (Due in 10 hours)

Introduction to Game Theory B BUS 490 and B BUS 590

Problem Set 2 Due: Monday July 3

1. ”Rock - Paper -Scissors” can be modeled as a strategic form game be- tween players 1 and 2 with strategy sets S1 = S2 = {R, P, S}. Assume that payoffs are 1, 0 or −1 depending on who wins the game.

(a) Create the corresponding payoff matrix.

(b) Evaluate u2(R, P ) and u1(S, S).

(c) Does any player have a dominant strategy?

(d) Is there a pure strategy NE?

(e) Evaluate u2([ 1 4 R, 3

4 S], P ).

(f) What is the best pure strategy response for player 2 to [ 1 2 R, 1

2 S]?

(g) What is the best response for player 2 to [ 1 2 R, 1

2 S]?

(h) Show that ([ 1 3 R, 1

3 P, 1

3 S], [ 1

3 R, 1

3 P, 1

3 S]) is a NE of the game.

2. Three persons, Ann, Bob and Chad, have to decide to work or shirk in a group project. Payoffs are given according to the following information.

• The three group members will receive the same grade. • If at least two group members work the grade assigned will be

good, giving each student four units of payoff.

• If only one group member works the grade assigned will be mediocre, giving each student two units of payoff.

• If no group member works the grade assigned will be bad, giving each student zero units of payoff.

• Working implies an effort worth losing one unit of payoff.

(a) Create the payoff matrix for this three-player game.

(b) Does any player have a dominant strategy?

(c) What are the pure strategy NE of the game?

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3. Find all the pure and mixed NE of the following games:

(a)

Player 2

l r

Player 1 U (3, 1) (1, 2)

D (1, 4) (4, 0)

(b)

Player 2

l r

Player 1 U (5, 10) (8, 8)

D (0, 0) (10, 5)

(c)

Player 2

l m r

Player 1

U (1, 5) (6, 7) (9, 8)

M (3, 5) (7, 6) (6, 2)

D (0, 8) (4, 4) (6, 2)

Extra Credit (10 pts.) Find all the pure and mixed NE of the following game:

Player 2

l m r

Player 1

U (1, 5) (6, 7) (9, 8)

M (3, 10) (7, 6) (6, 2)

D (0, 8) (4, 4) (6, 2)

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