Solutions for Homework 8
Chapter 10: Problem 1
a. Neither player has a dominant strategy.
b. Player 1’s secure strategy is B. Player 2’s secure strategy is E.
c. Nash equilibrium states, given the strategies of other players, no player can improve their payoff by unilaterally changing their own strategy.
Chapter 10: Problem 2
a.
Player 1Player 2
Strategy / A / B
A / $400, $400 / $100, $600
B / $600, $100 / $200, $200
b. B is dominant for each player.
c. (B, B).
d. Joint payoffs from (A, A) > joint payoffs from (A, B) = joint payoffs from (B, A) > joint payoffs from (B, B).
e. No; each firm’s dominant strategy is B. Therefore, since this is a one-shot game, each player would have an incentive to cheat on any collusive arrangement.
Chapter 10: Problem 3
a. Player 1’s optimal strategy is A. Player 1 does not have a dominant strategy. However, by putting herself in her rival’s shoes, Player 1 should anticipate that Player 2 will choose E (since E is Player 2’s dominant strategy). Player 1’s best response to E is A.
b. Player 1’s equilibrium payoff is 18.
Chapter 10: Problem 4
a. (A, C).
b. No.
c. Nash equilibrium results in suboptimal solution of πN = 30 for each firm. Cooperation (B, D) results in superior solution for both players πCoop = 60. Benefit of cheating πCheat - πCoop = 70 – 60 = 10 is earned only in the round in which player cheats. The cost of cheating is πCoop - πN = 60 – 30 = 30 is incurred in each subsequent round. Present value perpetuity is perpetual cash divided by the interest rate i. Any player cheats if benefit exceeds cost and cooperates if benefit is less than or equal to the cost. Condition for cooperative behavior (benefit is less than or equal to present value perpetuity of cost) can be written as benefit divided by perpetual cost is less than or equal to one over interest rate: Here, πCheat = 70, πCoop = 60, πN = 30 and i = .06. Since (70-60)/(60-30) = 1/3 < 1/0.06 = 16.67 each player can earn 60 via trigger strategies.
d. Yes.
Chapter 10: Problem 5
a. x > 4.
b. x < 5.
c. x < 5.
Chapter 10: Problem 6
a. See the accompanying figure.
($0, $25)
Right
Right ($20, $20)
Left
Left (-$5, $10)
b. ($0, $25) and ($20, $20).
c. ($20, $20) is the only subgame perfect equilibrium; the only reason ($0, $25) is a Nash equilibrium is because Player 2 threatens to play left if 1 plays left. This threat isn’t credible.
Problem Set: Problem 36
Firm BFirm A / 2-0 / 1-1 / 0-2 / Best response
Firm B
3-0 / (27,21) / (22.5,25.5) / (30,18) / 1-1
2-1 / (33,15) / (29,19) / (36,12) / 1-1
1-2 / (28,20) / (27,21) / (39,9) / 1-1
0-3 / (18,30) / (13.5,34.5) / (25.8,22.2) / 1-1
Best response
Firm A / 2-1 / 2-1 / 1-2
This is a constant-sum game (constant=48).
b. The strategy 1-1 is a dominant strategy for Firm B.
The predicted Nash Equilibrium in terms of strategies: (2-1,1-1)
The predicted Nash Equilibrium in terms of payoffs: (29,19).
Problem Set: Problem 37
TownMotorist / Enforce / Don't Enforce / Best response Town
Obey / (0,-15) / (0,0) / Don't Enforce
Disobey / (-20,-20) / (5,-10) / Don't Enforce
Best response Motorist / Obey / Disobey
a. The Nash Equilibrium in terms of strategies: (Disobey, Don't Enforce)
The Nash Equilibrium in terms of payoffs: (5,-10).
b. If the town precommits to enforcement, the motorist obeys. However, the town is not interested in such a commitment since its payoff with commitment is -15 and without commitment is -10.
Problem Set: Problem 38
Trump / 139 / 119 / 99 / Best response Pan-Am
139 / (34,38) / (15,42) / (6,32) / 119
119 / (42,20) / (22,22) / (10,25) / 99
99 / (35,7) / (27,9) / (18,16) / 99
Best response Trump /
119 / 99 / 99
a. (99,99).
b. With repeated play firms wish to coordinate on the better outcome (139,139).