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Answers to Back-of-the-Chapter Problems
1. In a Nash equilibrium, each player’s chosen strategy is optimal, given
2. It is never advantageous to move first in a zero-sum game. (The best
one can do is to choose one’s equilibrium pure strategy, if there is one.)
4. a. The completed table is:
Venezuela
Saudi Arabia 3 M barrels 4 M barrels
b. Saudi Arabia’s dominant strategy is to produce 8 million barrels.
.
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c. The basic asymmetry is in the size of the countries’ outputs. By cutting
price, Venezuela can expand output by 33.3 percent. For the same price
5. a. There are two equilibria: firm J develops E and firm K develops D,
6. a. The unique equilibrium outcome has firm A choosing High and firm B
b. If firm B chooses Low spending, firm A’s best response is Medium
spending. If firm B chooses Medium spending, firm A’s best response is
c. The firms should coordinate their advertising strategies by selecting
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7. a. Applying the method of “circles and squares” to the payoff table, we
see that there are two Nash equilibria: i) Both superpowers escalate
b. Yes, with the fall of the former Soviet Union, it appears that the
8. a. Applying the method of “circles and squares” to the payoff table, we see
b. With an indefinite number of moves, each should begin and continue to
implement Medium R&D as long as the other does. If either ever
c. If it’s hard to discern the rival’s R&D moves, it’s more difficult to
b. Once C3 is eliminated from consideration, R1 is dominated by R2.
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10. a. The payoff table below shows the total sales of the firms. (Carefully
Firm B
Firm A 2-0 1-1 0-2
b. For Firm A, the strategies 3-0 and 0-3 are dominated by both 2-1 and
1-2. But neither 2-1 nor 1-2 dominates the other. For Firm B, the
dominant strategy is 1-1. Regardless of Firm A’s action, 1-1 gives better
11. a. The town’s dominant strategy is non-enforcement. Anticipating this, the
b. If the town can make the “first move” by committing to 100 percent
enforcement, the situation changes. The typical motorist’s best response
c. Now the town enforces the law with probability p. The typical motorist
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equal to one another implies p = .2. As long as the enforcement
12. a. The buyer does not have a dominant strategy. She buys two units at P
b. With multiple rounds, the buyer could vary its purchases to encourage
c. Maximum total profits (32) are achieved at Q = 8 units. A negotiated
Discussion Question
a. In a one-time play, charging $139 is a dominated strategy; after its
b. The only equilibrium is for each airline to charge $99 in every period.
Obviously, this results in poor overall payoffs for both airlines. With
c. Top performance occurs when both airlines aim for a collusive outcome
($139 fares) and neither deviates to lower fares until near the end of the
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Spreadsheet Problems
S1. a. The complete payoff table is shown below.
Average Auto Inventory of the Other Dealers
My Inventory 175 200 225 250 275 300 325
b. If each competitor’s inventory averages 325, then Dealer 1 should stock
only 175 cars. For every 25 car decline in the average inventory per
c. The unique equilibrium has: Q1 = Q2 = Q3 = Q4 = 250. (Check this using
d. Total demand is 800 cars per year. Therefore, the four dealers could split
the market by holding 200 cars each in inventory. With all holding
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e. Free entry would drive profits down to zero. Given that P = $4000, Qd =
800, and AC = $2,400, industry profits fall to zero when total
S2. a. With Firm A splitting its spending $6 million/$4 million and Firm B
b. If Firm B spends $7 million in Market I, Firm A’s best response is to
c. Using the spreadsheet optimizer, we find the following best responses
for Firm A. Against $5 million spending, Firm A should spend $6.27.
Against $6 million spending, Firm A should spend $6.25 million.
Against $6.25 million spending, Firm A should spend $6.25 million.
Algebraic Solution.
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The optimality condition is:
Appendix Problems
1. Your proportions (x, 1-x) must leave your opponent indifferent between
his pure strategies, implying -16x + 8(1-x) = 24x - 16(1-x) or 24 = 64x
2. The Allies’ mixed strategy must satisfy:
The German’s mixed strategy must satisfy:
The respective solutions are x = 2/5 and y = 3/5. Germany is slightly
3. a) i) The payoff table is:
Dan
Bob Bet $6,000 Bet $0
In the upper-left entry, Dan only wins if Dan is correct and Bob is
incorrect (a .25 chance).
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ii) Dan’s dominant strategy is to bet $6,000 and Bob’s (weakly)
dominant strategy is to bet $2,001, so Bob will win 75 percent of the
b. Now the payoff table is:
Dan
Bob Bet $8,000 Bet $0
There is no longer a pure strategy equilibrium. The mixed strategy
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