Review Questions
1. What is a strategy in a game?
A player's strategy is a plan for actions in each possible situation
in the game.
2. What is a Nash equilibrium?
A Nash equilibrium is a situation in which each player makes his or her best response, that is, plays the strategy that maximizes his payoff, given the strategies of other players.
Thinking Exercises
15.3 Draw a decision tree for Clyde in the example from Table 1 and indicate his best strategy.
Redraw decision tree on page 378, but replace Bonnie's name with Clyde's and vice versa. In other words:
4. The police charge Rat and Skunk with shoplifting. If both plead not guilty, each will get a $300 fine. If both plead guilty, each will get a $500 fine. If Rat pleads guilty and implicates Skunk (who pleads not guilty), Rat will go free and Skunk will get a $1,000 fine. Skunk faces the same choice if Rat pleads not guilty. Draw a table to summarize the payoffs in this game and find each player's best response. What is the Nash equilibrium of this game?
See the table. The Nash equilibrium is for both players to confess and
each get a $500 fine.
| Skunk confesses | Skunk does NOT confess | |
| Rat confesses | both pay $500 | Rat goes free;
Skunk pays $1000 |
| Rat does NOT confess | Skunk goes free;
Rat pays $1000 |
both pay $300 |
Review Questions
5. What does it mean for a player to have no dominant strategy?
When a player's strategy depends on the strategies of others, they have no dominant strategy.
6. Explain why it is a Nash equilibrium in Table 15.6 for Carolyn and Mark to go to: (a) a party; (b) a movie.
(a) Given that Mark is going to the party, Carolyn gains more happiness by going to the party (2 units of happiness) rather than to the movie (loss of 2 units), so she has no incentive to switch to the movie. (She is playing her best response.) Similarly, given that Carolyn is going to the party, Mark gains more happiness by going to the party (2 units of happiness) rather than to the movie (loss of 2 units), so he has no incentive to switch to the movie. (He is playing her best response.)
(b) Same reasoning as in (a) with "party" and "movie" reversed
-- Given that Mark is going to the movie, Carolyn gains more happiness
by going to the move (2 units of happiness) rather than to the party (loss
of 2 units), so she has no incentive to switch to the party. Similarly,
given that Carolyn is going to the movie, Mark gains more happiness by
going to the movie (2 units of happiness) rather than to the party (loss
of 2 units), so he has no incentive to switch to the party.
Thinking Exercises
7. Draw decision trees for Folger's and Maxwell House in the example from Table 4.
8. Draw a decision tree for Pa in the example from Table 5.
Review Questions
9. What is a sequential game?
A sequential game is a game in which players make at least some of their decisions at different times.
10. What is a subgame perfect Nash equilibrium?
A subgame perfect Nash equilibrium is a Nash equilibrium in which every player's strategy is credible (no player makes incredible threats).
Thinking Exercises
11. Give examples of a credible threat and an incredible threat.
An example of a credible threat is when Microsoft says it is close to coming out with a new version of its computer operating system therefore discouraging competitors from coming out with their own products until after they see what Microsoft has developed. Some have accused Microsoft of using its new product announcements as a ploy to discourage competitors. This allows it to take longer than previously announced to actually complete the product. But since its announcements are a credible threat, they serve their intended purpose.
An example of an incredible threat would be for a large company to try to keep competitors out of its market by announcing that it will respond to new entrants by merging with the second biggest player in the market. However, in this case, competitors recognize it as an incredible threat because they know antitrust measures would prevent the merger.
12. Consider the example of Carolyn and Mark in Table 6 but suppose
that they play a sequential game. First, Carolyn chooses whether she will
to the party or the movie, then Mark chooses where he will go.
a. Draw a diagram like Figure 6 to describe this game.
b. Find the subgame perfect Nash equilibrium of the game.
a. Here is a description of the game:
b. The subgame perfect Nash equilibrium of the game is for Carolyn
to choose to go to the party followed by Mark's decision to go to the party.
Review Questions
13. Explain, with an example, why people can benefit from commitments.
If a person commits four hours of their day to volunteer to assist homeless persons after work, they can avoid the pitfall of other less beneficial ways of spending that time. For example, the person might simply go home and languish in front of the television. Or they might visit a pub after work, and stay there too long. But, because of the commitment, they instead end up helping people who need them.
14. Explain how a firm can deter entry by investing in excess capacity.
Consider a game like the one in Figure 7 (page 388). This is
like Figure 6 (page 385), except that the monopoly can choose to
invest in extra capacity by adding new machines and equipment before
playing the rest of the game. This extra capacity reduces the monopoly’s
marginal cost of raising output, so it raises the profit that the monopoly
would earn by cutting its price and raising sales. This allows the
monopoly to commit to cutting its price if the new firm enters.
By making its threat credible in this way, the monopoly can prevent the
new firm from entering the industry.
Further detail: If the monopoly does not invest in extra capacity, Figure 6 describes the rest of the game. The monopoly earns $3 million in the subgame perfect Nash equilibrium of that game. However, if the monopoly invests in extra capacity, then Figure 7 describes the rest of the game. The monopoly earns $5 million in the Nash equilibrium of the game in Figure 7 -- by (credibly) threatening to cut its price if the new firm enters. Consequently, the new firm does not enter and the monopoly earns $5 million.
15. Explain why repeated games can have different Nash equilibria than one-shot games.
In repeated games, people’s current actions can depend on the past behavior of other players. For example, in a repeated prisoner’s dilemma, each player can punish the other (in the future rounds of the game) for bad behavior (confessing) now. A second reason: when players have limited information, repeated games allow players to learn from experience what other players are likely to do in various situations.
16. Explain the tit-for-tat strategy. What evidence suggests that it is a good strategy in some games?
You follow a tit-for-tat strategy in a repeated game if you (1) cooperate on your first move and (2) for each later move, do what copy the other player’s action did in his previous move.
Studies of repeated prisoner’s dilemmas show that even when some players
choose very complicated strategies, tit-for-tat tends to be a winning strategy.
Even when players specifically try to beat tit-for-tat, tit-for-tat tends
to remain the best strategy.
Problems
17. What is the Nash equilibrium of the following game?
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Player 1 |
Heads | $5 each | $0 for Player 1, $9 for Player 2 |
| Tails | $10 for Player 1, $1 for Player 2 | $2 each | |
Both players choose tails and collect $2 each.
18. Explain how the prisoner’s dilemma may apply to:
(a). Picnickers in a park with competing boom boxes
(b). The nuclear arms race
(c). Children sharing a birthday cake or friends sharing a pizza
(a)
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Payoffs show units of picnickers' happiness. |
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1's boombox |
Very Loud | 0 each |
10 for Player 1, -5 for Player 2 |
| Soft | - 5 for
Player 1,
10 for Player 2 |
4 each | |
The Nash equilibrium is for both boomboxes to be very loud, with each player getting 0 units of happiness.
(b)
| Payoffs show units of happiness, after spending for weapons and living in fear of their use. |
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| USA | Lots of Nuclear Weapons | -5 for each |
10 for USA, - 10 for USSR |
| Few Nuclear Weapons |
- 10 for USA, 10 for USSR |
4 for each | |
The Nash equilibrium is for both countries to have lots of nuclear weapons,
with each getting 0 units of happiness.
(c)
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Payoffs show units of childrens' happiness. |
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| Josh | Eat the cake quickly, and then take more (etc.) to make sure that you get a lot. | 300 each |
600 for Josh, 100 for Jen |
| Eat the cake more slowly, even at the risk that others finish it off before you get more. |
100 for Josh, 600 for Jen |
500 each | |
The Nash equilibrium is for both to eat the cake quickly, with each player
getting 300 units of happiness.
19. Make up an example of a game with multiple equilibria.
See Table 6 on page 384.
20. Think of a real-life example of a repeated game.
Procter & Gamble and Kimberly-Clark have been engaged in a long-running diaper price war that resembles a repeated game. If P&G decides to cut the price of its Pampers diapers, K-C inevitably responds by changing the price of its Huggies diapers. Each company also tends to base its new product development decisions on the past actions of the other. ALSO: Pricing decisions by airlines, etc.
21. Find all the multiple equilibria in the following game. Two cars drive
toward each other on a road. If both drive on the left-hand side (as in England)
or on the right-hand side (as in most other countries), they pass each other
safely. If one drives on the left and the other on the right, however, they
crash.
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Natasha |
Drive on Left | Each gains 1 | Each loses 2 |
| Drive on Right | Each loses 2 | Each gains 1 | |
The two equilibria are for Natasha and Boris to either both drive on the right or both drive on the left.
15.23 Comment on this statement: "Committing is never a good idea because it takes away your flexibility to respond to a situation in whatever way is best at the time."
Although committing does limit flexibility, it can provide valuable
benefits by restricting future choices in a way that changes other people's
actions to one's benefit. Plus, commitments can, and are, broken, which
enables people to change their response at any given time.
15.14 Explain how a firm can deter entry by investing in excess capacity.
Excess capacity looms as a deterrent to entry by other firms because the firm controlling the ability to produce can do so to the point that the price of the good doesn't justify entering the industry.
15.15 Explain why repeated games can have different Nash equilibria than one-shot games.
If games are repeated, various subgame perfect Nash equilibria occur because firms decide to charge different prices, punish competitors, engage in tit-for-tat, etc.
15.16 Explain the tit-for-tat strategy. What evidence suggest that it is a good strategy in some games?
In the tit-for-tat strategy, players cooperate
unless one of them fails to cooperate in some round of the game, in which
case the others do in the next round what the uncooperative player did
in the current round. Tit-for-tat emphasized cooperation, encourages players
not to cheat, punishes with "an eye for an eye," but then forgives. Since
these are underlying social customs, this may explain why the strategy
is best.
15.18 Explain how the prisoner's dilemma may apply
to:
a. Traffic jams.
b. The nuclear arms race.
c. The Bertrand model of oligopoly in
chapter 15
d. Children sharing a birthday cake or
friends sharing a pizza.
e. Picnickers in a park with competing
boom boxes.
15.19 Make up an example of a game with multiple equilibria.
Joe and Mike both work for an advertising agency. Joe prefers to work on the ad campaign for a beer company. Mike prefers to work on the ad campaign of a regional shopping mall. Both do better, however, when they work on projects together. The following table illustrates their choices:
See attached table.
This game has two Nash equilibria, one in which both work on the beer campaign, and one in which they both work on the mall campaign.
15.20 Think of a real-life example of a repeated game.
Procter & Gamble and Kimberly-Clark have been engaged in a long-running diaper price war that resembles a repeated game. If P&G decides to cut the price of its Pampers diapers, K-C inevitably responds by changing the price of its Huggies diapers. Each company also tends to base its new product development decisions on the past actions of the other.15.22 Explain how a monopoly might deter entry of new competitors by accumulating large inventories of goods. (Hint: Review the example in which a monopoly invests in extra capacity to deter entry and follow a similar logic.)
The monopoly can threaten to charge a low price
if a competitor enters the market since it already has a large inventory
on hand, which it can use to quickly increase supply and slash prices.
The monopoly's threat to charge a low price is credible because it can
actually carry out the action.
15.24 Comment on this statement: "Nuclear deterrence is most effective if the government is willing to respond to a nuclear attack by completely destroying the other country, even if such destruction would be pointless because it would be too late to prevent damage from the initial attack."
By committing to the nuclear attack, a country can change the other country's actions to its own benefit because the commitment gives the other country an initiative not to attack.
15.25 What are the subgame perfect Nash equilibria of the game in Figure 15.8.
15.26 A Stanford professor of education and psychology,
Robert Calfee, recommends teaching tit-for-tat in grade school. "It's a
technology...(for how to deal with others in a society. What do you do
when you're mad at someone you love? Nothing in our schools teaches that."
How can you apply the tit-for-tat strategy
in everyday situations? Discuss examples and likely results. Is tit-for-tat
an effective way to behave in these situations? Is it a fair or moral way
to behave? Would other behaviors be better in some way (more likely to
produce good results or more ethical)? Be as specific as possible. Is tit-for-tat
a bad strategy in any situation?
15.27 In what recent real-life situation has your strategy for behavior depended ont he strategies or behavior of other people? Can you analyze the situion using game theory?
Students' answers to this question will vary.