Questions - Johan Lindén, Mälardalens högskola · Web view: A player’s best response is not...

Chapter 13Game Theory and Strategic Play

Questions1. What is a sequential game? How is it different from a simultaneous move game? Explain.

Answer: A sequential or an extensive-form game is one in which players make decisions at different times. It is different from a simultaneous game because one player can observe the decision of the other before making his/her own decision. This could result in a first-mover advantage, meaning the player who makes decision first gains more than the follower does. Moreover, the equilibrium of the game could be altered if a player can make a credible commitment to employ a particular strategy. These possibilities do not arise in a simultaneous-move game.

2. Is a player’s best response in a game the same as his dominant strategy? Explain.

Answer: A player’s best response is not the same as his dominant strategy. A best response is simply one player’s optimal choice, taking the other player’s action as given. In other words, a best response gives a player a payoff that is at least as large as the payoff from any other strategy she has available. However, when a player has the same best response to every possible strategy of the other player(s), then we say that the player has a dominant strategy. A dominant strategy is always a best response, but a best response is not necessarily a dominant strategy.

3. What is commitment? What is the difference between credible and incredible commitment? Give a real-life example.

Answer: Commitment refers to the ability to choose and stick with an action that might later be costly. A commitment is credible as long as the other players in the game rationally believe that a person will stick to her committed strategy. A commitment is incredible when the other player realizes that a rational person who tries to maximize payoffs will never take that action. One example of incredible commitment is when a student on academic probation informs the instructor that he will not study unless he gets the assurance of an A. This is not a credible commitment as the instructor knows that getting an F would be too costly for the student.

4. What is game theory? In what situation is the theory generally applicable?

Answer: Game theory is the study of strategic interactions. It generally applies to situations when an economic agent has to make strategic decisions by taking into account the possible response(s) of the other(s).

5. How can the tragedy of the commons be modeled as a prisoners’ dilemma game?

Answer: The tragedy of the commons refers to a situation where collective welfare is reduced by the actions of individuals who maximize their own welfare. Pollution is an example of the tragedy of the commons. The following matrix shows the payoffs that two firms face from polluting or not polluting. Each firm is affected by the pollution generated by the other firm while each firm benefits if the other firm does not pollute.

©2018 Pearson Education Ltd.

Chapter 13 | Game Theory and Strategic Play 148

Firm XPollute Don’t pollute

Firm YPollute Firm Y earns $10,000

Firm X earns $10,000Firm Y earns $15,000 Firm X earns $5,000

Don’t pollute

Firm Y earns $5,000 Firm X earns $15,000

Firm Y earns $12,000 Firm X earns $12,000

As shown in the payoff matrix, polluting is a dominant strategy for both firms and so in equilibrium both firms will pollute. This equilibrium leads to an outcome that is not best for both players. Both firms are worse off than they would be if both chose Don’t pollute because of this “tragedy of the commons” problem.

6. What is a zero-sum game? Can you think of any zero-sum games in real life?

Answer: A zero-sum game is one in which one player’s loss is another’s gain, so the sum of the payoffs of both the players is zero. Zero-sum games occur when two or more firms are competing for consumers. For example, when you buy a Toyota Prius, Toyota’s gain is another car manufacturer’s loss. Poker is a zero-sum game; my winnings are your losses.

7. What is the difference between a pure strategy and a mixed strategy?

Answer: A pure strategy in a game involves choosing a single action for a situation. A mixed strategy involves choosing between different actions randomly.

8. Suppose a player has a dominant strategy. Would she choose to play a mixed strategy (such as playing two strategies with probability 50-50)? Why or why not?

Answer: A dominant strategy is one best response to every possible strategy of the other player. A game with mixed strategies implies that the best action is to not choose any particular strategy. This means that, in a game with mixed strategies, neither player has a dominant strategy.

9. Although there are many examples of game theory in the real world, how well do you think specifics like payoff matrices, Nash equilibrium, and dominant strategies translate to reality?

Answer: It is difficult to assess the relevance of game theory to real-world problems. We may not always know the payoffs facing players in the game. In real-world situations, the payoffs are determined by the attitudes and feelings of individuals as well as by their monetary returns; emotional payoffs are difficult to observe. Even with dominant strategies, the same problem exists. Game theory often abstracts from important details. For example, in many strategic environments one player may be more cunning, wiser, or more experienced than another.

10. When can backward induction be used to arrive at the equilibrium for a game?

Answer: Backward induction can be used to find the equilibrium in games where players move sequentially. Backward induction involves starting at the end of the game and solving backwards to arrive at the equilibrium.

11. What is meant by the first-mover advantage? How does commitment matter in a game with a first-mover advantage?

a. Some games have a first-mover advantage and others do not. Suppose you were playing rock-paper-scissors as an extensive form game. First you choose rock, or paper, or scissors and then your opponent makes a choice. Is there a first-mover advantage in this game?

b. Two firms are thinking of entering a new market. If only one enters, it will make high profits. If two firms enter, then both will suffer losses. Suppose that the game is played sequentially, with Firm 1 deciding first. Does this game have a first-mover advantage?



Answer: A player can have a first-mover advantage in a game that is played sequentially rather than simultaneously. The sequential game features a first-mover advantage if, in equilibrium, the first mover earns more benefits than the second mover. However, not all games will have a first-mover advantage. A commitment is an action that one cannot turn back on later, even if it is costly. Using backward induction, we can choose the strategy that is optimal given the other player’s actions. However, a commitment device would be needed to ensure that the other player sticks to his or her strategy when push comes to shove.

a. No. You will choose either rock, paper, or scissors. Your opponent will see your move, and she can take advantage of your decision when it is her turn.

b. Yes. The firm that goes first can enter; the firm that goes second will then have no incentive to enter.

12. A trust game shown in Exhibit 13.15 is a sequential prisoners’ dilemma. This means that it is likely that the outcome of the game is not socially efficient. What factors could cause this equilibrium to be different in real life?

Answer: An important factor that makes the equilibrium socially efficient in trust games is reputational concerns; if the game is played several times, the players might attempt to develop a reputation as someone who can be trusted. If a game is repeated, it makes sense to play nicely as the other player may reciprocate your behavior. This long-run strategy may shed light on the kinds of interactions we often observe in the real world where people trust each other even when it’s not in their immediate best interest to do so.

13. Economic agents (for example, consumers or firms) often do things that at first glance seem to be inconsistent with their self-interest. People tip at restaurants and when they are on vacation even if they have no intention to return to the same place. Firms, sometimes, install costly pollution abatement equipment voluntarily. How can these deviations from Nash predictions be explained?

Answer: There are several possible explanations. Firms might voluntarily install pollution abatement equipment if they believe consumers will eventually learn about the firm’s polluting behavior and stop buying its products. In some cases, the payoffs may be more complex than just monetary payoffs. For example, people may care about the net benefits that someone else receives. If this is the case, then behavior may reflect social preferences, meaning that the individual’s benefits are defined not only by his own payoffs, but also by the payoffs of others. In other examples the implications of a repeated relationship are very different from a game that is played just once.



Problems1. Suppose there are cable TV companies in your city, Astounding Cable and Broadcast Cable. They

both must decide to on a high advertising budget, a moderate advertising budget, or a low advertising budget. They will make their decisions simultaneously. Their payoffs are as follows:

Astounding/Broadcast High Medium Low

High

Astounding earns $2 million.

Broadcast earns $5 million.

Astounding earns $5 million. Broadcast earns $7 million.


Medium





Low





a. Does Astounding have a dominant strategy? If so, what is it?

b. Does Broadcast have a dominant strategy? If so, what is it?

c. Is there a dominant strategy equilibrium? If so, what is it?

d. Are there any Nash equilibria in this game? If so, what are they?

Answer:

a. Medium is a dominant strategy for Astounding; it is a best response to any of Broadcast’s strategies. For example, if Broadcast chooses High, Astounding would earn $2 million by playing High, $6 million by playing Medium, or $1 million by playing Low. Therefore Astounding’s best response to High is medium.

b. Broadcast does not have a dominant strategy. A strategy is a dominant strategy if it is a best response to every strategy the other players could use. For Broadcast, Low is a best response to High but Medium is a best response to Medium or Low.

c. The game does not have a dominant strategy equilibrium. In a dominant strategy equilibrium each player uses a dominant strategy. We showed in part b that Broadcast does not have a dominant strategy.

d. The game has one Nash equilibrium: Astounding and Broadcast both play Medium. A strategy combination is a Nash equilibrium if each strategy is a best response to the strategies of others. If Broadcast plays Medium then Astounding would earn $5 million by playing High, $8 million by playing Medium, or $0 by playing Low; therefore Astounding’s best response to Medium is Medium. A similar argument would show that Broadcast’s best response to Medium is Medium.



2. Suppose Russia is deciding to Invade or Not Invade its neighbor Ukraine. The U.S. has to decide to be Tough or Make Concessions. They will make their decisions simultaneously. Their payoffs are as follows:

U.S. / Russia Not Invade Invade

Be Tough U.S. gets 5 Russia gets 4

U.S. gets 7 Russia gets 3

Make Concessions U.S. gets 3 Russia gets 5

U.S. gets 1 Russia gets 9

a. What is U.S.’s best response when Russia chooses Not Invade?

b. What is U.S.’s best response when Russia chooses Invade?

c. What is Russia’s best response when U.S. chooses Tough?

d. What is Russia’s best response when U.S. chooses Make Concessions?

e. What is the Nash equilibrium of this game?

Answer:

a. The best response to Not Invade is Tough (because 5 > 3).

b. The best response to Invade is Tough (because 7 > 1).

c. The best response to Tough is Not Invade (because 4 > 3).

d. The best response to Make Concessions is Invade (because 9 > 5).

e. In a Nash equilibrium every player chooses a best response to the other players’ strategies. In the Nash equilibrium for this game, the U.S. chooses Be Tough and Russia chooses Not Invade. We showed in part (a) that the best response to Not Invade is Tough. We showed in (c) that the best response to Tough is Not Invade.

3. Samsung and Sony have to decide whether they will increase the spending on research and development (R&D) in order to improve the features of their products sold worldwide. If they both increase the spending, the gains of doing so are zero for both. If only one of them increases the R&D budget while the other does not, the gain of the improved features is equal to the loss of the other company. If both of them do not change R&D spending, their customers will switch to other brand from the United States, and both will equally make huge losses.

a. Construct the pay-off matrix for the game. Is this a zero-sum game? Why or why not?

b. Is there a dominant strategy equilibrium? If so, what is it?.

Answers:

The pay-off matrix of the game is:

SonyIncrease R&D Do Not Increase R&D

Samsung

Increase R&D

Sony gains $0Samsung gains $0

Sony loses $XSamsung gains $X

Do Not Increase R&D

Sony gains $XSamsung loses $X

Sony loses $YSamsung loses $Y



Without the existence of the other brand from the United States, this game would constitute a zero-sum game as one company’s loss is the other company’s gain. Since there is the case when both companies could lose to the other brand (when both choose not to increase R&D spending), this is not a zero-sum game.

4. It is possible for 2-player games to be quite asymmetric: Each player might have a different set of options, and the payoffs may be quite different. Consider the following example between a large firm and a small firm (the first number in each box denotes the large firm’s payoff, the second number shows the small firm’s payoff):

Small Firm (player 2)

Expand Operation

StaySmall

Large Firm(player 1)

High Price 50, 20 60, 10

Medium Price 60, 20 70, 10

Low Price 40, 0 90, 10

a. Does either firm have a dominant strategy?

b. Find all Nash equilibria.

Answers:

a. No, neither player has a dominant strategy. For Large Firm, the best response to “Expand Operation” is “Medium Price”, while the best response to “Stay Small” is “Low Price”. Similarly, the best response for Small Firm depends on what Large Firm does.

b. (Medium Price, Expand Operation) and (Low Price, Stay Small) are both pure-strategy Nash equilibria. (It is possible for more than one Nash equilibrium to exist, and the theory makes no predictions about which of these two outcomes will be realized.)

5. A Beautiful Mind, a movie about John Nash, fails to properly demonstrate a Nash equilibrium. It attempts to do so in a bar scene where men at a bar (Nash and his friends) plan to ask women to dance. There is one beautiful woman that the men consider the most attractive, as well as several other women. Nash assumes women will only accept an offer to dance if the man has not first been rejected by the beautiful woman. In the movie, Nash proposes that all the men each agree to not ask the beautiful woman in the first place.

a. Nash’s proposal may lead to a good outcome for each man, but it is not a Nash equilibrium. Why not?

b. The movie initially shows all the men asking the beautiful woman to dance. To be fair, this is also not a Nash equilibrium. Why not?

c. Why is it a Nash equilibrium if exactly one man asks the beautiful woman to dance?

Answers:



a. The best response for each man is to deviate from the plan and to ask the beautiful woman to dance.

b. Most (if not all, depending on assumptions) of the men will get rejected by the beautiful woman as she can accept at most one offer to dance. Each man would be better off asking another woman to dance.

c. If exactly one man asks the beautiful woman to dance then this man is clearly best responding to the situation. Less obvious is that the other men are also choosing a best response since they cannot do any better than dancing with other women. If any of these men attempted to ask the beautiful woman to dance then she would have two offers, at which point one (or both) of the men might be rejected and end up dancing with nobody.

6. We might suppose a soccer player has three options when taking a penalty kick: Kick right (KR), kick left (KL), or kick down the center (KC). The goalie can choose to dive right (DR), dive left (DL) or stand in the center (SC). Assume the goalie blocks the kick whenever he guesses correctly (+1), but fails to make the save otherwise (-1). The payoff for the kicker is opposite. Write this game as a matrix. Are there any pure-strategy Nash equilibria?

Answer: There are no pure-strategy Nash equilibria in the following game. The first number in each cell gives the payoff for the Kicker, while the second number in each cell gives the payoff for the Goalie.

Goalie(Player 2)

DR DL SC

Kicker(Player 1)

KR

-1, 1 1, -1 1, -1

KL 1, -1 -1, 1 1, -1

KC

1, -1 1, -1 -1, 1

7. Use a matrix to model a two-player game of rock-paper-scissors with a payoff of 1 if you win, -1 if you lose, and 0 if you tie.

a. Draw the payoff matrix for this game.

b. Is there an equilibrium in this game where players use pure strategies?

c. Why should you use a mixed strategy to play this game?



Answer:

a. The payoff matrix for this game is shown below.

Player 1/Player 2 Rock Paper Scissors

RockPlayer 1 gets 0Player 2 gets 0

Player 1 gets -1Player 2 gets 1

Player 1 gets 1Player 2 gets -1

PaperPlayer 1 gets 1Player 2 gets -1

Player 1 gets 0Player 2 gets 0

Player 1 gets -1Player 2 gets 1

ScissorsPlayer 1 gets -1Player 2 gets 1

Player 1 gets 1Player 2 gets -1

Player 1 gets 0 Player 2 gets 0

b. No. In an equilibrium with pure strategies, each player would choose a strategy that is a best response to the other player’s strategy. But Rock is the best response to Scissors, Scissors is the best response to Paper, and Paper is the best response to Rock. So there is no combination of strategies where each player has chosen the best response to the other player’s strategy.

c. If you do not use a mixed strategy, you will be too predictable. Suppose, for example, you always play Rock. Then, your opponent will always play Paper and you will always lose.

8. Two gas stations, A and B, are locked in a price war. Each player has the option of raising its price (R) or continuing to charge the low price (C). They will choose strategies simultaneously. If both choose C, they will both suffer a loss of $100. If one chooses R and the other chooses C, (i) the one that chooses R loses many of its customers and earns $0, and (ii) the one that chooses C wins many new customers and earns $1,000. If they both choose R the price war ends and they each earn $500.

a. Draw the payoff matrix for this game.

b. Does either player have a dominant strategy? Explain.

c. How many Nash equilibria does this game have? Defend your answer carefully.

Answer:

a. The payoff matrix for this game is:

Gas Station A/Gas Station B Raise price (R) Continue low price

(C)

Raise price (R)A gets $500B gets $500

A gets $0B gets $1,000

Continue low price (C)A gets $1,000

B gets $0A gets -$100B gets -$100

b. Neither play has a dominant strategy. For both players C is the best response to R (because $1,000 > $500) and R is the best response to C (because $0 > -$100).

c. The game has two Nash equilibria. In one, A plays C and B plays R; in the other A plays R and B plays C. In a Nash both players choose a strategy that is a best response to the other player’s strategy. We argued in part (b) that C is the best response to R and R is the best response to C.



9. Consider a game with two players, 1 and 2. They play the extensive-form game summarized in the game tree below:

[Note that the textbook uses colors: Grey = Green; Black = Red]

a. Suppose Player 1 is choosing between the gray and black for his second move. Which will he choose if:

i. Gray, Gray has been played.

ii. Black, Black has been played

b. Suppose Player 2 is choosing between green and red, knowing the information above. Which will he choose if:

i. Gray has been played.

ii. Black has been played.

c. Finally, suppose Player 1 is choosing between gray and red in the first move. Given the information above, which will he choose?

d. Now describe the path that gives an equilibrium in this extensive game.

Answer:

a. Player 1’s second turn:

i. If both players have chosen Gray, then Player 1 would earn 1 by choosing Gray or 2 by choosing Black. Therefore, Player 1 would choose Black.

ii. If both players have chosen Black then Player 1 would earn 5 by choosing Gray or 0 by choosing Black. Therefore Player 1 will choose Gray.

b. Player 2’s turn:

i. If Player 1 has chosen Gray then 2 knows he will receive 0 if he chooses Gray (since, as shown above, Player 1 will choose Black on his second turn). Player 2 would earn 5 by playing Black, and so he should play Black if Player 1 has chosen Gray.



ii. If Player 1 has chosen Black Player 2 could earn 2 by choosing Gray. If Player 2 chooses Black, he knows that 1 will choose Gray and so 2 will earn 3. Therefore Player 2 should choose Black if Player 1 chose Black.

c. Player 1’s first turn:

If Player 1 chooses Black, he knows that Player 2 will choose Black, that he will choose Gray on his second turn, and therefore he will earn 5. If Player 1 chooses Gray, he knows that Player 2 will choose Black, and therefore he will earn 1. Therefore Player 1 should choose Black to begin the game.

d. On the equilibrium path Player 1 will choose Black to begin the game, Player 2 will choose Black, and Player 1 will choose Gray on his second turn.

10. Jones TV and Smith TV are the only two stores in your town that sell flat panel TV sets. First, Jones will choose whether to charge high prices or low prices. Smith will see Jones’s decision and then choose high or low prices. If they both choose High, each earns $10,000. If they both choose Low, each earns $8,000. If one chooses High and the other chooses Low, the one that chose High earns $6,000 and the one that chose Low earns $14,000.

a. Draw the game tree. Use backward induction to solve this game.

b. Suppose Smith goes to Jones and promises to choose High if Jones chooses High. Is this a credible promise?

c. Now suppose Jones starts a new policy that says it will always match or beat Smith’s price. It advertises the new policy heavily and so must choose Low if Smith chooses Low. So the game now has the following structure. First, Jones chooses High or Low. Second, Smith chooses High or Low. Third, if Jones has chosen High and Smith has chosen Low, Jones meets Smith’s price and chooses Low. Draw the game tree. Use backward induction to solve this game.

Answer:

a. If Jones chose High, Smith will choose Low since $14,000 > $10,000. If Jones chose Low, Smith will choose Low because $8,000 > $6,000. Now that Jones knows Smith will choose Low regardless of what it does, Jones will choose Low because $8,000 > $6,000.

b. It is not a credible promise. Once Jones has chosen High, Smith would earn $10,000 by choosing High or $14,000 by choosing Low. Therefore, Smith will never keep his promise to choose High.

c. Consider Smith’s decision if Jones has chosen High. If Smith chooses Low, Jones will match Smith’s price and switch to Low. If that happens, Smith will earn $8,000. If Smith chooses High,



if Jones has chosen High, Smith will earn $10,000. Therefore, if Jones chooses High, Smith will choose High. If Jones chooses Low, Smith will choose Low because $8,000 > $6,000. So, Jones knows that Smith will choose High if it chooses High and Smith will choose Low if it chooses Low. Therefore, Jones chooses High because $10,000 > $8,000.

11. While at the airport you hear over the loudspeaker an offer to be bumped off your current flight in exchange for $100 travel credit. After it becomes clear nobody will take this offer, the offer is increased to $200. A few minutes later, the airline offers $300; then $400, and so on. Individually, each passenger wants to take the offer, but collectively it is best for people to hold out. The strategic dynamic can be modeled with a two player “centipede game” (name is based on the shape of the game tree), shown below. Use backwards induction to determine the equilibrium in this game.

Answer: Based on backwards induction, passenger 1 will take $100 in the first node. To find this, start at the end and notice that player 2 prefers $600 to $0. Knowing this fact, in the second-to-last node, player 1 must choose between $500 and the $0 that will result from not taking; she will thus choose to take. This pattern continues and in each node we discover that it is always best to take rather than allow the pot of money to grow.



12. Consider a game with two players, China and Japan. They play the extensive-form game summarized in the following game tree:

The red Line indicates investing in Southeast Asia, the green indicates investing in South Asia, and the blue line indicates investing in Europe.

a. Suppose China is choosing the location of its next investment. Where will it choose if:

i. Japan chooses to invest in Southeast Asia.

ii. Japan chooses to invest in Europe.

b. Suppose Japan is choosing the location of its next investment. Where will it choose if:

i. China announces that it will follow the path by Japan.

ii. China announces that it will not follow the path by Japan.

c. Is your answer different when the two countries make decisions simultaneously?

d. Does Japan have the first-mover advantage here? Explain your answer.

Answer:

a. China’s decision:

i. If Japan chooses to invest in South-east Asia, the best response by China is to invest in Europe (300 > 150 > 80).

ii. If Japan chooses to invest in Europe, the best response by China is to invest in South-east Asia (150 > 80 > 50).

b. Japan’s decision:

i. China’s announcement is not credible here. If Japan chooses to invest in South-east Asia, it knows that China will choose to invest in Europe as the return is highest there and Japan will earn USD 400 billion. But if it chooses to invest in Europe, it knows that China will choose to invest in South-east Asia and Japan will earn USD 600 billion. So, Japan will choose to invest in Europe.

ii. China’s announcement is irrelevant here. If Japan chooses to invest in South-east Asia, it knows that China will choose to invest in Europe as the return is highest there and Japan will earn USD 400 billion. But if it chooses to invest in Europe, it knows that China will choose to invest in South-east Asia and Japan will earn USD 600 billion. So, Japan will choose to invest in Europe.



c. The equilibrium path in this sequential game is that Japan will choose to invest in Europe and China will choose to invest in South-east Asia.

The pay-off matrix if the game is played simultaneously is as follows.

Japan, China Europe South Asia South-east Asia

Southeast Asia USD 400 billion, USD 300 billion

USD 40 billion, USD 80 billion


Europe USD 200 billion, USD 50 billion



When China chooses to invest in Europe, Japan’s best response is to invest in South-east Asia (400 > 200). When China chooses to invest in South Asia, Japan’s best response is to invest in Europe (250 > 40) and if China invests in South-east Asia, Japan should invest in Europe (600 > 100).

When Japan chooses to invest in South-east Asia, China’s best response is to invest in Europe (300 > 150 > 80). If Japan invests in Europe, China’s best response is to invest in South-east Asia (150 > 80 > 50).

Thus, there are two Nash equilibria: Japan invests in Europe and China invests in South-east Asia, and Japan invests in South-east Asia and China invests in Europe.

d. In equilibrium path, Japan earns a lot higher than China (600 billion USD > 150 billion USD), so we say that Japan has the first-mover advantage here.

13. Two competing firms must choose their quantity of production simultaneously. Each firm can either choose a High quantity of 3 or a Low quantity of 2. The price for both firms is 9 - Q, where Q is the sum of both quantities. Costs are zero, the profit is simply price times quantity. For example, if firm 1 chooses High and firm 2 chooses Low then price is 9 - (3+2) = 4; payoff for firm 1 is 12 while payoff for firm 2 is 8.

a. Draw the complete matrix for this game.

b. Find all Nash equilibria.

c. If this game were instead played sequentially, would there be a first-mover advantage? Briefly explain.

Answers:

a. See game matrix. If both firms choose High, then the price is $3, and so payoff is $3(3) = $9 for both firms. If both firms choose Low, the price is $5, so payoff is $5(2) = $10 for both firms.

b. (High, High) is the unique Nash equilibrium. It is also a dominant-strategy equilibrium.



c. There is no first-mover advantage because the predicted outcome is (High, High) regardless of who moves first.

Firm 2

High Low

Firm 1 High

9, 9 12, 8

Low 8, 12 10, 10


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