Deck 6: Thinking Strategically

Question 1

Consider the following two player game.
Which one of the following statements is correct?

A) Bottom will be a dominant strategy for Player #1 if and only if e > a and g > c.
B) Bottom will be a dominant strategy for Player #1 if and only if e < a and g < c.
C) Top will be a dominant strategy for Player #1 if and only if e > a and g > c.
D) Top will be dominant strategy for Player #1 if and only if b > d and f > h.

Answer 1

Bottom will be a dominant strategy for Player #1 if and only if e > a and g > c.

Question 2

Consider the following two player game.
Which one of the following statements must be correct in order to make {Top, Left} an equilibrium of this game?

A) a > e and b > d.
B) c < a and g < c.
C) e > a and g > c.
D) b < d and f < h.

Answer 2

a > e and b > d.

Question 3

Consider the following two player game.
Which one of the following statements is correct?

A) a. Bottom is a dominant strategy for Player #1, while Player #2 does not have a dominant strategy.
B) Top is a dominant strategy for Player #1, while Left is a dominant strategy for Player #2.
C) Left is a dominant strategy for Player #2, while Player #1 does not have a dominant strategy.
D) Right is a dominant strategy for Player #2 while Bottom is a dominant strategy for Player #1.

Answer 3

a. Bottom is a dominant strategy for Player #1, while Player #2 does not have a dominant strategy.

Question 4

Consider the following two player game.
Which one of the following statements is correct?

A) There is a unique Nash Equilibrium in this game and that is {Top, Left}.
B) b. There is a unique Nash Equilibrium in this game and that is {Bottom, Right}.
C) There are two Nash equilibria in this game - {Top, Right} and {Bottom, Left}.
D) There are two Nash equilibria in this game - {Top, Left} and {Bottom, Right}.

Question 5

Consider the following 2-person sequential move game with Emilio and Janna. Emilio moves first and either chooses TAKE or PASS. If Emilio chooses TAKE, then the game ends. In this case, Emilio earns $35 and Janna earns $20. However, if Emilio chooses PASS then Janna gets to move. Janna can choose either LEFT or RIGHT. If Janna chooses LEFT, then Emilio gets $50 and Janna gets $10. If Janna chooses RIGHT, then both Emilio and Janna get $25 each. Assuming that players are payoff maximizers, if you solve this game using backward induction, then the subgame perfect equilibrium is one where:

A) a. Emilio Chooses TAKE at the very beginning and the game ends.
B) Emilio chooses PASS and Janna chooses LEFT.
C) Emilio chooses PASS and Janna chooses RIGHT.
D) Emilio chooses TAKE and Janna chooses LEFT.

Question 6

Consider the following 2-person sequential move game with Guo Chen and Nathan. Guo Chen moves first and either chooses TOP or BOTTOM. Nathan gets to see Guo Chen's choice and then responds with either LEFT or RIGHT. The game ends after Nathan's choice. If Guo Chen chooses TOP and Nathan chooses LEFT, then Guo Chen gets $8 and Nathan gets $2. If Guo Chen chooses TOP and Nathan chooses RIGHT, then Guo Chen gets $0 and Nathan gets $0. If Guo Chen chooses BOTTOM and Nathan chooses LEFT, then Guo Chen gets $A and Nathan gets $B. If Guo Chen chooses BOTTOM and Nathan chooses RIGHT, then Guo Chen gets $5 and Nathan gets $4. Assuming that players are payoff maximizers, which of the following must be true for {TOP, LEFT} to be the subgame perfect equilibrium using backward induction?

A) a. $B > $4; $A < $8.
B) $B > $4; $A > $8.
C) $B < $4; $A > $5.
D) $B < $4; $A < $5.

Question 7

Consider the following two player game.
Which one of the following statements must be correct for {Bottom, Right} to be Nash equilibrium in dominant strategies?

A) For Player #1, e > a and g > c; for Player #2, d > b and h > f.
B) For Player #1, e < a and g < c; for Player #2, d < b and h < f.
C) For Player #1, d > b and h > f; for Player #2, e > a and g > c.
D) For Player #1, b > f and d > h; for Player #2, a > c and e > g.

Question 8

Consider the following 2-person sequential move game with Elizabeth and Bridget. Elizabeth moves first and either chooses TOP or BOTTOM. Bridget gets to see Elizabeth's choice and then responds with either LEFT or RIGHT. The game ends after Bridget's choice. If Elizabeth chooses TOP and Bridget chooses LEFT, then Elizabeth gets $8 and Bridget gets $2. If Elizabeth chooses TOP and Bridget chooses RIGHT, then Elizabeth gets $0 and Bridget gets $0. If Elizabeth chooses BOTTOM and Bridget chooses LEFT, then Elizabeth gets $5 and Bridget gets $4. If Elizabeth chooses BOTTOM and Bridget chooses LEFT, then Elizabeth gets $A and Bridget gets $B. Assuming that players are payoff maximizers, which of the following must be true for {BOTTOM, RIGHT} to be the subgame perfect equilibrium using backward induction?

A) $B > $4; $A > $8.
B) $B > $4; $A < $8.
C) $B < $4; $A < $5.
D) $B < $4; $A < $8.

Question 9

Consider the following two player game.
If Player 1 moves first and Player 2 follows, then using the principle of backward induction the subgame perfect equilibrium of this game is

A) a. Top, Left.
B) Top, Right.
C) Bottom, Left.
D) Bottom, Right.

Question 10

Consider the following two player game.
If Player 2 moves first and Player 1 follows, then using the principle of backward induction the subgame perfect equilibrium of this game is

A) Top, Left.
B) Top, Right.
C) Bottom, Left.
D) d. Bottom, Right.

Question 11

Consider the following two player game.
Which one of the following statements is correct?

A) There is a unique Nash Equilibrium in this game and that is {Top, Left}
B) There is a unique Nash Equilibrium in this game and that is {Bottom, Right}
C) There are two Nash equilibria in this game - {Top, Right} and {Bottom, Left}.
D) There are two Nash equilibria in this game - {Top, Left} and {Bottom, Right}.

Question 12

The difference between a coordination problem and a prisoner's dilemma is that:

A) Coordination problems have multiple equilibria while prisoner's dilemma games have one unique dominant strategy equilibrium.
B) Prisoner's dilemma games have multiple equilibria while coordination games have one unique dominant strategy equilibrium.
C) Coordination problems have multiple equilibria, one of which must be payoff dominant, while prisoner's dilemma games have multiple equilibria which need not be payoff ranked.
D) Prisoner's dilemma games have multiple equilibria, one of which must be payoff dominant, while coordination problems have multiple equilibria which need not be payoff ranked.

Question 13

Consider the following two player game.
Which one of the following statements is correct?

A) There is a unique Nash Equilibrium in this game and that is {Top, Left}.
B) There is a unique Nash Equilibrium in this game and that is {Bottom, Right}
C) There are two Nash equilibria in this game - {Top, Right} and {Bottom, Left}.
D) There are two Nash equilibria in this game - {Top, Left} and {Bottom, Right}.

Question 14

Consider the following two player game.
If both players are strongly risk-averse then the most likely outcome of this game is:

A) There is a unique Nash Equilibrium in this game and that is {Top, Left}.
B) b. There is a unique Nash Equilibrium in this game and that is {Bottom, Right}.
C) There are two Nash equilibria in this game - {Top, Right} and {Bottom, Left}.
D) There are two Nash equilibria in this game - {Top, Left} and {Bottom, Right}.

Question 15

Consider the following two player game.
Which one of the following statements is correct?

A) There is a unique Nash Equilibrium in this game and that is {Top, Left}.
B) b. There is a unique Nash Equilibrium in this game and that is {Bottom, Right}.
C) There are two Nash equilibria in this game - {Top, Right} and {Bottom, Left}.
D) . There are two Nash equilibria in this game - {Top, Left} and {Bottom, Right}.

Question 16

Consider the following two player game.
Which one of the following statements is correct?

A) There is a unique Nash Equilibrium in this game and that is {Top, Left}.
B) b. There is a unique Nash Equilibrium in this game and that is {Bottom, Right}.
C) There are two Nash equilibria in this game - {Top, Right} and {Bottom, Left}.
D) There are two Nash equilibria in this game - {Top, Left} and {Bottom, Right}.

Question 17

It is often observed that some professionals prefer to drive expensive cars, while some others (who earn similar amounts) do not. What typically accounts for this difference?

A) This game has two Nash equilibria; one where everyone drives expensive cars and another where they do not.
B) This is because there is a dominant strategy Nash equilibrium here where everyone should drive expensive cars but some people do not understand that this is the equilibrium.
C) Those who are concerned about the environment often drive hybrid or electric cars that tend to be more expensive than petrol or diesel cars.
D) This is because for some people the car they drive serves as a signal of professional success; while some others signal success via different means.

Question 18

Consider the following two-player sequential move game. Player 1 moves first and chooses between Top and Bottom. Player 2 gets to see what Player 1 chose and then responds with Left or Right. If Player 1 chooses Top and Player 2 chooses Left, then Player 1 gets $3 and Player 2 gets $2. If Player 1 chooses Top and Player 2 chooses Right, then Player 1 gets $1 and Player 2 gets $1. If Player 1 chooses Bottom and Player 2 chooses Left, then Player 1 gets $0 and Player 2 gets $4. If Player 1 chooses Bottom and Player 2 chooses Right, then Player 1 gets $5 and Player 2 gets $3. The subgame perfect equilibrium of this game is:

A) {Top, Left}.
B) {Top, Right}.
C) {Bottom, Left}.
D) {Bottom, Right}.

Question 19

Consider the following two-player sequential move game. Player 2 is the incumbent firm in a market while Player 1 is considering entering this market. If Player 1 does not enter then the game ends. Player 1 earns $0 and Player 2 earns $80 million. If Player 1 chooses to enter, then Player 2 has two choices: To Accommodate Entry or to Fight Entry. If Player 2 chooses to accommodate entry, then Player 1 earns $10 million while Player #2 earns $50 million. If Player 2 chooses to fight entry, then Player 1 loses $1 million while Player 2 loses $2 million. The subgame perfect equilibrium of this game is:

A) Player 1 to not enter this market.
B) Player 1 to enter and Player 2 to accommodate entry. .
C) Player 1 to enter and Player 2 to fight entry.
D) Player 1 to not enter and Player 2 to fight entry.

Question 20

Consider the following two player game.
Which one of the following statements is correct?

A) There is a unique Nash Equilibrium in this game and that is {Top, Left}.
B) There is a unique Nash Equilibrium in this game and that is {Bottom, Right}
C) There are two Nash equilibria in this game - {Top, Right} and {Bottom, Left}.
D) There are two Nash equilibria in this game - {Top, Left} and {Bottom, Right}.

Question 21

Consider the following two player game.
Which one of the following statements is correct?

A) There is a unique Nash Equilibrium in this game and that is {Top, Left}.
B) {Top, Left} is the payoff dominant equilibrium while {Bottom, Right} is the secure equilibrium.
C) {Top, Left} is the secure equilibrium while {Bottom, Right} is the payoff dominant equilibrium.
D) There is a unique Nash Equilibrium in this game and that is {Bottom, Right}.

Question 22

Consider the following two player game.
Which one of the following statements is correct?

A) There is a unique Nash Equilibrium in this game and that is {Top, Left}.
B) There are two Nash Equilibria in this game: {Top, Left} and {Bottom, Right}.
C) Bottom is a dominant strategy for Player #1; Right is a dominant strategy for Player $2.
D) Bottom is a weakly dominant strategy for Player #1; Right is a weakly dominant strategy for Player $2.

Question 23

Consider the following two player simultaneous move game
Which one of the following statements is correct?

A) There is a unique Nash Equilibrium in this game and that is {Top, Left}.
B) Bottom is a dominant strategy for Player #1 but Player #2 does not have a dominant strategy.
C) Bottom is a dominant strategy for Player #1 while Right is a dominant strategy for Player #2.
D) There are two Nash equilibria in this game - {Top, Left} and {Bottom, Right}.

Question 24

Consider the following two player simultaneous move game
Which one of the following statements is correct?

A) There is a unique Nash Equilibrium in this game and that is {Top, Left}.
B) Bottom is a dominant strategy for Player #1 but Player #2 does not have a dominant strategy.
C) Bottom is a dominant strategy for Player #1 while Right is a dominant strategy for Player #2.
D) There are two Nash equilibria in this game - {Top, Left} and {Bottom, Right}.

Question 25

Consider the following two player simultaneous move game
Which one of the following statements is correct?

A) There is a unique Nash Equilibrium in this game and that is {Top, Left}.
B) Bottom is a dominant strategy for Player #1 but Player #2 does not have a dominant strategy.
C) Player #1 does not have a dominant strategy while Right is a weakly dominant strategy for Player #2.
D) There are two Nash equilibria in this game - {Top, Left} and {Bottom, Right}.

Question 26

Consider the following two player simultaneous move game
Which one of the following statements is correct?

A) There is a unique Nash Equilibrium in this game and that is {Top, Left}.
B) Bottom is a dominant strategy for Player #1 but Player #2 does not have a dominant strategy.
C) There is a unique Nash equilibrium in this game: {Top, Right}.
D) Left is a dominant is a dominant strategy for Player #1 but Player #1 does not have a dominant strategy.

Question 27

A key difference between the stag-hunt game and the battle-of-the-sexes game is that:

A) the stag-hunt game has one payoff-dominant equilibrium and one risk-dominant equilibrium; the battle-of-the-sexes game has two equilibria, none of which is either payoff or risk-dominant.
B) the battle-of-the-sexes game has one payoff-dominant equilibrium and one risk-dominant equilibrium; the stag-hunt game has two equilibria, none of which are either payoff or risk-dominant.
C) the battle-of-the-sexes game has one dominant strategy Nash equilibrium; the stag-hunt game has two equilibria, none of which are either payoff or risk-dominant.
D) the battle-of-the-sexes game has one payoff-dominant equilibrium and one risk-dominant equilibrium; the stag-hunt game has one dominant strategy Nash equilibrium.

Question 28

Consider the following payoff matrix for a 2-player game where the payoffs have the usual connotation. The first number in each cell represents the payoff to Player #1 and the second number the payoff to Player #2.
Which of the following statements is true?

A) This game has one unique Nash equilibrium: {Bottom, Right}.
B) This game has two Nash equilibria: {Top, Left} and {Middle, Centre}.
C) This game has two Nash equilibria: {Middle, Centre} and {Bottom, Right}.
D) This game has three Nash equilibria: {Top, Left}, {Middle, Centre} and {Bottom, Right}.

Question 29

Consider the following payoff matrix for a 2-player game where the payoffs have the usual connotation. The first number in each cell represents the payoff to Player #1 and the second number the payoff to Player #2.
Which of the following statements is true?

A) This game has one unique Nash equilibrium: {Bottom, Right}.
B) Bottom is dominated by Top for Player 1; Right is dominated by Left for Player 2.
C) Bottom is dominated by both Top and Middle for Player 1; Right is dominated by both Left and Centre for Player 2.
D) This game has three Nash equilibria: {Top, Left}, {Middle, Centre} and {Bottom, Right}.

Question 30

Consider the following payoff matrix for a 2-player game where the payoffs have the usual connotation. The first number in each cell represents the payoff to Player #1 and the second number the payoff to Player #2.
Which of the following statements is true?

A) This game has one unique Nash equilibrium: {Bottom, Right}.
B) The {Bottom, Right} outcome is a joint maximum in the sense that it yields the highest payoff to both players but {Bottom, Right} is not an equilibrium.
C) Bottom is dominated by both Top and Middle for Player 1; Right is dominated by both Left and Centre for Player 2.
D) This game has three Nash equilibria: {Top, Left}, {Middle, Centre} and {Bottom, Right}.

Question 31

Consider the following two player game. Player 1 moves first and chooses TOP or BOTTOM. If player 1 chooses TOP then the game ends and both Player 1 and Player 2 get $2 each. But if Player 1 chooses BOTTOM, then Player 2 gets the move and Player 2 can respond with LEFT or RIGHT. If Player 2 chooses LEFT then Player 1 gets $1 while Player 2 gets $4. But if Player 2 chooses RIGHT, then Player 1 gets $3 and Player 2 gets $1.
(a) Draw the game tree for this game. Clearly label your tree with the moves and the payoffs to each player. (b) If you solve this game using backward induction, then what would be the outcome of this game? That is, what is the subgame perfect equilibrium? Explain briefly how you arrive at your answer.

Question 32

Consider two players Yossarian and Nately. Each of them have to strategies: Hunt Stag or Hunt Rabbit. If both Yossarian and Nately choose to hunt stag, then they each get $8. If Yossarian hunts stag while Nately hunts rabbit, then Yossarian gets $0 and Nately gets $5. If Nately hunts stag but Yossarian hunts rabbit, then Yossarian gets $5 and Nately gets $0. Finally, if they both hunt rabbit, then they both get $5 each. (a) Depict the payoff matrix for this game. (b) What is/are the Nash equilibrium/equilibria of this game?

Question 33

Consider two players Yossarian and Nately. Each of them have to strategies: Hunt Stag or Hunt Rabbit. If both Yossarian and Nately choose to hunt stag, then they each get $3. If Yossarian hunts stag while Nately hunts rabbit, then Yossarian gets $0 and Nately gets $5. If Nately hunts stag but Yossarian hunts rabbit, then Yossarian gets $5 and Nately gets $0. Finally, if they both hunt rabbit, then they both get $1 each. (a) Depict the payoff matrix for this game. (b) What is/are the Nash equilibrium/equilibria of this game?

Question 34

Consider the following two player game.
What must be true of the values of a, b, c, d, e, f, g and h in order for {Bottom, Right} to be a dominant strategy Nash equilibrium

Question 35

Consider the following two player game.
What must be true of the values of a, b, c, d, e, f, g and h for {Top, Left} to be the payoff-dominant equilibrium of this game? What must be true of the values of a, b, c, d, e, f, g and h for {Bottom, Right} to be the risk-dominant equilibrium of this game?

Question 36

Consider the following two player game.
What must be true of the values of a, b, c, d, e, f, g and h for {Top, Left} to be an equilibrium of this game?

Question 37

Consider two rivals, say producers of camera film, Fuji and Kodak. Consumers want a film that accurately reproduces colour ad is not grainy. Assume that initially, they had products that were comparable. If one does R & D and improves its product, it will steal customers from its rival. If they both do R & D, and develop comparable products, then they will continue to share the market as before. Thus, the hypothetical payoff matrix (in millions of dollars) appears as below (the profits in the case of research take into account the expenditures on research). What is the Nash equilibrium of this game?

Question 38

Rosewood College and Elmwood College are the best private schools in the state. Each prides itself on its lacrosse team. If neither offers scholarships for promising players, then obviously the cost of scholarships will be zero. For $20,000 in scholarships, either could attract the best players (if the other did not offer scholarships), win the conference championship, and attract at least $50,000 in new donations. In this case, however, donations at the losing school would fall by $30,000. On the other hand, if both offered scholarships, there would be no advantage, no extra donations, and each would have spent the $20,000 for nothing. Clearly depict the payoff matrix of this game. What strategy will each school choose in the Nash equilibrium of this game?

Question 39

Consider two rival firms Fuji and Kodak. Each firm can choose to either cut price or not cut prices. If neither cuts prices, then they each make $7 billion. If both cut prices, then they each make $4 billion. If Kodak does not cut prices but Fuji does, then Fuji gets $9 billion while Kodak gets $3 billion. If Fuji does not cut prices but Kodak does then Kodak gets $9 billion and Fuji gets $3 billion. Draw the payoff matrix of this game and explain what do you think will happen in the Nash equilibrium of this game?

Question 40

Consider the following two-player sequential move game. Player 1 moves first and chooses between Top and Bottom. Player 2 gets to see what Player 1 chose and then responds with Left or Right. If Player 1 chooses Top and Player 2 chooses Left, then Player 1 gets $3 and Player 2 gets $2. If Player 1 chooses Top and Player 2 chooses Right, then Player 1 gets $1 and Player 2 gets $1. If Player 1 chooses Top and Player 2 chooses Left, then Player 1 gets $0 and Player 2 gets $4. If Player 1 chooses Top and Player 2 chooses Left, then Player 1 gets $5 and Player 2 gets $3. Draw the game tree for this game and then solve for the subgame perfect equilibrium.

Question 41

Consider the following two-player sequential move game. Player 2 is the incumbent firm in a market while Player 1 is considering entering this market. If Player 1 does not enter then the game ends. Player 1 earns $0 and Player 2 earns $80 million. If Player 1 chooses to enter, then Player 2 has two choices: To Accommodate Entry or to Fight Entry. IF Player 2 chooses to accommodate entry, then Player 1 earns $10 million while Player #2 earns $50 million. Draw the game tree for this game and solve for the subgame perfect equilibrium.

Question 42

Consider the following payoff matrix for a 2-player game where the payoffs have the usual connotation. The first number in each cell represents the payoff to Player #1 and the second number the payoff to Player #2.
(a) What is/are the Nash equilibrium/equilibria of this game? (b) Part B: Explain your answer very briefly.

Question 43

Consider the following game: Player #1 has three strategies Top, Middle and Bottom. Player 2 has four strategies A, B, C and D.

Are any of the strategies of the two players dominated? Which ones? Does this game have
any pure strategy Nash equilibrium? If yes, then how many and which ones? Briefly explain
your answers.