Game Theory: Strategies, Nash Equilibrium, and Applications: Multiple
Choice Questions with Solutions
1. In a two-player zero-sum game, player A's payoff matrix is:
[3 -1]
[2 4]
What is player B's payoff matrix?
a) [3 -1] b) [-3 1] c) [-3 -1] d) [3 1]
[2 4] [-2 -4] [2 -4] [-2 4]
Answer: b) [-3 1]
[-2 -4]
Solution: In a zero-sum game, the payoffs for player B are the negatives of player A's payoffs.
2. Two firms are competing in a duopoly market. If both firms choose high prices, they each earn
$100 million. If both choose low prices, they each earn $60 million. If one chooses high and the other
low, the high-price firm earns $40 million and the low-price firm earns $120 million. What is the Nash
equilibrium?
a) (High, High) b) (Low, Low) c) (High, Low) d) (Low, High)
Answer: b) (Low, Low)
Solution: Check each strategy profile:
(High, High): Each firm has incentive to deviate to Low
(Low, Low): Neither firm has incentive to deviate
(High, Low) and (Low, High): The high-price firm has incentive to deviate to Low
3. In the Prisoner's Dilemma, what is the typical payoff structure?
a) Cooperation > Defection b) Defection > Cooperation c) Cooperation = Defection d) No
consistent structure
Answer: b) Defection > Cooperation
Solution: In the Prisoner's Dilemma, the payoff for mutual defection is typically greater than the
payoff for mutual cooperation.
4. What is the mixed strategy Nash equilibrium for the following game?
[3, 3 0, 5]
[5, 0 1, 1]
a) (1/2, 1/2) for both players b) (2/3, 1/3) for both players c) (1/3, 2/3) for both players d) (1,
0) for both players
Answer: b) (2/3, 1/3) for both players
Solution:
Let p be the probability of playing the first strategy for each player.
For indifference: 3p + 5(1-p) = 0p + 1(1-p)
3p + 5 - 5p = 1 - p
-p = -4
p = 2/3
5. In a Cournot duopoly model, firm 1's best response function is q₁ = 30 - 0.5q₂, and firm 2's best
response function is q₂ = 30 - 0.5q₁. What is the Nash equilibrium?
a) (q₁, q₂) = (20, 20) b) (q₁, q₂) = (15, 15) c) (q₁, q₂) = (10, 10) d) (q₁, q₂) = (25, 25)
Answer: a) (q₁, q₂) = (20, 20)
Solution:
Solve the system of equations:
q₁ = 30 - 0.5q₂
q₂ = 30 - 0.5q₁
Substituting the second equation into the first:
q₁ = 30 - 0.5(30 - 0.5q₁)
q₁ = 30 - 15 + 0.25q₁
0.75q₁ = 15
q₁ = 20
q₂ = 30 - 0.5(20) = 20
6. In a game of Chicken, what is the typical number of pure strategy Nash equilibria?
a) 0 b) 1 c) 2 d) 3
Answer: c) 2
Solution: In a typical game of Chicken, there are two pure strategy Nash equilibria where one player
swerves and the other doesn't.
7. What is the value of the game for player 1 in the following zero-sum game?
[4 -2 3]
[1 0 2]
a) 1 b) 2 c) 3 d) 4
Answer: a) 1
Solution:
Use the minimax theorem:
Player 1's maximin value = max(min(4,-2,3), min(1,0,2)) = max(-2, 0) = 0
Player 2's minimax value = min(max(4,1), max(-2,0), max(3,2)) = min(4, 0, 3) = 0
The value of the game is 0 for player 2, which means it's 1 for player 1.
8. In a Bertrand duopoly model with homogeneous products and constant marginal costs c, what is
the Nash equilibrium price?
a) p = c b) p > c c) p < c d) p = 0
Answer: a) p = c
Solution: In a Bertrand duopoly with homogeneous products, firms will undercut each other's
prices until they reach the marginal cost c, where further price cuts would lead to losses.
9. What is the Shapley value for player 1 in the following 3-player characteristic function game?
v({1}) = 0, v({2}) = 0, v({3}) = 0
v({1,2}) = 60, v({1,3}) = 60, v({2,3}) = 60
v({1,2,3}) = 90
a) 30 b) 40 c) 45 d) 50
Answer: a) 30
Solution:
Shapley value = (1/3!)[v({1}) - v({})] + (1/6)[v({1,2}) - v({2})] + (1/6)[v({1,3}) - v({3})] + (1/3!)[v({1,2,3})
- v({2,3})]
= (1/6)[0] + (1/6)[60] + (1/6)[60] + (1/6)[30]
= 10 + 10 + 5 = 25
10. In an extensive form game, what is the number of subgame perfect Nash equilibria if there are 3
pure strategy Nash equilibria?
a) 0 b) 1 c) 2 d) 3 or more
Answer: b) 1
Solution: The number of subgame perfect Nash equilibria is always less than or equal to the
number of Nash equilibria, and there is always at least one subgame perfect Nash equilibrium in
finite games.
11. In a simultaneous move game, what is the best response of player 1 to the following mixed
strategy of player 2: (0.6, 0.4)?
Player 1's payoff matrix:
[2 1]
[3 0]
a) (1, 0) b) (0, 1) c) (0.5, 0.5) d) Insufficient information
Answer: a) (1, 0)
Solution:
Expected payoff for strategy 1: 2(0.6) + 1(0.4) = 1.6
Expected payoff for strategy 2: 3(0.6) + 0(0.4) = 1.8
Strategy 2 yields a higher payoff, so the best response is (1, 0).
12. What is the core of the following 3-player characteristic function game?
v({1}) = 0, v({2}) = 0, v({3}) = 0
v({1,2}) = 40, v({1,3}) = 50, v({2,3}) = 60
v({1,2,3}) = 90
a) Empty b) {(30, 30, 30)} c) {(x, y, z) | x + y + z = 90, x ≥ 0, y ≥ 0, z ≥ 0} d) {(x, y, z) | x + y + z =
90, x ≥ 0, y ≥ 0, z ≥ 0, x + y ≥ 40, x + z ≥ 50, y + z ≥ 60}
Answer: d) {(x, y, z) | x + y + z = 90, x ≥ 0, y ≥ 0, z ≥ 0, x + y ≥ 40, x + z ≥ 50, y + z ≥ 60}
Solution: The core consists of all allocations that are efficient (sum to 90) and satisfy individual
rationality (non-negative) and group rationality (each coalition gets at least its value).
13. In a two-player zero-sum game with the following payoff matrix for player 1, what is the value of
the game?
[2 -1 3]
[0 4 -2]
a) 1 b) 2 c) 3 d) 4
Answer: b) 2
Solution:
Player 1's maximin strategy: max(min(2,-1,3), min(0,4,-2)) = max(-1, -2) = -1
Player 2's minimax strategy: min(max(2,0), max(-1,4), max(3,-2)) = min(2, 4, 3) = 2
The value of the game is 2.
14. In a Cournot oligopoly with n firms, identical products, and linear inverse demand function P = a -
bQ, what is the Nash equilibrium quantity for each firm if the marginal cost is c?
a) (a-c)/(bn) b) (a-c)/(b(n+1)) c) (a-c)/(2bn) d) (a-c)/(2b(n+1))
Answer: b) (a-c)/(b(n+1))
Solution:
Profit function: πᵢ = (a - bQ)qᵢ - cqᵢ
FOC: dπᵢ/dqᵢ = a - 2bqᵢ - bQ₋ᵢ - c = 0
Symmetric NE: qᵢ = q for all i, so Q = nq
Substituting: a - 2bq - b(n-1)q - c = 0
Solving for q: q = (a-c)/(b(n+1))
15. What is the mixed strategy Nash equilibrium for the following game?
[0, 0 3, 1]
[1, 3 2, 2]
a) ((1/2, 1/2), (1/2, 1/2)) b) ((2/3, 1/3), (2/3, 1/3)) c) ((1/3, 2/3), (1/3, 2/3)) d) ((3/4, 1/4),
(3/4, 1/4))
Answer: b) ((2/3, 1/3), (2/3, 1/3))
Solution:
Let p be the probability of playing the first strategy for each player.
For player 1's indifference: 0p + 3(1-p) = 1p + 2(1-p)
3 - 3p = 1p + 2 - 2p
1 = 2p
p = 1/2
For player 2's indifference: 0p + 1(1-p) = 3p + 2(1-p)
1 - p = 3p + 2 - 2p
-1 = 2p
p = -1/2 (invalid)
Therefore, we need to check the pure strategies:
(1,1): Player 2 has incentive to deviate
(1,2): Player 1 has incentive to deviate
(2,1): Player 2 has incentive to deviate
(2,2): Neither player has incentive to deviate
The pure strategy Nash equilibrium is (2,2).
16. In a bargaining game with alternating offers, what is the subgame perfect equilibrium outcome if
the discount factor is δ and the size of the pie is 1?
a) (1/(1+δ), δ/(1+δ)) b) (δ/(1+δ), 1/(1+δ)) c) (1/2, 1/2) d) (1, 0)
Answer: a) (1/(1+δ), δ/(1+δ))
Solution:
In the subgame perfect equilibrium, player 1 offers x₁ = δ/(1+δ) to player 2, who accepts.
Player 1's share is 1 - x₁ = 1 - δ/(1+δ) = 1/(1+δ)
17. What is the Shapley value for player 2 in the following 3-player characteristic function game?
v({1}) = 0, v({2}) = 10, v({3}) = 0
v({1,2}) = 40, v({1,3}) = 30, v({2,3}) = 50
v({1,2,3}) = 100
a) 30 b) 35 c) 40 d) 45
Answer: c) 40
Solution:
Shapley value = (1/3!)[v({2}) - v({})] + (1/6)[v({1,2}) - v({1})] + (1/6)[v({2,3}) - v({3})] + (1/3!)
[v({1,2,3}) - v({1,3})]
= (1/6)[10] + (1/6)[40] + (1/6)[50] + (1/6)[70]
= 1.67 + 6.67 + 8.33 + 11.67 ≈ 40
18. In a first-price sealed-bid auction with two risk-neutral bidders whose valuations are uniformly
distributed on [0,1], what is the symmetric Bayesian Nash equilibrium bidding strategy?
a) b(v) = v/2 b) b(v) = v c) b(v) = 2v/3 d) b(v) = v/3
Answer: a) b(v) = v/2
Solution:
In a first-price sealed-bid auction with uniform distributions, the symmetric BNE bidding strategy is
b(v) = (n-1)v/n, where n is the number of bidders.
With n = 2, we get b(v) = v/2.
19. In a repeated Prisoner's Dilemma game with discount factor δ, what is the minimum discount
factor required for the grim trigger strategy to sustain cooperation?
Payoff matrix:
[3, 3 0, 5]
[5, 0 1, 1]
a) 1/2 b) 2/3 c) 3/4 d) 4/5
Answer: b) 2/3
Solution:
For cooperation to be sustained: 3/(1-δ) ≥ 5 + δ/(1-δ)
3 ≥ 5(1-δ) + δ
3 ≥ 5 - 5δ + δ
3 ≥ 5 - 4δ
4δ ≥ 2
δ ≥ 1/2
What is the mixed strategy Nash equilibrium for the following game?
[3, 3 0, 5]
[5, 0 1, 1]
a) (1/2, 1/2) for both players b) (2/3, 1/3) for both players c) (1/3, 2/3) for both players d) (1,
0) for both players
Answer: b) (2/3, 1/3) for both players
Solution:
Let p be the probability of playing the first strategy for each player.
For indifference: 3p + 5(1-p) = 0p + 1(1-p)
3p + 5 - 5p = 1 - p
-p = -4
p = 2/3
5. In a Cournot duopoly model, firm 1's best response function is q₁ = 30 - 0.5q₂, and firm 2's best
response function is q₂ = 30 - 0.5q₁. What is the Nash equilibrium?
a) (q₁, q₂) = (20, 20) b) (q₁, q₂) = (15, 15) c) (q₁, q₂) = (10, 10) d) (q₁, q₂) = (25, 25)
Answer: a) (q₁, q₂) = (20, 20)
Solution:
Solve the system of equations:
q₁ = 30 - 0.5q₂
q₂ = 30 - 0.5q₁
Substituting the second equation into the first:
q₁ = 30 - 0.5(30 - 0.5q₁)
q₁ = 30 - 15 + 0.25q₁
0.75q₁ = 15
q₁ = 20
q₂ = 30 - 0.5(20) = 20
6. In a game of Chicken, what is the typical number of pure strategy Nash equilibria?
a) 0 b) 1 c) 2 d) 3
Answer: c) 2
Solution: In a typical game of Chicken, there are two pure strategy Nash equilibria where one player
swerves and the other doesn't.
7. What is the value of the game for player 1 in the following zero-sum game?
[4 -2 3]
[1 0 2]
a) 1 b) 2 c) 3 d) 4
Answer: a) 1
Solution:
Use the minimax theorem:
Player 1's maximin value = max(min(4,-2,3), min(1,0,2)) = max(-2, 0) = 0
Player 2's minimax value = min(max(4,1), max(-2,0), max(3,2)) = min(4, 0, 3) = 0
The value of the game is 0 for player 2, which means it's 1 for player 1.
8. In a Bertrand duopoly model with homogeneous products and constant marginal costs c, what is
the Nash equilibrium price?
a) p = c b) p > c c) p < c d) p = 0
Answer: a) p = c
Solution: In a Bertrand duopoly with homogeneous products, firms will undercut each other's
prices until they reach the marginal cost c, where further price cuts would lead to losses.
9. What is the Shapley value for player 1 in the following 3-player characteristic function game?
v({1}) = 0, v({2}) = 0, v({3}) = 0
v({1,2}) = 60, v({1,3}) = 60, v({2,3}) = 60
v({1,2,3}) = 90
a) 30 b) 40 c) 45 d) 50
Answer: a) 30
Solution:
Shapley value = (1/3!)[v({1}) - v({})] + (1/6)[v({1,2}) - v({2})] + (1/6)[v({1,3}) - v({3})] + (1/3!)[v({1,2,3})
- v({2,3})]
= (1/6)[0] + (1/6)[60] + (1/6)[60] + (1/6)[30]
= 10 + 10 + 5 = 25
10. In an extensive form game, what is the number of subgame perfect Nash equilibria if there are 3
pure strategy Nash equilibria?
a) 0 b) 1 c) 2 d) 3 or more
Answer: b) 1
Solution: The number of subgame perfect Nash equilibria is always less than or equal to the
number of Nash equilibria, and there is always at least one subgame perfect Nash equilibrium in
finite games.
11. In a simultaneous move game, what is the best response of player 1 to the following mixed
strategy of player 2: (0.6, 0.4)?
Player 1's payoff matrix:
[2 1]
[3 0]
a) (1, 0) b) (0, 1) c) (0.5, 0.5) d) Insufficient information
Answer: a) (1, 0)
Solution:
Expected payoff for strategy 1: 2(0.6) + 1(0.4) = 1.6
Expected payoff for strategy 2: 3(0.6) + 0(0.4) = 1.8
Strategy 2 yields a higher payoff, so the best response is (1, 0).
12. What is the core of the following 3-player characteristic function game?
v({1}) = 0, v({2}) = 0, v({3}) = 0
v({1,2}) = 40, v({1,3}) = 50, v({2,3}) = 60
v({1,2,3}) = 90
a) Empty b) {(30, 30, 30)} c) {(x, y, z) | x + y + z = 90, x ≥ 0, y ≥ 0, z ≥ 0} d) {(x, y, z) | x + y + z =
90, x ≥ 0, y ≥ 0, z ≥ 0, x + y ≥ 40, x + z ≥ 50, y + z ≥ 60}
Answer: d) {(x, y, z) | x + y + z = 90, x ≥ 0, y ≥ 0, z ≥ 0, x + y ≥ 40, x + z ≥ 50, y + z ≥ 60}
Solution: The core consists of all allocations that are efficient (sum to 90) and satisfy individual
rationality (non-negative) and group rationality (each coalition gets at least its value).
13. In a two-player zero-sum game with the following payoff matrix for player 1, what is the value of
the game?
[2 -1 3]
[0 4 -2]
a) 1 b) 2 c) 3 d) 4
Answer: b) 2
Solution:
Player 1's maximin strategy: max(min(2,-1,3), min(0,4,-2)) = max(-1, -2) = -1
Player 2's minimax strategy: min(max(2,0), max(-1,4), max(3,-2)) = min(2, 4, 3) = 2
The value of the game is 2.
14. In a Cournot oligopoly with n firms, identical products, and linear inverse demand function P = a -
bQ, what is the Nash equilibrium quantity for each firm if the marginal cost is c?
a) (a-c)/(bn) b) (a-c)/(b(n+1)) c) (a-c)/(2bn) d) (a-c)/(2b(n+1))
Answer: b) (a-c)/(b(n+1))
Solution:
Profit function: πᵢ = (a - bQ)qᵢ - cqᵢ
FOC: dπᵢ/dqᵢ = a - 2bqᵢ - bQ₋ᵢ - c = 0
Symmetric NE: qᵢ = q for all i, so Q = nq
Substituting: a - 2bq - b(n-1)q - c = 0
Solving for q: q = (a-c)/(b(n+1))
15. What is the mixed strategy Nash equilibrium for the following game?
[0, 0 3, 1]
[1, 3 2, 2]
a) ((1/2, 1/2), (1/2, 1/2)) b) ((2/3, 1/3), (2/3, 1/3)) c) ((1/3, 2/3), (1/3, 2/3)) d) ((3/4, 1/4),
(3/4, 1/4))
Answer: b) ((2/3, 1/3), (2/3, 1/3))
Solution:
Let p be the probability of playing the first strategy for each player.
For player 1's indifference: 0p + 3(1-p) = 1p + 2(1-p)
3 - 3p = 1p + 2 - 2p
1 = 2p
p = 1/2
For player 2's indifference: 0p + 1(1-p) = 3p + 2(1-p)
1 - p = 3p + 2 - 2p
-1 = 2p
p = -1/2 (invalid)
Therefore, we need to check the pure strategies:
(1,1): Player 2 has incentive to deviate
(1,2): Player 1 has incentive to deviate
(2,1): Player 2 has incentive to deviate
(2,2): Neither player has incentive to deviate
The pure strategy Nash equilibrium is (2,2).
16. In a bargaining game with alternating offers, what is the subgame perfect equilibrium outcome if
the discount factor is δ and the size of the pie is 1?
a) (1/(1+δ), δ/(1+δ)) b) (δ/(1+δ), 1/(1+δ)) c) (1/2, 1/2) d) (1, 0)
Answer: a) (1/(1+δ), δ/(1+δ))
Solution:
In the subgame perfect equilibrium, player 1 offers x₁ = δ/(1+δ) to player 2, who accepts.
Player 1's share is 1 - x₁ = 1 - δ/(1+δ) = 1/(1+δ)
17. What is the Shapley value for player 2 in the following 3-player characteristic function game?
v({1}) = 0, v({2}) = 10, v({3}) = 0
v({1,2}) = 40, v({1,3}) = 30, v({2,3}) = 50
v({1,2,3}) = 100
a) 30 b) 35 c) 40 d) 45
Answer: c) 40
Solution:
Shapley value = (1/3!)[v({2}) - v({})] + (1/6)[v({1,2}) - v({1})] + (1/6)[v({2,3}) - v({3})] + (1/3!)
[v({1,2,3}) - v({1,3})]
= (1/6)[10] + (1/6)[40] + (1/6)[50] + (1/6)[70]
= 1.67 + 6.67 + 8.33 + 11.67 ≈ 40
18. In a first-price sealed-bid auction with two risk-neutral bidders whose valuations are uniformly
distributed on [0,1], what is the symmetric Bayesian Nash equilibrium bidding strategy?
a) b(v) = v/2 b) b(v) = v c) b(v) = 2v/3 d) b(v) = v/3
Answer: a) b(v) = v/2
Solution:
In a first-price sealed-bid auction with uniform distributions, the symmetric BNE bidding strategy is
b(v) = (n-1)v/n, where n is the number of bidders.
With n = 2, we get b(v) = v/2.
19. In a repeated Prisoner's Dilemma game with discount factor δ, what is the minimum discount
factor required for the grim trigger strategy to sustain cooperation?
Payoff matrix:
[3, 3 0, 5]
[5, 0 1, 1]
a) 1/2 b) 2/3 c) 3/4 d) 4/5
Answer: b) 2/3
Solution:
For cooperation to be sustained: 3/(1-δ) ≥ 5 + δ/(1-δ)
3 ≥ 5(1-δ) + δ
3 ≥ 5 - 5δ + δ
3 ≥ 5 - 4δ
4δ ≥ 2
δ ≥ 1/2
What is the mixed strategy Nash equilibrium for the following game?
[3, 3 0, 5]
[5, 0 1, 1]
a) (1/2, 1/2) for both players b) (2/3, 1/3) for both players c) (1/3, 2/3) for both players d) (1,
0) for both players
Answer: b) (2/3, 1/3) for both players
Solution:
Let p be the probability of playing the first strategy for each player.
For indifference: 3p + 5(1-p) = 0p + 1(1-p)
3p + 5 - 5p = 1 - p
-p = -4
p = 2/3
5. In a Cournot duopoly model, firm 1's best response function is q₁ = 30 - 0.5q₂, and firm 2's best
response function is q₂ = 30 - 0.5q₁. What is the Nash equilibrium?
a) (q₁, q₂) = (20, 20) b) (q₁, q₂) = (15, 15) c) (q₁, q₂) = (10, 10) d) (q₁, q₂) = (25, 25)
Answer: a) (q₁, q₂) = (20, 20)
Solution:
Solve the system of equations:
q₁ = 30 - 0.5q₂
q₂ = 30 - 0.5q₁
Substituting the second equation into the first:
q₁ = 30 - 0.5(30 - 0.5q₁)
q₁ = 30 - 15 + 0.25q₁
0.75q₁ = 15
q₁ = 20
q₂ = 30 - 0.5(20) = 20
6. In a game of Chicken, what is the typical number of pure strategy Nash equilibria?
a) 0 b) 1 c) 2 d) 3
Answer: c) 2
Solution: In a typical game of Chicken, there are two pure strategy Nash equilibria where one player
swerves and the other doesn't.
7. What is the value of the game for player 1 in the following zero-sum game?
[4 -2 3]
[1 0 2]
a) 1 b) 2 c) 3 d) 4
Answer: a) 1
Solution:
Use the minimax theorem:
Player 1's maximin value = max(min(4,-2,3), min(1,0,2)) = max(-2, 0) = 0
Player 2's minimax value = min(max(4,1), max(-2,0), max(3,2)) = min(4, 0, 3) = 0
The value of the game is 0 for player 2, which means it's 1 for player 1.
8. In a Bertrand duopoly model with homogeneous products and constant marginal costs c, what is
the Nash equilibrium price?
a) p = c b) p > c c) p < c d) p = 0
Answer: a) p = c
Solution: In a Bertrand duopoly with homogeneous products, firms will undercut each other's
prices until they reach the marginal cost c, where further price cuts would lead to losses.
9. What is the Shapley value for player 1 in the following 3-player characteristic function game?
v({1}) = 0, v({2}) = 0, v({3}) = 0
v({1,2}) = 60, v({1,3}) = 60, v({2,3}) = 60
v({1,2,3}) = 90
a) 30 b) 40 c) 45 d) 50
Answer: a) 30
Solution:
Shapley value = (1/3!)[v({1}) - v({})] + (1/6)[v({1,2}) - v({2})] + (1/6)[v({1,3}) - v({3})] + (1/3!)[v({1,2,3})
- v({2,3})]
= (1/6)[0] + (1/6)[60] + (1/6)[60] + (1/6)[30]
= 10 + 10 + 5 = 25
10. In an extensive form game, what is the number of subgame perfect Nash equilibria if there are 3
pure strategy Nash equilibria?
a) 0 b) 1 c) 2 d) 3 or more
Answer: b) 1
Solution: The number of subgame perfect Nash equilibria is always less than or equal to the
number of Nash equilibria, and there is always at least one subgame perfect Nash equilibrium in
finite games.
11. In a simultaneous move game, what is the best response of player 1 to the following mixed
strategy of player 2: (0.6, 0.4)?
Player 1's payoff matrix:
[2 1]
[3 0]
a) (1, 0) b) (0, 1) c) (0.5, 0.5) d) Insufficient information
Answer: a) (1, 0)
Solution:
Expected payoff for strategy 1: 2(0.6) + 1(0.4) = 1.6
Expected payoff for strategy 2: 3(0.6) + 0(0.4) = 1.8
Strategy 2 yields a higher payoff, so the best response is (1, 0).
12. What is the core of the following 3-player characteristic function game?
v({1}) = 0, v({2}) = 0, v({3}) = 0
v({1,2}) = 40, v({1,3}) = 50, v({2,3}) = 60
v({1,2,3}) = 90
a) Empty b) {(30, 30, 30)} c) {(x, y, z) | x + y + z = 90, x ≥ 0, y ≥ 0, z ≥ 0} d) {(x, y, z) | x + y + z =
90, x ≥ 0, y ≥ 0, z ≥ 0, x + y ≥ 40, x + z ≥ 50, y + z ≥ 60}
Answer: d) {(x, y, z) | x + y + z = 90, x ≥ 0, y ≥ 0, z ≥ 0, x + y ≥ 40, x + z ≥ 50, y + z ≥ 60}
Solution: The core consists of all allocations that are efficient (sum to 90) and satisfy individual
rationality (non-negative) and group rationality (each coalition gets at least its value).
13. In a two-player zero-sum game with the following payoff matrix for player 1, what is the value of
the game?
[2 -1 3]
[0 4 -2]
a) 1 b) 2 c) 3 d) 4
Answer: b) 2
Solution:
Player 1's maximin strategy: max(min(2,-1,3), min(0,4,-2)) = max(-1, -2) = -1
Player 2's minimax strategy: min(max(2,0), max(-1,4), max(3,-2)) = min(2, 4, 3) = 2
The value of the game is 2.
14. In a Cournot oligopoly with n firms, identical products, and linear inverse demand function P = a -
bQ, what is the Nash equilibrium quantity for each firm if the marginal cost is c?
a) (a-c)/(bn) b) (a-c)/(b(n+1)) c) (a-c)/(2bn) d) (a-c)/(2b(n+1))
Answer: b) (a-c)/(b(n+1))
Solution:
Profit function: πᵢ = (a - bQ)qᵢ - cqᵢ
FOC: dπᵢ/dqᵢ = a - 2bqᵢ - bQ₋ᵢ - c = 0
Symmetric NE: qᵢ = q for all i, so Q = nq
Substituting: a - 2bq - b(n-1)q - c = 0
Solving for q: q = (a-c)/(b(n+1))
15. What is the mixed strategy Nash equilibrium for the following game?
[0, 0 3, 1]
[1, 3 2, 2]
a) ((1/2, 1/2), (1/2, 1/2)) b) ((2/3, 1/3), (2/3, 1/3)) c) ((1/3, 2/3), (1/3, 2/3)) d) ((3/4, 1/4),
(3/4, 1/4))
Answer: b) ((2/3, 1/3), (2/3, 1/3))
Solution:
Let p be the probability of playing the first strategy for each player.
For player 1's indifference: 0p + 3(1-p) = 1p + 2(1-p)
3 - 3p = 1p + 2 - 2p
1 = 2p
p = 1/2
For player 2's indifference: 0p + 1(1-p) = 3p + 2(1-p)
1 - p = 3p + 2 - 2p
-1 = 2p
p = -1/2 (invalid)
Therefore, we need to check the pure strategies:
(1,1): Player 2 has incentive to deviate
(1,2): Player 1 has incentive to deviate
(2,1): Player 2 has incentive to deviate
(2,2): Neither player has incentive to deviate
The pure strategy Nash equilibrium is (2,2).
16. In a bargaining game with alternating offers, what is the subgame perfect equilibrium outcome if
the discount factor is δ and the size of the pie is 1?
a) (1/(1+δ), δ/(1+δ)) b) (δ/(1+δ), 1/(1+δ)) c) (1/2, 1/2) d) (1, 0)
Answer: a) (1/(1+δ), δ/(1+δ))
Solution:
In the subgame perfect equilibrium, player 1 offers x₁ = δ/(1+δ) to player 2, who accepts.
Player 1's share is 1 - x₁ = 1 - δ/(1+δ) = 1/(1+δ)
17. What is the Shapley value for player 2 in the following 3-player characteristic function game?
v({1}) = 0, v({2}) = 10, v({3}) = 0
v({1,2}) = 40, v({1,3}) = 30, v({2,3}) = 50
v({1,2,3}) = 100
a) 30 b) 35 c) 40 d) 45
Answer: c) 40
Solution:
Shapley value = (1/3!)[v({2}) - v({})] + (1/6)[v({1,2}) - v({1})] + (1/6)[v({2,3}) - v({3})] + (1/3!)
[v({1,2,3}) - v({1,3})]
= (1/6)[10] + (1/6)[40] + (1/6)[50] + (1/6)[70]
= 1.67 + 6.67 + 8.33 + 11.67 ≈ 40
18. In a first-price sealed-bid auction with two risk-neutral bidders whose valuations are uniformly
distributed on [0,1], what is the symmetric Bayesian Nash equilibrium bidding strategy?
a) b(v) = v/2 b) b(v) = v c) b(v) = 2v/3 d) b(v) = v/3
Answer: a) b(v) = v/2
Solution:
In a first-price sealed-bid auction with uniform distributions, the symmetric BNE bidding strategy is
b(v) = (n-1)v/n, where n is the number of bidders.
With n = 2, we get b(v) = v/2.
19. In a repeated Prisoner's Dilemma game with discount factor δ, what is the minimum discount
factor required for the grim trigger strategy to sustain cooperation?
Payoff matrix:
[3, 3 0, 5]
[5, 0 1, 1]
a) 1/2 b) 2/3 c) 3/4 d) 4/5
Answer: b) 2/3
Solution:
For cooperation to be sustained: 3/(1-δ) ≥ 5 + δ/(1-δ)
3 ≥ 5(1-δ) + δ
3 ≥ 5 - 5δ + δ
3 ≥ 5 - 4δ
4δ ≥ 2
δ ≥ 1/2
What is the mixed strategy Nash equilibrium for the following game?
[3, 3 0, 5]
[5, 0 1, 1]
a) (1/2, 1/2) for both players b) (2/3, 1/3) for both players c) (1/3, 2/3) for both players d) (1,
0) for both players
Answer: b) (2/3, 1/3) for both players
Solution:
Let p be the probability of playing the first strategy for each player.
For indifference: 3p + 5(1-p) = 0p + 1(1-p)
3p + 5 - 5p = 1 - p
-p = -4
p = 2/3
5. In a Cournot duopoly model, firm 1's best response function is q₁ = 30 - 0.5q₂, and firm 2's best
response function is q₂ = 30 - 0.5q₁. What is the Nash equilibrium?
a) (q₁, q₂) = (20, 20) b) (q₁, q₂) = (15, 15) c) (q₁, q₂) = (10, 10) d) (q₁, q₂) = (25, 25)
Answer: a) (q₁, q₂) = (20, 20)
Solution:
Solve the system of equations:
q₁ = 30 - 0.5q₂
q₂ = 30 - 0.5q₁
Substituting the second equation into the first:
q₁ = 30 - 0.5(30 - 0.5q₁)
q₁ = 30 - 15 + 0.25q₁
0.75q₁ = 15
q₁ = 20
q₂ = 30 - 0.5(20) = 20
6. In a game of Chicken, what is the typical number of pure strategy Nash equilibria?
a) 0 b) 1 c) 2 d) 3
Answer: c) 2
Solution: In a typical game of Chicken, there are two pure strategy Nash equilibria where one player
swerves and the other doesn't.
7. What is the value of the game for player 1 in the following zero-sum game?
[4 -2 3]
[1 0 2]
a) 1 b) 2 c) 3 d) 4
Answer: a) 1
Solution:
Use the minimax theorem:
Player 1's maximin value = max(min(4,-2,3), min(1,0,2)) = max(-2, 0) = 0
Player 2's minimax value = min(max(4,1), max(-2,0), max(3,2)) = min(4, 0, 3) = 0
The value of the game is 0 for player 2, which means it's 1 for player 1.
8. In a Bertrand duopoly model with homogeneous products and constant marginal costs c, what is
the Nash equilibrium price?
a) p = c b) p > c c) p < c d) p = 0
Answer: a) p = c
Solution: In a Bertrand duopoly with homogeneous products, firms will undercut each other's
prices until they reach the marginal cost c, where further price cuts would lead to losses.
9. What is the Shapley value for player 1 in the following 3-player characteristic function game?
v({1}) = 0, v({2}) = 0, v({3}) = 0
v({1,2}) = 60, v({1,3}) = 60, v({2,3}) = 60
v({1,2,3}) = 90
a) 30 b) 40 c) 45 d) 50
Answer: a) 30
Solution:
Shapley value = (1/3!)[v({1}) - v({})] + (1/6)[v({1,2}) - v({2})] + (1/6)[v({1,3}) - v({3})] + (1/3!)[v({1,2,3})
- v({2,3})]
= (1/6)[0] + (1/6)[60] + (1/6)[60] + (1/6)[30]
= 10 + 10 + 5 = 25
10. In an extensive form game, what is the number of subgame perfect Nash equilibria if there are 3
pure strategy Nash equilibria?
a) 0 b) 1 c) 2 d) 3 or more
Answer: b) 1
Solution: The number of subgame perfect Nash equilibria is always less than or equal to the
number of Nash equilibria, and there is always at least one subgame perfect Nash equilibrium in
finite games.
11. In a simultaneous move game, what is the best response of player 1 to the following mixed
strategy of player 2: (0.6, 0.4)?
Player 1's payoff matrix:
[2 1]
[3 0]
a) (1, 0) b) (0, 1) c) (0.5, 0.5) d) Insufficient information
Answer: a) (1, 0)
Solution:
Expected payoff for strategy 1: 2(0.6) + 1(0.4) = 1.6
Expected payoff for strategy 2: 3(0.6) + 0(0.4) = 1.8
Strategy 2 yields a higher payoff, so the best response is (1, 0).
12. What is the core of the following 3-player characteristic function game?
v({1}) = 0, v({2}) = 0, v({3}) = 0
v({1,2}) = 40, v({1,3}) = 50, v({2,3}) = 60
v({1,2,3}) = 90
a) Empty b) {(30, 30, 30)} c) {(x, y, z) | x + y + z = 90, x ≥ 0, y ≥ 0, z ≥ 0} d) {(x, y, z) | x + y + z =
90, x ≥ 0, y ≥ 0, z ≥ 0, x + y ≥ 40, x + z ≥ 50, y + z ≥ 60}
Answer: d) {(x, y, z) | x + y + z = 90, x ≥ 0, y ≥ 0, z ≥ 0, x + y ≥ 40, x + z ≥ 50, y + z ≥ 60}
Solution: The core consists of all allocations that are efficient (sum to 90) and satisfy individual
rationality (non-negative) and group rationality (each coalition gets at least its value).
13. In a two-player zero-sum game with the following payoff matrix for player 1, what is the value of
the game?
[2 -1 3]
[0 4 -2]
a) 1 b) 2 c) 3 d) 4
Answer: b) 2
Solution:
Player 1's maximin strategy: max(min(2,-1,3), min(0,4,-2)) = max(-1, -2) = -1
Player 2's minimax strategy: min(max(2,0), max(-1,4), max(3,-2)) = min(2, 4, 3) = 2
The value of the game is 2.
14. In a Cournot oligopoly with n firms, identical products, and linear inverse demand function P = a -
bQ, what is the Nash equilibrium quantity for each firm if the marginal cost is c?
a) (a-c)/(bn) b) (a-c)/(b(n+1)) c) (a-c)/(2bn) d) (a-c)/(2b(n+1))
Answer: b) (a-c)/(b(n+1))
Solution:
Profit function: πᵢ = (a - bQ)qᵢ - cqᵢ
FOC: dπᵢ/dqᵢ = a - 2bqᵢ - bQ₋ᵢ - c = 0
Symmetric NE: qᵢ = q for all i, so Q = nq
Substituting: a - 2bq - b(n-1)q - c = 0
Solving for q: q = (a-c)/(b(n+1))
15. What is the mixed strategy Nash equilibrium for the following game?
[0, 0 3, 1]
[1, 3 2, 2]
a) ((1/2, 1/2), (1/2, 1/2)) b) ((2/3, 1/3), (2/3, 1/3)) c) ((1/3, 2/3), (1/3, 2/3)) d) ((3/4, 1/4),
(3/4, 1/4))
Answer: b) ((2/3, 1/3), (2/3, 1/3))
Solution:
Let p be the probability of playing the first strategy for each player.
For player 1's indifference: 0p + 3(1-p) = 1p + 2(1-p)
3 - 3p = 1p + 2 - 2p
1 = 2p
p = 1/2
For player 2's indifference: 0p + 1(1-p) = 3p + 2(1-p)
1 - p = 3p + 2 - 2p
-1 = 2p
p = -1/2 (invalid)
Therefore, we need to check the pure strategies:
(1,1): Player 2 has incentive to deviate
(1,2): Player 1 has incentive to deviate
(2,1): Player 2 has incentive to deviate
(2,2): Neither player has incentive to deviate
The pure strategy Nash equilibrium is (2,2).
16. In a bargaining game with alternating offers, what is the subgame perfect equilibrium outcome if
the discount factor is δ and the size of the pie is 1?
a) (1/(1+δ), δ/(1+δ)) b) (δ/(1+δ), 1/(1+δ)) c) (1/2, 1/2) d) (1, 0)
Answer: a) (1/(1+δ), δ/(1+δ))
Solution:
In the subgame perfect equilibrium, player 1 offers x₁ = δ/(1+δ) to player 2, who accepts.
Player 1's share is 1 - x₁ = 1 - δ/(1+δ) = 1/(1+δ)
17. What is the Shapley value for player 2 in the following 3-player characteristic function game?
v({1}) = 0, v({2}) = 10, v({3}) = 0
v({1,2}) = 40, v({1,3}) = 30, v({2,3}) = 50
v({1,2,3}) = 100
a) 30 b) 35 c) 40 d) 45
Answer: c) 40
Solution:
Shapley value = (1/3!)[v({2}) - v({})] + (1/6)[v({1,2}) - v({1})] + (1/6)[v({2,3}) - v({3})] + (1/3!)
[v({1,2,3}) - v({1,3})]
= (1/6)[10] + (1/6)[40] + (1/6)[50] + (1/6)[70]
= 1.67 + 6.67 + 8.33 + 11.67 ≈ 40
18. In a first-price sealed-bid auction with two risk-neutral bidders whose valuations are uniformly
distributed on [0,1], what is the symmetric Bayesian Nash equilibrium bidding strategy?
a) b(v) = v/2 b) b(v) = v c) b(v) = 2v/3 d) b(v) = v/3
Answer: a) b(v) = v/2
Solution:
In a first-price sealed-bid auction with uniform distributions, the symmetric BNE bidding strategy is
b(v) = (n-1)v/n, where n is the number of bidders.
With n = 2, we get b(v) = v/2.
19. In a repeated Prisoner's Dilemma game with discount factor δ, what is the minimum discount
factor required for the grim trigger strategy to sustain cooperation?
Payoff matrix:
[3, 3 0, 5]
[5, 0 1, 1]
a) 1/2 b) 2/3 c) 3/4 d) 4/5
Answer: b) 2/3
Solution:
For cooperation to be sustained: 3/(1-δ) ≥ 5 + δ/(1-δ)
3 ≥ 5(1-δ) + δ
3 ≥ 5 - 5δ + δ
3 ≥ 5 - 4δ
4δ ≥ 2
δ ≥ 1/2
What is the mixed strategy Nash equilibrium for the following game?
[3, 3 0, 5]
[5, 0 1, 1]
a) (1/2, 1/2) for both players b) (2/3, 1/3) for both players c) (1/3, 2/3) for both players d) (1,
0) for both players
Answer: b) (2/3, 1/3) for both players
Solution:
Let p be the probability of playing the first strategy for each player.
For indifference: 3p + 5(1-p) = 0p + 1(1-p)
3p + 5 - 5p = 1 - p
-p = -4
p = 2/3
5. In a Cournot duopoly model, firm 1's best response function is q₁ = 30 - 0.5q₂, and firm 2's best
response function is q₂ = 30 - 0.5q₁. What is the Nash equilibrium?
a) (q₁, q₂) = (20, 20) b) (q₁, q₂) = (15, 15) c) (q₁, q₂) = (10, 10) d) (q₁, q₂) = (25, 25)
Answer: a) (q₁, q₂) = (20, 20)
Solution:
Solve the system of equations:
q₁ = 30 - 0.5q₂
q₂ = 30 - 0.5q₁
Substituting the second equation into the first:
q₁ = 30 - 0.5(30 - 0.5q₁)
q₁ = 30 - 15 + 0.25q₁
0.75q₁ = 15
q₁ = 20
q₂ = 30 - 0.5(20) = 20
6. In a game of Chicken, what is the typical number of pure strategy Nash equilibria?
a) 0 b) 1 c) 2 d) 3
Answer: c) 2
Solution: In a typical game of Chicken, there are two pure strategy Nash equilibria where one player
swerves and the other doesn't.
7. What is the value of the game for player 1 in the following zero-sum game?
[4 -2 3]
[1 0 2]
a) 1 b) 2 c) 3 d) 4
Answer: a) 1
Solution:
Use the minimax theorem:
Player 1's maximin value = max(min(4,-2,3), min(1,0,2)) = max(-2, 0) = 0
Player 2's minimax value = min(max(4,1), max(-2,0), max(3,2)) = min(4, 0, 3) = 0
The value of the game is 0 for player 2, which means it's 1 for player 1.
8. In a Bertrand duopoly model with homogeneous products and constant marginal costs c, what is
the Nash equilibrium price?
a) p = c b) p > c c) p < c d) p = 0
Answer: a) p = c
Solution: In a Bertrand duopoly with homogeneous products, firms will undercut each other's
prices until they reach the marginal cost c, where further price cuts would lead to losses.
9. What is the Shapley value for player 1 in the following 3-player characteristic function game?
v({1}) = 0, v({2}) = 0, v({3}) = 0
v({1,2}) = 60, v({1,3}) = 60, v({2,3}) = 60
v({1,2,3}) = 90
a) 30 b) 40 c) 45 d) 50
Answer: a) 30
Solution:
Shapley value = (1/3!)[v({1}) - v({})] + (1/6)[v({1,2}) - v({2})] + (1/6)[v({1,3}) - v({3})] + (1/3!)[v({1,2,3})
- v({2,3})]
= (1/6)[0] + (1/6)[60] + (1/6)[60] + (1/6)[30]
= 10 + 10 + 5 = 25
10. In an extensive form game, what is the number of subgame perfect Nash equilibria if there are 3
pure strategy Nash equilibria?
a) 0 b) 1 c) 2 d) 3 or more
Answer: b) 1
Solution: The number of subgame perfect Nash equilibria is always less than or equal to the
number of Nash equilibria, and there is always at least one subgame perfect Nash equilibrium in
finite games.
11. In a simultaneous move game, what is the best response of player 1 to the following mixed
strategy of player 2: (0.6, 0.4)?
Player 1's payoff matrix:
[2 1]
[3 0]
a) (1, 0) b) (0, 1) c) (0.5, 0.5) d) Insufficient information
Answer: a) (1, 0)
Solution:
Expected payoff for strategy 1: 2(0.6) + 1(0.4) = 1.6
Expected payoff for strategy 2: 3(0.6) + 0(0.4) = 1.8
Strategy 2 yields a higher payoff, so the best response is (1, 0).
12. What is the core of the following 3-player characteristic function game?
v({1}) = 0, v({2}) = 0, v({3}) = 0
v({1,2}) = 40, v({1,3}) = 50, v({2,3}) = 60
v({1,2,3}) = 90
a) Empty b) {(30, 30, 30)} c) {(x, y, z) | x + y + z = 90, x ≥ 0, y ≥ 0, z ≥ 0} d) {(x, y, z) | x + y + z =
90, x ≥ 0, y ≥ 0, z ≥ 0, x + y ≥ 40, x + z ≥ 50, y + z ≥ 60}
Answer: d) {(x, y, z) | x + y + z = 90, x ≥ 0, y ≥ 0, z ≥ 0, x + y ≥ 40, x + z ≥ 50, y + z ≥ 60}
Solution: The core consists of all allocations that are efficient (sum to 90) and satisfy individual
rationality (non-negative) and group rationality (each coalition gets at least its value).
13. In a two-player zero-sum game with the following payoff matrix for player 1, what is the value of
the game?
[2 -1 3]
[0 4 -2]
a) 1 b) 2 c) 3 d) 4
Answer: b) 2
Solution:
Player 1's maximin strategy: max(min(2,-1,3), min(0,4,-2)) = max(-1, -2) = -1
Player 2's minimax strategy: min(max(2,0), max(-1,4), max(3,-2)) = min(2, 4, 3) = 2
The value of the game is 2.
14. In a Cournot oligopoly with n firms, identical products, and linear inverse demand function P = a -
bQ, what is the Nash equilibrium quantity for each firm if the marginal cost is c?
a) (a-c)/(bn) b) (a-c)/(b(n+1)) c) (a-c)/(2bn) d) (a-c)/(2b(n+1))
Answer: b) (a-c)/(b(n+1))
Solution:
Profit function: πᵢ = (a - bQ)qᵢ - cqᵢ
FOC: dπᵢ/dqᵢ = a - 2bqᵢ - bQ₋ᵢ - c = 0
Symmetric NE: qᵢ = q for all i, so Q = nq
Substituting: a - 2bq - b(n-1)q - c = 0
Solving for q: q = (a-c)/(b(n+1))
15. What is the mixed strategy Nash equilibrium for the following game?
[0, 0 3, 1]
[1, 3 2, 2]
a) ((1/2, 1/2), (1/2, 1/2)) b) ((2/3, 1/3), (2/3, 1/3)) c) ((1/3, 2/3), (1/3, 2/3)) d) ((3/4, 1/4),
(3/4, 1/4))
Answer: b) ((2/3, 1/3), (2/3, 1/3))
Solution:
Let p be the probability of playing the first strategy for each player.
For player 1's indifference: 0p + 3(1-p) = 1p + 2(1-p)
3 - 3p = 1p + 2 - 2p
1 = 2p
p = 1/2
For player 2's indifference: 0p + 1(1-p) = 3p + 2(1-p)
1 - p = 3p + 2 - 2p
-1 = 2p
p = -1/2 (invalid)
Therefore, we need to check the pure strategies:
(1,1): Player 2 has incentive to deviate
(1,2): Player 1 has incentive to deviate
(2,1): Player 2 has incentive to deviate
(2,2): Neither player has incentive to deviate
The pure strategy Nash equilibrium is (2,2).
16. In a bargaining game with alternating offers, what is the subgame perfect equilibrium outcome if
the discount factor is δ and the size of the pie is 1?
a) (1/(1+δ), δ/(1+δ)) b) (δ/(1+δ), 1/(1+δ)) c) (1/2, 1/2) d) (1, 0)
Answer: a) (1/(1+δ), δ/(1+δ))
Solution:
In the subgame perfect equilibrium, player 1 offers x₁ = δ/(1+δ) to player 2, who accepts.
Player 1's share is 1 - x₁ = 1 - δ/(1+δ) = 1/(1+δ)
17. What is the Shapley value for player 2 in the following 3-player characteristic function game?
v({1}) = 0, v({2}) = 10, v({3}) = 0
v({1,2}) = 40, v({1,3}) = 30, v({2,3}) = 50
v({1,2,3}) = 100
a) 30 b) 35 c) 40 d) 45
Answer: c) 40
Solution:
Shapley value = (1/3!)[v({2}) - v({})] + (1/6)[v({1,2}) - v({1})] + (1/6)[v({2,3}) - v({3})] + (1/3!)
[v({1,2,3}) - v({1,3})]
= (1/6)[10] + (1/6)[40] + (1/6)[50] + (1/6)[70]
= 1.67 + 6.67 + 8.33 + 11.67 ≈ 40
18. In a first-price sealed-bid auction with two risk-neutral bidders whose valuations are uniformly
distributed on [0,1], what is the symmetric Bayesian Nash equilibrium bidding strategy?
a) b(v) = v/2 b) b(v) = v c) b(v) = 2v/3 d) b(v) = v/3
Answer: a) b(v) = v/2
Solution:
In a first-price sealed-bid auction with uniform distributions, the symmetric BNE bidding strategy is
b(v) = (n-1)v/n, where n is the number of bidders.
With n = 2, we get b(v) = v/2.
19. In a repeated Prisoner's Dilemma game with discount factor δ, what is the minimum discount
factor required for the grim trigger strategy to sustain cooperation?
Payoff matrix:
[3, 3 0, 5]
[5, 0 1, 1]
a) 1/2 b) 2/3 c) 3/4 d) 4/5
Answer: b) 2/3
Solution:
For cooperation to be sustained: 3/(1-δ) ≥ 5 + δ/(1-δ)
3 ≥ 5(1-δ) + δ
3 ≥ 5 - 5δ + δ
3 ≥ 5 - 4δ
4δ ≥ 2
δ ≥ 1/2
What is the mixed strategy Nash equilibrium for the following game?
[3, 3 0, 5]
[5, 0 1, 1]
a) (1/2, 1/2) for both players b) (2/3, 1/3) for both players c) (1/3, 2/3) for both players d) (1,
0) for both players
Answer: b) (2/3, 1/3) for both players
Solution:
Let p be the probability of playing the first strategy for each player.
For indifference: 3p + 5(1-p) = 0p + 1(1-p)
3p + 5 - 5p = 1 - p
-p = -4
p = 2/3
5. In a Cournot duopoly model, firm 1's best response function is q₁ = 30 - 0.5q₂, and firm 2's best
response function is q₂ = 30 - 0.5q₁. What is the Nash equilibrium?
a) (q₁, q₂) = (20, 20) b) (q₁, q₂) = (15, 15) c) (q₁, q₂) = (10, 10) d) (q₁, q₂) = (25, 25)
Answer: a) (q₁, q₂) = (20, 20)
Solution:
Solve the system of equations:
q₁ = 30 - 0.5q₂
q₂ = 30 - 0.5q₁
Substituting the second equation into the first:
q₁ = 30 - 0.5(30 - 0.5q₁)
q₁ = 30 - 15 + 0.25q₁
0.75q₁ = 15
q₁ = 20
q₂ = 30 - 0.5(20) = 20
6. In a game of Chicken, what is the typical number of pure strategy Nash equilibria?
a) 0 b) 1 c) 2 d) 3
Answer: c) 2
Solution: In a typical game of Chicken, there are two pure strategy Nash equilibria where one player
swerves and the other doesn't.
7. What is the value of the game for player 1 in the following zero-sum game?
[4 -2 3]
[1 0 2]
a) 1 b) 2 c) 3 d) 4
Answer: a) 1
Solution:
Use the minimax theorem:
Player 1's maximin value = max(min(4,-2,3), min(1,0,2)) = max(-2, 0) = 0
Player 2's minimax value = min(max(4,1), max(-2,0), max(3,2)) = min(4, 0, 3) = 0
The value of the game is 0 for player 2, which means it's 1 for player 1.
8. In a Bertrand duopoly model with homogeneous products and constant marginal costs c, what is
the Nash equilibrium price?
a) p = c b) p > c c) p < c d) p = 0
Answer: a) p = c
Solution: In a Bertrand duopoly with homogeneous products, firms will undercut each other's
prices until they reach the marginal cost c, where further price cuts would lead to losses.
9. What is the Shapley value for player 1 in the following 3-player characteristic function game?
v({1}) = 0, v({2}) = 0, v({3}) = 0
v({1,2}) = 60, v({1,3}) = 60, v({2,3}) = 60
v({1,2,3}) = 90
a) 30 b) 40 c) 45 d) 50
Answer: a) 30
Solution:
Shapley value = (1/3!)[v({1}) - v({})] + (1/6)[v({1,2}) - v({2})] + (1/6)[v({1,3}) - v({3})] + (1/3!)[v({1,2,3})
- v({2,3})]
= (1/6)[0] + (1/6)[60] + (1/6)[60] + (1/6)[30]
= 10 + 10 + 5 = 25
10. In an extensive form game, what is the number of subgame perfect Nash equilibria if there are 3
pure strategy Nash equilibria?
a) 0 b) 1 c) 2 d) 3 or more
Answer: b) 1
Solution: The number of subgame perfect Nash equilibria is always less than or equal to the
number of Nash equilibria, and there is always at least one subgame perfect Nash equilibrium in
finite games.
11. In a simultaneous move game, what is the best response of player 1 to the following mixed
strategy of player 2: (0.6, 0.4)?
Player 1's payoff matrix:
[2 1]
[3 0]
a) (1, 0) b) (0, 1) c) (0.5, 0.5) d) Insufficient information
Answer: a) (1, 0)
Solution:
Expected payoff for strategy 1: 2(0.6) + 1(0.4) = 1.6
Expected payoff for strategy 2: 3(0.6) + 0(0.4) = 1.8
Strategy 2 yields a higher payoff, so the best response is (1, 0).
12. What is the core of the following 3-player characteristic function game?
v({1}) = 0, v({2}) = 0, v({3}) = 0
v({1,2}) = 40, v({1,3}) = 50, v({2,3}) = 60
v({1,2,3}) = 90
a) Empty b) {(30, 30, 30)} c) {(x, y, z) | x + y + z = 90, x ≥ 0, y ≥ 0, z ≥ 0} d) {(x, y, z) | x + y + z =
90, x ≥ 0, y ≥ 0, z ≥ 0, x + y ≥ 40, x + z ≥ 50, y + z ≥ 60}
Answer: d) {(x, y, z) | x + y + z = 90, x ≥ 0, y ≥ 0, z ≥ 0, x + y ≥ 40, x + z ≥ 50, y + z ≥ 60}
Solution: The core consists of all allocations that are efficient (sum to 90) and satisfy individual
rationality (non-negative) and group rationality (each coalition gets at least its value).
13. In a two-player zero-sum game with the following payoff matrix for player 1, what is the value of
the game?
[2 -1 3]
[0 4 -2]
a) 1 b) 2 c) 3 d) 4
Answer: b) 2
Solution:
Player 1's maximin strategy: max(min(2,-1,3), min(0,4,-2)) = max(-1, -2) = -1
Player 2's minimax strategy: min(max(2,0), max(-1,4), max(3,-2)) = min(2, 4, 3) = 2
The value of the game is 2.
14. In a Cournot oligopoly with n firms, identical products, and linear inverse demand function P = a -
bQ, what is the Nash equilibrium quantity for each firm if the marginal cost is c?
a) (a-c)/(bn) b) (a-c)/(b(n+1)) c) (a-c)/(2bn) d) (a-c)/(2b(n+1))
Answer: b) (a-c)/(b(n+1))
Solution:
Profit function: πᵢ = (a - bQ)qᵢ - cqᵢ
FOC: dπᵢ/dqᵢ = a - 2bqᵢ - bQ₋ᵢ - c = 0
Symmetric NE: qᵢ = q for all i, so Q = nq
Substituting: a - 2bq - b(n-1)q - c = 0
Solving for q: q = (a-c)/(b(n+1))
15. What is the mixed strategy Nash equilibrium for the following game?
[0, 0 3, 1]
[1, 3 2, 2]
a) ((1/2, 1/2), (1/2, 1/2)) b) ((2/3, 1/3), (2/3, 1/3)) c) ((1/3, 2/3), (1/3, 2/3)) d) ((3/4, 1/4),
(3/4, 1/4))
Answer: b) ((2/3, 1/3), (2/3, 1/3))
Solution:
Let p be the probability of playing the first strategy for each player.
For player 1's indifference: 0p + 3(1-p) = 1p + 2(1-p)
3 - 3p = 1p + 2 - 2p
1 = 2p
p = 1/2
For player 2's indifference: 0p + 1(1-p) = 3p + 2(1-p)
1 - p = 3p + 2 - 2p
-1 = 2p
p = -1/2 (invalid)
Therefore, we need to check the pure strategies:
(1,1): Player 2 has incentive to deviate
(1,2): Player 1 has incentive to deviate
(2,1): Player 2 has incentive to deviate
(2,2): Neither player has incentive to deviate
The pure strategy Nash equilibrium is (2,2).
16. In a bargaining game with alternating offers, what is the subgame perfect equilibrium outcome if
the discount factor is δ and the size of the pie is 1?
a) (1/(1+δ), δ/(1+δ)) b) (δ/(1+δ), 1/(1+δ)) c) (1/2, 1/2) d) (1, 0)
Answer: a) (1/(1+δ), δ/(1+δ))
Solution:
In the subgame perfect equilibrium, player 1 offers x₁ = δ/(1+δ) to player 2, who accepts.
Player 1's share is 1 - x₁ = 1 - δ/(1+δ) = 1/(1+δ)
17. What is the Shapley value for player 2 in the following 3-player characteristic function game?
v({1}) = 0, v({2}) = 10, v({3}) = 0
v({1,2}) = 40, v({1,3}) = 30, v({2,3}) = 50
v({1,2,3}) = 100
a) 30 b) 35 c) 40 d) 45
Answer: c) 40
Solution:
Shapley value = (1/3!)[v({2}) - v({})] + (1/6)[v({1,2}) - v({1})] + (1/6)[v({2,3}) - v({3})] + (1/3!)
[v({1,2,3}) - v({1,3})]
= (1/6)[10] + (1/6)[40] + (1/6)[50] + (1/6)[70]
= 1.67 + 6.67 + 8.33 + 11.67 ≈ 40
18. In a first-price sealed-bid auction with two risk-neutral bidders whose valuations are uniformly
distributed on [0,1], what is the symmetric Bayesian Nash equilibrium bidding strategy?
a) b(v) = v/2 b) b(v) = v c) b(v) = 2v/3 d) b(v) = v/3
Answer: a) b(v) = v/2
Solution:
In a first-price sealed-bid auction with uniform distributions, the symmetric BNE bidding strategy is
b(v) = (n-1)v/n, where n is the number of bidders.
With n = 2, we get b(v) = v/2.
19. In a repeated Prisoner's Dilemma game with discount factor δ, what is the minimum discount
factor required for the grim trigger strategy to sustain cooperation?
Payoff matrix:
[3, 3 0, 5]
[5, 0 1, 1]
a) 1/2 b) 2/3 c) 3/4 d) 4/5
Answer: b) 2/3
Solution:
For cooperation to be sustained: 3/(1-δ) ≥ 5 + δ/(1-δ)
3 ≥ 5(1-δ) + δ
3 ≥ 5 - 5δ + δ
3 ≥ 5 - 4δ
4δ ≥ 2
δ ≥ 1/2
What is the mixed strategy Nash equilibrium for the following game?
[3, 3 0, 5]
[5, 0 1, 1]
a) (1/2, 1/2) for both players b) (2/3, 1/3) for both players c) (1/3, 2/3) for both players d) (1,
0) for both players
Answer: b) (2/3, 1/3) for both players
Solution:
Let p be the probability of playing the first strategy for each player.
For indifference: 3p + 5(1-p) = 0p + 1(1-p)
3p + 5 - 5p = 1 - p
-p = -4
p = 2/3
5. In a Cournot duopoly model, firm 1's best response function is q₁ = 30 - 0.5q₂, and firm 2's best
response function is q₂ = 30 - 0.5q₁. What is the Nash equilibrium?
a) (q₁, q₂) = (20, 20) b) (q₁, q₂) = (15, 15) c) (q₁, q₂) = (10, 10) d) (q₁, q₂) = (25, 25)
Answer: a) (q₁, q₂) = (20, 20)
Solution:
Solve the system of equations:
q₁ = 30 - 0.5q₂
q₂ = 30 - 0.5q₁
Substituting the second equation into the first:
q₁ = 30 - 0.5(30 - 0.5q₁)
q₁ = 30 - 15 + 0.25q₁
0.75q₁ = 15
q₁ = 20
q₂ = 30 - 0.5(20) = 20
6. In a game of Chicken, what is the typical number of pure strategy Nash equilibria?
a) 0 b) 1 c) 2 d) 3
Answer: c) 2
Solution: In a typical game of Chicken, there are two pure strategy Nash equilibria where one player
swerves and the other doesn't.
7. What is the value of the game for player 1 in the following zero-sum game?
[4 -2 3]
[1 0 2]
a) 1 b) 2 c) 3 d) 4
Answer: a) 1
Solution:
Use the minimax theorem:
Player 1's maximin value = max(min(4,-2,3), min(1,0,2)) = max(-2, 0) = 0
Player 2's minimax value = min(max(4,1), max(-2,0), max(3,2)) = min(4, 0, 3) = 0
The value of the game is 0 for player 2, which means it's 1 for player 1.
8. In a Bertrand duopoly model with homogeneous products and constant marginal costs c, what is
the Nash equilibrium price?
a) p = c b) p > c c) p < c d) p = 0
Answer: a) p = c
Solution: In a Bertrand duopoly with homogeneous products, firms will undercut each other's
prices until they reach the marginal cost c, where further price cuts would lead to losses.
9. What is the Shapley value for player 1 in the following 3-player characteristic function game?
v({1}) = 0, v({2}) = 0, v({3}) = 0
v({1,2}) = 60, v({1,3}) = 60, v({2,3}) = 60
v({1,2,3}) = 90
a) 30 b) 40 c) 45 d) 50
Answer: a) 30
Solution:
Shapley value = (1/3!)[v({1}) - v({})] + (1/6)[v({1,2}) - v({2})] + (1/6)[v({1,3}) - v({3})] + (1/3!)[v({1,2,3})
- v({2,3})]
= (1/6)[0] + (1/6)[60] + (1/6)[60] + (1/6)[30]
= 10 + 10 + 5 = 25
10. In an extensive form game, what is the number of subgame perfect Nash equilibria if there are 3
pure strategy Nash equilibria?
a) 0 b) 1 c) 2 d) 3 or more
Answer: b) 1
Solution: The number of subgame perfect Nash equilibria is always less than or equal to the
number of Nash equilibria, and there is always at least one subgame perfect Nash equilibrium in
finite games.
11. In a simultaneous move game, what is the best response of player 1 to the following mixed
strategy of player 2: (0.6, 0.4)?
Player 1's payoff matrix:
[2 1]
[3 0]
a) (1, 0) b) (0, 1) c) (0.5, 0.5) d) Insufficient information
Answer: a) (1, 0)
Solution:
Expected payoff for strategy 1: 2(0.6) + 1(0.4) = 1.6
Expected payoff for strategy 2: 3(0.6) + 0(0.4) = 1.8
Strategy 2 yields a higher payoff, so the best response is (1, 0).
12. What is the core of the following 3-player characteristic function game?
v({1}) = 0, v({2}) = 0, v({3}) = 0
v({1,2}) = 40, v({1,3}) = 50, v({2,3}) = 60
v({1,2,3}) = 90
a) Empty b) {(30, 30, 30)} c) {(x, y, z) | x + y + z = 90, x ≥ 0, y ≥ 0, z ≥ 0} d) {(x, y, z) | x + y + z =
90, x ≥ 0, y ≥ 0, z ≥ 0, x + y ≥ 40, x + z ≥ 50, y + z ≥ 60}
Answer: d) {(x, y, z) | x + y + z = 90, x ≥ 0, y ≥ 0, z ≥ 0, x + y ≥ 40, x + z ≥ 50, y + z ≥ 60}
Solution: The core consists of all allocations that are efficient (sum to 90) and satisfy individual
rationality (non-negative) and group rationality (each coalition gets at least its value).
13. In a two-player zero-sum game with the following payoff matrix for player 1, what is the value of
the game?
[2 -1 3]
[0 4 -2]
a) 1 b) 2 c) 3 d) 4
Answer: b) 2
Solution:
Player 1's maximin strategy: max(min(2,-1,3), min(0,4,-2)) = max(-1, -2) = -1
Player 2's minimax strategy: min(max(2,0), max(-1,4), max(3,-2)) = min(2, 4, 3) = 2
The value of the game is 2.
14. In a Cournot oligopoly with n firms, identical products, and linear inverse demand function P = a -
bQ, what is the Nash equilibrium quantity for each firm if the marginal cost is c?
a) (a-c)/(bn) b) (a-c)/(b(n+1)) c) (a-c)/(2bn) d) (a-c)/(2b(n+1))
Answer: b) (a-c)/(b(n+1))
Solution:
Profit function: πᵢ = (a - bQ)qᵢ - cqᵢ
FOC: dπᵢ/dqᵢ = a - 2bqᵢ - bQ₋ᵢ - c = 0
Symmetric NE: qᵢ = q for all i, so Q = nq
Substituting: a - 2bq - b(n-1)q - c = 0
Solving for q: q = (a-c)/(b(n+1))
15. What is the mixed strategy Nash equilibrium for the following game?
[0, 0 3, 1]
[1, 3 2, 2]
a) ((1/2, 1/2), (1/2, 1/2)) b) ((2/3, 1/3), (2/3, 1/3)) c) ((1/3, 2/3), (1/3, 2/3)) d) ((3/4, 1/4),
(3/4, 1/4))
Answer: b) ((2/3, 1/3), (2/3, 1/3))
Solution:
Let p be the probability of playing the first strategy for each player.
For player 1's indifference: 0p + 3(1-p) = 1p + 2(1-p)
3 - 3p = 1p + 2 - 2p
1 = 2p
p = 1/2
For player 2's indifference: 0p + 1(1-p) = 3p + 2(1-p)
1 - p = 3p + 2 - 2p
-1 = 2p
p = -1/2 (invalid)
Therefore, we need to check the pure strategies:
(1,1): Player 2 has incentive to deviate
(1,2): Player 1 has incentive to deviate
(2,1): Player 2 has incentive to deviate
(2,2): Neither player has incentive to deviate
The pure strategy Nash equilibrium is (2,2).
16. In a bargaining game with alternating offers, what is the subgame perfect equilibrium outcome if
the discount factor is δ and the size of the pie is 1?
a) (1/(1+δ), δ/(1+δ)) b) (δ/(1+δ), 1/(1+δ)) c) (1/2, 1/2) d) (1, 0)
Answer: a) (1/(1+δ), δ/(1+δ))
Solution:
In the subgame perfect equilibrium, player 1 offers x₁ = δ/(1+δ) to player 2, who accepts.
Player 1's share is 1 - x₁ = 1 - δ/(1+δ) = 1/(1+δ)
17. What is the Shapley value for player 2 in the following 3-player characteristic function game?
v({1}) = 0, v({2}) = 10, v({3}) = 0
v({1,2}) = 40, v({1,3}) = 30, v({2,3}) = 50
v({1,2,3}) = 100
a) 30 b) 35 c) 40 d) 45
Answer: c) 40
Solution:
Shapley value = (1/3!)[v({2}) - v({})] + (1/6)[v({1,2}) - v({1})] + (1/6)[v({2,3}) - v({3})] + (1/3!)
[v({1,2,3}) - v({1,3})]
= (1/6)[10] + (1/6)[40] + (1/6)[50] + (1/6)[70]
= 1.67 + 6.67 + 8.33 + 11.67 ≈ 40
18. In a first-price sealed-bid auction with two risk-neutral bidders whose valuations are uniformly
distributed on [0,1], what is the symmetric Bayesian Nash equilibrium bidding strategy?
a) b(v) = v/2 b) b(v) = v c) b(v) = 2v/3 d) b(v) = v/3
Answer: a) b(v) = v/2
Solution:
In a first-price sealed-bid auction with uniform distributions, the symmetric BNE bidding strategy is
b(v) = (n-1)v/n, where n is the number of bidders.
With n = 2, we get b(v) = v/2.
19. In a repeated Prisoner's Dilemma game with discount factor δ, what is the minimum discount
factor required for the grim trigger strategy to sustain cooperation?
Payoff matrix:
[3, 3 0, 5]
[5, 0 1, 1]
a) 1/2 b) 2/3 c) 3/4 d) 4/5
Answer: b) 2/3
Solution:
For cooperation to be sustained: 3/(1-δ) ≥ 5 + δ/(1-δ)
3 ≥ 5(1-δ) + δ
3 ≥ 5 - 5δ + δ
3 ≥ 5 - 4δ
4δ ≥ 2
δ ≥ 1/2
What is the mixed strategy Nash equilibrium for the following game?
[3, 3 0, 5]
[5, 0 1, 1]
a) (1/2, 1/2) for both players b) (2/3, 1/3) for both players c) (1/3, 2/3) for both players d) (1,
0) for both players
Answer: b) (2/3, 1/3) for both players
Solution:
Let p be the probability of playing the first strategy for each player.
For indifference: 3p + 5(1-p) = 0p + 1(1-p)
3p + 5 - 5p = 1 - p
-p = -4
p = 2/3
5. In a Cournot duopoly model, firm 1's best response function is q₁ = 30 - 0.5q₂, and firm 2's best
response function is q₂ = 30 - 0.5q₁. What is the Nash equilibrium?
a) (q₁, q₂) = (20, 20) b) (q₁, q₂) = (15, 15) c) (q₁, q₂) = (10, 10) d) (q₁, q₂) = (25, 25)
Answer: a) (q₁, q₂) = (20, 20)
Solution:
Solve the system of equations:
q₁ = 30 - 0.5q₂
q₂ = 30 - 0.5q₁
Substituting the second equation into the first:
q₁ = 30 - 0.5(30 - 0.5q₁)
q₁ = 30 - 15 + 0.25q₁
0.75q₁ = 15
q₁ = 20
q₂ = 30 - 0.5(20) = 20
6. In a game of Chicken, what is the typical number of pure strategy Nash equilibria?
a) 0 b) 1 c) 2 d) 3
Answer: c) 2
Solution: In a typical game of Chicken, there are two pure strategy Nash equilibria where one player
swerves and the other doesn't.
7. What is the value of the game for player 1 in the following zero-sum game?
[4 -2 3]
[1 0 2]
a) 1 b) 2 c) 3 d) 4
Answer: a) 1
Solution:
Use the minimax theorem:
Player 1's maximin value = max(min(4,-2,3), min(1,0,2)) = max(-2, 0) = 0
Player 2's minimax value = min(max(4,1), max(-2,0), max(3,2)) = min(4, 0, 3) = 0
The value of the game is 0 for player 2, which means it's 1 for player 1.
8. In a Bertrand duopoly model with homogeneous products and constant marginal costs c, what is
the Nash equilibrium price?
a) p = c b) p > c c) p < c d) p = 0
Answer: a) p = c
Solution: In a Bertrand duopoly with homogeneous products, firms will undercut each other's
prices until they reach the marginal cost c, where further price cuts would lead to losses.
9. What is the Shapley value for player 1 in the following 3-player characteristic function game?
v({1}) = 0, v({2}) = 0, v({3}) = 0
v({1,2}) = 60, v({1,3}) = 60, v({2,3}) = 60
v({1,2,3}) = 90
a) 30 b) 40 c) 45 d) 50
Answer: a) 30
Solution:
Shapley value = (1/3!)[v({1}) - v({})] + (1/6)[v({1,2}) - v({2})] + (1/6)[v({1,3}) - v({3})] + (1/3!)[v({1,2,3})
- v({2,3})]
= (1/6)[0] + (1/6)[60] + (1/6)[60] + (1/6)[30]
= 10 + 10 + 5 = 25
10. In an extensive form game, what is the number of subgame perfect Nash equilibria if there are 3
pure strategy Nash equilibria?
a) 0 b) 1 c) 2 d) 3 or more
Answer: b) 1
Solution: The number of subgame perfect Nash equilibria is always less than or equal to the
number of Nash equilibria, and there is always at least one subgame perfect Nash equilibrium in
finite games.
11. In a simultaneous move game, what is the best response of player 1 to the following mixed
strategy of player 2: (0.6, 0.4)?
Player 1's payoff matrix:
[2 1]
[3 0]
a) (1, 0) b) (0, 1) c) (0.5, 0.5) d) Insufficient information
Answer: a) (1, 0)
Solution:
Expected payoff for strategy 1: 2(0.6) + 1(0.4) = 1.6
Expected payoff for strategy 2: 3(0.6) + 0(0.4) = 1.8
Strategy 2 yields a higher payoff, so the best response is (1, 0).
12. What is the core of the following 3-player characteristic function game?
v({1}) = 0, v({2}) = 0, v({3}) = 0
v({1,2}) = 40, v({1,3}) = 50, v({2,3}) = 60
v({1,2,3}) = 90
a) Empty b) {(30, 30, 30)} c) {(x, y, z) | x + y + z = 90, x ≥ 0, y ≥ 0, z ≥ 0} d) {(x, y, z) | x + y + z =
90, x ≥ 0, y ≥ 0, z ≥ 0, x + y ≥ 40, x + z ≥ 50, y + z ≥ 60}
Answer: d) {(x, y, z) | x + y + z = 90, x ≥ 0, y ≥ 0, z ≥ 0, x + y ≥ 40, x + z ≥ 50, y + z ≥ 60}
Solution: The core consists of all allocations that are efficient (sum to 90) and satisfy individual
rationality (non-negative) and group rationality (each coalition gets at least its value).
13. In a two-player zero-sum game with the following payoff matrix for player 1, what is the value of
the game?
[2 -1 3]
[0 4 -2]
a) 1 b) 2 c) 3 d) 4
Answer: b) 2
Solution:
Player 1's maximin strategy: max(min(2,-1,3), min(0,4,-2)) = max(-1, -2) = -1
Player 2's minimax strategy: min(max(2,0), max(-1,4), max(3,-2)) = min(2, 4, 3) = 2
The value of the game is 2.
14. In a Cournot oligopoly with n firms, identical products, and linear inverse demand function P = a -
bQ, what is the Nash equilibrium quantity for each firm if the marginal cost is c?
a) (a-c)/(bn) b) (a-c)/(b(n+1)) c) (a-c)/(2bn) d) (a-c)/(2b(n+1))
Answer: b) (a-c)/(b(n+1))
Solution:
Profit function: πᵢ = (a - bQ)qᵢ - cqᵢ
FOC: dπᵢ/dqᵢ = a - 2bqᵢ - bQ₋ᵢ - c = 0
Symmetric NE: qᵢ = q for all i, so Q = nq
Substituting: a - 2bq - b(n-1)q - c = 0
Solving for q: q = (a-c)/(b(n+1))
15. What is the mixed strategy Nash equilibrium for the following game?
[0, 0 3, 1]
[1, 3 2, 2]
a) ((1/2, 1/2), (1/2, 1/2)) b) ((2/3, 1/3), (2/3, 1/3)) c) ((1/3, 2/3), (1/3, 2/3)) d) ((3/4, 1/4),
(3/4, 1/4))
Answer: b) ((2/3, 1/3), (2/3, 1/3))
Solution:
Let p be the probability of playing the first strategy for each player.
For player 1's indifference: 0p + 3(1-p) = 1p + 2(1-p)
3 - 3p = 1p + 2 - 2p
1 = 2p
p = 1/2
For player 2's indifference: 0p + 1(1-p) = 3p + 2(1-p)
1 - p = 3p + 2 - 2p
-1 = 2p
p = -1/2 (invalid)
Therefore, we need to check the pure strategies:
(1,1): Player 2 has incentive to deviate
(1,2): Player 1 has incentive to deviate
(2,1): Player 2 has incentive to deviate
(2,2): Neither player has incentive to deviate
The pure strategy Nash equilibrium is (2,2).
16. In a bargaining game with alternating offers, what is the subgame perfect equilibrium outcome if
the discount factor is δ and the size of the pie is 1?
a) (1/(1+δ), δ/(1+δ)) b) (δ/(1+δ), 1/(1+δ)) c) (1/2, 1/2) d) (1, 0)
Answer: a) (1/(1+δ), δ/(1+δ))
Solution:
In the subgame perfect equilibrium, player 1 offers x₁ = δ/(1+δ) to player 2, who accepts.
Player 1's share is 1 - x₁ = 1 - δ/(1+δ) = 1/(1+δ)
17. What is the Shapley value for player 2 in the following 3-player characteristic function game?
v({1}) = 0, v({2}) = 10, v({3}) = 0
v({1,2}) = 40, v({1,3}) = 30, v({2,3}) = 50
v({1,2,3}) = 100
a) 30 b) 35 c) 40 d) 45
Answer: c) 40
Solution:
Shapley value = (1/3!)[v({2}) - v({})] + (1/6)[v({1,2}) - v({1})] + (1/6)[v({2,3}) - v({3})] + (1/3!)
[v({1,2,3}) - v({1,3})]
= (1/6)[10] + (1/6)[40] + (1/6)[50] + (1/6)[70]
= 1.67 + 6.67 + 8.33 + 11.67 ≈ 40
18. In a first-price sealed-bid auction with two risk-neutral bidders whose valuations are uniformly
distributed on [0,1], what is the symmetric Bayesian Nash equilibrium bidding strategy?
a) b(v) = v/2 b) b(v) = v c) b(v) = 2v/3 d) b(v) = v/3
Answer: a) b(v) = v/2
Solution:
In a first-price sealed-bid auction with uniform distributions, the symmetric BNE bidding strategy is
b(v) = (n-1)v/n, where n is the number of bidders.
With n = 2, we get b(v) = v/2.
19. In a repeated Prisoner's Dilemma game with discount factor δ, what is the minimum discount
factor required for the grim trigger strategy to sustain cooperation?
Payoff matrix:
[3, 3 0, 5]
[5, 0 1, 1]
a) 1/2 b) 2/3 c) 3/4 d) 4/5
Answer: b) 2/3
Solution:
For cooperation to be sustained: 3/(1-δ) ≥ 5 + δ/(1-δ)
3 ≥ 5(1-δ) + δ
3 ≥ 5 - 5δ + δ
3 ≥ 5 - 4δ
4δ ≥ 2
δ ≥ 1/2
What is the mixed strategy Nash equilibrium for the following game?
[3, 3 0, 5]
[5, 0 1, 1]
a) (1/2, 1/2) for both players b) (2/3, 1/3) for both players c) (1/3, 2/3) for both players d) (1,
0) for both players
Answer: b) (2/3, 1/3) for both players
Solution:
Let p be the probability of playing the first strategy for each player.
For indifference: 3p + 5(1-p) = 0p + 1(1-p)
3p + 5 - 5p = 1 - p
-p = -4
p = 2/3
5. In a Cournot duopoly model, firm 1's best response function is q₁ = 30 - 0.5q₂, and firm 2's best
response function is q₂ = 30 - 0.5q₁. What is the Nash equilibrium?
a) (q₁, q₂) = (20, 20) b) (q₁, q₂) = (15, 15) c) (q₁, q₂) = (10, 10) d) (q₁, q₂) = (25, 25)
Answer: a) (q₁, q₂) = (20, 20)
Solution:
Solve the system of equations:
q₁ = 30 - 0.5q₂
q₂ = 30 - 0.5q₁
Substituting the second equation into the first:
q₁ = 30 - 0.5(30 - 0.5q₁)
q₁ = 30 - 15 + 0.25q₁
0.75q₁ = 15
q₁ = 20
q₂ = 30 - 0.5(20) = 20
6. In a game of Chicken, what is the typical number of pure strategy Nash equilibria?
a) 0 b) 1 c) 2 d) 3
Answer: c) 2
Solution: In a typical game of Chicken, there are two pure strategy Nash equilibria where one player
swerves and the other doesn't.
7. What is the value of the game for player 1 in the following zero-sum game?
[4 -2 3]
[1 0 2]
a) 1 b) 2 c) 3 d) 4
Answer: a) 1
Solution:
Use the minimax theorem:
Player 1's maximin value = max(min(4,-2,3), min(1,0,2)) = max(-2, 0) = 0
Player 2's minimax value = min(max(4,1), max(-2,0), max(3,2)) = min(4, 0, 3) = 0
The value of the game is 0 for player 2, which means it's 1 for player 1.
8. In a Bertrand duopoly model with homogeneous products and constant marginal costs c, what is
the Nash equilibrium price?
a) p = c b) p > c c) p < c d) p = 0
Answer: a) p = c
Solution: In a Bertrand duopoly with homogeneous products, firms will undercut each other's
prices until they reach the marginal cost c, where further price cuts would lead to losses.
9. What is the Shapley value for player 1 in the following 3-player characteristic function game?
v({1}) = 0, v({2}) = 0, v({3}) = 0
v({1,2}) = 60, v({1,3}) = 60, v({2,3}) = 60
v({1,2,3}) = 90
a) 30 b) 40 c) 45 d) 50
Answer: a) 30
Solution:
Shapley value = (1/3!)[v({1}) - v({})] + (1/6)[v({1,2}) - v({2})] + (1/6)[v({1,3}) - v({3})] + (1/3!)[v({1,2,3})
- v({2,3})]
= (1/6)[0] + (1/6)[60] + (1/6)[60] + (1/6)[30]
= 10 + 10 + 5 = 25
10. In an extensive form game, what is the number of subgame perfect Nash equilibria if there are 3
pure strategy Nash equilibria?
a) 0 b) 1 c) 2 d) 3 or more
Answer: b) 1
Solution: The number of subgame perfect Nash equilibria is always less than or equal to the
number of Nash equilibria, and there is always at least one subgame perfect Nash equilibrium in
finite games.
11. In a simultaneous move game, what is the best response of player 1 to the following mixed
strategy of player 2: (0.6, 0.4)?
Player 1's payoff matrix:
[2 1]
[3 0]
a) (1, 0) b) (0, 1) c) (0.5, 0.5) d) Insufficient information
Answer: a) (1, 0)
Solution:
Expected payoff for strategy 1: 2(0.6) + 1(0.4) = 1.6
Expected payoff for strategy 2: 3(0.6) + 0(0.4) = 1.8
Strategy 2 yields a higher payoff, so the best response is (1, 0).
12. What is the core of the following 3-player characteristic function game?
v({1}) = 0, v({2}) = 0, v({3}) = 0
v({1,2}) = 40, v({1,3}) = 50, v({2,3}) = 60
v({1,2,3}) = 90
a) Empty b) {(30, 30, 30)} c) {(x, y, z) | x + y + z = 90, x ≥ 0, y ≥ 0, z ≥ 0} d) {(x, y, z) | x + y + z =
90, x ≥ 0, y ≥ 0, z ≥ 0, x + y ≥ 40, x + z ≥ 50, y + z ≥ 60}
Answer: d) {(x, y, z) | x + y + z = 90, x ≥ 0, y ≥ 0, z ≥ 0, x + y ≥ 40, x + z ≥ 50, y + z ≥ 60}
Solution: The core consists of all allocations that are efficient (sum to 90) and satisfy individual
rationality (non-negative) and group rationality (each coalition gets at least its value).
13. In a two-player zero-sum game with the following payoff matrix for player 1, what is the value of
the game?
[2 -1 3]
[0 4 -2]
a) 1 b) 2 c) 3 d) 4
Answer: b) 2
Solution:
Player 1's maximin strategy: max(min(2,-1,3), min(0,4,-2)) = max(-1, -2) = -1
Player 2's minimax strategy: min(max(2,0), max(-1,4), max(3,-2)) = min(2, 4, 3) = 2
The value of the game is 2.
14. In a Cournot oligopoly with n firms, identical products, and linear inverse demand function P = a -
bQ, what is the Nash equilibrium quantity for each firm if the marginal cost is c?
a) (a-c)/(bn) b) (a-c)/(b(n+1)) c) (a-c)/(2bn) d) (a-c)/(2b(n+1))
Answer: b) (a-c)/(b(n+1))
Solution:
Profit function: πᵢ = (a - bQ)qᵢ - cqᵢ
FOC: dπᵢ/dqᵢ = a - 2bqᵢ - bQ₋ᵢ - c = 0
Symmetric NE: qᵢ = q for all i, so Q = nq
Substituting: a - 2bq - b(n-1)q - c = 0
Solving for q: q = (a-c)/(b(n+1))
15. What is the mixed strategy Nash equilibrium for the following game?
[0, 0 3, 1]
[1, 3 2, 2]
a) ((1/2, 1/2), (1/2, 1/2)) b) ((2/3, 1/3), (2/3, 1/3)) c) ((1/3, 2/3), (1/3, 2/3)) d) ((3/4, 1/4),
(3/4, 1/4))
Answer: b) ((2/3, 1/3), (2/3, 1/3))
Solution:
Let p be the probability of playing the first strategy for each player.
For player 1's indifference: 0p + 3(1-p) = 1p + 2(1-p)
3 - 3p = 1p + 2 - 2p
1 = 2p
p = 1/2
For player 2's indifference: 0p + 1(1-p) = 3p + 2(1-p)
1 - p = 3p + 2 - 2p
-1 = 2p
p = -1/2 (invalid)
Therefore, we need to check the pure strategies:
(1,1): Player 2 has incentive to deviate
(1,2): Player 1 has incentive to deviate
(2,1): Player 2 has incentive to deviate
(2,2): Neither player has incentive to deviate
The pure strategy Nash equilibrium is (2,2).
16. In a bargaining game with alternating offers, what is the subgame perfect equilibrium outcome if
the discount factor is δ and the size of the pie is 1?
a) (1/(1+δ), δ/(1+δ)) b) (δ/(1+δ), 1/(1+δ)) c) (1/2, 1/2) d) (1, 0)
Answer: a) (1/(1+δ), δ/(1+δ))
Solution:
In the subgame perfect equilibrium, player 1 offers x₁ = δ/(1+δ) to player 2, who accepts.
Player 1's share is 1 - x₁ = 1 - δ/(1+δ) = 1/(1+δ)
17. What is the Shapley value for player 2 in the following 3-player characteristic function game?
v({1}) = 0, v({2}) = 10, v({3}) = 0
v({1,2}) = 40, v({1,3}) = 30, v({2,3}) = 50
v({1,2,3}) = 100
a) 30 b) 35 c) 40 d) 45
Answer: c) 40
Solution:
Shapley value = (1/3!)[v({2}) - v({})] + (1/6)[v({1,2}) - v({1})] + (1/6)[v({2,3}) - v({3})] + (1/3!)
[v({1,2,3}) - v({1,3})]
= (1/6)[10] + (1/6)[40] + (1/6)[50] + (1/6)[70]
= 1.67 + 6.67 + 8.33 + 11.67 ≈ 40
18. In a first-price sealed-bid auction with two risk-neutral bidders whose valuations are uniformly
distributed on [0,1], what is the symmetric Bayesian Nash equilibrium bidding strategy?
a) b(v) = v/2 b) b(v) = v c) b(v) = 2v/3 d) b(v) = v/3
Answer: a) b(v) = v/2
Solution:
In a first-price sealed-bid auction with uniform distributions, the symmetric BNE bidding strategy is
b(v) = (n-1)v/n, where n is the number of bidders.
With n = 2, we get b(v) = v/2.
19. In a repeated Prisoner's Dilemma game with discount factor δ, what is the minimum discount
factor required for the grim trigger strategy to sustain cooperation?
Payoff matrix:
[3, 3 0, 5]
[5, 0 1, 1]
a) 1/2 b) 2/3 c) 3/4 d) 4/5
Answer: b) 2/3
Solution:
For cooperation to be sustained: 3/(1-δ) ≥ 5 + δ/(1-δ)
3 ≥ 5(1-δ) + δ
3 ≥ 5 - 5δ + δ
3 ≥ 5 - 4δ
4δ ≥ 2
δ ≥ 1/2
20
