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Question 2 MCQ · 1.0 marks
A man goes bankrupt and his entire estate at the time of bankruptcy is worth 300 units. The man
has two creditors; one’s claim is 250 units, the other’s, 130 units. The division of the estate
between them is carried out according to the contested-garment principle.
Based on the above data, answer the given subquestions.
Creditor 1 with claim 250 units gets
6406537366740.
210
6406537366741.
170
6406537366742.
250
6406537366743.
125
Published solution
**1. Allocate uncontested amounts:** Creditor 1 concedes \(300-250=50\) to creditor 2; creditor 2 concedes \(300-130=170\) to creditor 1.
**2. Divide the contested part:** The remaining estate is \(300-170-50=80\); give half, 40, to each creditor. Allocations are \(170+40=210\) and \(50+40=90\).
**Answer: A — 210.**
A: **Correct:** This is 170 uncontested units plus half the 80 contested units.
B: **Incorrect:** This includes only creditor 1’s uncontested part.
C: **Incorrect:** The claim is not the contested-garment allocation.
D: **Incorrect:** Half the claim is not the rule for this estate.
A man goes bankrupt and his entire estate at the time of bankruptcy is worth 300 units. The man
has two creditors; one’s claim is 250 units, the other’s, 130 units. The division of the estate
between them is carried out according to the contested-garment principle.
Based on the above data, answer the given subquestions.
Creditor 2 with claim 130 units gets
6406537366744.
130
6406537366745.
100
6406537366746.
90
6406537366747.
65
Published solution
**1. Allocate uncontested amounts:** Creditor 1 concedes \(300-250=50\) to creditor 2; creditor 2 concedes \(300-130=170\) to creditor 1.
**2. Divide the contested part:** The remaining estate is \(300-170-50=80\); give half, 40, to each creditor. Allocations are \(170+40=210\) and \(50+40=90\).
**Answer: C — 90.**
A: **Incorrect:** The creditor does not receive the full claim.
B: **Incorrect:** Equalizing the shortfall without accounting for the uncontested portions gives the wrong allocation.
C: **Correct:** This is 50 uncontested units plus half the 80 contested units.
D: **Incorrect:** Half the claim is not the allocation for this estate.
A cable TV station wants to connect six new customers to its network. The connections to the
network are described in the following tree graph:
Calculate the distribution of the costs among the six players if all six decide to connect to the
network.
Based on the above data, answer the given subquestions.
The cost to be paid by player 1 will be ________
Published solution
**1. Share each necessary edge:** The station–1 edge (cost 60) serves all six players. Edge 1–2 (18) serves players 2,4,5; edge 1–3 (16) serves players 3,6. Leaf edges 2–4 (7), 2–5 (12), 3–6 (5) serve only their leaf players.
**2. Allocate player 1’s path:** Under the Shapley tree-cost rule, divide each edge cost equally among players downstream of it. Thus \[c_1=60/6=10.\]
**Answer: 10.**
A cable TV station wants to connect six new customers to its network. The connections to the
network are described in the following tree graph:
Calculate the distribution of the costs among the six players if all six decide to connect to the
network.
Based on the above data, answer the given subquestions.
The cost to be paid by player 2 will be _________
Published solution
**1. Share each necessary edge:** The station–1 edge (cost 60) serves all six players. Edge 1–2 (18) serves players 2,4,5; edge 1–3 (16) serves players 3,6. Leaf edges 2–4 (7), 2–5 (12), 3–6 (5) serve only their leaf players.
**2. Allocate player 2’s path:** Under the Shapley tree-cost rule, divide each edge cost equally among players downstream of it. Thus \[c_2=60/6+18/3=16.\]
**Answer: 16.**
A cable TV station wants to connect six new customers to its network. The connections to the
network are described in the following tree graph:
Calculate the distribution of the costs among the six players if all six decide to connect to the
network.
Based on the above data, answer the given subquestions.
The cost to be paid by player 3 will be ________
Published solution
**1. Share each necessary edge:** The station–1 edge (cost 60) serves all six players. Edge 1–2 (18) serves players 2,4,5; edge 1–3 (16) serves players 3,6. Leaf edges 2–4 (7), 2–5 (12), 3–6 (5) serve only their leaf players.
**2. Allocate player 3’s path:** Under the Shapley tree-cost rule, divide each edge cost equally among players downstream of it. Thus \[c_3=60/6+16/2=18.\]
**Answer: 18.**
A cable TV station wants to connect six new customers to its network. The connections to the
network are described in the following tree graph:
Calculate the distribution of the costs among the six players if all six decide to connect to the
network.
Based on the above data, answer the given subquestions.
The cost to be paid by player 4 will be _________
Published solution
**1. Share each necessary edge:** The station–1 edge (cost 60) serves all six players. Edge 1–2 (18) serves players 2,4,5; edge 1–3 (16) serves players 3,6. Leaf edges 2–4 (7), 2–5 (12), 3–6 (5) serve only their leaf players.
**2. Allocate player 4’s path:** Under the Shapley tree-cost rule, divide each edge cost equally among players downstream of it. Thus \[c_4=60/6+18/3+7=23.\]
**Answer: 23.**
A cable TV station wants to connect six new customers to its network. The connections to the
network are described in the following tree graph:
Calculate the distribution of the costs among the six players if all six decide to connect to the
network.
Based on the above data, answer the given subquestions.
The cost to be paid by player 5 will be ________
Published solution
**1. Share each necessary edge:** The station–1 edge (cost 60) serves all six players. Edge 1–2 (18) serves players 2,4,5; edge 1–3 (16) serves players 3,6. Leaf edges 2–4 (7), 2–5 (12), 3–6 (5) serve only their leaf players.
**2. Allocate player 5’s path:** Under the Shapley tree-cost rule, divide each edge cost equally among players downstream of it. Thus \[c_5=60/6+18/3+12=28.\]
**Answer: 28.**
A cable TV station wants to connect six new customers to its network. The connections to the
network are described in the following tree graph:
Calculate the distribution of the costs among the six players if all six decide to connect to the
network.
Based on the above data, answer the given subquestions.
The cost to be paid by player 6 will be ________
Published solution
**1. Share each necessary edge:** The station–1 edge (cost 60) serves all six players. Edge 1–2 (18) serves players 2,4,5; edge 1–3 (16) serves players 3,6. Leaf edges 2–4 (7), 2–5 (12), 3–6 (5) serve only their leaf players.
**2. Allocate player 6’s path:** Under the Shapley tree-cost rule, divide each edge cost equally among players downstream of it. Thus \[c_6=60/6+16/2+5=23.\]
**Answer: 23.**
A game is split into two different games below. Supply the missing values.
Based on the above data, answer the given subquestions.
a = ______
Published solution
**1. Use pointwise addition:** For every coalition \(S\), the original game satisfies \(v(S)=v_1(S)+v_2(S)\). Match the same position in all three diagrams.
**2. Solve the corresponding entry:** \[a=5+5=10.\]
**Answer: 10.**
A game is split into two different games below. Supply the missing values.
Based on the above data, answer the given subquestions.
b = ______
Published solution
**1. Use pointwise addition:** For every coalition \(S\), the original game satisfies \(v(S)=v_1(S)+v_2(S)\). Match the same position in all three diagrams.
**2. Solve the corresponding entry:** \[b=20+30=50.\]
**Answer: 50.**
A game is split into two different games below. Supply the missing values.
Based on the above data, answer the given subquestions.
c = ______
Published solution
**1. Use pointwise addition:** For every coalition \(S\), the original game satisfies \(v(S)=v_1(S)+v_2(S)\). Match the same position in all three diagrams.
**2. Solve the corresponding entry:** \[c=20-12=8.\]
**Answer: 8.**
A game is split into two different games below. Supply the missing values.
Based on the above data, answer the given subquestions.
d = ______
Published solution
**1. Use pointwise addition:** For every coalition \(S\), the original game satisfies \(v(S)=v_1(S)+v_2(S)\). Match the same position in all three diagrams.
**2. Solve the corresponding entry:** \[d=15-5=10.\]
**Answer: 10.**
A game is split into two different games below. Supply the missing values.
Based on the above data, answer the given subquestions.
e = _____
Published solution
**1. Use pointwise addition:** For every coalition \(S\), the original game satisfies \(v(S)=v_1(S)+v_2(S)\). Match the same position in all three diagrams.
**2. Solve the corresponding entry:** \[e=10-3=7.\]
**Answer: 7.**
A game is split into two different games below. Supply the missing values.
Based on the above data, answer the given subquestions.
f = ______
Published solution
**1. Use pointwise addition:** For every coalition \(S\), the original game satisfies \(v(S)=v_1(S)+v_2(S)\). Match the same position in all three diagrams.
**2. Solve the corresponding entry:** \[f=30-10=20.\]
**Answer: 20.**
Based on the above data, answer the given subquestions.
S1 = ___________
Published solution
**1. Weight predecessor coalitions:** With three players, the empty predecessor set and the two-player predecessor set each have weight \(1/3\); each singleton predecessor set has weight \(1/6\). These weights average all six arrival orders.
**2. Sum marginal contributions:** \[\phi_1=\tfrac13(0)+\tfrac16[(12-6)+(6-6)]+\tfrac13(24-18)=3.\]
**3. Check efficiency:** The three Shapley values sum to \(3+12+9=24=v(N)\).
**Answer: 3.**
Based on the above data, answer the given subquestions.
S2 = ___________
Published solution
**1. Weight predecessor coalitions:** With three players, the empty predecessor set and the two-player predecessor set each have weight \(1/3\); each singleton predecessor set has weight \(1/6\). These weights average all six arrival orders.
**2. Sum marginal contributions:** \[\phi_2=\tfrac13(6)+\tfrac16[(12-0)+(18-6)]+\tfrac13(24-6)=12.\]
**3. Check efficiency:** The three Shapley values sum to \(3+12+9=24=v(N)\).
**Answer: 12.**
Based on the above data, answer the given subquestions.
S3 = ____________
Published solution
**1. Weight predecessor coalitions:** With three players, the empty predecessor set and the two-player predecessor set each have weight \(1/3\); each singleton predecessor set has weight \(1/6\). These weights average all six arrival orders.
**2. Sum marginal contributions:** \[\phi_3=\tfrac13(6)+\tfrac16[(6-0)+(18-6)]+\tfrac13(24-12)=9.\]
**3. Check efficiency:** The three Shapley values sum to \(3+12+9=24=v(N)\).
**Answer: 9.**
Based on the above data, answer the given subquestions.
What is the best response of roommate 2 if roommate 1 chooses t1=0?
t2 = ________
Published solution
**1. Maximize roommate 2’s payoff:** With \(t_1=0\), \(u_2=30t_2-5t_2^2\).
**2. Differentiate and verify:** \(30-10t_2=0\) gives \(t_2=3\). The second derivative is \(-10<0\), so this feasible solution is the maximum.
**Answer: 3.**
Suppose that you receive a piece of e-mail whose subject line contains the phrase “check this out”
(a popular phrase among scammers). Based just on this and without looking at the sender or the
message content, what is the chance that the message is spam? Suppose that 40% of all your
mails are spam and the remaining 60% are emails you want to receive. Suppose that 1% of all
spam messages contain the phrase “check this out” in their subject lines, while 0.4% of all non-
spam messages contain this phrase.
Based on the above data, answer the given subquestions.
What is the total probability of having the phrase “check this out” in the subject line of an email?
Published solution
**1. Weight by mail type:** The phrase appears with probability 0.01 in spam and 0.004 in non-spam; priors are 0.4 and 0.6.
**2. Use total probability:** \[P(\text{phrase})=0.4(0.01)+0.6(0.004)=0.0064.\]
**Answer: 0.0064.**
Suppose that you receive a piece of e-mail whose subject line contains the phrase “check this out”
(a popular phrase among scammers). Based just on this and without looking at the sender or the
message content, what is the chance that the message is spam? Suppose that 40% of all your
mails are spam and the remaining 60% are emails you want to receive. Suppose that 1% of all
spam messages contain the phrase “check this out” in their subject lines, while 0.4% of all non-
spam messages contain this phrase.
Based on the above data, answer the given subquestions.
What is the probability that the message is spam if the subject contains the phrase “check this
out”?
Published solution
**1. Calculate the joint and total probabilities:** \(P(\text{spam and phrase})=0.4(0.01)=0.004\), while \(P(\text{phrase})=0.004+0.6(0.004)=0.0064\).
**2. Apply Bayes’ rule:** \[P(\text{spam}\mid\text{phrase})=\frac{0.004}{0.0064}=0.625.\]
**Answer: 0.625.**
Given the following preference structure, find a stable matching system using the Gale–Shapley
algorithm when the men propose:
Women: A,B,C,D.
Men: a,b,c,d.
Based on the above data, answer the given subquestions.
Ms. D will be paired with
6406537366774.
a
6406537366775.
d
6406537366776.
b
6406537366777.
c
Published solution
**1. Interpret ranks:** Take smaller numbers as better ranks, with equal numbers denoting ties. a proposes to B, b to B, c to A, and d to C. B retains b over a; A retains c; C retains d.
**2. Continue deferred acceptance:** a next proposes to A, which prefers a to c. Rejected c proposes to B, which prefers c to b. Rejected b then tries A (rejected) and C, which prefers b to d. d tries B and A (both reject him), then D.
**3. Read the result:** (A,a), (B,c), (C,b), (D,d). No encountered acceptance comparison is tied, so unused tied ranks do not change this run.
**Answer: B — d.**
**Review note:** The PDF does not define the direction of numerical preferences and contains tied entries. Under the usual smaller-rank-is-better convention, the final matching is (A,a), (B,c), (C,b), (D,d). The source key lists (A,b), (B,a), (C,c), (D,d), which has blocking pairs under this convention.
A: **Incorrect:** He is matched to another woman in that run.
B: **Correct:** This is her partner in the men-proposing run under the stated rank convention.
C: **Incorrect:** He is matched to another woman in that run.
D: **Incorrect:** He is matched to another woman in that run.
Mohan and Charan, both students of game theory at IIT Chennai, are fans of old board games.
This year the Indian-Board-Game-Day, a big fair, is held in Chennai and naturally both go to see
what is offered. They both end up at one stand where the ancient board game “Chaturanga” is
offered in a second price auction. If they both bid the same, the winner is decided with the toss of
a coin (and the winner pays the bid, as the second bid coincides with the first). While both of them
are of course very interested, nobody else cares about the old game and participates in the
auction. Mohan and Charan know each other well. In particular, they both know the willingness to
pay for the game of the other. Mohan is willing to pay ₹70000, while Charan’s maximum
willingness to pay is ₹90000. The only admissible bids are 50, 70, and 90 thousand of INR.
Represent the situation as a normal form game and answer the given subquestions:
There is no strictly dominated strategy for Mohan and Charan both.
6406537366790.
True
6406537366791.
False
Published solution
Mohan owns rows; Charan owns columns. Bids and payoffs are in thousands of INR; each pair is (Mohan payoff, Charan payoff). Ties split winning probability equally and the winner pays the tied bid.
\[\begin{array}{c|ccc} & 50 & 70 & 90 \\ 50 & (10,20) & (0,40) & (0,40) \\ 70 & (20,0) & (0,10) & (0,20) \\ 90 & (20,0) & (0,0) & (-10,0)\end{array}\]
**1. Check strict dominance:** Mohan’s bids 70 and 90 tie against Charan’s 50 and 70. Against Charan’s 90, bids 50 and 70 tie. No bid beats another at every column.
**2. Check Charan:** Against Mohan’s 90, all Charan bids pay 0. Therefore no strategy can be strictly better at every row, even though weak dominance exists.
**Answer: A — True.**
A: **Correct:** Neither player has a strictly dominated strategy.
B: **Incorrect:** Weak dominance does not imply strict dominance at every opponent action.
Mohan and Charan, both students of game theory at IIT Chennai, are fans of old board games.
This year the Indian-Board-Game-Day, a big fair, is held in Chennai and naturally both go to see
what is offered. They both end up at one stand where the ancient board game “Chaturanga” is
offered in a second price auction. If they both bid the same, the winner is decided with the toss of
a coin (and the winner pays the bid, as the second bid coincides with the first). While both of them
are of course very interested, nobody else cares about the old game and participates in the
auction. Mohan and Charan know each other well. In particular, they both know the willingness to
pay for the game of the other. Mohan is willing to pay ₹70000, while Charan’s maximum
willingness to pay is ₹90000. The only admissible bids are 50, 70, and 90 thousand of INR.
Represent the situation as a normal form game and answer the given subquestions:
No. of Nash equilibria in pure strategies is
6406537366792.
1
6406537366793.
3
6406537366794.
5
6406537366795.
7
Published solution
Mohan owns rows; Charan owns columns. Bids and payoffs are in thousands of INR; each pair is (Mohan payoff, Charan payoff). Ties split winning probability equally and the winner pays the tied bid.
\[\begin{array}{c|ccc} & 50 & 70 & 90 \\ 50 & (10,20) & (0,40) & (0,40) \\ 70 & (20,0) & (0,10) & (0,20) \\ 90 & (20,0) & (0,0) & (-10,0)\end{array}\]
**1. Mark mutual best responses:** Equal payoffs can make several bids best responses. Include ties rather than keeping only one maximizer.
**2. List equilibria:** In thousands, the five pairs are (50,70), (50,90), (70,90), (90,50), and (90,70).
**Answer: C — 5 pure Nash equilibria.**
A: **Incorrect:** Tied best responses produce more than one equilibrium.
B: **Incorrect:** There are five mutual-best-response cells.
C: **Correct:** All five listed cells satisfy the Nash condition.
D: **Incorrect:** Only five cells, not seven, satisfy both players’ conditions.
Mohan and Charan, both students of game theory at IIT Chennai, are fans of old board games.
This year the Indian-Board-Game-Day, a big fair, is held in Chennai and naturally both go to see
what is offered. They both end up at one stand where the ancient board game “Chaturanga” is
offered in a second price auction. If they both bid the same, the winner is decided with the toss of
a coin (and the winner pays the bid, as the second bid coincides with the first). While both of them
are of course very interested, nobody else cares about the old game and participates in the
auction. Mohan and Charan know each other well. In particular, they both know the willingness to
pay for the game of the other. Mohan is willing to pay ₹70000, while Charan’s maximum
willingness to pay is ₹90000. The only admissible bids are 50, 70, and 90 thousand of INR.
Represent the situation as a normal form game and answer the given subquestions:
For Mohan, playing 70000 weakly dominates playing 50000 or 90000.
6406537366796.
True
6406537366797.
False
Published solution
Mohan owns rows; Charan owns columns. Bids and payoffs are in thousands of INR; each pair is (Mohan payoff, Charan payoff). Ties split winning probability equally and the winner pays the tied bid.
\[\begin{array}{c|ccc} & 50 & 70 & 90 \\ 50 & (10,20) & (0,40) & (0,40) \\ 70 & (20,0) & (0,10) & (0,20) \\ 90 & (20,0) & (0,0) & (-10,0)\end{array}\]
**1. Compare payoff vectors:** Mohan’s bid 70 gives \((20,0,0)\), bid 50 gives \((10,0,0)\), and bid 90 gives \((20,0,-10)\).
**2. Apply weak dominance:** Bid 70 never pays less than either alternative and pays strictly more in at least one column for each comparison.
**Answer: A — True.**
A: **Correct:** Truthful bid 70 weakly dominates each other bid.
B: **Incorrect:** The comparison is weak dominance because some columns give equal payoffs.
Mohan and Charan, both students of game theory at IIT Chennai, are fans of old board games.
This year the Indian-Board-Game-Day, a big fair, is held in Chennai and naturally both go to see
what is offered. They both end up at one stand where the ancient board game “Chaturanga” is
offered in a second price auction. If they both bid the same, the winner is decided with the toss of
a coin (and the winner pays the bid, as the second bid coincides with the first). While both of them
are of course very interested, nobody else cares about the old game and participates in the
auction. Mohan and Charan know each other well. In particular, they both know the willingness to
pay for the game of the other. Mohan is willing to pay ₹70000, while Charan’s maximum
willingness to pay is ₹90000. The only admissible bids are 50, 70, and 90 thousand of INR.
Represent the situation as a normal form game and answer the given subquestions:
For Charan, playing 90000 weakly dominates playing 50000 or 70000.
6406537366798.
True
6406537366799.
False
Published solution
Mohan owns rows; Charan owns columns. Bids and payoffs are in thousands of INR; each pair is (Mohan payoff, Charan payoff). Ties split winning probability equally and the winner pays the tied bid.
\[\begin{array}{c|ccc} & 50 & 70 & 90 \\ 50 & (10,20) & (0,40) & (0,40) \\ 70 & (20,0) & (0,10) & (0,20) \\ 90 & (20,0) & (0,0) & (-10,0)\end{array}\]
**1. Compare payoff vectors:** Charan’s bid 90 gives \((40,20,0)\), bid 50 gives \((20,0,0)\), and bid 70 gives \((40,10,0)\).
**2. Apply weak dominance:** Bid 90 never pays less than either alternative and pays strictly more in at least one row for each comparison.
**Answer: A — True.**
A: **Correct:** Truthful bid 90 weakly dominates each other bid.
B: **Incorrect:** Equalities in some rows prevent strict dominance but do not prevent weak dominance.
A game v describes a four-player market in which there are two sellers and two buyers. Where the
first two players are potential sellers and the other two players are potential buyers. Each seller
has a good that he offers for sale and whose worth for him is $100. Each buyer wants to buy a
single good whose worth for him is $150. The potential buyers do not have a good like the one
offered for sale. Describe the game in coalition function form.
Based on the above data, answer the given subquestions.
v(1) = _____
Published solution
**1. Count endowments and trades:** This coalition contains 1 seller(s) and 0 buyer(s). It can transfer 0 good(s) to buyers; each transfer increases gross coalition worth by 150−100=50.
**2. Calculate total worth:** Retained seller goods are worth 100 each. Thus \[v(S)=100(1)+50(0)=100.\]
**Answer: 100.**
A game v describes a four-player market in which there are two sellers and two buyers. Where the
first two players are potential sellers and the other two players are potential buyers. Each seller
has a good that he offers for sale and whose worth for him is $100. Each buyer wants to buy a
single good whose worth for him is $150. The potential buyers do not have a good like the one
offered for sale. Describe the game in coalition function form.
Based on the above data, answer the given subquestions.
v(2) = _______
Published solution
**1. Count endowments and trades:** This coalition contains 1 seller(s) and 0 buyer(s). It can transfer 0 good(s) to buyers; each transfer increases gross coalition worth by 150−100=50.
**2. Calculate total worth:** Retained seller goods are worth 100 each. Thus \[v(S)=100(1)+50(0)=100.\]
**Answer: 100.**
A game v describes a four-player market in which there are two sellers and two buyers. Where the
first two players are potential sellers and the other two players are potential buyers. Each seller
has a good that he offers for sale and whose worth for him is $100. Each buyer wants to buy a
single good whose worth for him is $150. The potential buyers do not have a good like the one
offered for sale. Describe the game in coalition function form.
Based on the above data, answer the given subquestions.
v(3) = ________
Published solution
**1. Count endowments and trades:** This coalition contains 0 seller(s) and 1 buyer(s). It can transfer 0 good(s) to buyers; each transfer increases gross coalition worth by 150−100=50.
**2. Calculate total worth:** Retained seller goods are worth 100 each. Thus \[v(S)=100(0)+50(0)=0.\]
**Answer: 0.**
A game v describes a four-player market in which there are two sellers and two buyers. Where the
first two players are potential sellers and the other two players are potential buyers. Each seller
has a good that he offers for sale and whose worth for him is $100. Each buyer wants to buy a
single good whose worth for him is $150. The potential buyers do not have a good like the one
offered for sale. Describe the game in coalition function form.
Based on the above data, answer the given subquestions.
v(1,3) = ________
Published solution
**1. Count endowments and trades:** This coalition contains 1 seller(s) and 1 buyer(s). It can transfer 1 good(s) to buyers; each transfer increases gross coalition worth by 150−100=50.
**2. Calculate total worth:** Retained seller goods are worth 100 each. Thus \[v(S)=100(1)+50(1)=150.\]
**Answer: 150.**
A game v describes a four-player market in which there are two sellers and two buyers. Where the
first two players are potential sellers and the other two players are potential buyers. Each seller
has a good that he offers for sale and whose worth for him is $100. Each buyer wants to buy a
single good whose worth for him is $150. The potential buyers do not have a good like the one
offered for sale. Describe the game in coalition function form.
Based on the above data, answer the given subquestions.
v(1,4) = ________
Published solution
**1. Count endowments and trades:** This coalition contains 1 seller(s) and 1 buyer(s). It can transfer 1 good(s) to buyers; each transfer increases gross coalition worth by 150−100=50.
**2. Calculate total worth:** Retained seller goods are worth 100 each. Thus \[v(S)=100(1)+50(1)=150.\]
**Answer: 150.**
A game v describes a four-player market in which there are two sellers and two buyers. Where the
first two players are potential sellers and the other two players are potential buyers. Each seller
has a good that he offers for sale and whose worth for him is $100. Each buyer wants to buy a
single good whose worth for him is $150. The potential buyers do not have a good like the one
offered for sale. Describe the game in coalition function form.
Based on the above data, answer the given subquestions.
v(2,3) = _______
Published solution
**1. Count endowments and trades:** This coalition contains 1 seller(s) and 1 buyer(s). It can transfer 1 good(s) to buyers; each transfer increases gross coalition worth by 150−100=50.
**2. Calculate total worth:** Retained seller goods are worth 100 each. Thus \[v(S)=100(1)+50(1)=150.\]
**Answer: 150.**
A game v describes a four-player market in which there are two sellers and two buyers. Where the
first two players are potential sellers and the other two players are potential buyers. Each seller
has a good that he offers for sale and whose worth for him is $100. Each buyer wants to buy a
single good whose worth for him is $150. The potential buyers do not have a good like the one
offered for sale. Describe the game in coalition function form.
Based on the above data, answer the given subquestions.
v(2,4) = _______
Published solution
**1. Count endowments and trades:** This coalition contains 1 seller(s) and 1 buyer(s). It can transfer 1 good(s) to buyers; each transfer increases gross coalition worth by 150−100=50.
**2. Calculate total worth:** Retained seller goods are worth 100 each. Thus \[v(S)=100(1)+50(1)=150.\]
**Answer: 150.**
A game v describes a four-player market in which there are two sellers and two buyers. Where the
first two players are potential sellers and the other two players are potential buyers. Each seller
has a good that he offers for sale and whose worth for him is $100. Each buyer wants to buy a
single good whose worth for him is $150. The potential buyers do not have a good like the one
offered for sale. Describe the game in coalition function form.
Based on the above data, answer the given subquestions.
v(3,4) = _______
Published solution
**1. Count endowments and trades:** This coalition contains 0 seller(s) and 2 buyer(s). It can transfer 0 good(s) to buyers; each transfer increases gross coalition worth by 150−100=50.
**2. Calculate total worth:** Retained seller goods are worth 100 each. Thus \[v(S)=100(0)+50(0)=0.\]
**Answer: 0.**
A game v describes a four-player market in which there are two sellers and two buyers. Where the
first two players are potential sellers and the other two players are potential buyers. Each seller
has a good that he offers for sale and whose worth for him is $100. Each buyer wants to buy a
single good whose worth for him is $150. The potential buyers do not have a good like the one
offered for sale. Describe the game in coalition function form.
Based on the above data, answer the given subquestions.
v(1,2,3) = ______
Published solution
**1. Count endowments and trades:** This coalition contains 2 seller(s) and 1 buyer(s). It can transfer 1 good(s) to buyers; each transfer increases gross coalition worth by 150−100=50.
**2. Calculate total worth:** Retained seller goods are worth 100 each. Thus \[v(S)=100(2)+50(1)=250.\]
**Answer: 250.**
A game v describes a four-player market in which there are two sellers and two buyers. Where the
first two players are potential sellers and the other two players are potential buyers. Each seller
has a good that he offers for sale and whose worth for him is $100. Each buyer wants to buy a
single good whose worth for him is $150. The potential buyers do not have a good like the one
offered for sale. Describe the game in coalition function form.
Based on the above data, answer the given subquestions.
v(2,3,4) = _______
Published solution
**1. Count endowments and trades:** This coalition contains 1 seller(s) and 2 buyer(s). It can transfer 1 good(s) to buyers; each transfer increases gross coalition worth by 150−100=50.
**2. Calculate total worth:** Retained seller goods are worth 100 each. Thus \[v(S)=100(1)+50(1)=150.\]
**Answer: 150.**
A game v describes a four-player market in which there are two sellers and two buyers. Where the
first two players are potential sellers and the other two players are potential buyers. Each seller
has a good that he offers for sale and whose worth for him is $100. Each buyer wants to buy a
single good whose worth for him is $150. The potential buyers do not have a good like the one
offered for sale. Describe the game in coalition function form.
Based on the above data, answer the given subquestions.
v(1,3,4) =______
Published solution
**1. Count endowments and trades:** This coalition contains 1 seller(s) and 2 buyer(s). It can transfer 1 good(s) to buyers; each transfer increases gross coalition worth by 150−100=50.
**2. Calculate total worth:** Retained seller goods are worth 100 each. Thus \[v(S)=100(1)+50(1)=150.\]
**Answer: 150.**
A game v describes a four-player market in which there are two sellers and two buyers. Where the
first two players are potential sellers and the other two players are potential buyers. Each seller
has a good that he offers for sale and whose worth for him is $100. Each buyer wants to buy a
single good whose worth for him is $150. The potential buyers do not have a good like the one
offered for sale. Describe the game in coalition function form.
Based on the above data, answer the given subquestions.
v(1,2,3,4) = _______
Published solution
**1. Count endowments and trades:** This coalition contains 2 seller(s) and 2 buyer(s). It can transfer 2 good(s) to buyers; each transfer increases gross coalition worth by 150−100=50.
**2. Calculate total worth:** Retained seller goods are worth 100 each. Thus \[v(S)=100(2)+50(2)=300.\]
**Answer: 300.**
**1. Evaluate a unilateral deviation:** If nobody learns, one type A person gains communication with 700 type B people by learning B, at cost \(C_A\).
**2. Prevent that deviation:** \(700-C_A\le0\), hence \(C_A\ge700\). The full no-learning equilibrium also requires \(C_B\ge300\).
**Answer: B — \(C_A\ge700\).**
A: **Incorrect:** This permits profitable learning when the cost is below 700.
B: **Correct:** This prevents a type A speaker’s profitable learning deviation.
C: **Incorrect:** 500 is not the number of type B speakers.
D: **Incorrect:** A cost between 500 and 700 still allows a profitable deviation.
**1. Evaluate a type B deviation:** Learning A permits communication with 300 additional type A people.
**2. Prevent that deviation:** \(300-C_B\le0\), so \(C_B\ge300\). The full no-learning equilibrium also requires \(C_A\ge700\).
**Answer: D — \(C_B\ge300\).**
A: **Incorrect:** The condition must be a lower bound tied to 300, not an upper bound at 700.
B: **Incorrect:** This is sufficient but unnecessarily restrictive; it is not the exact threshold.
C: **Incorrect:** This allows learning to be profitable below 300.
D: **Correct:** This is the exact no-deviation threshold for type B speakers.
Based on the above data, answer the given subquestions.
An outcome is said to be socially efficient if it maximizes total benefits minus total costs. For what
values of CA and CB would it be socially efficient for type A’s to learn language B?
The condition on CA and CB should be
**1. Count social gains:** If all 300 A speakers learn B, each of the \(300\times700\) cross-group pairs can communicate, benefiting both members. The total gain is \(2(300)(700)=420000\).
**2. Compare with no learning:** The cost is \(300C_A\). Nonnegative social gain requires \(300C_A\le420000\), or \(C_A\le1400\).
**3. Compare who should learn:** Having all B speakers learn instead costs \(700C_B\). A speakers learning is at least as good if \(300C_A\le700C_B\), or \(C_B\ge(3/7)C_A\). Mixed learning does not lower the minimum full-connectivity cost: each cross-group pair requires at least one of its members to learn.
**Answer: A — \(C_A\le1400,\ C_B\ge(3/7)C_A\).**
A: **Correct:** These conditions account for both sides’ communication gains and the cheaper complete-learning group.
B: **Incorrect:** Both inequality directions are reversed.
C: **Incorrect:** 700 is the private threshold, not the social threshold 1400; this is sufficient but too restrictive.
D: **Incorrect:** This reverses the cheaper-group comparison and uses the private threshold.
Based on the above data, answer the given subquestions.
For what values of CA and CB, would there be an equilibrium in which the type A’s learned
language B and the type B’s did not learn language A.
CA should be
**1. Evaluate type A’s choice:** When B speakers do not learn, an A speaker obtains 700 extra communication links by learning. Staying with learning requires \(C_A\le700\).
**2. Check type B:** With all A speakers bilingual, a B speaker gets no extra links from learning A, so any \(C_B\ge0\) is consistent with not learning.
**Answer: A — \(C_A\le700\).**
A: **Correct:** This is the exact threshold for A speakers to learn.
B: **Incorrect:** This makes learning unattractive above 700.
C: **Incorrect:** This is sufficient but too restrictive; costs up to 700 also sustain the equilibrium.
D: **Incorrect:** This lower bound does not prevent costs above 700.
Based on the above data, answer the given subquestions.
For what values of CA and CB, would there be a Nash equilibrium in which the type B's learned
language A and the type A’s did not learn language B.
CB should be
**1. Evaluate type B’s choice:** When A speakers do not learn, a B speaker gains 300 links by learning A. Thus learning is a best response when \(C_B\le300\).
**2. Check type A:** Since all B speakers become bilingual, an A speaker obtains no additional links by learning B; any \(C_A\ge0\) supports not learning.
**Answer: D — \(C_B\le300\).**
A: **Incorrect:** This allows costs between 300 and 700, where learning is unprofitable.
B: **Incorrect:** This makes learning unprofitable.
C: **Incorrect:** The learning condition is an upper bound, not a lower bound.
D: **Correct:** This is the exact threshold for B speakers to learn.
Three roommates need to vote on whether they will adopt a new rule and clean their room once a
week, or stick to the current once a month rule. Each votes “yes” for the new rule or “no” for the
current rule. Imagine that players 1 and 2 prefer the new rule while player 3 prefers the old rule.
Imagine that the players require a unanimous vote to adopt the new rule. Player 1 votes first, then
player 2, and then player 3, each one observing the previous votes. Draw this as an extensive form
game and answer the given subquestions.
No. of subgames in this game is
6406537366825.
5
6406537366826.
4
6406537366827.
7
6406537366828.
8
Published solution
\[\begin{array}{ccccccc}&&&P_1&&&\\ &\swarrow Y&&&&\searrow N&\\ &P_2&&&&P_2&\\ \swarrow Y&&\searrow N&&\swarrow Y&&\searrow N\\ P_3&&P_3&&P_3&&P_3\\ \swarrow Y\quad\searrow N&&\swarrow Y\quad\searrow N&&\swarrow Y\quad\searrow N&&\swarrow Y\quad\searrow N\\ \text{new}\quad\text{old}&&\text{old}\quad\text{old}&&\text{old}\quad\text{old}&&\text{old}\quad\text{old}\end{array}\]
Terminal outcomes show the adopted rule. Ordinal payoffs may be normalized to (1,1,0) for new and (0,0,1) for old, in player order 1,2,3. This normalization represents the stated preferences.
**1. Count nonterminal nodes:** There is 1 root, 2 player-2 nodes, and 4 player-3 nodes. Each is a singleton information set because earlier votes are observed.
**2. Count subgames:** Every decision node starts a subgame, including the whole game: \(1+2+4=7\).
**Answer: C — 7 subgames.**
A: **Incorrect:** There are seven decision nodes.
B: **Incorrect:** Four counts only player 3’s decision nodes.
C: **Correct:** All seven nodes start subgames.
D: **Incorrect:** Eight is the number of terminal vote histories.
Three roommates need to vote on whether they will adopt a new rule and clean their room once a
week, or stick to the current once a month rule. Each votes “yes” for the new rule or “no” for the
current rule. Imagine that players 1 and 2 prefer the new rule while player 3 prefers the old rule.
Imagine that the players require a unanimous vote to adopt the new rule. Player 1 votes first, then
player 2, and then player 3, each one observing the previous votes. Draw this as an extensive form
game and answer the given subquestions.
No. of pure strategies player 1 has
6406537366829.
2
6406537366830.
3
6406537366831.
4
6406537366832.
6
Published solution
\[\begin{array}{ccccccc}&&&P_1&&&\\ &\swarrow Y&&&&\searrow N&\\ &P_2&&&&P_2&\\ \swarrow Y&&\searrow N&&\swarrow Y&&\searrow N\\ P_3&&P_3&&P_3&&P_3\\ \swarrow Y\quad\searrow N&&\swarrow Y\quad\searrow N&&\swarrow Y\quad\searrow N&&\swarrow Y\quad\searrow N\\ \text{new}\quad\text{old}&&\text{old}\quad\text{old}&&\text{old}\quad\text{old}&&\text{old}\quad\text{old}\end{array}\]
Terminal outcomes show the adopted rule. Ordinal payoffs may be normalized to (1,1,0) for new and (0,0,1) for old, in player order 1,2,3. This normalization represents the stated preferences.
**1. Specify every contingency:** Player 1 has 1 information set(s), each with votes Yes and No.
**2. Count complete plans:** A pure strategy chooses at every information set, giving \(2^{1}=2\) plans.
**Answer: A — 2 pure strategies.**
A: **Correct:** This is the number of complete contingent voting plans.
B: **Incorrect:** The count must be 2^1=2, not this number.
C: **Incorrect:** The count must be 2^1=2, not this number.
D: **Incorrect:** The count must be 2^1=2, not this number.
Three roommates need to vote on whether they will adopt a new rule and clean their room once a
week, or stick to the current once a month rule. Each votes “yes” for the new rule or “no” for the
current rule. Imagine that players 1 and 2 prefer the new rule while player 3 prefers the old rule.
Imagine that the players require a unanimous vote to adopt the new rule. Player 1 votes first, then
player 2, and then player 3, each one observing the previous votes. Draw this as an extensive form
game and answer the given subquestions.
No. of pure strategies player 2 has
6406537366833.
2
6406537366834.
3
6406537366835.
4
6406537366836.
6
Published solution
\[\begin{array}{ccccccc}&&&P_1&&&\\ &\swarrow Y&&&&\searrow N&\\ &P_2&&&&P_2&\\ \swarrow Y&&\searrow N&&\swarrow Y&&\searrow N\\ P_3&&P_3&&P_3&&P_3\\ \swarrow Y\quad\searrow N&&\swarrow Y\quad\searrow N&&\swarrow Y\quad\searrow N&&\swarrow Y\quad\searrow N\\ \text{new}\quad\text{old}&&\text{old}\quad\text{old}&&\text{old}\quad\text{old}&&\text{old}\quad\text{old}\end{array}\]
Terminal outcomes show the adopted rule. Ordinal payoffs may be normalized to (1,1,0) for new and (0,0,1) for old, in player order 1,2,3. This normalization represents the stated preferences.
**1. Specify every contingency:** Player 2 has 2 information set(s), each with votes Yes and No.
**2. Count complete plans:** A pure strategy chooses at every information set, giving \(2^{2}=4\) plans.
**Answer: C — 4 pure strategies.**
A: **Incorrect:** The count must be 2^2=4, not this number.
B: **Incorrect:** The count must be 2^2=4, not this number.
C: **Correct:** This is the number of complete contingent voting plans.
D: **Incorrect:** The count must be 2^2=4, not this number.
Three roommates need to vote on whether they will adopt a new rule and clean their room once a
week, or stick to the current once a month rule. Each votes “yes” for the new rule or “no” for the
current rule. Imagine that players 1 and 2 prefer the new rule while player 3 prefers the old rule.
Imagine that the players require a unanimous vote to adopt the new rule. Player 1 votes first, then
player 2, and then player 3, each one observing the previous votes. Draw this as an extensive form
game and answer the given subquestions.
No. of pure strategies player 3 has
6406537366837.
4
6406537366838.
6
6406537366839.
12
6406537366840.
16
Published solution
\[\begin{array}{ccccccc}&&&P_1&&&\\ &\swarrow Y&&&&\searrow N&\\ &P_2&&&&P_2&\\ \swarrow Y&&\searrow N&&\swarrow Y&&\searrow N\\ P_3&&P_3&&P_3&&P_3\\ \swarrow Y\quad\searrow N&&\swarrow Y\quad\searrow N&&\swarrow Y\quad\searrow N&&\swarrow Y\quad\searrow N\\ \text{new}\quad\text{old}&&\text{old}\quad\text{old}&&\text{old}\quad\text{old}&&\text{old}\quad\text{old}\end{array}\]
Terminal outcomes show the adopted rule. Ordinal payoffs may be normalized to (1,1,0) for new and (0,0,1) for old, in player order 1,2,3. This normalization represents the stated preferences.
**1. Specify every contingency:** Player 3 has 4 information set(s), each with votes Yes and No.
**2. Count complete plans:** A pure strategy chooses at every information set, giving \(2^{4}=16\) plans.
**Answer: D — 16 pure strategies.**
A: **Incorrect:** The count must be 2^4=16, not this number.
B: **Incorrect:** The count must be 2^4=16, not this number.
C: **Incorrect:** The count must be 2^4=16, not this number.
D: **Correct:** This is the number of complete contingent voting plans.
Three roommates need to vote on whether they will adopt a new rule and clean their room once a
week, or stick to the current once a month rule. Each votes “yes” for the new rule or “no” for the
current rule. Imagine that players 1 and 2 prefer the new rule while player 3 prefers the old rule.
Imagine that the players require a unanimous vote to adopt the new rule. Player 1 votes first, then
player 2, and then player 3, each one observing the previous votes. Draw this as an extensive form
game and answer the given subquestions.
Imagine now that the players put their votes in a hat, so that the votes of earlier movers are not
observed by the later movers. No. of subgames in the new extensive form game will be
6406537366841.
1
6406537366842.
4
6406537366843.
7
6406537366844.
16
Published solution
**1. Merge hidden histories:** Player 2’s two nodes lie in one information set, and player 3’s four nodes lie in one information set. Their preceding votes are unobserved.
**2. Apply the subgame condition:** A proper subgame cannot start inside or cut an information set. Neither later player has a singleton starting information set, so only the whole game is a subgame.
**Answer: A — 1 subgame.**
A: **Correct:** Only the entire game qualifies as a subgame.
B: **Incorrect:** Hidden-history nodes cannot separately start subgames.
C: **Incorrect:** Seven applies to the observed-vote game.
D: **Incorrect:** Sixteen is a strategy count in the observed-vote game, not a subgame count.
