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Question 2 MCQ · 1.0 marks
Consider the following prisoners’ dilemma game and answer the given subquestions
Unique NE of this game is
6406537135411.
(Defect, Defect )
6406537135412.
(Cooperate, Cooperate )
6406537135413.
Both
6406537135414.
None
Published solution
Player 1 owns rows; player 2 owns columns. Each pair is (player 1 payoff, player 2 payoff).
\[\begin{array}{c|cc} & Cooperate & Defect \\ Cooperate & (4,4) & (1,5) \\ Defect & (5,1) & (3,3)\end{array}\]
**1. Compare actions:** For each player, Defect pays more against either opponent action (5>4 and 3>1). It is strictly dominant.
**2. Intersect best responses:** Both players defect in the unique stage-game Nash equilibrium.
**Answer: A — (Defect, Defect ).**
A: **Correct:** Defect is strictly dominant for both players.
B: **Incorrect:** Each player can gain by defecting against cooperation.
C: **Incorrect:** Mutual cooperation is not a Nash equilibrium.
D: **Incorrect:** The dominant-strategy equilibrium exists.
Consider the following prisoners’ dilemma game and answer the given subquestions
Suppose, this game is repeatedly played finite times
6406537135415.
In the last period Cooperate is a dominant strategy irrespective of history of
the game
6406537135416.
In the last period Defect is a dominant strategy irrespective of history of the
game
6406537135417.
In the last period Cooperate is a dominant strategy for a particular game
history
6406537135418.
In the last period Defect is a dominant strategy for a particular game history
Published solution
Player 1 owns rows; player 2 owns columns. Each pair is (player 1 payoff, player 2 payoff).
\[\begin{array}{c|cc} & Cooperate & Defect \\ Cooperate & (4,4) & (1,5) \\ Defect & (5,1) & (3,3)\end{array}\]
**1. Examine the final period:** With a known finite horizon, no future reward or punishment follows the last action. The final period therefore has the stage-game incentives.
**2. Compare actions:** Defect remains strictly dominant after every possible history.
**Answer: B — In the last period Defect is a dominant strategy irrespective of history of the .**
A: **Incorrect:** Cooperation is not dominant in the final stage.
B: **Correct:** Defection is dominant after every history.
C: **Incorrect:** No history reverses the final-stage payoff comparison.
D: **Incorrect:** This understates the result: defection is dominant after every history, not merely one.
Consider the following prisoners’ dilemma game and answer the given subquestions
Suppose, this game is repeatedly played for 2 times and in the first period, if both players choose
their dominant strategy. Derive the payoff matrix for second period and find out the value of
following unknowns that represent payoff at the end of second period:
j= ____________
Published solution
Source layout: player 2 owns rows and player 1 owns columns; source pairs are (row-player payoff, column-player payoff). Thus j,k=(7,7), l,m=(4,8), p,q=(8,4), r,s=(6,6).
\[\begin{array}{c|cc} & Cooperate & Defect \\ Cooperate & (7,7) & (4,8) \\ Defect & (8,4) & (6,6)\end{array}\]
**1. Carry forward the first period:** Both players defect, so each already has payoff 3. The question asks for undiscounted cumulative payoffs after period 2.
**2. Add the relevant payoff:** The source entry j has second-period payoff 4. Thus \(j=3+4=7\).
**Answer: 7.**
Consider the following prisoners’ dilemma game and answer the given subquestions
Suppose, this game is repeatedly played for 2 times and in the first period, if both players choose
their dominant strategy. Derive the payoff matrix for second period and find out the value of
following unknowns that represent payoff at the end of second period:
l=____________
Published solution
Source layout: player 2 owns rows and player 1 owns columns; source pairs are (row-player payoff, column-player payoff). Thus j,k=(7,7), l,m=(4,8), p,q=(8,4), r,s=(6,6).
\[\begin{array}{c|cc} & Cooperate & Defect \\ Cooperate & (7,7) & (4,8) \\ Defect & (8,4) & (6,6)\end{array}\]
**1. Carry forward the first period:** Both players defect, so each already has payoff 3. The question asks for undiscounted cumulative payoffs after period 2.
**2. Add the relevant payoff:** The source entry l has second-period payoff 1. Thus \(l=3+1=4\).
**Answer: 4.**
Consider the following prisoners’ dilemma game and answer the given subquestions
Suppose, this game is repeatedly played for 2 times and in the first period, if both players choose
their dominant strategy. Derive the payoff matrix for second period and find out the value of
following unknowns that represent payoff at the end of second period:
m=____________
Published solution
Source layout: player 2 owns rows and player 1 owns columns; source pairs are (row-player payoff, column-player payoff). Thus j,k=(7,7), l,m=(4,8), p,q=(8,4), r,s=(6,6).
\[\begin{array}{c|cc} & Cooperate & Defect \\ Cooperate & (7,7) & (4,8) \\ Defect & (8,4) & (6,6)\end{array}\]
**1. Carry forward the first period:** Both players defect, so each already has payoff 3. The question asks for undiscounted cumulative payoffs after period 2.
**2. Add the relevant payoff:** The source entry m has second-period payoff 5. Thus \(m=3+5=8\).
**Answer: 8.**
Consider the following prisoners’ dilemma game and answer the given subquestions
Suppose, this game is repeatedly played for 2 times and in the first period, if both players choose
their dominant strategy. Derive the payoff matrix for second period and find out the value of
following unknowns that represent payoff at the end of second period:
r =____________
Published solution
Source layout: player 2 owns rows and player 1 owns columns; source pairs are (row-player payoff, column-player payoff). Thus j,k=(7,7), l,m=(4,8), p,q=(8,4), r,s=(6,6).
\[\begin{array}{c|cc} & Cooperate & Defect \\ Cooperate & (7,7) & (4,8) \\ Defect & (8,4) & (6,6)\end{array}\]
**1. Carry forward the first period:** Both players defect, so each already has payoff 3. The question asks for undiscounted cumulative payoffs after period 2.
**2. Add the relevant payoff:** The source entry r has second-period payoff 3. Thus \(r=3+3=6\).
**Answer: 6.**
(Bayesian Game ) Consider a variant of the situation modelled by Battle of sexes (BOS) (as discussed
in lectures) in which player 1 is unsure whether player 2 prefers to go out with her or prefers to
avoid her, whereas player 2, as before, knows player 1’s preferences. Specifically, suppose player 1
thinks that with probability 1/2 player 2 wants to go out with her, and with probability 1/2 player 2
wants to avoid her. The situation is depicted in the following matrix of a static game with
incomplete information:
With the help of this game matrix, we may come up with the following table depicting expected
payoff of player 1 corresponding to different actions of both types of player 2.
Based on the above data, answer the given subquestions.
Find out the value of following unknown:
a = __________
Published solution
Player 1 owns rows; columns are player 2’s (meet-type action, avoid-type action). Entries are player 1’s expected payoffs, not payoff pairs.
\[\begin{array}{c|cccc} & BB & BS & SB & SS \\ B & 1 & 0.5 & 0.5 & 0 \\ S & 0 & 1.0 & 1.0 & 2\end{array}\]
**1. Weight types:** The meet and avoid types each have probability \(1/2\). For this cell, player 1’s type-contingent payoffs are 1 and 1.
**2. Average:** \[a=\tfrac12(1)+\tfrac12(1)=1.\]
**Answer: 1.**
(Bayesian Game ) Consider a variant of the situation modelled by Battle of sexes (BOS) (as discussed
in lectures) in which player 1 is unsure whether player 2 prefers to go out with her or prefers to
avoid her, whereas player 2, as before, knows player 1’s preferences. Specifically, suppose player 1
thinks that with probability 1/2 player 2 wants to go out with her, and with probability 1/2 player 2
wants to avoid her. The situation is depicted in the following matrix of a static game with
incomplete information:
With the help of this game matrix, we may come up with the following table depicting expected
payoff of player 1 corresponding to different actions of both types of player 2.
Based on the above data, answer the given subquestions.
Find out the value of following unknown:
b =__________
Published solution
Player 1 owns rows; columns are player 2’s (meet-type action, avoid-type action). Entries are player 1’s expected payoffs, not payoff pairs.
\[\begin{array}{c|cccc} & BB & BS & SB & SS \\ B & 1 & 0.5 & 0.5 & 0 \\ S & 0 & 1.0 & 1.0 & 2\end{array}\]
**1. Weight types:** The meet and avoid types each have probability \(1/2\). For this cell, player 1’s type-contingent payoffs are 1 and 0.
**2. Average:** \[b=\tfrac12(1)+\tfrac12(0)=0.5.\]
**Answer: 0.5.**
(Bayesian Game ) Consider a variant of the situation modelled by Battle of sexes (BOS) (as discussed
in lectures) in which player 1 is unsure whether player 2 prefers to go out with her or prefers to
avoid her, whereas player 2, as before, knows player 1’s preferences. Specifically, suppose player 1
thinks that with probability 1/2 player 2 wants to go out with her, and with probability 1/2 player 2
wants to avoid her. The situation is depicted in the following matrix of a static game with
incomplete information:
With the help of this game matrix, we may come up with the following table depicting expected
payoff of player 1 corresponding to different actions of both types of player 2.
Based on the above data, answer the given subquestions.
Find out the value of following unknown:
c =__________
Published solution
Player 1 owns rows; columns are player 2’s (meet-type action, avoid-type action). Entries are player 1’s expected payoffs, not payoff pairs.
\[\begin{array}{c|cccc} & BB & BS & SB & SS \\ B & 1 & 0.5 & 0.5 & 0 \\ S & 0 & 1.0 & 1.0 & 2\end{array}\]
**1. Weight types:** The meet and avoid types each have probability \(1/2\). For this cell, player 1’s type-contingent payoffs are 0 and 1.
**2. Average:** \[c=\tfrac12(0)+\tfrac12(1)=0.5.\]
**Answer: 0.5.**
(Bayesian Game ) Consider a variant of the situation modelled by Battle of sexes (BOS) (as discussed
in lectures) in which player 1 is unsure whether player 2 prefers to go out with her or prefers to
avoid her, whereas player 2, as before, knows player 1’s preferences. Specifically, suppose player 1
thinks that with probability 1/2 player 2 wants to go out with her, and with probability 1/2 player 2
wants to avoid her. The situation is depicted in the following matrix of a static game with
incomplete information:
With the help of this game matrix, we may come up with the following table depicting expected
payoff of player 1 corresponding to different actions of both types of player 2.
Based on the above data, answer the given subquestions.
Find out the value of following unknown:
d =__________
Published solution
Player 1 owns rows; columns are player 2’s (meet-type action, avoid-type action). Entries are player 1’s expected payoffs, not payoff pairs.
\[\begin{array}{c|cccc} & BB & BS & SB & SS \\ B & 1 & 0.5 & 0.5 & 0 \\ S & 0 & 1.0 & 1.0 & 2\end{array}\]
**1. Weight types:** The meet and avoid types each have probability \(1/2\). For this cell, player 1’s type-contingent payoffs are 0 and 0.
**2. Average:** \[d=\tfrac12(0)+\tfrac12(0)=0.\]
**Answer: 0.**
(Bayesian Game ) Consider a variant of the situation modelled by Battle of sexes (BOS) (as discussed
in lectures) in which player 1 is unsure whether player 2 prefers to go out with her or prefers to
avoid her, whereas player 2, as before, knows player 1’s preferences. Specifically, suppose player 1
thinks that with probability 1/2 player 2 wants to go out with her, and with probability 1/2 player 2
wants to avoid her. The situation is depicted in the following matrix of a static game with
incomplete information:
With the help of this game matrix, we may come up with the following table depicting expected
payoff of player 1 corresponding to different actions of both types of player 2.
Based on the above data, answer the given subquestions.
Find out the value of following unknown:
p =__________
Published solution
Player 1 owns rows; columns are player 2’s (meet-type action, avoid-type action). Entries are player 1’s expected payoffs, not payoff pairs.
\[\begin{array}{c|cccc} & BB & BS & SB & SS \\ B & 1 & 0.5 & 0.5 & 0 \\ S & 0 & 1.0 & 1.0 & 2\end{array}\]
**1. Weight types:** The meet and avoid types each have probability \(1/2\). For this cell, player 1’s type-contingent payoffs are 0 and 0.
**2. Average:** \[p=\tfrac12(0)+\tfrac12(0)=0.\]
**Answer: 0.**
(Bayesian Game ) Consider a variant of the situation modelled by Battle of sexes (BOS) (as discussed
in lectures) in which player 1 is unsure whether player 2 prefers to go out with her or prefers to
avoid her, whereas player 2, as before, knows player 1’s preferences. Specifically, suppose player 1
thinks that with probability 1/2 player 2 wants to go out with her, and with probability 1/2 player 2
wants to avoid her. The situation is depicted in the following matrix of a static game with
incomplete information:
With the help of this game matrix, we may come up with the following table depicting expected
payoff of player 1 corresponding to different actions of both types of player 2.
Based on the above data, answer the given subquestions.
Find out the value of following unknown:
q =__________
Published solution
Player 1 owns rows; columns are player 2’s (meet-type action, avoid-type action). Entries are player 1’s expected payoffs, not payoff pairs.
\[\begin{array}{c|cccc} & BB & BS & SB & SS \\ B & 1 & 0.5 & 0.5 & 0 \\ S & 0 & 1.0 & 1.0 & 2\end{array}\]
**1. Weight types:** The meet and avoid types each have probability \(1/2\). For this cell, player 1’s type-contingent payoffs are 0 and 2.
**2. Average:** \[q=\tfrac12(0)+\tfrac12(2)=1.\]
**Answer: 1.**
(Bayesian Game ) Consider a variant of the situation modelled by Battle of sexes (BOS) (as discussed
in lectures) in which player 1 is unsure whether player 2 prefers to go out with her or prefers to
avoid her, whereas player 2, as before, knows player 1’s preferences. Specifically, suppose player 1
thinks that with probability 1/2 player 2 wants to go out with her, and with probability 1/2 player 2
wants to avoid her. The situation is depicted in the following matrix of a static game with
incomplete information:
With the help of this game matrix, we may come up with the following table depicting expected
payoff of player 1 corresponding to different actions of both types of player 2.
Based on the above data, answer the given subquestions.
Find out the value of following unknown:
r =__________
Published solution
Player 1 owns rows; columns are player 2’s (meet-type action, avoid-type action). Entries are player 1’s expected payoffs, not payoff pairs.
\[\begin{array}{c|cccc} & BB & BS & SB & SS \\ B & 1 & 0.5 & 0.5 & 0 \\ S & 0 & 1.0 & 1.0 & 2\end{array}\]
**1. Weight types:** The meet and avoid types each have probability \(1/2\). For this cell, player 1’s type-contingent payoffs are 2 and 0.
**2. Average:** \[r=\tfrac12(2)+\tfrac12(0)=1.\]
**Answer: 1.**
(Bayesian Game ) Consider a variant of the situation modelled by Battle of sexes (BOS) (as discussed
in lectures) in which player 1 is unsure whether player 2 prefers to go out with her or prefers to
avoid her, whereas player 2, as before, knows player 1’s preferences. Specifically, suppose player 1
thinks that with probability 1/2 player 2 wants to go out with her, and with probability 1/2 player 2
wants to avoid her. The situation is depicted in the following matrix of a static game with
incomplete information:
With the help of this game matrix, we may come up with the following table depicting expected
payoff of player 1 corresponding to different actions of both types of player 2.
Based on the above data, answer the given subquestions.
Find out the value of following unknown:
t =__________
Published solution
Player 1 owns rows; columns are player 2’s (meet-type action, avoid-type action). Entries are player 1’s expected payoffs, not payoff pairs.
\[\begin{array}{c|cccc} & BB & BS & SB & SS \\ B & 1 & 0.5 & 0.5 & 0 \\ S & 0 & 1.0 & 1.0 & 2\end{array}\]
**1. Weight types:** The meet and avoid types each have probability \(1/2\). For this cell, player 1’s type-contingent payoffs are 2 and 2.
**2. Average:** \[t=\tfrac12(2)+\tfrac12(2)=2.\]
**Answer: 2.**
(Bayesian Game ) Consider a variant of the situation modelled by Battle of sexes (BOS) (as discussed
in lectures) in which player 1 is unsure whether player 2 prefers to go out with her or prefers to
avoid her, whereas player 2, as before, knows player 1’s preferences. Specifically, suppose player 1
thinks that with probability 1/2 player 2 wants to go out with her, and with probability 1/2 player 2
wants to avoid her. The situation is depicted in the following matrix of a static game with
incomplete information:
With the help of this game matrix, we may come up with the following table depicting expected
payoff of player 1 corresponding to different actions of both types of player 2.
Based on the above data, answer the given subquestions.
What is the Bayesian Nash equilibrium Strategy of player 1 in this game?
6406537135431.
B
6406537135432.
S
6406537135433.
Either of the two
6406537135434.
None
Published solution
Meet type: player 1 owns rows, player 2 owns columns; pairs are (player 1 payoff, player 2 payoff).
\[\begin{array}{c|cc} & B & S \\ B & (1,2) & (0,0) \\ S & (0,0) & (2,1)\end{array}\]
Avoid type: player 1 owns rows, player 2 owns columns; pairs are (player 1 payoff, player 2 payoff).
\[\begin{array}{c|cc} & B & S \\ B & (1,0) & (0,1) \\ S & (0,2) & (2,0)\end{array}\]
Player 1 owns rows; columns are player 2’s (meet-type action, avoid-type action). Entries are player 1’s expected payoffs, not payoff pairs.
\[\begin{array}{c|cccc} & BB & BS & SB & SS \\ B & 1 & 0.5 & 0.5 & 0 \\ S & 0 & 1.0 & 1.0 & 2\end{array}\]
**1. Respond by type:** Against S, the meet type selects S and the avoid type selects B.
**2. Check player 1:** Against (S,B), player 1 earns 1.0 from S and 0.5 from the other action. Therefore S is the strict best response.
**Answer: B — S.**
A: **Incorrect:** B gives the lower expected payoff at the candidate equilibrium.
B: **Correct:** S gives the higher expected payoff at the pure equilibrium.
C: **Incorrect:** The two actions are not tied at a pure equilibrium.
D: **Incorrect:** A pure Bayesian equilibrium exists.
(Bayesian Game ) Consider a variant of the situation modelled by Battle of sexes (BOS) (as discussed
in lectures) in which player 1 is unsure whether player 2 prefers to go out with her or prefers to
avoid her, whereas player 2, as before, knows player 1’s preferences. Specifically, suppose player 1
thinks that with probability 1/2 player 2 wants to go out with her, and with probability 1/2 player 2
wants to avoid her. The situation is depicted in the following matrix of a static game with
incomplete information:
With the help of this game matrix, we may come up with the following table depicting expected
payoff of player 1 corresponding to different actions of both types of player 2.
Based on the above data, answer the given subquestions.
What is the Bayesian Nash equilibrium Strategy of both types of player 2 in this game?
6406537135435.
(B,S)
6406537135436.
(B,B)
6406537135437.
(S,B)
6406537135438.
(S,S)
Published solution
Meet type: player 1 owns rows, player 2 owns columns; pairs are (player 1 payoff, player 2 payoff).
\[\begin{array}{c|cc} & B & S \\ B & (1,2) & (0,0) \\ S & (0,0) & (2,1)\end{array}\]
Avoid type: player 1 owns rows, player 2 owns columns; pairs are (player 1 payoff, player 2 payoff).
\[\begin{array}{c|cc} & B & S \\ B & (1,0) & (0,1) \\ S & (0,2) & (2,0)\end{array}\]
Player 1 owns rows; columns are player 2’s (meet-type action, avoid-type action). Entries are player 1’s expected payoffs, not payoff pairs.
\[\begin{array}{c|cccc} & BB & BS & SB & SS \\ B & 1 & 0.5 & 0.5 & 0 \\ S & 0 & 1.0 & 1.0 & 2\end{array}\]
**1. Respond by type:** Against S, the meet type selects S and the avoid type selects B.
**2. Check player 1:** Against (S,B), player 1 earns 1.0 from S and 0.5 from the other action. Therefore S is the strict best response.
**Answer: C — (S,B).**
A: **Incorrect:** This plan makes at least one type choose an action that is not a best response to player 1’s equilibrium action.
B: **Incorrect:** This plan makes at least one type choose an action that is not a best response to player 1’s equilibrium action.
C: **Correct:** The meet type coordinates and the avoid type chooses the opposite action.
D: **Incorrect:** This plan makes at least one type choose an action that is not a best response to player 1’s equilibrium action.
Based on the above data, answer the given subquestions.
Choose the correct alternative
6406537135439.
(Slow, Slow) is an NE
6406537135440.
(Slow, Fast) is an NE
6406537135441.
Both
6406537135442.
None
Published solution
Player 1 owns rows; player 2 owns columns. Each pair is (player 1 payoff, player 2 payoff).
\[\begin{array}{c|cc} & Slow & Fast \\ Slow & (4,4) & (2,1) \\ Fast & (1,2) & (5,5)\end{array}\]
**1. Compare best responses:** Against Slow, each player strictly prefers Slow; against Fast, each strictly prefers Fast. Thus (Slow,Slow) and (Fast,Fast) are pure Nash equilibria.
**2. Check the offered profiles:** (Slow,Fast) is not a mutual best response, so only the first listed statement is true.
**Answer: A — (Slow, Slow) is an NE.**
A: **Correct:** Both players best respond at (Slow,Slow).
B: **Incorrect:** At least one player gains by switching to match the other.
C: **Incorrect:** The asymmetric profile is not a Nash equilibrium.
D: **Incorrect:** The first listed profile is an equilibrium.
Based on the above data, answer the given subquestions.
ESS in this game is
6406537135443.
Slow
6406537135444.
Fast
6406537135445.
Both
6406537135446.
None
Published solution
Player 1 owns rows; player 2 owns columns. Each pair is (player 1 payoff, player 2 payoff).
\[\begin{array}{c|cc} & Slow & Fast \\ Slow & (4,4) & (2,1) \\ Fast & (1,2) & (5,5)\end{array}\]
**1. Apply the strict criterion:** A pure strategy is evolutionarily stable when its payoff against itself strictly exceeds a rare mutant’s payoff against it.
**2. Check both:** Slow gives 4 against Slow, while a Fast mutant gives 1; Fast gives 5 against Fast, while a Slow mutant gives 2. Both inequalities are strict.
**Answer: C — Both Slow and Fast.**
A: **Incorrect:** Slow is stable, but Fast is stable too.
B: **Incorrect:** Fast is stable, but Slow is stable too.
C: **Correct:** Both pure strategies resist invasion by the other.
D: **Incorrect:** Both pure strategies satisfy the strict ESS criterion.
Given the following preference structure, find a stable matching system using the Gale- Shapley
algorithm when the men propose.
Based on the above data, answer the given subquestions.
Ms. A will be paired with
6406537135483.
a
6406537135484.
c
6406537135485.
d
6406537135486.
b
Published solution
**1. First proposals:** a proposes to B, b and c to A, and d to C. A retains c and rejects b; B retains a; C retains d.
**2. Resolve rejections:** b proposes to C, which prefers b to d. Rejected d then tries A, which retains c, and finally D, which accepts d.
**3. Read the final matching:** (A,c), (B,a), (C,b), (D,d). Lower ranks denote higher preference.
**Answer: B — c.**
A: **Incorrect:** He finishes matched to a different woman after deferred acceptance.
B: **Correct:** This is her partner at termination.
C: **Incorrect:** He finishes matched to a different woman after deferred acceptance.
D: **Incorrect:** He finishes matched to a different woman after deferred acceptance.
Given the following preference structure, find a stable matching system using the Gale- Shapley
algorithm when the men propose.
Based on the above data, answer the given subquestions.
Ms. B will be paired with
6406537135487.
b
6406537135488.
c
6406537135489.
a
6406537135490.
d
Published solution
**1. First proposals:** a proposes to B, b and c to A, and d to C. A retains c and rejects b; B retains a; C retains d.
**2. Resolve rejections:** b proposes to C, which prefers b to d. Rejected d then tries A, which retains c, and finally D, which accepts d.
**3. Read the final matching:** (A,c), (B,a), (C,b), (D,d). Lower ranks denote higher preference.
**Answer: C — a.**
A: **Incorrect:** He finishes matched to a different woman after deferred acceptance.
B: **Incorrect:** He finishes matched to a different woman after deferred acceptance.
C: **Correct:** This is her partner at termination.
D: **Incorrect:** He finishes matched to a different woman after deferred acceptance.
Given the following preference structure, find a stable matching system using the Gale- Shapley
algorithm when the men propose.
Based on the above data, answer the given subquestions.
Ms. C will be paired with
6406537135491.
b
6406537135492.
d
6406537135493.
c
6406537135494.
a
Published solution
**1. First proposals:** a proposes to B, b and c to A, and d to C. A retains c and rejects b; B retains a; C retains d.
**2. Resolve rejections:** b proposes to C, which prefers b to d. Rejected d then tries A, which retains c, and finally D, which accepts d.
**3. Read the final matching:** (A,c), (B,a), (C,b), (D,d). Lower ranks denote higher preference.
**Answer: A — b.**
A: **Correct:** This is her partner at termination.
B: **Incorrect:** He finishes matched to a different woman after deferred acceptance.
C: **Incorrect:** He finishes matched to a different woman after deferred acceptance.
D: **Incorrect:** He finishes matched to a different woman after deferred acceptance.
Given the following preference structure, find a stable matching system using the Gale- Shapley
algorithm when the men propose.
Based on the above data, answer the given subquestions.
Ms. D will be paired with
6406537135495.
a
6406537135496.
d
6406537135497.
b
6406537135498.
c
Published solution
**1. First proposals:** a proposes to B, b and c to A, and d to C. A retains c and rejects b; B retains a; C retains d.
**2. Resolve rejections:** b proposes to C, which prefers b to d. Rejected d then tries A, which retains c, and finally D, which accepts d.
**3. Read the final matching:** (A,c), (B,a), (C,b), (D,d). Lower ranks denote higher preference.
**Answer: B — d.**
A: **Incorrect:** He finishes matched to a different woman after deferred acceptance.
B: **Correct:** This is her partner at termination.
C: **Incorrect:** He finishes matched to a different woman after deferred acceptance.
D: **Incorrect:** He finishes matched to a different woman after deferred acceptance.
Consider the example of cascade discussed in the lecture but with a modification that now the urn
is containing either 4 blue balls and 2 red balls or 4 red balls and 2 blue balls with equal
probability. That simply means that the urn is equally likely to be blue-majority urn or red-majority
urn. We assume everyone hears what the previous individuals report and get their own private
signal by randomly drawing one ball from the urn.
Based on the above data, answer the given subquestions.
Suppose one ball is picked from the red majority urn. What is the probability that the ball is blue?
6406537135447.
1/3
6406537135448.
1/4
6406537135449.
1/2
6406537135450.
2/3
Published solution
**1. Count blue balls:** In a red-majority urn the blue fraction is \(1/3\).
**Answer: A — 1/3.**
A: **Correct:** This equals the required probability.
B: **Incorrect:** This differs from the required probability 1/3.
C: **Incorrect:** This differs from the required probability 1/3.
D: **Incorrect:** This differs from the required probability 1/3.
Consider the example of cascade discussed in the lecture but with a modification that now the urn
is containing either 4 blue balls and 2 red balls or 4 red balls and 2 blue balls with equal
probability. That simply means that the urn is equally likely to be blue-majority urn or red-majority
urn. We assume everyone hears what the previous individuals report and get their own private
signal by randomly drawing one ball from the urn.
Based on the above data, answer the given subquestions.
What is the probability that the urn is red majority if the first student draws a red ball?
6406537135451.
2/4
6406537135452.
1/3
6406537135453.
2/3
6406537135454.
1/4
Published solution
**1. Apply Bayes’ rule:** The prior probabilities of the two urn types are equal.
**2. Update on red:** \[P(R\text{-majority}\mid R)=\frac{2/3}{2/3+1/3}=2/3.\]
**Answer: C — 2/3.**
A: **Incorrect:** This differs from the required probability 2/3.
B: **Incorrect:** This differs from the required probability 2/3.
C: **Correct:** This equals the required probability.
D: **Incorrect:** This differs from the required probability 2/3.
Consider the example of cascade discussed in the lecture but with a modification that now the urn
is containing either 4 blue balls and 2 red balls or 4 red balls and 2 blue balls with equal
probability. That simply means that the urn is equally likely to be blue-majority urn or red-majority
urn. We assume everyone hears what the previous individuals report and get their own private
signal by randomly drawing one ball from the urn.
Based on the above data, answer the given subquestions.
Now suppose, the first three students draw the ball in the sequence (Red, Red, Blue).
What is the probability of getting this sequence if the urn is a blue majority urn?
6406537135455.
2/27
6406537135456.
4/27
6406537135457.
1/27
6406537135458.
6/27
Published solution
**1. Use independent draws:** Under blue majority, red has probability \(1/3\) and blue \(2/3\).
**2. Multiply in order:** \[P(RRB\mid B\text{-majority})=(1/3)^2(2/3)=2/27.\]
**Answer: A — 2/27.**
**Review note:** This uses independent draws with replacement, as in the standard information-cascade model. The passage does not explicitly specify replacement; sampling without replacement gives different probabilities. The supplied scoring key follows replacement.
A: **Correct:** This equals the required probability.
B: **Incorrect:** This differs from the required probability 2/27.
C: **Incorrect:** This differs from the required probability 2/27.
D: **Incorrect:** This differs from the required probability 2/27.
Consider the example of cascade discussed in the lecture but with a modification that now the urn
is containing either 4 blue balls and 2 red balls or 4 red balls and 2 blue balls with equal
probability. That simply means that the urn is equally likely to be blue-majority urn or red-majority
urn. We assume everyone hears what the previous individuals report and get their own private
signal by randomly drawing one ball from the urn.
Based on the above data, answer the given subquestions.
Now suppose, the first three students draw the ball in the sequence (Red, Red, Blue).
What is the probability of getting this sequence if the urn is a red majority urn?
6406537135459.
1/27
6406537135460.
4/27
6406537135461.
6/27
6406537135462.
2/27
Published solution
**1. Use independent draws:** Under red majority, red has probability \(2/3\) and blue \(1/3\).
**2. Multiply in order:** \[P(RRB\mid R\text{-majority})=(2/3)^2(1/3)=4/27.\]
**Answer: B — 4/27.**
**Review note:** This uses independent draws with replacement, as in the standard information-cascade model. The passage does not explicitly specify replacement; sampling without replacement gives different probabilities. The supplied scoring key follows replacement.
A: **Incorrect:** This differs from the required probability 4/27.
B: **Correct:** This equals the required probability.
C: **Incorrect:** This differs from the required probability 4/27.
D: **Incorrect:** This differs from the required probability 4/27.
Consider the example of cascade discussed in the lecture but with a modification that now the urn
is containing either 4 blue balls and 2 red balls or 4 red balls and 2 blue balls with equal
probability. That simply means that the urn is equally likely to be blue-majority urn or red-majority
urn. We assume everyone hears what the previous individuals report and get their own private
signal by randomly drawing one ball from the urn.
Based on the above data, answer the given subquestions.
Now suppose, the first three students draw the ball in the sequence (Red, Red, Blue).
What is the total probability of getting this sequence?
6406537135463.
1/27
6406537135464.
1/9
6406537135465.
4/27
6406537135466.
4/9
Published solution
**1. Obtain conditional likelihoods:** They are \(2/27\) for blue majority and \(4/27\) for red majority.
**2. Average over the prior:** \[P(RRB)=\tfrac12(2/27)+\tfrac12(4/27)=1/9.\]
**Answer: B — 1/9.**
**Review note:** This uses independent draws with replacement, as in the standard information-cascade model. The passage does not explicitly specify replacement; sampling without replacement gives different probabilities. The supplied scoring key follows replacement.
A: **Incorrect:** This differs from the required probability 1/9.
B: **Correct:** This equals the required probability.
C: **Incorrect:** This differs from the required probability 1/9.
D: **Incorrect:** This differs from the required probability 1/9.
Consider the example of cascade discussed in the lecture but with a modification that now the urn
is containing either 4 blue balls and 2 red balls or 4 red balls and 2 blue balls with equal
probability. That simply means that the urn is equally likely to be blue-majority urn or red-majority
urn. We assume everyone hears what the previous individuals report and get their own private
signal by randomly drawing one ball from the urn.
Based on the above data, answer the given subquestions.
Now suppose, the first three students draw the ball in the sequence (Red, Red, Blue).
What is the probability of the urn being red majority in this scenario?
6406537135467.
3/6
6406537135468.
1/3
6406537135469.
2/3
6406537135470.
1/6
Published solution
**1. Compare likelihoods:** The sequence RRB has likelihood \(4/27\) under red majority and \(2/27\) under blue majority.
**2. Normalize:** Equal priors cancel, so \[P(R\text{-majority}\mid RRB)=\frac{4/27}{4/27+2/27}=2/3.\]
**Answer: C — 2/3.**
**Review note:** This uses independent draws with replacement, as in the standard information-cascade model. The passage does not explicitly specify replacement; sampling without replacement gives different probabilities. The supplied scoring key follows replacement.
A: **Incorrect:** This differs from the required probability 2/3.
B: **Incorrect:** This differs from the required probability 2/3.
C: **Correct:** This equals the required probability.
D: **Incorrect:** This differs from the required probability 2/3.
**Grim Trigger:** Consider the infinitely repeated game with discount factor δ<1 of the following
variant of the Prisoner’s Dilemma:
Based on the above data, answer the given subquestions.
Unique NE of this game is
6406537135471.
(T,L)
6406537135472.
(B,R)
6406537135473.
(M,C)
6406537135474.
None
Published solution
Player 1 owns rows; player 2 owns columns. Each pair is (player 1 payoff, player 2 payoff).
\[\begin{array}{c|ccc} & L & C & R \\ T & (6,6) & (-1,7) & (-2,8) \\ M & (7,-1) & (4,4) & (-1,5) \\ B & (8,-2) & (5,-1) & (0,0)\end{array}\]
**1. Interpret the equilibrium question:** In the stage game, B strictly dominates T and M, while R strictly dominates L and C.
**2. Intersect dominant strategies:** The unique stage-game Nash equilibrium is (B,R). The infinitely repeated game can also sustain cooperation for sufficiently high discount factors.
**Answer: B — (B,R).**
**Review note:** The question introduces an infinitely repeated game but asks for a unique NE. This answer refers to the stage game; the repeated game need not have a unique equilibrium.
A: **Incorrect:** Both can gain by deviating in the stage game.
B: **Correct:** This is the unique stage-game equilibrium.
C: **Incorrect:** Both can gain by deviating in the stage game.
D: **Incorrect:** The stage-game equilibrium exists.
**Grim Trigger:** Consider the infinitely repeated game with discount factor δ<1 of the following
variant of the Prisoner’s Dilemma:
Based on the above data, answer the given subquestions.
For which values of the discount factor δ can the players support the pair of actions (M,C) played in
every period?
6406537135475.
(δ ≥ 1/5)
6406537135476.
(δ ≥ 1/3)
6406537135477.
(δ ≥ 3/7)
6406537135478.
(δ ≥ 1/2)
Published solution
Player 1 owns rows; player 2 owns columns. Each pair is (player 1 payoff, player 2 payoff).
\[\begin{array}{c|ccc} & L & C & R \\ T & (6,6) & (-1,7) & (-2,8) \\ M & (7,-1) & (4,4) & (-1,5) \\ B & (8,-2) & (5,-1) & (0,0)\end{array}\]
**1. Specify grim trigger:** Cooperate forever unless a deviation occurs; thereafter play the stage equilibrium (B,R), yielding 0 each period.
**2. Compare payoffs:** Cooperation gives \(4/(1-\delta)\). A best one-period deviation gives 5, followed by zero, so
\[\frac{4}{1-\delta}\ge 5\quad\Longleftrightarrow\quad\delta\ge 1/5.\]
**3. Check credibility:** The punishment is a stage-game Nash equilibrium, so no player gains by deviating during punishment.
**Answer: A — \(\delta\ge 1/5\).**
A: **Correct:** This is the exact incentive threshold.
B: **Incorrect:** This is not the exact threshold; the incentive constraint gives 1/5.
C: **Incorrect:** This is not the exact threshold; the incentive constraint gives 1/5.
D: **Incorrect:** This is not the exact threshold; the incentive constraint gives 1/5.
**Grim Trigger:** Consider the infinitely repeated game with discount factor δ<1 of the following
variant of the Prisoner’s Dilemma:
Based on the above data, answer the given subquestions.
For which values of the discount factor δ can the players support the pair of actions (T,L) played in
every period?
6406537135479.
(δ ≥ 1/7)
6406537135480.
(δ ≥ 0)
6406537135481.
(δ ≥ 1/4)
6406537135482.
(δ ≥ 1/2)
Published solution
Player 1 owns rows; player 2 owns columns. Each pair is (player 1 payoff, player 2 payoff).
\[\begin{array}{c|ccc} & L & C & R \\ T & (6,6) & (-1,7) & (-2,8) \\ M & (7,-1) & (4,4) & (-1,5) \\ B & (8,-2) & (5,-1) & (0,0)\end{array}\]
**1. Specify grim trigger:** Cooperate forever unless a deviation occurs; thereafter play the stage equilibrium (B,R), yielding 0 each period.
**2. Compare payoffs:** Cooperation gives \(6/(1-\delta)\). A best one-period deviation gives 8, followed by zero, so
\[\frac{6}{1-\delta}\ge 8\quad\Longleftrightarrow\quad\delta\ge 1/4.\]
**3. Check credibility:** The punishment is a stage-game Nash equilibrium, so no player gains by deviating during punishment.
**Answer: C — \(\delta\ge 1/4\).**
A: **Incorrect:** This is not the exact threshold; the incentive constraint gives 1/4.
B: **Incorrect:** This is not the exact threshold; the incentive constraint gives 1/4.
C: **Correct:** This is the exact incentive threshold.
D: **Incorrect:** This is not the exact threshold; the incentive constraint gives 1/4.
