Question 1 Question 2

Problem Set 2

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The due date for this homework is Mon 3 Feb 2014 9:05 AM CET.

In accordance with the Coursera Honor Code, I (Dang Quang Vinh) certify that the answers here are my own work.

Question 1 Part 1. Implementing a Voting Mechanism

Consider a voting with 2 alternative outcomes (a, b). When designing a voting mechanism, the designer can NOT specify: The actions are voting over three choices (x, y, z). The mapping from votes to outcomes. The voting rule is a simple majority rule. Player 1 must prefer a.

Question 2 Part 1. Implementing a Voting Mechanism

Suppose the designer uses the following voting mechanism: each player submits one vote, vi ∈ a, b ,

and the winning alternative

w

by all players. For instance, if there are probability of

1 n

is uniformly randomly chosen from all votes submitted

n

players, each of their votes will be chosen with a

.

Do players have dominant strategies in this voting game? Yes No

Question 3 Part 1. Implementing a Voting Mechanism

Suppose the designer uses the following voting mechanism: each player submits one vote, vi ∈ a, b ,

and the winning alternative

w

by all players. For instance, if there are probability of

1 n

is uniformly randomly chosen from all votes submitted

n

players, each of their votes will be chosen with a

.

There exists a Bayes-Nash equilibrium in this voting game. Yes No

Question 4 Part 2: Impossibility of General, Dominant-Strategy Implementation

Consider the following mechanisms, in which one truthful reporting of preferences is NOT a dominant strategy? A voting with player 1 being the dictator. A buyer chooses between buying or not, when the price is fixed. A voting under Borda rule. None of the above.

Question 5 Part 3: Transferable Utility

Here is a game with 2 players, θ

= (θ1 , θ2 ).

Let the outcome be

o = (x, p) = ((x1 , x2 ), (p1 , p2 )) .Consider

the following 3 utilities:

(A)

ui (θ, o) = (x1 + x2 ) θi pi

(B)

ui (θ, o) = (x1 + x2 )(θ1 + θ2 ) − p1 − p2

(C)

ui (θ, o) = x1 x2 (θ1 + θ2 ) − pi

Which of the utilities is quasilinear preferences with transferable utility? A B C None

Question 6 Part 3: Transferable Utility

Here is a game with 2 players, θ

= (θ1 , θ2 ).

Let the outcome be

o = (x, p) = ((x1 , x2 ), (p1 , p2 )) .Consider

the following 3 utilities:

(A)

ui (θ, o) = (x1 + x2 ) θi pi

(B)

ui (θ, o) = (x1 + x2 )(θ1 + θ2 ) − p1 − p2

(C)

ui (θ, o) = x1 x2 (θ1 + θ2 ) − pi

Which of the utilities is quasilinear preferences with transferable utility and private values? A B C None

Question 7 Part 4: Mechanism Design as an Optimization Problem

Consider a voting with 3 players (1, 2, 3) and 2 alternatives (a, b). Each player has a strict preference, and prefers a or

b

with equal probabilities. Each player gets 5 if his/her top choice is

selected and 0 otherwise.

Consider the following 2 voting rules with transferable utility: in both cases, each player submits one vote

vi ∈ a, b

for

i ∈ 1, 2, 3.

(i) The winning alternative

w

is uniformly randomly chosen from the 3 votes submitted by the 3

players, and there is no transfer. (ii)

w

is chosen under the same rule as in (i). Players with

vi ≠ w

vi = w

pay 1, and players with

get 1.

For example under (ii), if player 1 prefers a , votes a and winning alternative, player 1 gets 5 because

a

w = v1 = a

wins, pays 1 because

is selected to be the

v1 = a ,

and thus gets a

net payoff 4.

Which voting rule(s) is truthful? Both (i) and (ii) Only (i) Only (ii) Neither

Question 8 Part 4: Mechanism Design as an Optimization Problem

Consider a voting with 3 players (1, 2, 3) and 2 alternatives (a, b). Each player has a strict preference, and prefers a or

b

with equal probabilities. Each player gets 5 if his/her top choice is

selected and 0 otherwise.

Consider the following 2 voting rules with transferable utility: in both cases, each player submits one vote

vi ∈ a, b

for

i ∈ 1, 2, 3.

(i) The winning alternative

w

is uniformly randomly chosen from the 3 votes submitted by the 3

players, and there is no transfer. (ii)

w

is chosen under the same rule as in (i). Players with

vi ≠ w

vi = w

pay 1, and players with

get 1.

For example under (ii), if player 1 prefers a , votes a and winning alternative, player 1 gets 5 because

a

w = v1 = a

wins, pays 1 because

is selected to be the

v1 = a ,

and thus gets a

net payoff 4.

Which voting rule(s) is efficient? Both (i) and (ii) Only (i) Only (ii) Neither

Question 9 Part 4: Mechanism Design as an Optimization Problem

Consider a voting with 3 players (1, 2, 3) and 2 alternatives (a, b). Each player has a strict preference, and prefers a or

b

with equal probabilities. Each player gets 5 if his/her top choice is

selected and 0 otherwise.

Consider the following 2 voting rules with transferable utility: in both cases, each player submits one vote

vi ∈ a, b

for

i ∈ 1, 2, 3.

(i) The winning alternative

w

is uniformly randomly chosen from the 3 votes submitted by the 3

players, and there is no transfer. (ii)

w

is chosen under the same rule as in (i). Players with

vi ≠ w

vi = w

pay 1, and players with

get 1.

For example under (ii), if player 1 prefers a , votes a and winning alternative, player 1 gets 5 because

a

w = v1 = a

wins, pays 1 because

is selected to be the

v1 = a ,

and thus gets a

net payoff 4.

Which voting rule(s) is budget balanced? Both (i) and (ii) Only (i) Only (ii) Neither

Question 10 Part 4: Mechanism Design as an Optimization Problem

Consider a voting with 3 players (1, 2, 3) and 2 alternatives (a, b). Each player has a strict preference, and prefers a or

b

with equal probabilities. Each player gets 5 if his/her top choice is

selected and 0 otherwise.

Consider the following 2 voting rules with transferable utility: in both cases, each player submits one vote

vi ∈ a, b

for

i ∈ 1, 2, 3.

(i) The winning alternative

w

is uniformly randomly chosen from the 3 votes submitted by the 3

players, and there is no transfer. (ii)

w

is chosen under the same rule as in (i). Players with

vi ≠ w

vi = w

pay 1, and players with

get 1.

For example under (ii), if player 1 prefers a , votes a and winning alternative, player 1 gets 5 because

a

wins, pays 1 because

net payoff 4.

Which voting rule(s) is ex interim individual rational? Both (i) and (ii) Only (i) Only (ii) Neither

w = v1 = a

is selected to be the

v1 = a ,

and thus gets a

Question 11 Part 4: Mechanism Design as an Optimization Problem

Consider a voting with 3 players (1, 2, 3) and 2 alternatives (a, b). Each player has a strict preference, and prefers a or

b

with equal probabilities. Each player gets 5 if his/her top choice is

selected and 0 otherwise.

Consider the following 2 voting rules with transferable utility: in both cases, each player submits one vote

vi ∈ a, b

for

i ∈ 1, 2, 3.

(i) The winning alternative

w

is uniformly randomly chosen from the 3 votes submitted by the 3

players, and there is no transfer. (ii)

w

is chosen under the same rule as in (i). Players with

vi ≠ w

vi = w

pay 1, and players with

get 1.

For example under (ii), if player 1 prefers a , votes a and winning alternative, player 1 gets 5 because

a

w = v1 = a

wins, pays 1 because

is selected to be the

v1 = a ,

and thus gets a

net payoff 4.

Which voting rule earns a higher revenue? (i) (ii) Their Revenues are the same

Question 12 Part 4: Mechanism Design as an Optimization Problem

Consider a voting with 3 players (1, 2, 3) and 2 alternatives (a, b). Each player has a strict preference, and prefers a or

b

with equal probabilities. Each player gets 5 if his/her top choice is

selected and 0 otherwise.

Consider the following 2 voting rules with transferable utility: in both cases, each player submits one vote

vi ∈ a, b

for

i ∈ 1, 2, 3.

(i) The winning alternative

w

is uniformly randomly chosen from the 3 votes submitted by the 3

players, and there is no transfer. (ii)

w

is chosen under the same rule as in (i). Players with

vi ≠ w

vi = w

pay 1, and players with

get 1.

For example under (ii), if player 1 prefers a , votes a and winning alternative, player 1 gets 5 because

a

w = v1 = a

wins, pays 1 because

is selected to be the

v1 = a ,

and thus gets a

net payoff 4.

Which voting rule gives a higher maxmin fairness? (i) (ii) Their maxmin fairnesses are the same

In accordance with the Coursera Honor Code, I (Dang Quang Vinh) certify that the answers here are my own work.

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