Deck 10: Problems With Group Decision Making

Question 1

Imagine that three city council members are trying to decide how to spend a surplus. The options currently being debated are (i) spend it on improving primary education in the municipality, (ii) spend it on improving the level of medical care offered by the local hospital, or (iii) lower local taxes and use the surplus to cover the costs of existing programs. The council employs majority rule to make its decisions. The councillors have the following preference orderings over the spending choices:
Councillor 1: Education $\succ$ Medical $\succ$ Tax cut
Councillor 2: Medical $\succ$ Tax cut $\succ$ Education
Councillor 3: Tax cut $\succ$ Education $\succ$ Medical
Assume that the councillors hold a round-robin tournament that pits each alternative against every other alternative in a series of pair-wise votes. The winner is the alternative that wins the most contests. Based on this information, answer the following four questions.
-Given the preference orderings listed above, what would the result be of a pair-wise contest between the spending choices education and medical?

A) education
B) medical
C) tax cut
D) it would be a tie

Answer 1

A

Question 2

Imagine that three city council members are trying to decide how to spend a surplus. The options currently being debated are (i) spend it on improving primary education in the municipality, (ii) spend it on improving the level of medical care offered by the local hospital, or (iii) lower local taxes and use the surplus to cover the costs of existing programs. The council employs majority rule to make its decisions. The councillors have the following preference orderings over the spending choices:
Councillor 1: Education $\succ$ Medical $\succ$ Tax cut
Councillor 2: Medical $\succ$ Tax cut $\succ$ Education
Councillor 3: Tax cut $\succ$ Education $\succ$ Medical
Assume that the councillors hold a round-robin tournament that pits each alternative against every other alternative in a series of pair-wise votes. The winner is the alternative that wins the most contests. Based on this information, answer the following four questions.
-Given the preference orderings listed above, what would the result be of a pair-wise contest between the spending choices medical and tax cut?

A) education
B) medical
C) tax cut
D) it would be a tie

Answer 2

B

Question 3

Imagine that three city council members are trying to decide how to spend a surplus. The options currently being debated are (i) spend it on improving primary education in the municipality, (ii) spend it on improving the level of medical care offered by the local hospital, or (iii) lower local taxes and use the surplus to cover the costs of existing programs. The council employs majority rule to make its decisions. The councillors have the following preference orderings over the spending choices:
Councillor 1: Education $\succ$ Medical $\succ$ Tax cut
Councillor 2: Medical $\succ$ Tax cut $\succ$ Education
Councillor 3: Tax cut $\succ$ Education $\succ$ Medical
Assume that the councillors hold a round-robin tournament that pits each alternative against every other alternative in a series of pair-wise votes. The winner is the alternative that wins the most contests. Based on this information, answer the following four questions.
-Given the preference orderings listed above, what would the result be of a pair-wise contest between the spending choices tax cut and education?

A) education
B) medical
C) tax cut
D) it would be a tie

Answer 3

C

Question 4

Imagine that three city council members are trying to decide how to spend a surplus. The options currently being debated are (i) spend it on improving primary education in the municipality, (ii) spend it on improving the level of medical care offered by the local hospital, or (iii) lower local taxes and use the surplus to cover the costs of existing programs. The council employs majority rule to make its decisions. The councillors have the following preference orderings over the spending choices:
Councillor 1: Education $\succ$ Medical $\succ$ Tax cut
Councillor 2: Medical $\succ$ Tax cut $\succ$ Education
Councillor 3: Tax cut $\succ$ Education $\succ$ Medical
Assume that the councillors hold a round-robin tournament that pits each alternative against every other alternative in a series of pair-wise votes. The winner is the alternative that wins the most contests. Based on this information, answer the following four questions.
-Which of the outcomes, if any, is a Condorcet winner?

A) education
B) medical
C) tax cut
D) there is no Condorcet winner in this example

Question 5

assume that Councillor 3 has to take a family member to the emergency room and has a very poor experience with the level of medical care provided at the local hospital. This leads her to view improving medical care as a higher priority than improving education in the municipality, if money is to be spent improving any programs. Councillors 1 and 2 do not change their preference orderings. Thus, the councillors now have the following preference orderings over the spending choices:
Councillor 1: Education $\succ$ Medical $\succ$ Tax cut
Councillor 2: Medical $\succ$ Tax cut $\succ$ Education
Councillor 3: Tax cut $\succ$ Medical $\succ$ Education
Assume that the councillors hold a round-robin tournament that pits each alternative against every other alternative in a series of pair-wise votes. The winner is the alternative that wins the most contests. Based on this information, answer the following four questions.
-Given the preference orderings listed above, what would the result be of a pair-wise contest between the spending choices education and medical?

A) education
B) medical
C) tax cut
D) it would be a tie

Question 6

assume that Councillor 3 has to take a family member to the emergency room and has a very poor experience with the level of medical care provided at the local hospital. This leads her to view improving medical care as a higher priority than improving education in the municipality, if money is to be spent improving any programs. Councillors 1 and 2 do not change their preference orderings. Thus, the councillors now have the following preference orderings over the spending choices:
Councillor 1: Education $\succ$ Medical $\succ$ Tax cut
Councillor 2: Medical $\succ$ Tax cut $\succ$ Education
Councillor 3: Tax cut $\succ$ Medical $\succ$ Education
Assume that the councillors hold a round-robin tournament that pits each alternative against every other alternative in a series of pair-wise votes. The winner is the alternative that wins the most contests. Based on this information, answer the following four questions.
-Given the preference orderings listed above, what would the result be of a pair-wise contest between the spending choices medical and tax cut?

A) education
B) medical
C) tax cut
D) it would be a tie

Question 7

assume that Councillor 3 has to take a family member to the emergency room and has a very poor experience with the level of medical care provided at the local hospital. This leads her to view improving medical care as a higher priority than improving education in the municipality, if money is to be spent improving any programs. Councillors 1 and 2 do not change their preference orderings. Thus, the councillors now have the following preference orderings over the spending choices:
Councillor 1: Education $\succ$ Medical $\succ$ Tax cut
Councillor 2: Medical $\succ$ Tax cut $\succ$ Education
Councillor 3: Tax cut $\succ$ Medical $\succ$ Education
Assume that the councillors hold a round-robin tournament that pits each alternative against every other alternative in a series of pair-wise votes. The winner is the alternative that wins the most contests. Based on this information, answer the following four questions.
-Given the preference orderings listed above, what would the result be of a pair-wise contest between the spending choices tax cut and education?

A) education
B) medical
C) tax cut
D) it would be a tie

Question 8

assume that Councillor 3 has to take a family member to the emergency room and has a very poor experience with the level of medical care provided at the local hospital. This leads her to view improving medical care as a higher priority than improving education in the municipality, if money is to be spent improving any programs. Councillors 1 and 2 do not change their preference orderings. Thus, the councillors now have the following preference orderings over the spending choices:
Councillor 1: Education $\succ$ Medical $\succ$ Tax cut
Councillor 2: Medical $\succ$ Tax cut $\succ$ Education
Councillor 3: Tax cut $\succ$ Medical $\succ$ Education
Assume that the councillors hold a round-robin tournament that pits each alternative against every other alternative in a series of pair-wise votes. The winner is the alternative that wins the most contests. Based on this information, answer the following four questions.
-Which of the outcomes, if any, is a Condorcet winner?

A) education
B) medical
C) tax cut
D) there is no Condorcet winner in this example

Question 9

What is Codorcet's paradox?

A) An option that beats all other options in a series of pair-wise contests.
B) A situation in which an actor has cyclical (non-transitive) preferences over alternatives.
C) A situation in which the collective preferences of a group are not guaranteed to be rational, even though each of the actors in the group individually have rational preferences.

Question 10

As the number of individuals and/or the number of alternatives involved in any decision-making situation increase, what happens to the likelihood of group intransitivity?

A) It stays the same.
B) It increases.
C) It decreases.

Question 11

From the point of view of someone trying to design the ideal set of decision-making rules, the key characteristic of the Borda Count that is troubling is that:

A) It does not allow individuals to express their opinions over all of the alternatives.
B) Group choices can be influenced by the introduction of irrelevant alternatives.
C) It provides group members with incentives not to vote.

Question 12

If a voter chooses an alternative that is not her most preferred one because by doing so she can produce a more preferred final outcome than might otherwise be the case, then she is engaging in:

A) sincere voting.
B) strategic voting.

Question 13

Consider the following preference orderings.
Councillor 1: Education $\succ$ Medical $\succ$ Tax cut
Councillor 2: Medical $\succ$ Tax cut $\succ$ Education
Councillor 3: Tax cut $\succ$ Education $\succ$ Medical
If Councillor 1 gets to be the agenda setter and choose the ordering in which the three councillors vote over the alternatives, which of the following agendas should she set in order to get her most-preferred outcome?

A) It wouldn't make a difference; the final choice would be the same regardless of the agenda.
B) First a vote between education and medical, and then the winner against tax cuts.
C) First a vote between medical and tax cuts, and then the winner against education.
D) First a vote between tax cuts and education, and then the winner against medical.

Question 14

What's the difference between a preference ordering and a utility function?

A) They are the same.
B) A utility function is a numerical scale that represents an individual's preference ordering.

Question 15

Do voters need to have single-peaked preferences in order for the Median Voter Theorem to hold?

Question 16

According to the logic of the median voter theorem, where should we expect candidates (or parties) to locate in the policy space in two-candidate (or two-party) races?

A) Anywhere in the ideological space.
B) They should each locate near the opposing extremes.
C) They should each locate in the middle of the ideological space.
D) They should each locate at the median voter's ideal point.

Question 17

Figure 1 illustrates an election in which there are seven voters (A, B, C, D, E, F, G) arrayed along a single left-right issue dimension that runs from 0 (most left) to 10 (most right). Each voter is assumed to have a single-peaked preference ordering over the issue dimension and to vote for the party that is located closest to her ideal point. The voters are participating in a majority rule election in which there are two parties, P1 and P2, competing for office. These parties can be thought of as "office-seeking" parties since they only care about winning the election and getting into office.
Figure 1: Illustrating the Median Voter Theorem

-What is the ideological position of the median voter in Figure 1?

A) 1.5
B) 2
C) 3
D) 4
E) 5
F) 5.5
G) 7
H) 8

Question 18

Figure 1 illustrates an election in which there are seven voters (A, B, C, D, E, F, G) arrayed along a single left-right issue dimension that runs from 0 (most left) to 10 (most right). Each voter is assumed to have a single-peaked preference ordering over the issue dimension and to vote for the party that is located closest to her ideal point. The voters are participating in a majority rule election in which there are two parties, P1 and P2, competing for office. These parties can be thought of as "office-seeking" parties since they only care about winning the election and getting into office.
Figure 1: Illustrating the Median Voter Theorem

-Let's suppose that P₁ locates at Position 2 on the left-right issue dimension and that P₂ locates at Position 7. Who wins the election in the situation illustrated by Figure 1?

A) The two parties tie.
B) P₁ wins.
C) P₂ wins.

Question 19

Figure 1 illustrates an election in which there are seven voters (A, B, C, D, E, F, G) arrayed along a single left-right issue dimension that runs from 0 (most left) to 10 (most right). Each voter is assumed to have a single-peaked preference ordering over the issue dimension and to vote for the party that is located closest to her ideal point. The voters are participating in a majority rule election in which there are two parties, P1 and P2, competing for office. These parties can be thought of as "office-seeking" parties since they only care about winning the election and getting into office.
Figure 1: Illustrating the Median Voter Theorem

-Now suppose that P₁ locates at position 4 on the left-right issue dimension and that P₂ locates at position 4. Who wins the election in the situation illustrated by Figure 1?

A) The two parties tie.
B) P₁ wins.
C) P₂ wins.

Question 20

Suppose that some event occurs that causes several voters to adopt more centrist positions on the left-right issue dimension. The new distribution of voters is shown in Figure 2.
Figure 2: Illustrating the Median Voter Theorem-A Centrist Electorate

-Given the centrist nature of the distribution of voters in Figure 2, where will parties P₁ and P₂ locate in the left-right space?

A) 3
B) 3.5
C) 4
D) 4.5
E) 5
F) 5.5
G) 6
H) 6.5

Question 21

Suppose now that some polarizing event occurs that causes several voters to adopt more extreme positions on the left-right issue dimension. The new distribution of voters is shown in Figure 3.
Figure 3: Illustrating the Median Voter Theorem-A Polarized Electorate

-Where will parties P₁ and P₂ locate in the left-right space given the polarized nature of the electorate shown in Figure 3?

A) 0
B) 1
C) 3
D) 5
E) 8
F) 9
G) 10

Question 22

What happens if we extend the logic of the median voter theorem to a multidimensional ideological space?

A) Nothing changes.
B) The most likely outcome is that there will be no Condorcet winner.
C) The most likely outcome is that there will be cyclical majorities.
D) Both (B) and (C) are true.

Question 23

What is the fundamental implication of Arrow's theorem?

A) No alternative can beat the one preferred by the median voter in pair-wise majority-rule elections if the number of voters is odd, voter preferences are single-peaked over a single policy dimension, and voters vote sincerely.
B) If there are two or more issue dimensions and three or more voters with preferences in the issue space who all vote sincerely, then it is likely (except in very extreme case) that there will be no Condorcet winner.
C) There is no possible decision-making rule satisfying a minimal standard of fairness that is guaranteed to produce a rational decision for a group.

Question 24

Does Arrow's theorem apply to majority rule decision-making procedures?

Question 25

Which of the following conditions are included in the set of fairness conditions that Arrow thought any minimally fair decision-making procedure should satisfy?

A) No single individual should fully determine the outcome regardless of the preferences of other group members.
B) The choice made by the group should not be affected by the rankings of irrelevant alternatives.
C) If everyone in the group prefers one alternative over another, the group choice should reflect these preferences.
D) Individuals' personal judgments are not restricted.
E) All of these are part of Arrow's criteria.
F) Only (A) and (B) are mentioned by Arrow.

Question 26

Describe, in your own words, each of the conditions in Arrow's Impossibility Theorem.

Question 27

According to Arrow, if a decision-making procedure met conditions P, I, and D, it could not also meet both the rationality assumption and condition U. What does this say about the fundamental trade-off that all political institutions must face?

Question 28

Discuss the theoretical constraints on democracy. What tensions necessarily arise if democracy is to be understood as a minimally fair method of converting the preferences of individual citizens into stable social outcomes?