Unit 2

Question 1

What does public choice theory study?

Answer 1

The aggregation of individual preferences

Question 2

What is the goal of voting theory?

Answer 2

To make the greatest number of people happy whilst allowing all to vote honestly

Question 3

What does a voting method do?

Answer 3

It contains rules for valid voting and how votes are aggregated to yield a final result

Question 4

What is a Condorcet voting method?

Answer 4

It is an election method that elects the candidate that would win a majority of the vote in all of the head-to-head elections against each of the other candidates, whenever there is such a candidate. A candidate with this property is called the Condorcet winner.

Question 5

Explain the steps in the Condorcet voting algorithm?

Answer 5

1) Rank all candidates (elements) in order of preference, 1 best through to n, worst
2) Pairwise comparison between each candidate; the winner is determined by winning the majority of the pairwise competitions (1 v 1 contests)

Question 6

How do we deal with an incomplete preference list in the CVA?

Answer 6

Ranked candidates assumed to be better than unranked ones

Question 7

What type of method is a CVA?

Answer 7

Majoritarian

Question 8

Note about CVA:

Answer 8

The Condorcet voting algorithm is a majoritarian method which specifies that the winner of the election is the candidate(s) that beats or ties with every other candidate in a pair-wise comparison.

Question 9

How can we generalise these pairwise competitions? Why is this rarely done in reality?

Answer 9

Model elections with a digraph

BUT this is too slow for any real life applications
check I know what a digraph is!

Question 10

See

Answer 10

Example 2.1 on doing a CVA

Question 11

2 problems with using Condorcet voting?

Answer 11

1) Is too slow with any significant number of voters/candidates
2) Doesn’t always generate a winner; can observe a Condorcet cycle

Question 12

What is a Condorcet cycle? What does it lead to? Define this?

Answer 12

Situation when cyclic voting preferences lead to inability to determine a winner

Causes the Condorcet paradox = situation in which collective preferences are cyclic

Question 13

Why does the Condorcet paradox occur?

Answer 13

It occurs when there are conflicting majorities made up of different groups of individuals

eg. 1) ABC, 2) BCA, 3) CAB

Question 14

If a majority rule such as the CVA leads to no solution, what has happened? What does this mean for voters?

Answer 14

The rule has generated socially intransitive preferences!

Therefore, voters may not have an incentive to truthfully reveal their preferences!

Question 15

EV point regarding the Condorcet paradox?

Answer 15

Quite unlikely with small no. of Vs and Cs: eg) 3 voters, 3 candidates (with strong pref. orderings) -> 12/216 chance of cyclic preferences

But the larger the no. candidates and voters, less more likely it becomes!!!

Question 16

Explain a hypothetical political application of the Condorcet voting method?

Answer 16

3 MPs must divide ‘m’ budget between their 3 districts, with an objective of maximising the funds for their home district (see notes for rest of example, check I understand it!)

Question 17

Why may someone change their vote in CV method?(2)

Answer 17

1) To win

2) To avoid their worst outcome!!!

Question 18

A CV method is very slow to find a winner in a large set, what can be done about this?

Answer 18

It can be modified to increase its speed and therefore usefulness: eg. Debian voting system/Condorcet-fuse algorithm

Question 19

What is Arrows impossibility theorem?

Answer 19

NO method of aggregating individual preferences will produce coherent results if 4 minimal conditions are met: universal domain, unanimity, independence and non-dictatorship (can I explain all these?)

Question 20

What did May’s theorem show?

Answer 20

It proved that the only satisfaction of Arrows conditions is Simple Majority Voting (ie. Majoritarian) (note: must have odd no. voters and indecision is not allowed!)

Question 21

How can we obtain coherent results whilst abiding to AIT?

Answer 21

Placing restrictions on universality

Question 22

What are single-peaked preferences?

Answer 22

The restriction that every individual is assumed to have one most preferred alternative; this is a restriction on the domain!

Question 23

Draw diagram showing SPPs vs non SPPs

Answer 23

see notes

Question 24

Give a political example of SPPs?

Answer 24

eg. interest rates set by CB

Question 25

What do SPPs imply?

Answer 25

That each group member has a most preferred point, and their utility DECREASES as they move away from this point!

Question 26

Given any point ‘x’, SPPs mean we can always determine a set of points that are preferred to x, P(x). What does it mean if P(x) is empty?

Answer 26

Means that no point is preferred to x tf P(x)=P(x*), denoting the most preferred alternative!

Question 27

How does Drüner et al. (2016) define the winset?

Answer 27

The set of points that all or a majority of decision makers prefer over the status quo

Question 28

See

Answer 28

Winset and SPPs in notes top of second page! (important!!!)

Question 29

What does Black’s Median Voter Theorem state?

Answer 29

If all voter’s preferences are single-peaked on a single dimension, then the most preferred point of the median voter is a Condorcet winner!

Question 30

Draw a 3 person and a 2 person SPPs diagram showing the median point/region?

Answer 30

see notes!!!

Question 31

4 assumptions of the Median Voter Theorem/

Answer 31

P-S3

1) Participation of all in the vote
2) Single-peaked preferences
3) Single dimension
4) Sincere voting from all agents

Question 32

Prove MVT with 3 person diagram.

Answer 32

See notes

Question 33

Explain how MVT may be applied to redistribution in a democracy?

Answer 33

see notes!!!

Question 34

Why is the median voter decisive?

Answer 34

Because it is necessary to reach the majority!

Question 35

Tax rate example MVT?

Answer 35

Rank individuals based on income, find median income earner, ask all people to vote on a tax rate, and the median income earner will choose the tax rate (assuming identical voters who are rational)

Question 36

What is a proportional tax rate?

Answer 36

One that is the same rate regardless of income (eg. always 40%)

Question 37

Given 3 individuals with incomes x1, x2, x3, and proportional tax rate, t. What is the total tax revenue? If there is uniform redistributive transfer, how much does each receive back? What is person i’s net tax? Who gains and who loses? By how much? How will the median voter vote?

Answer 37

1) t(x1+x2+x3)
2) T=t(x1+x2+x3)/3
3) tx(i) - t(x1+x2+x3)/3
4) Gainers: anyone below the avg.income and vice versa; the larger the distance between, the larger the gain/loss
5) Median voter will vote for a positive proportional tax rate if his income is below the average and will oppose any taxation if his income is above the average!