Unit 2 Flashcards
What does public choice theory study?
The aggregation of individual preferences
What is the goal of voting theory?
To make the greatest number of people happy whilst allowing all to vote honestly
What does a voting method do?
It contains rules for valid voting and how votes are aggregated to yield a final result
What is a Condorcet voting method?
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.
Explain the steps in the Condorcet voting algorithm?
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)
How do we deal with an incomplete preference list in the CVA?
Ranked candidates assumed to be better than unranked ones
What type of method is a CVA?
Majoritarian
Note about CVA:
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.
How can we generalise these pairwise competitions? Why is this rarely done in reality?
Model elections with a digraph
BUT this is too slow for any real life applications
check I know what a digraph is!
See
Example 2.1 on doing a CVA
2 problems with using Condorcet voting?
1) Is too slow with any significant number of voters/candidates
2) Doesn’t always generate a winner; can observe a Condorcet cycle
What is a Condorcet cycle? What does it lead to? Define this?
Situation when cyclic voting preferences lead to inability to determine a winner
Causes the Condorcet paradox = situation in which collective preferences are cyclic
Why does the Condorcet paradox occur?
It occurs when there are conflicting majorities made up of different groups of individuals
eg. 1) ABC, 2) BCA, 3) CAB
If a majority rule such as the CVA leads to no solution, what has happened? What does this mean for voters?
The rule has generated socially intransitive preferences!
Therefore, voters may not have an incentive to truthfully reveal their preferences!
EV point regarding the Condorcet paradox?
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!!!
Explain a hypothetical political application of the Condorcet voting method?
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!)
Why may someone change their vote in CV method?(2)
1) To win
2) To avoid their worst outcome!!!
A CV method is very slow to find a winner in a large set, what can be done about this?
It can be modified to increase its speed and therefore usefulness: eg. Debian voting system/Condorcet-fuse algorithm
What is Arrows impossibility theorem?
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?)
What did May’s theorem show?
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!)
How can we obtain coherent results whilst abiding to AIT?
Placing restrictions on universality
What are single-peaked preferences?
The restriction that every individual is assumed to have one most preferred alternative; this is a restriction on the domain!
Draw diagram showing SPPs vs non SPPs
see notes
Give a political example of SPPs?
eg. interest rates set by CB
What do SPPs imply?
That each group member has a most preferred point, and their utility DECREASES as they move away from this point!
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?
Means that no point is preferred to x tf P(x)=P(x*), denoting the most preferred alternative!
How does Drüner et al. (2016) define the winset?
The set of points that all or a majority of decision makers prefer over the status quo
See
Winset and SPPs in notes top of second page! (important!!!)
What does Black’s Median Voter Theorem state?
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!
Draw a 3 person and a 2 person SPPs diagram showing the median point/region?
see notes!!!
4 assumptions of the Median Voter Theorem/
P-S3
1) Participation of all in the vote
2) Single-peaked preferences
3) Single dimension
4) Sincere voting from all agents
Prove MVT with 3 person diagram.
See notes
Explain how MVT may be applied to redistribution in a democracy?
see notes!!!
Why is the median voter decisive?
Because it is necessary to reach the majority!
Tax rate example MVT?
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)
What is a proportional tax rate?
One that is the same rate regardless of income (eg. always 40%)
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?
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!
