1. Social Choice Lecture 19
John Hey

2. Voting This and the next two lectures concern voting systems.
Voting is used to choose between alternatives.
We note that we have exactly the same problems (though rather more obviously manifested here) as we had when we discussed Arrow.
Basically: if people have different preferences then it is difficult/impossible to aggregate them.
If we are trying to choose the most preferred, for example, President, if people have different preferences, what do we mean by ‘most preferred’?
I note, without shame, that most of this material I have taken from Wikipedia:
and linked pages.

3. Lectures on Voting Systems
Lecture 19, Thursday the 3th of December: Single-Winner Voting Systems.
Lecture 20, Friday the 4th of December: Historical and Factual description of voting systems.
Lecture 21, Thursday the 10th of December: Multiple-Winner Voting Systems.

4. Lecture 19: Single-Winner Voting Systems
Plurality (first-past-the-post)
Multiple-round systems
Two round
Exhaustive ballot
Preferential systems (Ranking methods)
Condorcet criterion
Bucklin voting
Coombs method
Borda count
Non-ranking methods
Approval voting
Range voting

5. Candidates (primary election first?). Weight of votes. Status quo.
Preamble
The ballot.
Candidates (primary election first?).
Weight of votes.
Status quo.
Constituencies.

6. Single or Sequential vote methods
Plurality (or “first-past-the-post” or “relative majority” or “winner-take-all”) – the one with the most votes wins, even if not an absolute majority.
Runoff methods have multiple rounds, eliminating weak candidates – ensuring that winner gets an absolute majority of votes cast.
Primary election is also a two round system.

7. Ranked voting methods (also known as preferential voting)
Voters asked to rank candidates (perhaps not all and perhaps the same rank for more than 1 candidate).
... and then one of ...
Instant-Runoff Voting (IRV) (candidates eliminated sequentially and rankings transferred, but without mutliple rounds of voting).
Borda count method (ranks are summed)
Condorcet method (pairwise comparisons)
Coombs method.
Supplementary voting.
Bucklin voting.

8. Condorcet Method
Rank the candidates in order (1st, 2nd, 3rd, etc.) of preference. Tie rankings, which express no preference between the tied candidates, are allowed.
For each ballot paper, compare the ranking of each candidate on the ballot paper to every other candidate, one pair at a time (pairwise), and tally a "win" for the higher-ranked candidate.
Sum these wins for all ballots cast, maintaining separate tallies for each pairwise combination.
The candidate who wins every one of their pairwise contests is the most preferred over all other candidates, and hence the winner of the election.
In the event no single candidate wins all pairwise contests, use a resolution method described later. (This is the case of ‘No Condorcet Winner’.)
A particular point of interest is that it is possible for a candidate to be the most preferred overall without being the first preference of any voter. In a sense, the Condorcet method yields the "best compromise" candidate, the one that the largest majority will find to be least disagreeable, even if not their favorite.

9. An example, 1
Imagine that Tennessee is having an election on the location of its capital. The population of Tennessee is concentrated around its four major cities, which are spread throughout the state. For this example, suppose that the entire electorate lives in these four cities, and that everyone wants to live as near the capital as possible. The candidates for the capital are:
Memphis, the state's largest city, with 42% of the voters, but located far from the other cities;
Nashville, with 26% of the voters, near the center of Tennessee;
Knoxville, with 17% of the voters;
Chattanooga, with 15% of the voters.

10. An example, 2 Preferences of voters: 42% of voters (close to Memphis)
26% of voters (close to Nashville)
15% of voters (close to Chattanooga)
17% of voters (close to Knoxville)
Memphis
Nashville
1. Chattanooga
1. Knoxville
2. Nashville
2, Chattanooga
2. Knoxville
2. Chattanooga
3. Chattanooga
3. Knoxville
3. Nashville
4. Knoxville
4. Memphis

11. An example 3 Now see who is winner in all pairwise comparisons: Pair
Memphis (42%) vs. Nashville (58%)
Nashville
Memphis (42%) vs. Chattanooga (58%)
Chattanooga
Memphis (42%) vs. Knoxville (58%)
Knoxville
Nashville (68%) vs. Chattanooga (32%)
Nashville (68%) vs. Knoxville (32%)
Chattanooga (83%) vs. Knoxville (17%)

12. An example 4
We note that Nashville wins in each pairwise competition and thus is the Condorcet Winner.
On the contrary, first-past-the-post (or majority) voting would make Memphis the winner… (look at first places: 42% for Memphis, 26% Nashville, 15% Chattanooga and 17% Knoxville).
… and Instant-Runoff Voting (IRV) would make Knoxville the winner. (Chattanooga (15%) is eliminated in the first round; votes transfer to Knoxville. Nashville (26%) eliminated in the second around; votes transfer to Knoxville. Knoxville wins with 58%.) See end.
As said before: “. In a sense, the Condorcet method yields the "best compromise" candidate, the one that the largest majority will find to be least disagreeable, even if not their favorite.”

13. No Condorcet Winner
For example, suppose we have three candidates, A, B, and C, and that there are three voters with preferences as follows:
Voter 1: A preferred to B preferred to C,
Voter 2: B preferred to C preferred to A,
Voter 3: C preferred to A preferred to B.
We note that.
A majority of voters (1 and 3) prefer A to B
A majority of voters (1 and 2) prefer B to C
A majority of voters (2 and 3) prefer C to A.
Thus there is no Condorcet Winner.
There are various methods for getting a winner in such a case.

14. Coombs method
Each voter rank-orders all of the candidates on their ballot. If at any time one candidate is ranked first (among non-eliminated candidates) by an absolute majority of the voters, then this is the winner. As long as this is not the case, the candidate which is ranked last (again among non-eliminated candidates) by the most (or a plurality of) voters is eliminated. (Conversely, in Instant Runoff Voting the candidate ranked first (among non-eliminated candidates) by the least amount of voters is eliminated.)

15. Bucklin voting
Voters are allowed rank preference ballots (first, second, third, etc.).
First choice votes are first counted. If one candidate has a majority, that candidate wins. Otherwise the second choices are added to the first choices. Again, if a candidate with a majority vote is found, the winner is the candidate with the most votes in that round. Lower rankings are added as needed.
A majority is determined based on the number of valid ballots. Since, after the first round, there may be more votes cast than voters, it is possible for more than one candidate to have majority support. This makes Bucklin a variation of approval voting. (No example in public elections of such a multiple majority is known.)
Since preferences are counted one rank at a time, organized voters who agree on one candidate before the election would be at an advantage to voters who support several similar candidates. If there are two candidates who appeal to the Yellow party, and five who split the vote of the Purple party, it may take four preference rounds before each of the Purple party voters support each other enough to elect a Purple party candidate, whereas the Yellow party would have caste all of its votes for the two Yellow party candidates after just the second round. This makes Bucklin voting in between plurality voting and approval voting.

16. Rated and range voting methods
Rating voting method
Range voting: voters give
numerical ratings.
Approval voting

17. Range Voting
Range voting uses a ratings ballot; that is, each voter rates each candidate with a number within a specified range, such as 0 to 99 or 1 to 5. Although in cumulative voting voters are not permitted to provide scores for more than some number of candidates, in range voting all candidates can be and should be rated.
The scores for each candidate are summed, and the candidate with the highest sum is the winner. If voters are explicitly allowed to abstain from rating certain candidates, as opposed to implicitly giving the lowest number of points to unrated candidates, then a candidate's score would be the average rating from voters who did rate this candidate.

18. Approval Voting
Approval voting is a single-winner voting system used for elections. Each voter may vote for (approve of) as many of the candidates as they wish. The winner is the candidate receiving the most votes. Each voter may vote for any combination of candidates and may give each candidate at most one vote.
Approval voting is a form of range voting with the range restricted to two values, 0 and 1. Approval voting can be compared to plurality voting without the rule that discards ballots which vote for more or less than one candidate.

19. Criteria for judging voting systems
Majority criterion: If there exists a majority that ranks (or rates) a single candidate higher than all other candidates, does that candidate always win?
Monotonicity criterion: Is it impossible to cause a winning candidate to lose by ranking him higher, or to cause a losing candidate to win by ranking him lower?
Consistency criterion: If the electorate is divided in two and a particular candidate wins in both parts, does that candidate always win overall?
Participation criterion: Is voting honestly always better than not voting at all? (This is grouped with the distinct but similar Consistency Criterion in the table below.)
Condorcet criterion: If a candidate beats every other candidate in pairwise comparison, does that candidate always win? (This implies the majority criterion, above)
Condorcet loser criterion: If a candidate loses to every other candidate in pairwise comparison, does that candidate always lose?
Independence of irrelevant alternatives: Is the outcome the same after adding or removing non-winning candidates?
Independence of clone candidates: Is the outcome the same if candidates identical to existing candidates are added?
Reversal symmetry: If individual preferences of each voter are inverted, does the original winner never win?
Polynomial time: Can the winner be calculated in a runtime that is polynomial in the number of candidates and in the number of voters?

20. Consistency & Participation Condorcet Condorcet loser IIA
Majority
Monotone
Consistency & Participation
Condorcet
Condorcet loser
IIA
Clone independence
Reversal symmetry
Polynomial time
Approval
Ambiguous
Yes No
Borda count
No (teaming)
IRV
Kemeny-Young
Minimax
No (vote-splitting)
Plurality
Range voting
Ranked Pairs
Runoff voting
Schulze

21. Conclusions There does not appear to be a perfect voting system.
Is that surprising?
What is the factual position? See tomorrow’s lecture.
Might going to a Multiple-Winner System help?
See Lecture 21.

22. C

23. Instant Runoff Voting Preferences of voters:
42% of voters (close to Memphis)
26% of voters (close to Nashville)
15% of voters (close to Chattanooga)
17% of voters (close to Knoxville)
Memphis
Nashville
1. Chattanooga
1. Knoxville
2. Nashville
2, Chattanooga
2. Knoxville
2. Chattanooga
3. Chattanooga
3. Knoxville
3. Nashville
4. Knoxville
4. Memphis

24. Eliminating Chattanooga (only 15% of first votes)
Preferences of voters:
42% of voters (close to Memphis)
26% of voters (close to Nashville)
15% of voters (close to Chattanooga)
17% of voters (close to Knoxville)
Memphis
Nashville
1. Knoxville
2. Nashville
2. Knoxville
3. Knoxville
3. Memphis

25. Now we eliminate Nashville (only 26% of first votes) – Knoxville wins
Preferences of voters:
42% of voters (close to Memphis)
26% of voters (close to Nashville)
15% of voters (close to Chattanooga)
17% of voters (close to Knoxville)
Memphis
1. Knoxville
2. Knoxville
2. Memphis