MGF 1107

Mathematics of Social Choice

Part 1b – Fairness Criteria and Methods of Voting

How is this math?

• Mathematics is essentially the application of deductive reasoning to the study relations among patterns, structures, shapes, forms and change.

• We begin with general principles inherent in methods and forms of voting and deduce specific conclusions representing new knowledge about the process of voting.

• Our textbook refers to this as an axiomatic approach. This means we will begin with certain assumptions. These assumptions are generally called axioms or postulates. Some of these assumptions will also be referred to a fairness criteria.

The Axiomatic Approach

A simple example: public transportation

• What criteria might we establish as our “axioms” ?

1. Safety2. Reliability3. Speed

• We could begin with these criteria as our requirements for any method public transportation and then find or create a method of public transportation that satisfies one or more of these criteria.

• What systems of public transportation satisfy these criteria ?

• Does any method satisfy all criteria?

Axiomatic Approach

• Establish axioms first, then make deductive conclusions based on those axioms.

• Some of the axioms we study are called fairness criteria or simply criteria. Note that the plural form is criteria (as in we will study many criteria of fairness in voting) and also note that the singular form of the word is criterion (as in we will learn one criterion at a time).

• What methods of voting already in use will satisfy a given set of criteria?

• Which criteria do various voting methods fail to satisfy?

• Do any systems satisfy all criteria?

Fairness Criteria for Voting

1. If a candidate receives a majority of first place votes, that candidate should be the winner. (Majority Criterion)

2. If a candidate beats all other candidates in one-on-one comparisons, that candidate should be the winner. (Condorcet Winner Criterion)

3. If a re-election is held with the same ballots and non-winning candidates are removed, the previous winner should still win. (Independence of Irrelevant Alternatives Criterion)

Fairness Criteria for Voting - Continued

4. If there is at least one candidate (say candidate A) that every voter prefers to another (say candidate B) then it should be impossible for B to win. (Pareto Criterion)

5. It should be impossible for a winning candidate to lose in a re-election if the only changes in the votes where changes that were favorable to that candidate. (Monotonicity Criterion)

6. All voters should be treated equally. No voter has special influence, only the ballot counts. If voters exchange ballots, the result of the election should still be the same.

7. All candidates should be treated equally. No candidate has more privilege than any other. In the case of two candidates, this means if every voter reversed their vote, the election result would be reversed as well.

Majority Rules• Majority Rules satisfies all 7 of these criteria.

• It is the best method of voting but only guarantees a winner if there are two candidates and an odd number of voters.

• If there are more than two candidates, it is possible that none of the candidates receives a majority and thus no winner could be determined by majority rules.

• When we have more than 2 candidates, we must use other methods of voting, even though a majority could still occur with more than 2 candidates.

• Unfortunately, there is no method of voting that will satisfy all 7 of the fairness criteria stated. An important theorem in voting states that is impossible to make a method of voting that will satisfy all the voting criteria.

Fairness Criteria – Shorter versions for the first 5

1. MC – If a candidate has a majority of first place votes, that candidate should win.

2. CWC – If a candidate beats all others head-to-head, that candidate should win.

3. IIA – If a re-election is held and non-winning candidates drop out,

then the previous winner should still win

4. Pareto – It should be impossible for a candidate to win if there is always at least one other candidate which every voter prefers to that candidate.

5. Monotonicity – In a re-election, it should be impossible for a winning candidate to become a loser when all vote changes are favorable to that candidate.

Voting Methods

• Majority Rules

• Plurality Method

• Borda Count Method

• Condorcet Method

• The Hare System

• Sequential Pairwise Voting

• Approval Voting

Elections with Two Candidates

• May’s theorem (1952) – If the number of voters is odd, the election is for only two candidates, and we require a voting system that never results in a tie, then majority rules is the only voting system that satisfies the following three criteria:

– All voters are treated equally.– Both candidates are treated equally.– If a re-election were held and a single voter changed his or her vote from a

vote for the previous losing candidate to a vote for the previous winning candidate, the outcome would still be the same (monotonicity).

• May’s theorem states that majority rules is the only method of voting that satisfies all of the above three fairness criteria.

• In fact, we could prove that majority rules is the only method of voting that satisfies all seven of the fairness criteria we have identified.

• Majority Rules satisfies all seven criteria! What else is there to say?

Why is this chapter not done?

• Why is there more to say? Who needs anything else?

• If majority rules actually can be shown to satisfy all seven of our criteria of fairness, why bother with anything else – why can’t we just use majority rules all the time and be done with it?!

The problem is…

• We do not consider “majority rules” to be a valid method of voting when there are more than 2 candidates. This is because, if there are more than two candidates, it can happen that none of the candidates receives a majority of the votes.

• For example, the vote can be split 25%, 30% and 45% and then no candidate has a majority of the vote. Then there would be no winner under majority rules. We will assume a method of voting must always produce at least one winner. Ties are ok, but there must be at least one winner.

• In an election where the vote is split 25%, 30% and 45%, the candidate receiving 45% would be the winner only if we are using the plurality method, because that candidate has the most votes. But that candidate does not have a majority of the vote and is not a winner by majority rules.

Plurality Versus Majority

• To win a plurality of the vote means to have the most votes (more than any other candidate).

• To win a majority means to have more than 50% of the vote.

• Clearly a majority is automatically a plurality but a plurality is not always a majority.

This is REALLY IMPORTANT!!

• Why is majority rules not a legitimate method of voting with 3 or more candidates?

• CORRECT – Because in some outcomes there may be no winner at all.• CORRECT – Because it is possible that no candidate receives a

majority of votes.

• INCORRECT – Because majority rules is only used for two candidate elections.

• INCORRECT – Because it is difficult or impossible for one candidate to receive a majority of votes when there are three or more candidates.

• INCORRECT – Because it is possible that more than one candidate receives a majority of votes.

• INCORRECT – Because the winner would not have a majority of the votes.

• INCORRECT – Because it is possible that some candidate does not get a majority of votes.

This is REALLY IMPORTANT!!

• VERY IMPORTANT CONCEPT: A method of voting is not “legitimate” if it is possible that there would be no winner at all.

• Notice that with three or more candidates it is still possible that one candidate receives a majority of the votes, however, with three candidates, the point is – it might happen that none receive a majority of votes.

A Majority Winner Can Always Occur

• With 3 or more candidates we can still have a majority winner. The majority winner is the candidate that has a majority of votes (that is, a majority of first place votes).

• An important concept is that we can not say at the beginning of an election with three or more candidates, that the winner will be decided by majority rules – of course this is because there might not be a majority winner.

• But in any election with 2 or 3 or more candidates, it is always possible that one candidate receives a majority of the first place votes.

Social Choice Procedures

• We define a “social choice procedure” to be a method of voting. Any method of voting must produce at least one winner.

• A method of voting can produce two or more winners (ties) and we will still say that it is a legitimate method of voting. Producing two winners means it still produced at least one winner and we can deal with ties in some previously agreed upon manner.

• We assume voters can rank given candidates in the form of a preference list and that each individual preference list will have no ties.

• An example of a preference schedule is shown below – this example indicates individual preferences for a total of 10 individual voters regarding three candidates.

Number of Voters (10 total)

Rank 9 1

First Dominoes Pizza Hut

Second Pizza Hut Poppa Johns

Third Poppa Johns Dominoes

Voting with Two or More Candidates

• There are 5 voting methods (other than majority rules) that we will study in depth in this chapter. These social choice procedures can be used for elections with 3 or more candidates.

– Plurality– Borda Count– Sequential Pairwise Voting– Hare System– Approval Voting

The Plurality Voting Method

• Consider only the first preference (a voter can only vote for his or her first preference.)

• The election winner is determined as the candidate receiving the most votes. There can be ties for winner.

• The election winner does not necessarily have a majority of votes.

• With three or more candidates, there are potential problems.

Plurality Method

• In the example below, if we use plurality voting, then the voters second and third choice are not considered. Each voter can only vote for their first choice.

• In this example, the winner would be Dominoes because that candidate has more votes than the other candidates.

Number of Voters (10 total)

Rank 9 1

First Dominoes Pizza Hut

Second Pizza Hut Poppa Johns

Third Poppa Johns Dominoes

Plurality Method

As an example of a potential problem with the plurality method ...

Consider three candidates: George Bush, John Kerry and Ralph Nader.

Suppose Bush receives 40% of the vote, Kerry receives 30% and Naderreceives 30% of the vote. Who wins? Bush has a plurality of the votes(but not a majority) and therefore he wins by the plurality method and notby majority rules. ( Remember to have a plurality means only to have the most votes.)

Notice that a majority of voters do not even want Bush elected. 60% of the electorate, that is a majority, chose a candidate other than Bush and yet Bush wins because he does have more votes than anyone else.

Because of an apparent discrepancy between the outcome of the election and the preference of a majority of voters, we have encountered an apparent problem with the plurality method of voting.

Because of this problem, we may be more motivated to consider other alternative methods of voting…

More Problems …

Borda Count Voting Method

• Assign points in a non-increasing manner to the ranked candidates on each individual voter’s preference list.

• If there are n candidates, a first place vote is worth n - 1 points, a second place vote is worth n - 2 points, and so on, down to 0 points for last place.

• Add the total points received for each candidate from all voters. The winner is the candidate with the most points. There can be ties for winner. The result of the election is a group ranking of the candidates (a sort of “social choice” of preference with regards to the candidates.)

• This is the method of voting used, for example, to determine the ranking of football teams (as in the AP Poll) and also award the Heisman Trophy.

Sequential Pairwise Voting

• With sequential pairwise voting we must have a given agenda before determining a winner. The agenda is the order in which the candidates are compared one-on-one.

• The name derives from the fact that the candidates are compared sequentially – one after another – in the order determined by the agenda. The candidates are compared “pairwise” which means one-on-one or “head-to-head”.

• The winner of each head-to-head comparison is determined by the relative rankings of each of the candidates within the preference lists of the voters. For example: Comparing X and Y, X wins if more voters rank X above Y.

• The winner of each pairwise comparison is then considered head-to-head with the next candidate on the agenda. There can be ties at any stage in the sequence of head-to-head comparisons. These ties may be carried over to the next comparison or eliminated. A single winner may emerge or there may be a tie of two or more candidates declared the winners by this method.

The Hare System

• A winner is determined by repeatedly deleting candidates, in stages, that are “least preferred” in the sense of being at the top of the fewest number of preference lists.

• A single winner may emerge after all other candidates have been deleted or there may be a tie among two or more candidates.

Approval Voting

• Each voter gives one vote to as many of the candidates as he or she finds acceptable.

• The only limit on the number of votes a voter can cast is the number of candidates in contention. That is, a voter can approve of none of the candidates, one candidate, more than one, or all of the candidates.

• A voter indicates disapproval by not casting an approval vote for a particular candidate.

• The winner of the election is the candidate with the largest number of votes. There can be ties.

• This method is used to elect the secretary general of the United Nations.

Real Life Examples • Plurality – Municipal, State, Federal elections in

the U.S.• Borda Count – Voting for the best college football

team, such as AP College Football Poll; voting for the Heisman Trophy

• Sequential Pairwise – Legislative process • Hare System – Choosing the site of the Olympics,

the Academy Awards, elections in Australia and Ireland

• Approval Voting – U.N. Secretary General, academic and professional societies

Comment on Condorcet Method

• The Condorcet Method is introduced again later in the notes. It is not a valid method of voting because, like majority rules, it may not produce a winner when there are more than 2 candidates. (Actually, if there are 2 candidates, the Condorcet Method is exactly the same as majority rules)

• The Condorcet method is still important for theoretical reasons having to do with the Condorcet criterion as we will see later.

• Essentially, the Condorcet method states that each candidate is compared one-on-one with every other candidate. The candidate that is preferred to every other candidate is the winner. As will be shown in many examples, there may not be one candidate that is preferred to all the others and so there may not be a winner by this method.

Which voting method is the best?

We could return to the axiomatic approach…

Can we find a voting method that will satisfy all of the fairness criteria?

If we assume general principles of fairness, what would this imply is the “best method”?