Transcript MGF 1107 Mathematics of Social Choice

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. Safety
2. Reliability
3. 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-onone 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
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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
Poppa Johns
Dominoes
Third
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
Poppa Johns
Dominoes
Third
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 Nader
receives 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 not
by 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.
More Problems …
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…
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-tohead 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”?