Announcements

• John Allen Paulos will be coming to speak at Penn on Tuesday, April 22, at 7:00 pm, in Meyerson Hall.

Current assignments:

• Read Sections 5.4--5.5 of Mathematics and Politics.

Properties of voting systems

• We still don't know how to determine which voting systems could be considered better than others.
• We will discuss some of the properties that characterize them.
• • Pareto condition: If everyone prefers candidate A over candidate B, then candidate B can't win.
  • Monotonicity: If A wins the election and your ballot changes to move A higher on your preference list, then A is still the winner.
  • Independence of irrelevant alternatives (IIA): If B would lose an election between A and B, and a new candidate C is added to the election, then B is still the loser. (i.e. you can always vote your conscience).
  • Condorcet condition: If A is a Condorcet winner (i.e. beats every other candidate in a pairwise contest), then A is the winner.
• Which voting systems satisfy which properties?
• Notice that these are properties about every possible election that could occur in the system.
• So to prove a system has one of these properties, you have to prove it happens for every possible election (no matter how many candidates or voters or what their preferences are).
• But to prove a system doesn't have one of these properties, you only have to show that the property fails for one particular election (so you can choose the number of candidates and the number of voters and their preferences however you want).
• The Pareto condition:
  If everyone prefers candidate A over candidate B, then candidate B can't win.
  • Plurality?
    Yes.
    Proof: If everyone prefers candidate A over candidate B, then candidate B is not at the top of anyone's preference list. Since plurality involves choosing the candidate who is at the top of the most preference lists, candidate B can't win.
  • Instant runoff (Hare)?
    Yes.
    Proof: As before, if everyone prefers candidate A over candidate B, then candidate B is not at the top of anyone's preference list. So candidate B will be the first one eliminated from consideration since s/he got the fewest number of first-place votes. Thus candidate B can't win.
  • Borda count?
    Yes.
    Proof: Since A is higher on every voter's preference list than B, A gets at least one more point than B from every voter. So B cannot have more points than A and thus cannot win.
  • Sequential pairwise voting?
    No!
    Proof: Find an example where every voter prefers B over D, but D wins. (from textbook)
    Say there are four alternatives: A, B, C, D; and three voters.
    

    Voter 1	Voter 2	Voter 3
    A	C	B
    B	A	D
    D	B	C
    C	D	A

    
    Every voter prefers B over D.
    With agenda ABCD, the contests are:
    1. A vs. B (A wins)
    2. A vs. C (C wins)
    3. C vs. D (D wins)
    4. So the winner is D.
  • Dictatorship?
    Yes.
    Proof: If every voter prefers A over B, then the dictator also prefers A over B, so B is not the winner.
• Monotonicity:
  If A is one of the winners, and a voter changes his/her preference list to move A higher, then A is still one of the winners.
  • Plurality?
    Yes.
    Proof: If A is one of the winners, then A must have gotten the highest number of first-place votes. If a voter moves A higher, then A might get another first-place vote from that voter (while another candidate loses a first-place vote). In this case, A still wins. If the voter doesn't move A all the way up to first place, then that voter's first choice stays the same and the outcome of the election is unaffected. So A still wins.
  • Instant runoff (Hare)?
    No!
    Proof: Find an example where A wins, then someone changes his/her preference list to move A up, and A loses as a result. (from textbook)
    Suppose there are three candidates: A, B, and C.

    Voters 1-7	Voters 8-12	Voters 13-16	Voter 17
    A	C	B	B
    B	A	C	A
    C	B	A	C

    A candidate would need 9 first-place votes to win.
    A has 7, B has 5, and C has 5.
    We delete B and C since they both have the lowest number of first-place votes.
    A is the winner.
    Now keep all preferences the same and change voter 17's list. Put A at the top.

    Voters 1-7	Voters 8-12	Voters 13-16	Voter 17
    A	C	B	A
    B	A	C	B
    C	B	A	C

    
    Now A has 8 votes, B has 4 votes, and C has 5 votes.
    B gets eliminated. Now the lists are:

    Voters 1-7	Voters 8-12	Voters 13-16	Voter 17
    A	C	C	A
    C	A	A	C

    Now A has 8 first-place votes and C has 9 first-place votes.
    C is now the winner!
  • Borda count?
    Yes.
    Proof: Suppose A is a winner under the Borda count. Then A has the highest number of points. If one voter moves A higher, then A will get more points from that voter. Other candidates will either move lower or stay the same, and thus no other candidate will get more points from that voter. So A still has more points than any other candidate, and A still wins.
  • Sequential pairwise voting?
    Yes.
    Proof: Suppose A was one of the winners in sequential pairwise voting with a particular agenda. This means that A beat or tied any candidate that A had to compete with (which wasn't necessarily all of them). If A moves up in the preference list of one voter, then A still beats all the same candidates that A beat before. And the other pairwise elections between candidates are unaffected by the change.
    So the election proceeds as before: until A comes up in the agenda, all the other elections have exactly the same outcome as before. Whoever wins just before A comes up in the agenda now competes against A (and loses). Then A competes with everyone afterward. A still wins all of these contests. So A is still the winner.
  • Dictatorship?
    Yes.
    Proof: Suppose A was the winner and somebody moves A up higher on the preference list. Then this person can't be the dictator (because the dictator already had A as high as possible). So this person's list has no effect on the election, and A still wins.
• Independence of Irrelevant Alternatives (IIA):
  Here is an easy way to think of the (IIA) condition.
  • Suppose there are two "major-party" candidates, a Democrat and a Republican.
  • Let's say 51% of the voters are "Democrat-leaning." They prefer the Democrat over the Republican (although if they had a third-party choice, they might prefer that to either one).
  • Let's say 49% of the voters are "Republican-leaning."
  • Clearly if these are the only two candidates, the Republican loses.
  • The (IIA) condition says that if a minor-party candidate enters the race, then no matter how many people decide to vote for that candidate (whether in first place, second place, or third place), the Republican still loses.
  • In other words, there is no "spoiler effect."
  • The Democrat-leaning voters can vote their true preferences without worrying about electing the Republican.
  • Plurality?
    No!
    Proof: Just find an explicit example, e.g. the 2000 Presidential election.
    • Bush, Nader, Gore: 49%
    • Gore, Nader, Bush: 48%
    • Nader, Gore, Bush: 3%
    In a two-person contest between Bush and Gore, Bush loses, 49-51.
    If Nader runs, then Bush wins.
  • Instant runoff (Hare)?
    No!
    (This is surprising because the main advantage of instant runoff voting is touted as enabling people to vote their true preferences.)
    (In the Bush-Gore-Nader election just cited, Gore wins in instant-runoff, regardless of whether Nader is included or not.)
    (But if Nader actually got a higher percentage of first-place votes, Bush could still end up winning even in instant-runoff.)
    Proof: Suppose the preferences in the 2000 Presidential election were as follows:
    • Bush, Nader, Gore: 49%
    • Gore, Nader, Bush: 23%
    • Gore, Bush, Nader: 2%
    • Nader, Gore, Bush: 26%
    In a two-person contest between Gore and Bush, Gore gets 51% and Bush gets 49% of the vote.
    If Nader enters the race, all the Bush voters prefer Nader over Gore.
    All the Nader voters prefer Gore over Bush.
    Most of the Gore voters prefer Nader over Bush, but a few prefer Bush over Nader.
    Gore gets eliminated since he only got 25% of the first-choice votes.
    Now it's Bush vs. Nader.
    Bush gets 51% of the vote, and Nader gets 49% of the vote.
    So Nader entering the race has caused Bush to win.
  • Borda count?
    No!
    Proof: We use the example from the textbook.

    Voters 1-3	Voters 4 and 5
    A	C
    B	B
    C	A

    A wins and B loses.
    Suppose voters 4 and 5 move B up to first place and keep A in last place. Then:

    Voters 1-3	Voters 4 and 5
    A	B
    B	C
    C	A

    
    Then B is the winner.
    However, in either case, A beats B in a pairwise contest by 60% to 40%.
  • Sequential pairwise voting?
    No!
    Proof: Find an example where a majority prefers A over B, but with the introduction of a candidate C, B wins. (from textbook)
    Consider three candidates A, B, C, and three voters 1, 2, 3. Suppose the agenda is CBA.

    Voter 1	Voter 2	Voter 3
    C	A	B
    B	C	A
    A	B	C

    C defeats B, and then A defeats C.
    Voter 1 now moves B up to first place, keeping A in last place.

    Voter 1	Voter 2	Voter 3
    B	A	B
    C	C	A
    A	B	C

    Now C loses to B, and then B defeats A. So B becomes the winner.
    
  • Dictatorship?
    Yes.
    Proof: If B loses, then B was not the dictator's first choice. If a new candidate C gets introduced, and the dictator still prefers some other candidate A over B, then B still loses.