Announcements

• John Allen Paulos will be coming to speak at Penn on Tuesday, April 22, at 7:00 pm, in Meyerson Hall.
• If you cannot take the quiz on Thursday, April 24, you must bring a note in order to qualify for the makeup quiz.

Current assignments:

• Properties of voting systems: problem set
• Read Sections 5.6--5.8 of Mathematics and Politics.

Strategic voting

• We saw on Tuesday that any voting system which elects the majority-winner in any two-person election cannot satisfy the axiom of independence of irrelevant alternatives (IIA).
• Example:
  • Suppose a majority of voters are liberal.
  • Then a conservative would lose any two-candidate election against a liberal.
  • But there is no voting system which would guarantee that if there were two liberals against one conservative, one of the liberals would win.
  • The consequence is that the liberals cannot necessarily vote honestly.
  • They need to somehow coordinate and get behind a single candidate.
• Gibbard-Satterthwaite Theorem: In any voting system except a dictatorship, there will be an election for which a single voter (who knows what every other voter is going to do) would benefit by voting dishonestly.
• (This is closely related to the Arrow Theorem.)
• We will explore some of the ways in which strategic voting happens.
• (Some material borrowed from this very good article.)
• Example: Plurality voting
  • By far the most well-known...
  • Suppose 49% prefer Bush (then Nader and Gore, in some order which doesn't matter).
  • And 48% are known to prefer Gore (then Nader and Bush).
  • And 3% prefer Nader, then Gore, then Bush.
  • Suppose the 3% Nader voters want to change the outcome of the election (and all can agree on a strategy).
  • They cannot hope to change anyone else's votes.
  • So they cannot make Nader win.
  • If they vote Nader, Gore, Bush, then Bush will win (their last "honest" preference).
  • If they vote Gore, Nader, Bush, then Gore will win (their second "honest" preference).
  • So if they actually prefer Gore over Bush, their only rational choice is to vote dishonestly.
  • Question: In general, when is strategic voting a good idea for a small group of voters in the plurality system?
  • Answer: When the top two candidates have close vote totals.
• Example: Instant runoff voting
  • In instant runoff, Nader's votes get thrown away first, then they all go to Gore anyway, so the strategic trick from before won't work, and the Nader voters should vote honestly.
  • Let's consider something like the 1992 Presidential election (Clinton-Bush-Perot).
  • Suppose the preferences look like this:
    • 40%: Clinton, Bush, Perot
    • 29%: Bush, Clinton, Perot
    • 31%: Perot, Bush, Clinton
    (Perot voters are more conservative and prefer Bush, though the Democrats and Republicans both think Perot is too wacky.)
  • How does instant runoff work here?
  • Bush gets eliminated. Now the preferences are:
    • 69%: Clinton, Perot
    • 31%: Perot, Clinton
    And Clinton easily wins.
  • Who would vote strategically?
  • Not the Clinton voters; Clinton already wins.
  • Not the Bush voters; Clinton is their second choice.
  • Perot voters could, though. Perot won't win, but they would want to elect Bush.
  • They could do this by switching their votes to Bush, Perot, Clinton.
  • Not all of them have to, either, just 2 out of 31:
    • 40%: Clinton, Bush, Perot
    • 29%: Bush, Clinton, Perot
    • 29%: Perot, Bush, Clinton
    • 2%: Bush, Perot, Clinton
  • Then Perot gets eliminated first, and Bush gets 60% of the vote against Clinton.
  • Question: In general, when might strategic voting be a good idea for a small group of voters in the instant runoff system?
  • Answer: When the bottom two candidates have close vote totals and nonsimilar preferences (here Perot voters like Bush but Bush voters don't like Perot).
• Example: Instant runoff voting (again)
  • Here's an even weirder example.
  • Suppose a few of the Perot voters voted dishonestly, as before.
    • 40%: Clinton, Bush, Perot
    • 29%: Bush, Clinton, Perot
    • 29%: Perot, Bush, Clinton
    • 2%: Bush, Perot, Clinton
    This makes Clinton lose and Bush win.
  • Clinton voters are not happy.
  • Suppose 3% of the Clinton voters change their preferences and put Perot first.
  • Now the preferences are:
    • 3%: Perot, Clinton, Bush
    • 37%: Clinton, Bush, Perot
    • 29%: Bush, Clinton, Perot
    • 29%: Perot, Bush, Clinton
    • 2%: Bush, Perot, Clinton
  • Now Clinton has 37% first-place votes, Bush has 31% first-place votes, and Perot has 32% first-place votes.
  • Bush gets eliminated first.
  • What's left is:
    • 34%: Perot, Clinton
    • 66%: Clinton, Perot
  • So now Clinton wins again.
  • By voting against Clinton and for Perot (their last-place candidate), the Clinton voters have gotten Clinton to win.
  • (This is non-monotonicity!)
  • Question: In general, when might strategic voting be a good idea for a small group of voters in the instant runoff system?
  • Answer: When the small group supports the plurality winner, and there is a third-party candidate with a strong following among a significant minority but a very weak following among the rest of the public.
• Example: Borda count
  • Consider the following election.
    • 45%: Bush, Nader, Gore
    • 3%: Gore, Bush, Nader
    • 46%: Gore, Nader, Bush
    • 6%: Nader, Gore, Bush
    (We used this example to show that the Borda count doesn't satisfy the Condorcet property.)
  • The Borda counts are:
    • Bush: 45 × 2 + 3 × 1 + 46 × 0 + 6 × 0 = 93
    • Gore: 45 × 0 + 3 × 2 + 46 × 2 + 6 × 1 = 106
    • Nader: 45 × 1 + 3 × 0 + 46 × 1 + 6 × 2 = 103
  • Gore is the winner in the Borda count.
  • The 45% of people who voted Bush, Nader, Gore are most unhappy.
  • If a few of them change their preference to Nader, Bush, Gore, they can make Nader win:
    • 4%: Nader, Bush, Gore
    • 41%: Bush, Nader, Gore
    • 3%: Gore, Bush, Nader
    • 46%: Gore, Nader, Bush
    • 6%: Nader, Gore, Bush
  • Now the Borda counts are:
    • Bush: 4 × 1 + 41 × 2 + 3 × 1 + 46 × 0 + 6 × 0 = 89
    • Gore: 4 × 0 + 41 × 0 + 3 × 2 + 46 × 2 + 6 × 1 = 106
    • Nader: 4 × 2 + 41 × 1 + 3 × 0 + 46 × 1 + 6 × 2 = 107
  • So Nader wins the Borda count.
  • Question: In general, when might strategic voting be a good idea for a small group of voters in the Borda count system?
  • Answer: (Subtle!) When there is a Condorcet winner and the Borda count does not pick it.
• Example: Sequential pairwise voting
  • Consider the model we've been using of the 2000 election:
    • 49%: Bush, Nader, Gore
    • 48%: Gore, Nader, Bush
    • 3%: Nader, Gore, Bush
  • In this election, Nader is the Condorcet winner. So no matter what the agenda is, Nader will win sequential pairwise voting.
  • Bush voters have no chance to manipulate the election, since Bush loses to both Gore and Nader.
  • Gore voters, however, could manipulate the election.
  • Since Gore beats Bush, Gore voters could try to eliminate Nader first.
  • Suppose the agenda is Nader, Bush, Gore.
  • Suppose a small number of Gore voters change their preference lists:
    • 49%: Bush, Nader, Gore
    • 2%: Gore, Bush, Nader
    • 46%: Gore, Nader, Bush
    • 3%: Nader, Gore, Bush
  • Then in the contest between Nader and Bush, Bush wins 51% to 49%.
  • The next (and last) contest is between Bush and Gore; Gore wins 51% to 49%.
  • So Gore voters have made Gore win.
  • Notice that unlike all the other examples so far, nobody had to change their first-place votes!
  • Question: In general, when might strategic voting be a good idea for a small group of voters in the sequential pairwise system?
  • Answer: When the pairwise elections are all very close. A candidate who loses pairwise to one and wins pairwise over another can manipulate the election to win.

Other voting systems and properties

• Some other systems are described on the methods page of Condorcet.org.
• Alternative criteria for voting systems are described on the criteria page of Condorcet.org.