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Current assignments:

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

Properties of voting systems

• We still have one more condition to check for each of the four voting systems.
• Condorcet condition:
  If A is a Condorcet winner (i.e. beats every other candidate in a pairwise contest), then A is the winner.
  • Plurality?
    No!
    Proof: We just need to construct a single example. We use the hypothetical Bush-Gore-Nader election of 2000, with preferences given as
    • Bush, Nader, Gore: 49%
    • Gore, Nader, Bush: 48%
    • Nader, Gore, Bush: 3%
    We already know that Nader is the Condorcet winner of this election.
    Since Bush wins in the plurality system, plurality does not satisfy the Condorcet condition.
  • Instant runoff (Hare)?
    No!
    Proof: We again use the Bush-Gore-Nader election.
    Nader is the Condorcet winner.
    Gore wins the instant runoff election (after Nader gets eliminated).
    So the Hare system does not satisfy the Condorcet condition.
  • Borda count?
    No!
    Proof: We need to change the numbers slightly. Suppose instead the results were
    • Bush, Nader, Gore: 45%
    • Gore, Bush, Nader: 3%
    • Gore, Nader, Bush: 46%
    • Nader, Gore, Bush: 6%
    Then in a pairwise contest between Bush and Nader, the result is Bush--48%, Nader--52%.
    In a contest between Gore and Nader, the result is Gore--49%, Nader--51%.
    So Nader is the Condorcet winner.
    Now for the Borda count totals (convert percentages to numbers):
    • 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
    So Gore is the Borda count winner.
  • Sequential pairwise voting?
    Yes.
    Proof: This was homework problem 11 in Chapter 5.
  • Dictatorship?
    No!
    Proof: The dictator might vote for someone who is not the Condorcet winner.

The Condorcet voting paradox

• One problem with the election of 2000 (in our model) is that the results were so close.
• This doesn't just cause technical problems with recounts...
• It makes it difficult to decide which candidate should rightfully be declared the winner.
• The problem is that the pairwise election results are very close:
  • Bush vs. Gore: 49-51
  • Bush vs. Nader: 49-51
  • Gore vs. Nader: 48-52
• Nader is the Condorcet winner, but just barely.
• Gore wins instant runoff, but just barely.
• Bush wins plurality, but just barely.
• No matter how a winner is selected, about half the people will be unhappy and prefer another candidate.
• It gets worse...
• Condorcet paradox: In a three-candidate election, it is possible that no matter who wins the election, a 2/3-majority of people would prefer another candidate.
• Suppose the voter preferences were:
  • one-third: A, B, C
  • one-third: B, C, A
  • one-third: C, A, B
• (To remember this, think of "rock-paper-scissors". Rock=A, Scissors=B, Paper=C. Rock beats scissors, scissors beats paper, and paper beats rock.)
• Between A and B, A gets two-thirds of the vote.
• Between B and C, B gets two-thirds of the vote.
• Between C and A, C gets two-thirds of the vote.
• If A was declared the winner, two-thirds of the people would say "Please let's have B."
• So you declare B the winner, and then two-thirds of the people say "No, C is really a better choice."
• Then you declare C the winner, and two-thirds of the people say "Put A back in there."
• Of course, in this example the only proper way to decide is to make it a three-way tie.
• But if the numbers were slightly different (say 35%, 33%, 32%), you'd still have the same problem, although each of our methods would pick a winner.
• You could avoid this if people voted transitively.
• That is, if a majority prefers A over B, and a majority prefers B over C, then a majority should prefer A over C.
• But this doesn't necessarily happen.
• This phenomenon leads to most of the paradoxes in voting theory.
• Here's the basic reason:
• • "Independence of irrelevant alternatives" means that in a three-person election, any candidate who loses at least one pairwise election must lose the race.
  • (Losers don't become winners if a third-party candidate enters the race.)
  • So if every candidate loses at least one pairwise election, there is no winner.

An easy impossibility theorem

Theorem:
There is no social choice procedure that satisfies independence of irrelevant alternatives and that always elects a majority winner in a two-person race.

Notes:

• The textbook does a very special case which looks kind of suspicious, since it involves ties. I will change the numbers around so that the paradox does not come from a tie.
• Notice that the Condorcet criterion implies that in a two-person election, whoever gets the majority of the vote wins. (Not all voting systems satisfy this! e.g. dictatorship).
• So if we write it this way, it's more surprising than the theorem in the book (though we use the same technique).

• If IIA is satisfied, and if the outcome of a unanimous two-person election is that the winner is the social choice, then the voting system must be a dictatorship. (Arrow's theorem!)
• So IIA is not satisfied in any reasonable voting system.
• Thus there is always the possibility of "strategic voting," where people don't vote their true preferences in order to prevent the most hated candidate from winning.