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Voting

Voting. Part 3. Condorcet Winner (again). 1980 New York senate race 45% Al D’Amato 44% Elizabeth Holtzman 11% Jacob Javitz So D’Amato was the (plurality) winner, and won the senate seat. This is all the information that could be found in ballots.

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Voting

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  1. Voting Part 3

  2. Condorcet Winner (again) • 1980 New York senate race 45% Al D’Amato 44% Elizabeth Holtzman 11% Jacob Javitz • So D’Amato was the (plurality) winner, and won the senate seat. • This is all the information that could be found in ballots. • But exit polls revealed something interesting…

  3. Condorcet Winner (review) To find the Condorcet Winner when there are 3 choices A,B,C: • Put each pair of candidates in a head-to-head match-up. • For each pair, decide which candidate most voters prefer. • If one candidate defeats all other candidates, then it is the Condorcet Winner.

  4. Use the following exit polls to find the Condorcet winner: • [D’Amato] • [Holtzman] • [Javitz]

  5. Condorcet Winner Criterion • A voting system satisfies the Condorcet Winner Criterion if the Condorcet winner, when there is one, wins the election under the given voting system. • So the 1980 New York Senate race shows plurality voting does not satisfy the Condorcet Winner Criterion. • But a “fair” voting system should satisfy Condorcet Winner Criterion. • Conclusion: Plurality voting is unfair. Marquis de Condorcet, 1743-1794

  6. Who is the Condorcet Winner in the election represented in the table? • [Amy] • [Ben] • [Cal] • [There is none.]

  7. Condorcet’s Paradox • What we’ve just seen is that it is possible that there is no Condorcet Winner. 1. Amy beats Ben in head-to-head race. 2. Ben beats Cal in head-to-head race. 3. Cal beats Amy in head-to-head race. • In this election, 2/3 of the public prefers somebody else to the any particular candidate. • So no matter which candidate is picked, the majority opinion is thwarted. • Condorcet discovered this paradox in 18th century.

  8. Borda Count (revisted) • Recall that in the Borda Count, when we have 3 candidates, we give: 2 points for each 1st place vote 1 point for each 2nd place vote 0 points for each 3rd place vote • Used in the French Academy of Sciences to elect members until Napoleon became its president in 1801. • Used today in many situations, including AP Football Poll. Jean Charles Borda 1733-1799

  9. Use the Borda Count to find the winner. • [Amy] • [Ben] • [Cal] • [No winner]

  10. Manipulating the Borda Count • Amy won our election, leaving the 4 supporters of Cal disappointed because they really, really don’t like Amy. • These supporters know they can’t change the votes of the other 6 people. • So they know that Cal couldn’t have won. • But they really wish Amy had not won. • Could they have changed the outcome just by changing their own 4 votes?

  11. Suppose the voting bloc of 4 changes votes from 1st table to 2nd table. Who wins in the 2nd table under the Borda count? • [Amy] • [Ben] • [Cal] • [No winner]

  12. Independence of Irrelevant Alternatives • The 4 people who didn’t want Amy to win, voted against their interests (Cal) and put Ben at the top of the ballot. This caused Amy to lose. • So their candidate Cal lost, but so did their least favorite candidate. • Thus the Borda Count fails the following test: Independence of Irrelevant Alternatives. In a fair election it should be impossible for a nonwinning candidate (Ben) to change to winner unless at least one voter reverses the order in which they listed Ben and the original winner (Amy). • So the Borda count is unfair (it does not satisfy the Independence of Irrelevant Alternatives).

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