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VOTING

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BINARY METHODS Choosing between only two alternatives at time Majority Rule Pairwise voting Condorcet Method Agenda Paradox

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BINARY METHODS Majority Rule Method Requires that the alternative with a majority of votes wins. Strategy Simply Vote for the person you prefer to win.

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BINARY METHODS Pairwise Voting Method Used to deal with more than two alternatives, which involves a repetition of binary votes These procedures are multistaged, which means they entail voting on pairs of alternatives in a series of majority votes to determine who is preferred. A number of pairwise procedure exist, distinguished by the order of votes or the method of pairing alternatives

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BINARY METHODS Pairwise Voting Condorcet Method Complete round robin of majority votes putting each alternative against all of the others. Shortcomings The Condorcet Paradox arises when no clear winner emerges from this process, so there is no majority winner This paradox is most likely to occur when there is a large group of people are considering large groups of alternatives.

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BINARY METHODS Condorcet Paradox Starburst Example There are 12 people ranking from 1 to 3 which flavor they prefer for a new Starburst flavor, either grape, sour apple, kiwi. If 4 people each pick one of the rankings then there is no clear winner because no one has the majority of votes. GrapeKiwiSour apple Sour Apple GrapeKiwi Sour Apple Grape 4 People

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BINARY METHODS Pairwise Voting Measures alternative win- loss record in a round robin contest. This method eliminates candidates and produces “scores.” Example is World Cup for Soccar Copeland Index

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BINARY METHODS Pairwise Voting Amendment Procedure When there are three alternatives, pairing two in a first round and then putting the third up against the winner in a second round. Strategy When you are bound by a pairwise method such as the amendment procedure you can use your prediction of the second round out come to determine your optimal voting strategy.

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BINARY METHODS Agenda Paradox Starburst example. This Paradox deals with the ordering of the alternatives in a binary voting procedure. The person that determines the ordering of the three alternatives has power over the final outcome. If the person determining the order wants grape to win they can pair sour apple and kiwi and then the winner vs grape. This could help give this person the out come they want. GrapeKiwiSour apple Sour Apple GrapeKiwi Sour Apple Grape 4 People

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PLURATIVE METHODS More than two alternatives are considered simultaneously by each voter Plurality rule Borda count Approval voting

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PLURATIVE METHODS PLURALITY RULE Pays no attention to ranking Example: 1994 Winter Olympics Highly manipulable Each voter casts a single vote for their most preferred alternative The candidate with the most votes wins Not necessarily majority METHOD SHORTCOMINGS STRATEGIC VOTING The misrepresentation of one’s preferences in order to achieve a more preferred outcome

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PLURATIVE METHODS PLURALITY RULE Strategic Voting SPOILER An additional candidate that diverts votes away from the leading candidate Usually has little chance of winning the election Example: Presidential Election of 2000 (Gore-Bush-Nader) Did Nader secure win for Bush? STRATEGY Vote for leader to prevent least favorite from winning

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PLURATIVE METHODS PLURALITY RULE Strategic Voting EXAMPLE – 1994 Maine Gubernatorial Election Election Results: King (I) – 36% Brennan (D) – 34% Collins (R) – 23% Carter (GP)* – 6% King (I) – 36% Brennan (D) – 37% Collins (R) – 23% Carter (GP) – 3% Strategic voting by Carter voters * Many of Carter’s voters next preferred the Democrat Brennan. BUT…What if everyone misrepresents their preferences? EXAMPLE – 1992 Presidential Election (Clinton-Bush-Perot) According to Newsweek: “exit polls on election day asked voters how they would have cast their ballots if they ‘believed Ross Perot had a chance to win.’ In this hypothetical contest, Perot had won 40 percent of the vote, against 31 percent for Clinton and 27 percent for Bush.”

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PLURATIVE METHODS BORDA COUNT METHOD Voters rank all of the alternatives, from most preferred to least preferred Points are assigned to each position Candidate with the most points wins SHORTCOMINGS REVERSAL PARADOX: When one of the candidates is removed from consideration after the votes have been tallied, recalculation is necessary. Fails to be “independent of irrelevant alternatives”: a change in the set of candidates can change the rankings of unaffected candidates.

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PLURATIVE METHODS BORDA COUNT Reversal Paradox EXAMPLE – (Hypothetical) All-Time Baseball Award Imagine a special commemorative honor to be awarded to the all- time best player in major-league baseball Candidates: Babe Ruth (BR) Ted Williams (TW) Pete Rose (PR) Willie Mays (WM) Seven leading baseball historians are to rank these candidates on their ballots. They vote in the following way:

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All-Time Baseball Award Rankings Apply (4-3-2-1) count: Ruth: 4+3+4+1+3+3+1=19 Williams: 1+4+1+2+4+4+2=18 Rose: 2+1+2+3+1+1+3=13 Mays: 3+2+3+4+2+2+4=20 Pete Rose is taken off the ballot Apply (3-2-1) count: Ruth: 3+2+3+1+2+2+1=14 Williams: 1+3+1+2+3+3+2=15 Mays: 2+1+2+3+1+1+3=13 The result of removing even the last place candidate results in a complete reversal of the rankings

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PLURATIVE METHODS BORDA COUNT STRATEGIC VOTING Place least preferred “favorite” last in an attempt to give them a lower overall score In other words, rank your second choice last If every coach follows such a strategy, the winner is the same as if each coach voted honestly Strategic voting, however, defeats the purpose of giving weight to the rankings. In an attempt to discourage such manipulation, many of these rankings are published, such as in pre-season rankings, sports awards, etc.

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PLURATIVE METHODS APPROVAL VOTING METHOD Voters cast one vote for each candidate of which they “approve” The candidate with the most approval votes wins When a group is being elected, winners are those who meet a certain number of votes SHORTCOMINGS Pays no attention to ranking The “approval” level is arbitrary

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Mixed Strategies Majority runoff Rounds Single transferable vote (Hare procedure) These use different methods based on binary and plurative methods Using different methods for the same voting aggregation may have different outcomes, and for this reason strategies are important for choosing which voting method to use

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Mixed Strategies Majority Runoff Method Used to decrease a large pool of possibilities to a binary decision Voters vote for most-preferred candidate and majority rules If there is a tie between two candidates, voters vote again for those two candidates in the same manner.

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Mixed Strategies Rounds Method Voters give single vote or preference ranking. Worst candidate is eliminated after each round of aggregation, and voters consider remaining candidates Last stage between two candidates is binary and majority determines winner. This method is used in determining sites for the Olympic Games

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Mixed Strategies Single Transferable Vote Method Uses voters’ full preference and proportional representation. Voter has one vote, ranking all candidates. Candidates below a set quota are eliminated and their votes are transferred to the next highest ranking candidate. Procedure continues until an appropriate number of candidates win.

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Arrow’s Impossibility Theorem Proves that no fair voting method can satisfy the 6 critical principles that Arrow identified, which are: The vote must be complete, where voters rank all alternatives. The vote must be transitive. It should satisfy positive responsiveness The voting should be independent of outside factors It cannot be dictatorial There should be no change in the set of candidates.

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CLASS VOTE RESULTS C = Increased course offerings D = Late-night dining hall W = New, larger workout facility Imagine that Holy Cross just received a generous donation. The money is to be spent on ONE thing – it cannot be divided and distributed to various places. If you could choose one place where this money would go, rank these three choices: Increased course offerings, with the addition of more faculty A late-night dining hall, so that legitimate eating options would be available after 8pm A new, larger workout facility, to reduce the waiting time to use exercise equipment.

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Voting Geometry: A Mathematical Study of Voting Methods and Their Properties Alan T. Sherman Dept. of CSEE, UMBC March 27, 2006.

Voting Geometry: A Mathematical Study of Voting Methods and Their Properties Alan T. Sherman Dept. of CSEE, UMBC March 27, 2006.

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