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Math 1010 ‘Mathematical Thought and Practice’ An active learning approach to a liberal arts mathematics course

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Nell Rayburn David Cochener Department of Mathematics Austin Peay State University Clarksville, Tennessee

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Decisions How do we make choices?

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Types of Decisions Individual—our opinion is our decision. Group—Individual opinions are expressed by voting(at least in a democratic society)and some procedure is used to combine these individual preferences for a group decision.

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Two Basic Questions What type of election decision procedure should we use to combine individual decisions (preferences) into a group decision? How can we be sure that what is decided is really what the group wants?

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Desirable Properties of Majority Rule (Two Alternative Case) All voters are treated equally. (Swapping marked ballots gives no change) Both alternatives are treated equally. (If all votes are reversed, so is the winner.) If a new election were held and a single voter changed from a vote for the previous loser to the previous winner, then the outcome would be the same as before.

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May’s Theorem If the number of voters is odd, and if we are interested only in voting procedures that never result in a tie, then majority rule is the only voting system for two alternatives that satisfies the conditions listed on the previous slide.

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Kenneth O. May (1915-1977) Mathematician, Political Activist PhD, Univ. of California, 1946 Mathematics plays a crucial role in social science!

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Ordinal Ballots Preference List

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Ordinal Ballots List your choices in order with the favorite on top and ‘least favorite’ on bottom Ballots must be Complete (you must rank all candidates) Linear (no ties) Transitive (If you prefer A to B and B to C, then you must prefer A to C.)

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Preference Schedule 181210942 AllenEvertDavisBakerChilds BakerChildsEvertDavisEvertDavis ChildsBakerChilds Baker Davis BakerEvertDavisEvert Allen

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Five Election Decision Methods Plurality Standard Runoff Sequential Runoff Borda Count Condorcet Winner Criterion

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Plurality Whoever has the most votes wins! Problems? Remember Jesse Ventura??

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1998 Minnesota Governor’s Race Jesse ‘The Body’ Ventura:38% Hubert Humphrey III:33% Norm Coleman:29% The latter two were highly experienced, but somewhat dull compared to Ventura.

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Standard Runoff If there is no majority, the two candidates receiving the most votes compete head to head.

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Sequential Runoff If no one has a majority, eliminate the candidate(s) with the fewest first place votes, and count again. Continue in this way until someone has a majority. Also known as Hare elimination.

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Borda Count If there are n candidates, assign n – 1 points to a first place choice, n – 2 points to a second place choice,…, 0 points to a last place choice. Sum the points for each candidate – the one with the most points wins. Problems? Borda Count can violate majority rule!

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A surprising result! Borda count may violate majority rule! A has a majority, but B wins Borda count by 21 – 19 over C, with A getting 18 points and D getting 8 points. 623 ABC BCD CDB DAA

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Condorcet Winner Criterion Conduct head – to – head contests between each pair of candidates. If any candidate can beat each of the others, he is a Condorcet candidate. Under this method a Condorcet candidate is declared the winner. May not always be decisive! (produce a winner)

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Fairness Criteria 1. Majority Criterion. If there is a choice that has a majority of the first- place votes, then that choice should be the winner of the election. 2. Condorcet Criterion. If there is a choice that is preferred by the voters over each of the other choices (in a head-to-head matchup), then that choice should be the winner of the election. 3. Monotonicity Criterion. If choice X is a winner of an election and, in a re-election, all the changes in the ballots are favorable to X, then X should still be a winner. 4. Independence-of-Irrelevant-Alternatives Criterion. If choice X is a winner of an election, and one(or more) of the other choices is disqualified and the ballots recounted, then X should still be a winner. (Also called Binary Independence.) IIA can also be stated as: It is impossible for an alternative B to move from non-winner to winner unless at least one voter reverses the order in which he/she had ranked B and the winning alterative. 5. Pareto Criterion. If every voter prefers alternative X over alternative Y, then the voting method should rank X above Y.

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Fairness Criteria **Every method we have studied can violate one or more of these! Plurality: violates Condorcet, IIA Standard Runoff: violates Monotonicity, Condorcet Hare Elimination(Sequential Runoff): violates Monotonicity, Condorcet and IIA Coombs’ Method: violates Monotonicity and Condorcet Sequential Pairwise Runoff: violates Pareto Condorcet’s Method: May not even produce a winner! Borda Count: violates Majority (and hence Condorcet) and IIA Is there any election decision procedure we could devise which satisfy these fairness criteria if we have 3 or more candidates and use ordinal ballots to rank the candidates?

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Arrow’s Theorem The search for the perfect election decision procedure

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The story so far… We have studied several election decision procedures designed to produce one or more winners from a slate of 3 or more candidates. Each procedure has had some desirable features and some undesirable ones (‘quirks’). We’ve even seen that these methods can give different winners using exactly the same set of ordinal ballots!

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Enter Kenneth Arrow Arrow, an economist, wanted to find a completely ‘fair’ election decision procedure. He began by making a list of a few basic properties that he believed any good election decision procedure should have:

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Arrow’s Properties Universality—The decision procedure must be any to process any set of ordinal ballots to produce a winner, and must be able to compare any two alternatives. Non-dictatorship (no one voter can determine the outcome) Independence-of-Irrelevant-Alternatives Criterion (Binary Independence) Pareto Criterion

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IIA or Binary Independence It is impossible for an alternative B to move from non-winner to winner unless at least one voter reverses the order in which he/she had ranked B and the winning alternative. In other words, whether A or B wins should depend only on how the voters compare A to B, and not on how other alternatives are ranked relative to A or B.

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Pareto Criterion If every voter prefers alternative X to alternative Y, then the decision procedure should rank X above Y.

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Asking the Impossible In 1951 Arrow published a book Social Values and Individual Choice in which he proved that there does not exist an election procedure which ranks for society 3 or more candidates based on individual preferences and which satisfies the fairness criteria we have listed.

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