Week One of Voting Systems
Week One of Voting Systems MATH 1014
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This 4 page Class Notes was uploaded by Amy Brogan on Sunday January 17, 2016. The Class Notes belongs to MATH 1014 at University of Cincinnati taught by Mary Koshar in Spring 2016. Since its upload, it has received 52 views. For similar materials see Mathematics of Social Choice in Mathematics (M) at University of Cincinnati.
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Date Created: 01/17/16
Koshar Amy Brogan January 11, 13, &15, 2016 Quantitative vs. Qualitative Examples: o Temperature: Qualitative: very cold Quantitative: 15° F o Steep Incline: Qualitative: pretty steep Quantitative: gradient (6°), slope o Intelligence: Qualitative: smart Quantitative: Intelligence Quota (IQ) Quantitative reasoning: using given data to compute a solution to a problem o Problem solving: arithmetical, algebraic, geometric, logarithmic, and/or statistical methods Analyze: examine methodically and in detail 4 Beliefs That Make You Stupidhttps://www.youtube.com/watch?v=RH95h36NChI Question Everything Chart 1 RANK (8) [# of votes](5) (4) (3) (1) st 1 choice A B B C A 2 choice B C A A C 3 choice C A C B B Borda Count: point system for ranking vote o Last place gets zero points, increasing by one up from there rd nd st o Using chart 1: [ 3 = 0, 2 = 1, 1 = 2 points] A: 16+0+4+3+2= 25 B: 8+10+8+0+0= 26 *winner C: 0+5+0+6+1= 12 Hare “Elimination” Method o Eliminate the choice with the least amount of 1 place votes and choose from there o Number of rounds = number of candidates o Using Chart 1: Eliminate fewest 1 place: c A vs. B [Hare method continued] Chart 1.2 (Eliminate C) RANK (8) (5) (4) (3) (1) 1 choice A B B C A nd 2rdhoice B C A A C 3 choice C A C B B Most 1 place: A: 12 * winner B: 9 Plurality o Most votes in first place o Can have a tie because it’s not for majority Majority = more than 50% o Using Chart 1: Most 1 place: A: 9 *tie B: 9 *tie C: 3 Plurality Run-Off o Only two rounds st Top two 1 place candidates (eliminate all the rest) Choose from the new most first place o Chart 1: A: 9 (eliminate C) 12 *winner B: 9 (See chart 1.2) 9 Condercet’s “Head-to-head” Method: o Compare two choices at a time, looking for one who is the best overall o When comparing two, eliminate all others, such as in chart 1 where in the 1 column A gets the 8 votes against B, but B gets the points against C o Chart 1: A (8+3+1=12) vs. B (5+4=9) A wins here B vs. C B already lost to A, so this round is unnecessary A (8+4+1=13) vs. C (5+3=8) A wins again A wins overall Rank Method o Spread out possible points Person 1 Person 2 A 4 points 3 points B 1 point 2 points C 0 0 Chart 2 Borda Points Rank (10) [votes] (7) (6) (5) 3 First W Y S S 2 Second Y S Y P 1 Third S W W Y 0 Fourth P P P W Condorcet’s Method o W (10) vs. Y (7+6+5=18) Y wins o Y (10 [excluding W] + 7=17) vs. S (6+5=11) Y wins o Y (10+7+6=23) vs. P (5) Y wins o Y wins overall Plurality: o W: 10 o Y: 7 o S: 11 *winner Hare o Eliminate P o Eliminate Y o W (10) vs S (7+6+5=18) S Wins Borda Count o W: (10*3)+(7*1)+(6*1)+(5*0) = 43 o Y: (10*2)+(7*3)+(6*2)+(5*1) = 58 * winner o S: (10*1)+(7*2)+(6*3)+(5*3) = 57 o P: (10*0)+(7*0)+(6*0)+(5*2) = 10 Desirable Traits of Voting - Should be a winner - Every Ballot is Equ- Equal candidates (no secret states) - Advantage for actual want vs. manipulation (voting only for political party, or just to beat one candidate) Condorcet winner: candidate beats all others in head to head matches Monotonicity: if B wins the election, and a recast makes B’s ranking higher, B should still win Independent Irrelevant Alternatives (IIA): change in rankings of irrelevant alternates should not change the outcome o If B wins election 1, and in the recast C beats A, B should still win Pareto Condition: everyone prefers A to B, then B should not win Drawbacks: Condorcet: Possible to have NO winner because no one beats everyone head-to-head Hare: winner doesn’t beat every other candidate Plurality Run-off: not the same outcome as Condorcet Plurality: If B&C switch ranking, the original winner (A) may not stay the winner Sequential Pairwise: does not satisfy Pareto condition Kenneth Arrow: Arrow’s Impossibility Theorem: 1951: Proved that finding an absolute fair and decisive voting system is impossible With 3+ candidates and any number of voters, there does not exist there never will exist a voting system that always produces a winner, satisfies Pareto condition, is independent of irrelevant alternatives, and is not a dictatorship Chart 3 (5) (4) (3) (3) (2) 1st E A C D B 2nd B B B B C rd 3 C C D C D 4th D D A A A 5th A E E E E st Hare Method: eliminate those with least amount of 1 place votes A B C D E 4 2 (least 1 place) 3 3 5 4 / 5 3 (least new 1 ) 5 st 4 (least 1 ) / 8 / 5 / / 12 *winner / 5 Condorcet: Chart 3: 17 votes = 9 votes for majority E (5) vs. B (4+3+3) B B (5+4) vs. C B has majority already, C eliminated B (5+4) vs D D Eliminated B (5+3+3+2) vs. A (4) B is winner overall Plurality Run-off: A B C D E 4* top 2 2 3 3 5* top 2 4+3+3+2 = 9* win 5 Plurality Problems: 6 4 3 6 4 3 1st A C B 1st A C C 2nd B A C 2nd B A B rd rd 3 C B A 3 C B A A wins with 6 votes, but If in the recast, 3 people vote C over B, then C wins with 7 votes total instead of A(6)
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