Week 2: Voting Systems Continued
Week 2: Voting Systems Continued MATH 1014
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This 4 page Class Notes was uploaded by Amy Brogan on Friday January 22, 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 43 views. For similar materials see Mathematics of Social Choice in Mathematics (M) at University of Cincinnati.
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Date Created: 01/22/16
Koshar Amy Brogan January 20, 2016 Further Understanding the Voting Systems (17) (12) (9) (4) (4) (2) (1) First Q T E W Q E Q Second W R Y R T W E Third E Q R Y R Q T Fourth R W W Q Y R Y Fifth T Y T T E T W Sixth Y E Q E W Y R First on your own, find the winner of these votes by Condorcet, Hare’s Method, Borda count, Plurality, and Plurality Run-Off. You can do this; I have faith in you! Okay, once you’re done, check your methods: Condorcet: Head to Head Comparisons Q W So let’s try it with R: 17 9 R E 12 4 12 17 4 2 4 9 1 4 3 Q wins against W’s 15 1 T Q R(20) does not beat E(29), votes with 34 who was already beaten by 12 17 Q E W, so R and E can’t be the 9 4 17 9 Condorcet winner. 4 2 12 2 4 T Y 1 1 17 9 T (21) was beaten by Q(28), Q (38) beats E (11) 12 4 and since Q was beaten 4 Q R earlier, there is not a 17 12 2 Condorcet winner. 4 9 1 T (36) beats Y(13) 2 4 1 Q(24) does not beat R(25) Hare’s Elimination Method: Top votes Q W E R T Y 22 4 11 0 (eliminate) 12 0 (eliminate) 22 4 (eliminate) 11 12 26 11 (eliminate) 12 28 21 Q wins with Hare’s method. Remember: We are only looking at who is the top vote per round, per column of votes. As candidates are eliminated their votes have to go to a new candidate because there will be a new candidate in the top ranking. Here’s a look with each candidate eliminated in each round: Round 1: R & Y (doesn’t affect any first-place ranks) Round 2: W Round 3: E Top Votes between last two (Q & T) highlighted: Borda Count: (17) (12) (9) (4) (4) (2) (1) First (5) Q T E W Q E Q Second (4) W R Y R T W E Third (3) E Q R Y R Q T Fourth (2) R W W Q Y R Y Fifth (1) T Y T T E T W Sixth (0) Y E Q E W Y R (Votes in Column x Borda count for rank) + (votes in next column x Borda count for rank) + etc. (17x_) + (12x_) + (9x_) + (4x_) + (4x_) + (2x_) + (1x_) = ___ Q: (17x5) + (12x3) + (9x0) + (4x2) + (4x5) + (2x3) + (1x5) = 160 W: (17x4) + (12x2) + (9x2) + (4x5) + (4x0) + (2x4) + (1x1) = 139 E: (17x3) + (12x0) + (9x5) + (4x0) + (4x1) + (2x5) + (1x4) = 114 R: (17x2) + (12x4) + (9x3) + (4x4) + (4x3) + (2x2) + (1x0) = 141 T: (17x1) + (12x5) + (9x1) + (4x1) + (4x4) + (2x1) + (1x3) = 111 Y: (17x0) + (12x1) + (9x4) + (4x3) + (4x2) + (2x0) + (1x2) = 70 To check your math, you can add up all the totals you got here^ and compare it with the total Borda points. st nd Borda points for 1 place + Borda Points for 2 + … all multiplied by total number of voters 5+4+3+2+1 = 15 Total Voters: 17+12+9+4+4+2+1= 49 15 x 49 = 735 Borda points 160+139+114+141+111+70 = 735 Plurality: You only need to look at the first place ranking: (17) (12) (9) (4) (4) (2) (1) First Q T E W Q E Q Q has 22 first place votes, and the most, so Q wins by Plurality. Plurality Run-Off As there wasn’t a tie with Plurality, there is no need for a run-off. We would only need to do a run-off if there was a tie of two or more candidates. Curious about how many ranking outcomes there could have been? (If each voter had voted a different way) 6x5x4x3x2x1, or 6! (known as a factorial) = 720 possible outcomes! Review True or False? (Answer these with an odd number of voters in mind) 1. Plurality can tie? 2. Condorcet can tie? 3. P candidates ranked and Borda counted, all first place received P points? 4. With 3+ candidates, plurality and majority mean the same? 5. With Hare Method, candidates are eliminated by least amount of last-place votes. 6. Candidate A beats others in Head-to-Head, then A is the Condorcet winner. 7. Plurality Run-off: The two candidates with the most first place are considered? Answers: 1. True 2. False, one candidate has to beat all of the other candidates 3. False: close, but no. The Borda count ranking gives first place P-1 points 4. False: plurality can have a tie, while gaining the majority vote means having more than 50% of the votes 5. False: candidates are eliminated based on the fewest amount of first place votes 6. True, if A beats all other candidates, False if A only beats most of the other candidates 7. True
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