Social Choice Theory [Election Theory] Voting Methods Plurality, Majority, Borda Count, Runoff, Sequential Runoff, Condorcet

Preference Schedules A list or table of all candidates/options ranked in order of preference, and the number of people who prefer them in that order more preferred moving in the direction the arrow points A B C D B C D A C B D A D B C A 8 5 6 7

PLURALITY & MAJORITY Plurality – winner chosen by most “1st place” votes Majority – requires winner to get MORE than ½ of “1st place” votes A B C D B C D A C B D A D B C A 8 5 6 7

BORDA COUNT 8 5 6 7 Winner chosen using a point system A B C D B C B B Common point assignments last place = 1 point, next to last = 2 points, etc… One could also assign point values of 0, 1, 2, etc. to make the math easier A B C D B C D A C B D A D B C A 8 5 6 7

BORDA COUNT - work 8 5 6 7 A: 8(3) + 5(0) + 6(0) + 7(0) = 24 assign points 3 2 1 A B C D B C D A C B D A D B C A 8 5 6 7

D is the winner by a vote of 18 to 8 RUNOFF The two candidates receiving the most 1st place votes are put in a head-to-head match up to determine winner. 1st place vote totals A: 8 eliminate candidates B & C B: 5 b/c they have the C: 6 fewest 1st place votes D: 7 A B C D B C B B Re-total the votes based on whether A or D was more preferred in each schedule A: 8 D: 5 + 6 + 7 = 18 C D D C D A A A 8 5 6 7 D is the winner by a vote of 18 to 8

C is the winner by a vote of 18 to 8 SEQUENTIAL RUNOFF The candidate with the fewest first place votes is eliminated in round 1, votes are re-tallied without that candidate, and again the candidate with the fewest 1st place votes is eliminated; continue until only one candidate remains, who we declare the winner. 1st place vote totals A: 8 eliminate candidate B (lowest #) B: 5 C: 6 D: 7 A B C D B C B B Re-total the votes based on whether A, C, or D was more preferred in each schedule A: 8 C: 5 + 6 = 11 eliminate candidate D D: 7 C D D C D A A A Re-total the votes based on whether A or C was more preferred in each schedule A: 8 C: 5 + 6 + 7 = 18 8 5 6 7 C is the winner by a vote of 18 to 8

CONDORCET METHOD A Condorcet winner is the candidate that would defeat all other candidates in a runoff election (head-to-head matchup) Condorcet Paradox – there is not always a Condorcet winner (rock, paper, scissors) A B C D B C D A C B D A D B C A 8 5 6 7

CONDORCET METHOD 8 5 6 7 A B C D B C D A C B D A D B C A Read Row vs. Col. A B C D A vs. B 8 to 18 L A vs. C A vs. D B vs. A 18 to 8 W B vs. C 20 to 6 B vs. D 19 to 7 C vs. A C vs. B 6 to 20 C vs. D D vs. A D vs. B 7 to 19 D vs. C 7 to 19 L A B C D B C D A C B D A D B C A 8 5 6 7 Since B is more preferred over all other candidates when paired head-to-head, B is the Condorcet Winner