1.3 The Borda Count Method 1 In the Borda Count Method each place on a ballot is assigned points. In an election with N candidates we give 1 point for last place, 2 points for second from last place, and so on. 2 1
Borda Count Method At the top of the ballot, a first-place vote is worth N points. The points are tallied for each candidate separately, and the candidate with the highest total is the winner. We call such a candidate the Borda winner. 3 Borda Count Method 4 A gets 56 + 10 + 8 + 4 + 1 = 81 points B gets 42 + 30 + 16 + 16 + 2 = 106 points C gets 28 + 40 + 24 + 8 + 4 = 104 points D gets 14 + 20 + 32 + 12 + 3 = 81 points 2
1.4 The Pluralitywith-elimination Method 5 Plurality-with with-elimination Method Round 1. Count the first-place votes for each candidate, just as you would in the plurality method. If a candidate has a majority of firstplace votes, that candidate is the winner. Otherwise, eliminate the candidate (or candidates if there is a tie) with the fewest firstplace votes. 6 3
7 Plurality-with with-elimination Method Round 2. Cross out the name(s) of the candidates eliminated from the preference and recount the first-place votes. (Remember that when a candidate is eliminated from the preference schedule, in each column the candidates below it move up a spot.) 8 4
Plurality-with with-elimination Method Round 2 (continued). If a candidate has a majority of first-place votes, declare that candidate the winner. Otherwise, eliminate the candidate with the fewest first-place votes. 9 10 5
Plurality-with with-elimination Method Round 3, 4, etc. Repeat the process, each time eliminating one or more candidates until there is a candidate with a majority of firstplace votes. That candidate is the winner of the election. 11 12 6
So what is wrong with the plurality-with with- elimination method? The Monotonicity Criterion If candidate X is a winner of an election and, in a reelection, the only changes in the ballots are changes that favor X (and only X), then X should remain a winner of the election. 13 1.5 The Method of Pairwise Comparisons 14 7
The Method of Pairwise Comparisons In a pairwise comparison between between X and Y every vote is assigned to either X or Y, the vote got in to whichever of the two candidates is listed higher on the ballot. The winner is the one with the most votes; if the two candidates split the votes equally, it ends in a tie. 15 The Method of The Method of Pairwise Comparisons The winner of the pairwise comparison gets 1 point and the loser gets none; in case of a tie each candidate gets ½ point. The winner of the election is the candidate with the most points after all the pairwise comparisons are tabulate. 16 8
The Method of Pairwise Comparisons 17 There are 10 possible pairwise comparisons: A vs. B, A vs. C, A vs. D, A vs. E, B vs. C, B vs. D, B vs. E, C vs. D, C vs. E, D vs. E The Method of Pairwise Comparisons 18 A vs. B: B wins 15-7. B gets 1 point. A vs. C: A wins 16-6. C gets 1 point. etc. Final Tally: A-3, B-2.5, C-2, D-1.5, E-1. A wins. 9
So what is wrong with the method of pairwise comparisons? 19 The Independence-of of-irrelevant-alternatives Criterion (IIA) If candidate X is a winner of an election and in a recount one of the non-winning candidates is removed from the ballots, then X should still be a winner of the election. 20 Eliminate C (an irrelevant alternative) from this election and B wins (rather than A). 10
How Many Pairwise Comparisons? In an election between 5 candidates, there were 10 pairwise comparisons. How many comparisons will be needed for an election having 6 candidates? Ans. 5 + 4 + 3 + 2 + 1 = 15 21 The Number of Pairwise Comparisons In an election with N candidates the total number of pairwise comparisons between candidates is (N - 1) N 2 22 11
Rankings 23 Extended Ranking Extended Plurality Extended Borda Count Extended Plurality with Elimination Extended Pairwise Comparisons Recursive Ranking Recursive Plurality Recursive Plurality with Elimination Rankings Recursive Ranking Step 1: [Determine first place] Choose winner using method and remove that candidate. Step 2: [Determine second place] Choose winner of new election (without candidate removed in step 1) and remove that candidate. Steps 3, 4, etc.: [Determine third, fourth, etc. places] Continue in same manner using method on remaining candidates yet to be ranked. 24 12
Rankings- Recursive Plurality First-place: A Second-place: B 25 Third-place: C Fourth-place: D Conclusion Methods of Vote Counting Fairness Criteria Arrow s Impossibility Theorem It is mathematically impossible for a democratic voting method to satisfy all of the fairness criteria. 26 13