Excursions in Modern Mathematics Sixth Edition[精品].ppt

* The Mathematics of Voting So what is wrong with the plurality-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. * The Mathematics of Voting 1.5 The Method of Pairwise Comparisons * The Mathematics of Voting 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. * The Mathematics of Voting 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 ? point. The winner of the election is the candidate with the most points after all the pairwise comparisons are tabulate. * The Mathematics of Voting The Method of Pairwise Comparisons 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 Mathematics of Voting The Method of Pairwise Comparisons 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. * The Mathematics of Voting So what is wrong with the method of pairwise comparisons? The Independence-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. * The Mathematics of Voting Eliminate C (an irrelevant alternative) from this election and B wins (rather than A). * The Mathematics of Voting How Many Pairwise Comparisons? In an election between 5 candidates, there were 10 pairwise comparisons.