Voting Worksheet We have asked a group of 55 people to rank their chocolate preferences out of Kit Kats, Snickers, Reese's Cups, and M&M's. Suppose we have the following voter preference schedule once the results have been tallied. 18 15 11 9 2 1st K S R M S 2nd M M S R R 3rd S R M S M 4 th R K K K K 1. Is there a majority rule candidate? In other words, did any one candidate receive more than 50% of the 1st place votes? 2. The plurality method states the candidate with the most 1st place votes wins the election. What candidate wins by the plurality method? 3. The Borda count method assigns each rank a point with more points awarded to higher rankings. We will assign 1 point for fourth place, 2 points for third, 3 points for second, and 4 points for first. Calculate the points for each candidate. K: M: S: R: What candidate wins by the Borda count method? 4. With the plurality with elimination method eliminates the candidate with the fewest 1st place votes. Each candidate then is bumped up until a candidate receives more than 50% of 1st place votes. Which candidate gets eliminated first?

MAT 202 Quantitative Reasoning Fill out the new preference schedule by bumping up each remaining candidate. 1st 2nd 3 rd There is still not a plurality winner. Now which candidate gets eliminated? Fill out the new preference schedule by bumping up each remaining candidate. 1st 2nd Who wins by the plurality elimination method? 5. With Copeland's method, each head-to-head matchup is considered. In a head-to-head matchup, a winner receives 1 point, a loser 0 points, and a tie point. Calculate the points of each head-to-head matchup. Point Point Kit Kat vs M&M's M&M's vs Snickers Kit Kat vs Snickers M&M's vs Reese's Kit Kat vs Reese's Snickers vs Reese's How many points does each candidate receive? Kit Kat: M&M's: Snickers: Reese's: Who wins by Copeland's method? 6. The Condorcet Criterion says that if one candidate is preferred over each of the others in a head-to-head comparison, that candidate should win the election. Such a candidate is the Condorcet candidate.

MAT 202 Quantitative Reasoning Which candidate is the Condorcet candidate? Answer "none" if there is not one. There are several Fairness Criteria that ensure a fair result. (1) The Condorcet Criterion (CC) is satisfied. (2) The Monotonicity Criterion (MC) says that if voters switch their vote from candidate 1 to candidate 2, this switch in voting should not hurt candidate 2's chances of winning. (3) The Irrelevant Alternatives Criterion (IAC) says that adding or removing a non-winning candidate should not affect the outcome of the election. Let's now test which of our methods from before are fair! 7. Which candidate won the plurality method from before? Is (CC) satisfied? Let M&M's (a losing candidate) drop out. Then which candidate wins with plurality? Is (IAC) satisfied? Is this a fair method? 8. Which candidate won the Borda count method from before? Is (CC) satisfied? Let Reese's (a losing candidate) drop out. Now find the Borda count points of each candidate. K: M: S: Now what candidate wins by the Borda count method? Is (IAC) satisfied? Is this a fair method?

MAT 202 Quantitative Reasoning In a similar way, plurality with elimination and Copeland's method are also unfair. Arrow's Impossibility Theorem states that there is (or ever will be) a voting method that can satisfy all of the fairness criterion. 9. Is this surprising to you?