8.2 Voting Possibilities and Fairness Criteria Unit 4- Discrete Mathematics

The Fairness Criteria of Voting What do we mean by fair? Over the years, those who study voting theory have proposed numerous criteria which most people would expect a 'fair' preferential election method to satisfy. There are four Fairness Criteria: The Majority Criterion The Condorcet Criterion The Irrelevant Alternatives Criterion The Monotonicity Criterion Kenneth Arrow Mathematical economist who proved in 1952 that there is NO consistent method of making a fair choice among three or more candidates with preferential voting. Borda Winner = Boston

The Majority Criterion The voting methods we’ve been practicing are all used in real world situations but none are perfect. Majority Criterion- if a candidate has majority of the first place votes, then that candidate should be the winner Lets show that the BORDA METHOD fails the majority criterion. Look at the following voter profile with 20 members below voting on Boston, Chicago, Philly, and Orlando. Borda Winner = Boston 8 4 1st C B 2nd P O 3rd 4th

The Majority Criterion Borda Winner = Boston This method does not select the candidate with the “majority” of the votes, who should be Chicago. The Plurality, Pairwise, and Hare Methods do satisfy the Majority criterion. 8 4 1st C B 2nd P O 3rd 4th

The Majority Criterion Plurality- always satisfies Pairwise- always satisfies Borda-does not satisfy Hare- always satisfies

The Condorcet Criterion The candidate who can win a pairwise comparison with EVERY other candidate is called the Condorcet candidate. Condorcet Criterion- should be the winner of an election if they win in pairwise comparison. (This is sometimes called HEAD TO HEAD) Lets show that the PLURALITY method fails the Condorcet criterion.

The Condorcet Criterion Let’s try to prove the Condorcet Criterion again with an example. Remember, the Condorcet Winner is the winner of the pairwise comparison voting method.

The Condorcet Criterion Condorcet Criterion (aka Head to Head) Plurality- may not satisfy Pairwise- always satisfies Borda- may not satisfy Hare- may not satisfy

The Monotonicity Criterion Monotonicity Criterion- If a candidate wins an election, and, in a reelection, the only changes are changes that favor the candidate, then the same candidate should win the reelection. For example, if a candidate wins a straw vote (pre-election poll), then gains additional support in the actual election without losing any original support, that candidate should win the actual election. This criterion can actually be violated by the Hare Method, which can produce a different winner from the straw vote winner.

The Monotonicity Criterion Monotonicity Criterion- If a candidate wins an election, and, in a reelection, the only changes are changes that favor the candidate, then the same candidate should win the reelection. Ex: Using the Hare Method, C is the winner In the actual election, the 3 votes in the last column were changed to C, A, B, adding to the first column. Hare makes B the winner, even though C actually gained more votes through the switch.

The Monotonicity Criterion Plurality- always satisfies Pairwise- always satisfies Borda- always satisfies Hare- does not satisfy

The Irrelevant Alternatives Criterion Irrelevant Alternatives Criterion- If a candidate wins an election and, before the results are announced, one or more candidates withdraw from the running, the same candidate should be the winner on the recount All four voting methods may violate this criterion. Ex: Borda – C wins If A drops out, then B wins by Borda.

The Irrelevant Alternatives Criterion Plurality- may not satisfy Pairwise- may not satisfy Borda- may not satisfy Hare- may not satisfy

Summary! Criteria → Majority Condorcet Monotonicity Irrelevant Alt Plurality Satisfies VIOLATES Pairwise Borda Hare

Do Now Please! Practice Problems Worksheet Please complete #9!