Syllabus update: Now keeping best 3 of 4 tests

Transcription

1 Syllabus update: Now keeping best 3 of 4 tests The answer was 22. Recall order of operations: Parentheses, exponents, multiplication/division, addition/subtraction. PEMDAS Please Excuse My Dear Aunt Sally Voting method = way to pick a winner -Plurality (most 1 st place votes wins) -Borda (weighted sum)

2 -Pairwise comparison -Plurality with elimination -for each method, we can find the winner, or find the ranking # of voters 3 1 st R H F 2 nd H S H 3 rd F O S

3 Choosing which football game to go to. Rose Bowl, Fiesta Bowl, Hula Bowl, Sugar Bowl, Orange Bowl Pairwise comparison: 4 th O F O 5 th S R R Pairwise comparison points: R= 0 O=1 H= 4 S=2 F=3 Recall that a Condorcet candidate will always win in the pairwise comparison method. -Check that the pairwise comparison points add up to the total number of comparisons. Ranking: H, F, S, O, R

4 Back to counting. N candidates requires N(N 1) 2 head to head comparisons. Each of the N candidates runs against N-1 other people. This gives N(N-1) comparisons. Each comparison is between two people, so this is counting each comparison twice. Thus there are N(N 1) 2 different comparisons. Another way is to look at the sum: (N 1) = N(N 1) 2

5 How many pairwise comparisons for 13 candidates? 13(12) = voters, 5 candidates, how many ballots are possible? 5 candidates = how many preference ballots possible? Same as arranging 5 objects/letters/numbers etc in some order. Subtasks: Subtask 1: pick 1 st person Subtask 2: pick 2 nd person Subtask 3: pick 3 rd person Subtask 4: pick 4 th person Subtask 5: pick 5 th person Total ways to arrange 5 candidates: 5! = (5)(4)(3)(2)(1) = 120 For N candidates, you can fill out the preference ballot in N! ways.

6 Recall the original question: 10 voters, 5 candidates, how many ballots are possible? 120 ways to fill out the ballot, but only ten voters filling out a ballot so there can only be 10 different ballots. (Can t have more ballots than voters.) Now, 6 candidates and 500 voters. How many different ballots possible? 6 candidates gives 6! ways to complete a ballot. 6! = 720 But only 500 voters, so only 500 possible ballots. Question 2: 6 candidates, 5000 voters. How many ballots possible? Again, 6 candidates gives 6! ways to fill out the ballot. 6! = 720, and we have more than 720 voters, so all 720 possible ballots could occur.

7 #of voters st A C D B C 2 nd B B C D D 3 rd C D B C B 4 th D A A A A Math club. # of voters 3 1 st R H F 2 nd H S H 3 rd F O S 4 th O F O 5 th S R R

8 Fairness/Ethics -important to decide the voting method before the voting starts (otherwise you can choose the method that gives the most favorable outcome for you) # of voters 3 1 st R H F 2 nd H S H 3 rd F O S 4 th O F O 5 th S R R (Bowl games) Suppose the voters know beforehand that the winner will be chosen by plurality. -may encourage insincere voting (abandon candidate you prefer in favor of someone with stronger support) (-you see this in primaries frequently)

9 The four fairness criteria (singular of criteria is criterion) -A principle that says in a given situation, person x should win -Majority criterion: if a candidate has a majority (more than half) of the first place votes, that candidate should win -A voting method satisfies a given fairness criterion if it always chooses the winner the criterion suggests. It is common for a criterion to not apply, but when it does, it must be agreed with. -A voting method violates a given fairness criterion if there is any election where the method fails to choose the candidate suggested by the criterion. -satifies = always agrees -violates = disagrees even once

10 6 2 3 A B C B C D C D B D A A (Hiring committee, 11 voters, 4 candidates) A majority requires more than 11/2=5.5, so a majority requires at least 6 votes -therefore A is a majority candidate -According to the majority criterion, A should win Plurality: Winner = A Plurality can only give one majority candidate. If there were two candidates with a majority (more than half the 1 st place votes) the total votes would exceed the number of voters, so this is impossible. Therefore a majority candidate will always have the most votes, and will win in the plurality method. Because the plurality method always agrees with the majority criterion, we say this method satisfies the majority criterion. -The same is true for plurality by elimination.

11 -Pairwise comparison also satisfies the majority criterion -this is because a majority candidate is always a Condorcet candidate (wins every head-to-head comparison) Plurality, Plurality with elimination, Pairwise comparison A B C B C D C D B D A A

12 Borda method? A= 4( )+3( )+2( )+1( ) = 29 B= 4( )+3( )+2( )+1( ) = 32 C=4( )+3( )+2( )+1( ) = 30 D=4( )+3( )+2( )+1( ) = 19 Borda winner is B B is NOT the majority candidate, so Borda method violates the majority criterion. -since we found at least one example of an election where a majority candidate is not the Borda winner, Borda violates the majority criterion