Fairness Criteria and Arrow’s Theorem Section 1.4 Animation

1.4 Summary, Arrow's Theorem 2 Hey, Dr. Arrow. I’ve found a cool voting system that I think is really fair. I call it the “PERFECT” method and here’s how it works... That’s great, son. Now here are the 4 things you have to show me.

1.4 Summary, Arrow's Theorem criterion Candidate B Candidate A If one candidate has. Then the “PERFECT” method should always choose that candidate Order is IMPORTANT

1.4 Summary, Arrow's Theorem criterion If one candidate can. then the “PERFECT” method should always choose that candidate Animation

1.4 Summary, Arrow's Theorem 5 3.Monotonicity criterion Animation Candidate A wins an election using the “PERFECT” method For some reason, possibly a court order, a. must be held In between elections all the voters who change their preference ballots do so such that A in their ballots Then A should win the second election using “PERFECT”

1.4 Summary, Arrow's Theorem 6 4. Independence of Irrelevant Alternatives criterion Candidate A wins an election using the “PERFECT” method For some reason a second election must be held In between elections. of the first election drops out Then A should win the second election using “PERFECT”

1.4 Summary, Arrow's Theorem 7 Arrow’s Theorem Any voting system that one can devise must. at least one of the fairness criteria in election Important Theme #2

1.4 Summary, Arrow's Theorem 8 Borda violates the Majority criterion 522Pts KCL3 CLD2 LDC1 DKK0 IF. THEN. Order is Important

1.4 Summary, Arrow's Theorem 9 Plurality violates the Condorcet criterion KCD CDL LLC DKK IF. THEN. Order is Important

1.4 Summary, Arrow's Theorem 10 IRV violates the Monotonicity criterion (After election 1, the 5 voters in column 2 reverse O and B) BBOG GOBO OGGB ELECTION 1. ELECTION 2.

1.4 Summary, Arrow's Theorem 11 The Hare method violates the Irrelevant Alternative criterion. (Assume that candidate C drops out of the race after the 1 st election) ELECTION 1. ELECTION ABCD BDDA CABC DCAB

1.4 Summary, Arrow's Theorem 12 Approval voting violates the Majority criterion The candidates in RED are the “approved” candidates ABACA DCDDD CABAB BDCBC ELECTION 1. ELECTION 2.

1.4 Summary, Arrow's Theorem 13 The Plurality voting method always satisfies the Majority criterion 1.True 2.False

1.4 Summary, Arrow's Theorem 14 IRV always satisfies the Majority criterion 1.True 2.False

1.4 Summary, Arrow's Theorem 15 The Condorcet method always satisfies the Condorcet criterion 1.True 2.False

1.4 Summary, Arrow's Theorem 16 To show that a voting method (call it X) does not satisfy the Majority criterion you must show there is 1. No majority winner 2. A majority winner and voting method X picks that winner 3. A majority winner but voting method X picks a different candidate. 4. None of the above

1.4 Summary, Arrow's Theorem 17 To show that a voting method (call it X) does not satisfy the Monotonicity criterion you must show that a candidate 1. Wins 1 st election using method X, but does not win 2 nd 2. Wins neither election using method X. 3. Loses the 1 st election using X, but wins 2 nd 4. None of the above

1.4 Summary, Arrow's Theorem 18 Arrow’s Theorem says that 1. At least one of the 4 criteria is violated in every election 2. At lest one of the 4 criteria is violated in some election 3. All of the 4 criteria are violated in some election 4. None of the above

1.4 Summary, Arrow's Theorem 19 End of 1.4

1.4 Summary, Arrow's Theorem 20 Mirror, Mirror on the Wall the Fairest of Them All? Who’s What’s

1.4 Summary, Arrow's Theorem 21 Kenneth Arrow (1952) Won the Nobel Prize in Economics Constructed four criteria that any "fair" voting system should satisfy Known as Arrow’s Fairness Criteria

1.4 Summary, Arrow's Theorem 22 Election 1 using “PERFECT” Election 2 using “PERFECT” Candidates I win! Not so fast. There were some shenanigans in Precinct 6. We’re doing this one again In the meantime only one voter changes her preference ballot and she moves the winner up in her ballot The winner of election 1 wins again But of course

1.4 Summary, Arrow's Theorem 23 Election 1 using “PERFECT” Election 2 using “PERFECT” Candidates I win! Not so fast folks. There were some shenanigans in Precinct 6. We’re doing this one again In the meantime one of the losing candidates drops out of the race The winner of election 1 wins again No surprise

1.4 Summary, Arrow's Theorem 24 Meta - Material