Elections with Only 2 Alternatives
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1 Math 203: Chapter 12: Voting Systems and Drawbacks: How do we decide the best voting system? Elections with Only 2 Alternatives What is an individual preference list? Majority Rules: Pick 1 of 2 candidates and one with most votes wins. (1) All voters treated equally. (2) Both candidates treated equally. (3) New election held and single voter changed vote from previous loser to previous winner. Then, the outcome is the same if all other voters kept their original vote. Other Options (1) Dictatorship (2) Imposed Rule (3) Minority Rule May s Theorem: If the number of voters is odd and we are only interested in a voting system that never ends in a tie, then the majority rule is the ONLY system for 2 alternatives that satisfy the three conditions above.
2 Elections with 3 or More Alternatives Assume the ballot is individual preference list. Allow ties, but assume in the real-world, this wouldn t happen, or can be broken by some random decision. (1) Plurality Voting and the Condorcet Winner Criterion Here, only the first place votes count. Example 1: 2000 Election, Part 1 What does it mean to be a Condorcet winner? Who was the Condorcet winner in the example? The Condorcet Winner Criterion (CWC) is satisfied in a voting procedure if for every possible sequence of preference lists, either (1) there is no CW (often the case), or (2) there is a CW, they re unique, and they are the unique winner of the election. Can we satisfy CWC in plurality voting? Plurality ignores 2nd choice, if one exists. There may be a candidate which is no one s first choice, but everyone s close second choice. Example 2: 2000 Election, Part 2 Another drawback: Manipulability: There are situations where there is an advantage to vote for someone who misrepresents your true preference. Vote Bush or Gore, rather than Nader because Nader will lose anyway.
3 (2) The Borda Count Rank all contestants/candidates/alternatives along with deciding a winner, and who finished 2nd, 3rd, etc. Assume points to each voter s rankings are in a nonincreasing order, that is, a voter s first choice receives more points than a voter s second choice, etc. The Borda Count is a rank method of voting where n alternatives and a first place vote receives n 1 points, a second place vote receives n 2 votes, and so on. The last place receives 0 points. Example 1: Other ranking methods: Track Meet Calculate the Borda Score of candidate A by counting the number of occurrences of other candidate names below candidate A s name. Example 2: Football Teams What is the Borda Score of each team? Modified Example 2: Football Teams What is the independence of irrelevant alternatives?
4 A voting system is said to satisfy the independence of irrelevant alternatives (IIA) if it is impossible for an alternative B to move from nonwinner status to winner status unless at least one voter reverses the order in which they had B and the winning alternative ranked. Example 3: Candidates A,B,C What is sincere voting? Burda said My scheme is only intended for honest men!
5 (3) Sequential Pairwise Voting and the Pareto Condition What is an agenda? How is it different from a preference list? Sequential Pairwise Voting starts with an agenda and pits the first alternative against the second in a one on one contest. The winner (or both if tie) moves on to confront third alternative and so on. Losers are deleted. Continue to those remaining, and hence the winners. Does the agenda affect the outcome given a preference list? Example 1: Exercise 18 from Text: Other ranking methods: Track Meet Does Agenda 1 bother you? What is the Pareto Condition?
6 (4) The Hare System Where is this system used? The Hare System proceeds to arrive at a winner by repeatedly deleting those alternatives that are least preferred or appear at the opt of the fewest preference lists. Repeat. If two or more alternatives remain and all would be eliminated in next round, then there s a tie. Example 1: 13 Voters A voting system has monotonicity if... Read and make sure you understand the example on page
7 Insurmountable Difficulties Problems with any system with 3 or more alternatives, just just the four we ve discussed Condorcet s Example: Condorcet s Voting Paradox is that it s possible for 2/3 society to prefer A over B, 2/3 prefer B over C, and 2/3 prefer C over A. Arrow s Impossibility Theorem: There does not exist and never will exist any social choice procedure that satisfies both CWC and IIA. Can We Do Better? We re OK with 2 alternatives. We have problems with 3 or more alternatives. What do we do? What is approval voting? When is approval voting appropriate? Have we eliminated problems if we use approval voting?
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