Public Choice. Slide 1

Transcription

1 Public Choice We investigate how people can come up with a group decision mechanism. Several aspects of our economy can not be handled by the competitive market. Whenever there is market failure, there is a rationale for government intervention. For example, universal health care provides a solution to the adverse selection problem. the government also plays a role in redistributing income more equitably. Slide 1

2 On whose behalf should the government act? We will talk about desirable features of a group decision mechanism (a voting procedure). We ll first ask what we want the voting rule to accomplish if everybody votes sincerely (votes according to his or her preferences). Next, we ll talk about how different voting procedures may be manipulated by strategic voters. We ll find two central messages comprised in Arrow s Impossibility Theorem and the Gibbard-Satterthwaite Theorem. Slide 2

3 Voting Rules in Canada Canada's electoral system is referred to as a "singlemember plurality" or "first-past-the-post" system. In every electoral district, the candidate with the most votes wins a seat in the House of Commons and represents that electoral district as its member of Parliament. An absolute majority (more than 50 percent of the votes in the electoral district) is not required for a candidate to be elected. Slide 3

4 Electoral reform in Canada I have listed a few editorials from the Globe and Mail on this topic. Note that editorials are opinions voiced by editors of the newspaper. I don t share the view of all that is said in these pieces, but they contain a lot of food for thought. Slide 4

5 Electoral reform in Canada article / Slide 5

6 Electoral reform in Canada Slide 6

7 Voting Rules - Examples Plurality Rule (first-past-the-post): Each voter casts a vote for his or her preferred candidate. Elect the candidate who is named most often. Pair-wise Majority Rule: two candidates are put against each other in a vote. Whichever candidate is preferred by more people than the other, this candidate is preferred by society to the other candidate. Candidate who receives a majority of votes when put against every other candidate (the Condorcet winner) is elected. Slide 7

8 Voting Rules - Examples Scoring Method: Assigns numbers to ranks, then sums up the numbers a candidate gets from all individuals based on their ranking. E.g. assign 3 to first rank, 2 to second, 1 to third and 0 to fourth. Then we sum up the ranks for each candidate and then order the candidates according to their descending scores. Elect the candidate ranked highest. Slide 8

9 Modified Plurality Rules Plurality rule with run-off (e.g. how French president is elected): If in the first round using the plurality no candidate gains a majority, the two candidates with the highest vote count go into a second round. Whoever wins then is the winner of the election. Ranked Ballot (e.g. used in Australia): Voters rank all candidates. In first stage, only first-ranked candidates are considered. If nobody receives a majority, the candidate with lowest vote count is eliminated and the second-ranked candidate of those voting for the eliminated candidate is moved up to first rank. Apply plurality and see if now one of the remaining candidates gains a majority. If not, proceed with eliminating candidates until one candidate receives a Slide 9 majority.

10 Let s vote! Table 1 # of voters: Preference Rankings st A D B C 2 nd B A D A 3 rd C C C D 4 th D B A B Slide 10

11 Questions Which candidate is elected if we use the plurality rule? Which candidate is elected if we use the plurality rule with run off? Which candidate is elected if we use the ranked ballot? Which candidate is elected if we use the pair-wise majority rule? Which candidate is elected if we use a scoring method? Slide 11

12 Answers Plurality Rule # of voters: Preference Rankings st A D B C 2 nd B A D A 3 rd C C C D 4 th D B A B D gets 3 votes, A gets 5 votes, B gets 6 votes, C gets 7 votes, thus C is elected. Slide 12

13 Answers Plurality w/ runoff Preference Rankings # of voters: st A D B C 2 nd B A D A 3 rd C C C D 4 th D B A B D gets 3 votes, A gets 5 votes, C gets 7 votes, B gets 6 votes, thus B and C go into the run-off. Then B gets elected with 11 votes. Slide 13

14 Answers Ranked Ballot # of voters: Preference Rankings st A D 1. B 2. C 2 nd B A D A 3 rd C C C D 4 th D B A B D gets 3 votes, A gets 5 votes, B gets 6 votes, C gets 7 votes, thus D is eliminated. Then B is eliminated and C wins with 13 votes. Slide 14

15 Answers Pairwise Majority Rule Preference Rankings # of voters: st A D B C 2 nd B A D A 3 rd C C C D 4 th D B A B Majority rule: A:B = 15:6, A:C=8:13, A:D = 12:9; B:C=11:10, B:D = 11:10, C:D=12:9. No Condorcet winner; D loses against every other candidate. Slide 15

16 Answers Scoring Method # of voters: Preference Rankings 5 Score 3 Score 6 Score 7 Score 1 st A 3 D 3 B 3 C 3 2 nd B 2 A 2 D 2 A 2 3 rd C 1 C 1 C 1 D 1 4 th D 0 B 0 A 0 B 0 A: 5*3+(3+7)*2+6*0=35 B: 5*2 +6*3 + (3+7)*0= 28 C: 7*3 + 14*1= 35, D: 3*3 + 6*2 + 7*1+5*0 = 28, A and C tie at first place. Slide 16

17 Any lessons? Different voting rules potentially elect different candidates, so not just voters preferences matter in who gets elected but also the procedure by which we elect candidates. Some voting rules are more decisive than others. Should we avoid ambiguity? Slide 17

18 First-past-the post vs Ranked Ballot In previous example, both voting rules elected the same candidate. Of course this is not always the case. Next example illustrates this. Note that the next example has a Condorcet winner; D wins against any other candidate in a pair-wise election. Slide 18

19 Let s vote! Table 2 # of voters: Preference Rankings st A D B C 2 nd D A D D 3 rd B C A B 4 th C B C A Slide 19

20 Plurality vs. Ranked Ballot With plurality rule, B wins. With ranked ballot, D wins. D is the candidate most often ranked second. It is also the Condorcet winner. So first-past-the-post fails to elect the Condorcet winner. In this example, ranked ballot doesn t. Does ranked ballot always elect the Condorcet winner? Slide 20

21 Let s vote (again)! Table 3 # of voters: Preference Rankings st A B C D 2 nd B A D C 3 rd C C B B 4 th D D A A Slide 21

22 Condorcet winner not elected The Condorcet winner is C. First-past-the post, plurality with run-off, and ranked ballot all fail to elect the Condorcet winner sometimes. These rules result in B or D being elected. I d argue that B is a better candidate than D, so ranked ballot seems to do worse here than first-past-the-post. Slide 22

23 Electoral Reform Is going from first-past-the-post to ranked ballot an improvement? If a candidate wins a majority of the votes in the first round, all of these voting rules would elect the Condorcet winner. If no candidate wins a majority, it s not clear which of the two rules is better. Slide 23

24 In search of the ideal voting rule No voting cycles (see first example and outcome under pair-wise majority rule) Pareto Optimality Every vote counts (unrestricted domain and non-dictatorship) Independence Slide 24

25 Transitivity One desirable feature of a voting mechanism is to prevent voting cycles. This idea is reflected in transitivity. Transitivity means that if X is preferred to Y and Y is preferred to Z, then X must be preferred to Z. For example, the relation greater equal is transitive. Slide 25

26 More Conditions for an Ideal Voting Mechanism Unrestricted Domain: no matter what preference ordering people might have, they should have an equal say in the voting process. That is, we cannot exclude a person, because we think it is weird to prefer Y to Z and Z to X. Nondictatorship: no individual in society should be so powerful that the voting mechanism reflects only his or her preferences over every set of alternatives put up for a vote. Slide 26

27 More Conditions Pareto Optimality: If there is one alternative that everybody prefers to another alternative, say everybody prefers X to Y, then Y should not be elected. Slide 27

28 Question: Do the majority rule and the scoring methods satisfy Pareto optimality? PO is satisfied by both pair-wise MR, and the scoring method (SM). If an alternative is preferred by every voter to another alternative this alternative has a clear majority over the other, hence the dominated alternative cannot be the Condorcet winner. For SM the dominated alternative always gets less points than the other alternative and hence is higher up in group ranking due to a higher score. Slide 28

29 More Conditions Independence: the social ranking of two alternatives X and Y should only depend on these two alternatives. Slide 29

30 Plurality Rules and Independence First-past-the-post, plurality with run-off, and ranked ballot do not satisfy independence. To see this, check out Table 1 and then see if removing one of the candidates that didn t get elected by the rules will yield a different winner of the election. Slide 30

31 Independence Use Table 1 # of voters: Preference Rankings st A D B C 2 nd B A D A 3 rd C C C D 4 th D B A B Slide 31

32 Arrow s Impossibility Theorem No voting mechanism exists that satisfies all conditions simultaneously. That s unfortunate, but it also makes our lives more interesting J Slide 32

33 Single Peaked Preferences and the Median Voter If we restrict preferences to be single peaked, the pair-wise majority rule always generates a transitive group preference, so there is a Condorcet winner. Think of it as ranking of political parties from left to right. Order political parties along the left-right scale as follows: A, B, C, D. Define the median voter as the voter whose preferences lie in the middle of the set of all voters preferences; half the voters are located to the right of the median voter and half of the voters are located to the left. Table 3 is an example of single-peaked preferences. Tables 1 and 2 violate single-peakedness. Slide 33

34 Not Single-peaked Preferences Table 1 # of voters: Preference Rankings st A D B C 2 nd B A D A 3 rd C C C D 4 th D B A B Slide 34

35 The Median Voter Theorem As long as all preferences are singlepeaked, the outcome of pair-wise majority voting reflects the preferences of the median voter. Moreover, it is only necessary for the voting mechanism to know the peak of each voter in order to compute the Condorcet winner of the election. Slide 35

36 Median Voter Theorem applied Table 3 # of voters: Preference Rankings st A B C D 2 nd B A D C 3 rd C C B B 4 th D D A A The 11 th voter prefers C. Slide 36

37 The Median Voter Theorem implies that a party close to the center will receive a majority of votes. Pierre Trudeau once said: We are in the extreme centre, the radical middle. That is our position. Pair-wise MR is as easy to administer as plurality rule, but has better properties. Pair-wise MR and plurality rule are the same when there are only two candidates. Slide 37

38 Criticism of MVTh Political beliefs may not always be ranked along a single spectrum. The median voter on the issue of subsidizing day care may not be the same person as the median voter on the issue of provincial versus federal rights. If people have multi-peaked preferences over political parties, the median voter theorem does not hold either. Implies that all politicians would adopt the preferred policies of the median voter to get elected. However, even if politicians want to be elected, ideology, personality and leadership play a role in their decisions and political positions. Slide 38

39 Vote Manipulation Thus far we have assumed that people vote in such a way that reflects their preferences. If a voter thinks that her first choice has no chance of being selected, she may decide to vote for her second choice or even third choice to prevent an alternative she considers disastrous from being chosen. This process is called strategic voting. There are advocates of the ranked ballot who claim voters wouldn t vote strategically with this method, but they do with first-past-thepost. Slide 39

40 First-past-the-post and strategic voting Table 3 # of voters: Preference Rankings st A B C D 2 nd B A D C 3 rd C C B B 4 th D D A A Slide 40

41 Ranked Ballot and strategic voting Table 1 # of voters: Preference Rankings st A D B C 2 nd B A D A 3 rd C C C D 4 th D B A B Slide 41

42 The Gibbard-Satterthwaite Theorem When a single outcome is to be chosen from more than two alternatives, the only voting rule that cannot be manipulated is a dictatorial one. Slide 42

43 Is there a way out of GSTh? Again this theorem seems to be rather pessimistic. However, once again by restricting individual preferences to be singlepeaked, we have a positive result. Given single-peaked preferences, the pairwise majority rule cannot be manipulated. (Moulin 1988). Slide 43

44 Single-peakedness and Median Voter To see why the pair-wise majority rule cannot be manipulated with single-peaked preferences, recall that no voter would claim a peak past the median voter s peak, because that would mean voting for a candidate that is worse than the one based on sincere voting. But since any claimed peak between a voter s actual peak and the median voter s peak would not change the outcome of the election, nobody has an incentive to vote strategically. Slide 44

45 Median Voter Theorem applied Table 3 # of voters: Preference Rankings st A B C D 2 nd B A D C 3 rd C C B B 4 th D D A A The 11 th voter prefers C. Slide 45

46 Conclusion Although the Arrow Impossibility Theorem and the Gibbard-Satterthwaite Theorem are rather pessimistic, the conclusion is that society has to live with some imperfections in the voting mechanism, and NOT that we should have a dictatorship. And so I conclude with Winston Churchill who once said that Democracy is the worst form of government except for all those others that have been tried. Slide 46