Economics 470 Some Notes on Simple Alternatives to Majority Rule

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1 Economics 470 Some Notes on Simple Alternatives to Majority Rule Some of the voting procedures considered here are not considered as a means of revealing preferences on a public good issue, but as a means of choosing a candidate for a given office. Majority Rule: Choose the candidate who is ranked first by (n/2) + 1 voters. Plurality Rule: Choose the alternative ranked first by the largest number of voters. Condorcet criterion: Choose the alternative that defeats all others in pairwise elections using majority rule. Majority Rule, Runoff: If one of m candidates receives a majority of first-place votes, this candidate is the winner. If not, a second election is held between the two topvote-getting candidates. The candidate receiving the most votes on the second ballot is the winner. The Hare System: Each voter indicates the candidate he ranks highest of the m candidates. Remove from the list of the candidates the one ranked highest by the fewest voters. Repeat the procedure for the remaining m - 1 candidates. Continue until just one candidate is left. The Coombs System: Each voter indicates the candidate he ranks lowest of the m candidates. Remove from the list of candidates the one ranked lowest by the most voters. Repeat the procedure for remaining m - 1 candidates. Continue until only one candidate remains, who is the winner. Approval Voting: Each voter votes for the k candidates (1 # k # m) he ranks highest of the m candidates, where k can vary from voter to voter. The candidate with the most votes is the winner. Borda Count: Give each of the m candidates a score of 1 to m based on the candidate's ranking in a voter's preference ordering; that is, the candidate ranked first receives m points, the second one m - 1,..., the lowest-ranked candidate one point. The candidate with the highest number of points is declared the winner.

2 2 Table 7.1. X X Y Z W Y Y Z Y Y Z Z W W Z W W X X X _ For above table, show that Condorcet winner is Y, Plurality winner is X, Approval winner (if cast single vote) is X. This example illustrates that a Condorcet winner may exist and is not picked by Plurality or Approval voting. The Borda count is as follows Y = 16 X = 11 Z = 13 W = 10 so that Y is Borda Count winner Table 7.2. X X X Y Y Y Y Y Z Z Z Z Z X X In Table 7.2, X is majority rule winner, X is Condorcet winner, but Y is Borda count winner.

3 3 Table 7.3. Y W X Y W X Z Z Z X Z X W X Z W Y Y W Y _ In Table 7.3, X is Condorcet winner, W wins under Hare System, X and Z are Borda count winners. Things to Note 1. Although the other procedures always pick a winner, even when a Condorcet winner does exist, the alternative rules do not always choose the Condorcet winner. 2. One way to evaluate the different procedures is to compute the percentages of the time that a Condorcet winner exists and is selected by a given procedure. Simulations of an electorate of 25 voters with randomly allocated utility functions and various numbers of candiates. (Voters are assumed to max Expected Utility under approval voting by voting for all candidates whose utilities exceed the mean of the candidates for that voter). Condorcet efficiency is not very sensitive to the number of voters. Table 7.4. Condorcet efficiency for a random society (25 voters) Number of Candidates Voting System Runoff Plurality Hare Coombs Approval Borda Social utility maximizer

4 Definition of Choice Set: An element x in S is a best element of S with respect to the binary relation R iff for every y in S, xry. The set of best elements in S is called the choice set C(S, R). Axioms of Borda Count (1) Neutrality: a form of impartiality with respect to issues and candidates. (2) Cancellation property: a form of impartiality toward voters. What determines the social ordering of x and y is the number of voters who prefer x to y versus the number preferring y to x. The identities of the voters do not matter. (3) Faithfulness: the voting procedure, when applied to a society consisting of only one individual, chooses as a best element that voter's most preferred element, that is, is faithful to voter's preferences. * (4) Consistency: Let N and N be two groups of voters who are to select an alternative from the set S. Let C and C be the respective sets of alternatives that the two groups select using voting procedure B. Then if C and C have any elements in common (C intersection C is not the null set, then the winning issue under procedure B when these two subgroups are merged is contained in this common set C = C intersection C. T If two groups of voters agree on an alternative when choosing separately from a set of alternatives, they should agree on the same alternative when they are combined. Table 7.5 N N V6 V7 z x y z z x x x y z x x y z y z x y y z y N we get C = {x, y, z} z = x = y = N we get C = {x} x = 10, z = 9, y = N union N we get C = {x} x = 16, z = 15, y = 11 T 4

5 5 For pairwise majority rule C 1 = {x, y, z} C 2 = {x, z} C T = {z} hence, violates consistency. Other examples Table 7.6 Voters x a b x a b x c x a c x a c b c x b c x b a b c a b c a Borda counts: x = 22 a = 17 b = 16 c = 15 now delete x a = 13 b = 14 c = 15 Preferences for inverted-order paradox in Table 7.6 Table 7.7 Voters a b c a b c a b c x b c x b c x a c x a c x a b x a b x Borda count: x = 13 a = 18 b = 19 c = 20 Suppose x dies, then a = 15 b = 14 c = 13 Preferences for winner-turns-loser paradox