Theory of Public Choice
Theory of Public Choice
1.Majority voting: the typical mechanism used to aggregate individual votes into a social decision, whereby
individual policy options are put to a vote and the option that receives the majority of votes is chosen.
2.TO BE CONSISTENT, an aggregation mechanism must satisfy three goals:
Dominance: If one choice is preferred by all voters, then the aggregation mechanism must be such that this choice is made by society. E.g. if every individual prefers building an additional refectory to building
a swimming pool in our university, the aggregation mechanism must yield a decision to build a refectory.
Transitivity: Choices must satisfy the mathematical property of transitivity: if A is preferred to B, and B is preferred to C, then A must be preferred to C.
Independence of irrelevant alternatives: choices must satisfy the condition that if one choice is preferred to another, then the introduction of a third independent choice will not change that ranking. E.g. if building a refectory is preferred to building a swimming pool, then the introduction of an option to build a new library will not suddenly cause building a swimming pool to be preferred to building a refectory.
3.Majority voting can produce a consistent aggregation of individual preferences, or Condorcet winner, only if
preferences are restricted to take a certain form. E.g. PPT (medium school spending and taxes)
4.However, when majority voting does not deliver a consistent aggregation of individual preferences, or a
Condorcet winner, the problem of cycling rises. This problem, ultimately, give dictatorial power to the agenda setter, the person who decides how voting is to be done, since the mechanism and the order of voting can significantly influence the outcome.
5.Arrow’s Impossibili ty theorem: there is no social decision (voting) rule that converts individual preferences
into a consistent aggregate decision without either (a) restricting preferences or (b) imposing a dictatorship.
That is, every method of aggregation, or every voting rule, has examples failing to turn individual preferences into a clear, socially preferred outcome.
6.The most common restriction of preferences that is used to solve the impossibility problem is to impose
single-peaked preferences. A peak in preferences is a point that is preferred to all its immediate neighbors and utility falls as choices move away in any direction from that peak. With single-peak preferences there is a clear winner.
7.Median voter theorem: if preferences are single-peaked, majority voting will yield the outcome preferred by
the median voter whose tastes are in the middle of the set of voters.
8.Alternatives to majority rules:
Plurality voting:
1.Definition: In contrast majority voting in which one candidate has to take at least 51% of the vote, the
plurality vote winner takes more votes than any other candidates without winning the majority of votes. Under Plurality voting only first choices matter and get one point. Next choices do not count at all, which implies that much information is lost. It is widely applied in elections in numerous countries such as the US.
2. E.g. 193 students are investigated and their preferences of three types of food are listed in Table 1. For
example, 43 students prefer Cantonese food to Sichuan food, which is preferred to Japanese food.
Using majority voting mechanism, Cantonese is Condorcet winner, beating Sichuan and Japanese by 129 to 64 and 107 to 86 respectively. Japanese, however, is the plurality vote winner (Cantonese=43, Sichuan=64, Japanese=86), despite the fact that the majority consider it as the worst option. Hence, plurality voting can generate a result different from Condorcet winner
since it allows candidates to win without a clear mandate from the people. Another example is the US president election in 2000, in which Gore lost the election although his total vote is more than Bush. Borda voting:
1. Definition: The Borda voting, or Borda count, is a single-winner election method in which voters rank
candidates in order of preference. It determines the winner of an election by giving each candidate a certain number of points corresponding to the position in which he or she is ranked by each voter. One instance is that a voter’s first choice receives n votes and second choice gets n -1 points and so on. Once all votes have been counted, the candidate with the most point is the winner. It is common in award competition such as the election of MVP of NBA games.
2. E.g. according to the information of Table 2, the final point of Cantonese, Sichuan, and Japanese is
414 (=83*3+55*2+55*1), 386 (=83*2+55*1+55*3), and 358 (=83*1+55*3+55*2) respectively. Therefore, the Borda vote winner is Cantonese, followed by Sichuan and Japanese. However, if adding another item, swimming pool, to the set of choices. After calculation, swimming pool (607) becomes the winner, followed by Japanese (468), Sichuan (441) and Cantonese (414). The order among the initial three choices changes, violating the Independence of Irrelevant Alternatives.
Runoff voting
1. Definition: Runoff voting is a voting system used to elect one winner from a pool of candidates using
preferential voting. Specifically, voters rank candidates in order of preference, and their votes are initially allocated to their first choice candidate. If after this initial count no candidate has a majority of votes cast, the candidate with the fewest votes is eliminated and votes for that candidate are redistributed according to the voters’ second preferences. This process continues until one candidate receives more than 50% of the votes, upon which a winner is declared. This voting method is used in elections in some European countries as well as some US states.
(43) (64) (86) Cantonese Sichuan Japanese Sichuan Cantonese Cantonese Japanese
Japanese
Sichuan
(83)
(55)
(55) Cantonese Japanese Sichuan
Sichuan Cantonese Japanese Japanese Sichuan
Cantonese
(83)
(55)
(55) Swimming pool Japanese Sichuan Cantonese Swimming pool Japanese Sichuan Cantonese Swimming pool Japanese
Sichuan
Cantonese
2. E.g. Again use data in Table 1, since only first-place votes are counted under runoff voting,
Cantonese (43) will be eliminated, leaving Sichuan (64) and Japanese (86) in the second round. Then, after considering the voters’ second preference s, Sichuan beats Japanese by 107 to 86. This result, however, is different from Condorcet winner, which should be Cantonese in this example. In other words, runoff voting may fail to select a Condorcet winner when it exists. Moreover, runoff voting can violate positive responsiveness, which means increasing the vote for the wining option should not lead to declare another option a winner. To illustrate the violation of positive responsiveness, consider the example in Table 4, which has 4 options and 17 voters. Under runoff voting, the result of the first election is a tie between options a and b, with 6 votes each, while c is eliminated, with only 5 votes. In the second round, the supporters of c more to their second choice, a, giving a an extra 5 votes and a decisive victory for a over b. Now suppose that preferences are changed so that the wining option a attracts extra support from the 2 voters in the last column who switch their first-choice from b to a. It is surprised to find that in round 2, a will lose and c will win.