Chapter 9: Social Choice: The Impossible Dream The application of mathematics to the study of human beings their behavior, values, interactions, conflicts, and methods of making decisions is generally considered to be a recent thing. Yet the study of voting and social choice goes back several centuries. One primary example of a social choice problem is the selection of a good voting system. Voting is a subject that lies at the very heart of representative government and participatory democracy. Problem to be addressed: determine a good procedure that will turn individual preferences for different candidates into a single choice by the whole group. The goal is to find such procedures that will result in an outcome that reflects the will of the people. We begin in Chapter 9 with the question of how a group of individuals, each with his or her own set of values, selects one outcome from a list of possibilities. While majority rule is a good system for deciding an election with just two candidates, it turns out that there is no perfect way of deciding an election in which there are three or more candidates. 1
We start in section 9.1 with An Introduction to Social Choice. Most elections with which we are familiar involve two candidates; however, there are times when three or more candidates are vying for an office. Often, as in a recent mayoral election in Russellville or in the last senatorial election in Arkansas, when more than two candidates are vying for a position, we are required to have a runoff, since one candidate does not capture a majority of the votes. This is costly in terms of both time and resources, so it would be helpful to find a better method. Also, as in the case of the 2000 presidential election (arguably the most controversial presidential election in U.S. history), the fact that there were numerous candidates on the ballot actually changed the outcome of the election, due to the situation in Florida, where George W. Bush beat Al Gore by only a few hundred votes, which caused Bush to win the state of Florida and win the presidential election! There are several methods that can be used to elect a single candidate from a choice of three or more, and many of them involve the use of a Preference List Ballot, which we will examine more closely in a few minutes. 2
A ballot consisting of a rank ordering of candidates (which we often picture as a vertical list with the most preferred candidate on top and the least preferred on the bottom) is called a preference list ballot because it is a statement of the preferences of the individual who is voting. Preference list ballots allow voters to make a much clearer statement of their preferences than ballots allowing only a single vote. They are used in situations such as rating football teams and scoring track meets. Although we do not allow ties in a preference list ballot, most voting rules of interest will result in a tie at times. In the real world, the number of voters is often so large that ties seldom occur. However, to simplify what we do in this chapter, we make the following assumption. The number of voters assumption: Throughout this chapter, we consider only elections in which there is an odd number of voters. 3
Suppose we were to vote on something in this class. If there is an odd number of students, then we have no problem. If there is an even number, I could vote to make it an odd number. That way, there is no possibility of a tie. A majority will definitely occur, which will be a minimum of half of the votes plus 1. Let's do a simple vote. Consider Coke vs. Pepsi or Rock vs. Country. 4
9.2 Majority Rule and Condorcet's Method When a choice is being made between two candidates, the first type of voting system to suggest itself is majority rule. Each voter indicates a preference for one of the two, and the one with the most votes wins. There are three good things about majority rule. 1) All voters are treated equally. (Everyone's vote counts the same.) 2) Both candidates are treated equally. (If a new election were held and every voter reversed his or her vote, the outcome would be reversed as well.) 3) It is monotone. (If candidate X is the winner and a new election is held in which the only change made is for one more person to vote for X, then X will remain the winner.) These things seem obvious to us, but consider situations in which each of these properties would not hold. For example, condition 1) is not satisfied in a dictatorship (such as in Cuba), where the voters are not treated equally. The ballots of all voters except the dictator are ignored. Condition 2) is not satisfied in a situation of imposed rule (such as in a small town good old boy network) where the candidates are not treated equally. It may be a matter of "who you know" rather than "what you know." And condition 3) is not satisfied in minority rule (such as in a golf game, where the lowest score wins), because if the person with the lowest score (the winner) got another point (vote), they may not still be the winner. 5
May's Theorem (proved in 1952 by Kenneth May) tells us that among all twocandidate voting systems that never result in a tie, majority rule is the only one that treats all voters equally, treats both candidates equally, and is monotone. But what if there are three or more candidates? Is there a way to build on the strengths of majority rule? It turns out that there is a system that does just that, and it is called Condorcet's method. Our description of Condorcet's method begins with the observation that if we have a sequence of preference list ballots, then, for each pair of candidates, we can determine who the winner would have been had the election involved only these two in a one one one contest using majority rule. Consider the following example involving candidates A, B, and C: Rank Number of Voters (3) First A B C Second B C A Third C A B The 1st voter ranked A first, B second, and C third, the 2nd voter ranked B first, C second, and A third, and the 3rd voter ranked C first, A second, and B third. 6
To help us better understand a preference list ballot, let's do one of our own. We can rank holidays, ice cream flavors, restaurants, or presidential candidates, to name just a few. 7
Another example of a preference list ballot is shown below. This time there are 15 voters. 8
Description of Condorcet's Method: With the voting system known as Condorcet's method, a candidate is a winner precisely when he or she would, on the basis of the ballots cast, defeat every other candidate in a one on one contest using majority rule. (Historically, the voting system attributed to the Marquis de Condorcet in the 18th century was actually developed by Ramon Llull in the 13th century.) Example: Determine the winner in each preference list ballot below using Condorcet's method. 9
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Condorcet s voting paradox can occur with three or more candidates in an election where Condorcet s method yields no winners. For example, in a three candidate race, two thirds of voters could favor A over B, two thirds of voters could favor B over C, and two thirds of voters could favor C over A. This is the example given in the text. With three or more candidates, there are elections in which Condorcet s method yields no winners. Does Condorcet s voting paradox occur in the following tables? 11
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Ex: A committee of five people is looking for a new math department head, and they have narrowed it down to four candidates, whose names are Bill, Carol, Dennis, and Evelyn. They have expressed their preferences using the preference list ballot below. Use the Condorcet method to determine who will get the job. Homework: Read pp. 327 332 and do hw #9. 13
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