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Voting and Apportionment 14.3 Apportionment Methods 14.4 Apportionment One of the Most Precious Rights in Our Democracy Is the Right to Vote

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Voting and Apportionment 14.3 Apportionment Methods 14.4 Apportionment One of the Most Precious Rights in Our Democracy Is the Right to Vote 304470-ch14_pV1-V59 11/7/06 10:01 AM Page 1 CHAPTER 14 The concept of apportionment or fair division plays a vital role in the operation of corporations, politics, and educational institutions. For example, colleges and universities deal with large issues of apportionment such as the allocation of funds. In Section 14.3, you will encounter many different types of apportionment problems. David Butow/Corbis SABA 14.1 Voting Systems* 14.2 Voting Objections Voting and Apportionment 14.3 Apportionment Methods 14.4 Apportionment One of the most precious rights in our democracy is the right to vote. Objections We have elections to select the president of the United States, senators and representatives, members of the United Nations General Assembly, baseball players to be inducted into the Baseball Hall of Fame, and even “best” performers to receive Oscar and Grammy awards. There are many *Portions of this section were developed by ways of making the final decision in these elections, some simple, some Professor William Webb of Washington more complex. State University and funded by a National Electing senators and governors is simple: Have some primary elec- Science Foundation grant (DUE-9950436) tions and then a final election. The candidate with the most votes in the awarded to Professor V. S. Manoranjan. final election wins. Elections for president, as attested by the controver- sial 2000 presidential election, are complicated by our Electoral College system. Under this system, each state is allocated a number of electors selected by their political parties and equal to the number of its U.S. senators (always two), plus the number of its U.S. representatives (which may change each decade according to the size of each state’s population as determined in the census). These state electors cast their electoral votes (one for president and one for vice president) and send them to the president of the Senate who, on the following January 6, opens and reads them before both houses of Congress. The candidate for president with the most electoral votes, provided that it is an absolute majority (one over half the total), is declared president. Similarly, the vice presidential can- didate with the absolute majority of electoral votes is declared vice pres- ident. At noon on January 20, the duly elected president and vice presi- dent are finally sworn into office. In this chapter we will look at several voting methods, the “fairness” Online Study Center of these methods, how votes are apportioned or divided among voters or For links to Internet sites related states, and the fairness of these apportionments. to Chapter 14, please access college.hmco.com/PIC/bello9e and click on the Online Study Center icon. V1 304470-ch14_pV1-V59 11/7/06 10:01 AM Page 2 V2 14 Voting and Apportionment 14.1 Voting Systems Monsieur Butterfly and Pizza Too G S T A R N T I E When you vote in a presidential election, you are not directly voting for the pres- T D T ident! You are actually voting for electors, individuals who cast the electoral E votes on behalf of their party and states. They are the ones who elect the presi- G dent. Originally, electors were free to cast their votes as they pleased, but many of today’s electors are “bound” or “committed” by state law (25 states have such laws) to vote for the candidate who received the most popular votes in their state. HUMAN SIDE OF MATH In a typical U.S. election, voters vote for their first choices by using a ballot. A Marie-Jean so-called butterfly ballot used in Palm Beach County, Florida, during the 2000 Antoine presidential election is shown. There was some confusion about votes cast for Nicolas de Pat Buchanan (second hole) or Al Gore (third hole). Caritat, Marquis de Condorcet, was born (1743–1794) The Granger Collection September 17, 1743, in Ribemont, France. Condorcet distinguished himself as a writer, administrator, and politician. His most important work was the Essay on the Application of Analysis to the Probability of Majority Decisions (1785), in which he tried to combine mathematics and philosophy to apply to Robert Duyos/South Florida Sun-Sentinel social phenomena. One of A butterfly ballot used in Palm Beach County, Florida, during the 2000 presidential election. the major developments in this work is known as the About 460,000 votes were cast in Palm Beach County, and of those, 3400 Condorcet paradox, a topic were for Buchanan. Assuming that the remaining precincts in Florida would covered in this chapter. yield the same proportion of votes for Buchanan, how many of the approxi- mately 6 million votes cast in Florida would you project for Buchanan? Think Looking Ahead about it before you answer! In this chapter we shall study The proportion of votes for Buchanan in Palm Beach was different voting systems, the “flaws” or objections that can be 3400 ϭ 34 raised about such systems, the 460,000 4600 methods used to fairly apportion resources among different If the same proportion applies to Florida, groups, and the objections to these apportionment methods. F ϭ 34 6,000,000 4600 or equivalently 4600F ϭ 34 ϫ 6,000,000 F Ϸ 44,348 304470-ch14_pV1-V59 11/7/06 10:01 AM Page 3 14.1 Voting Systems V3 Thus, you would expect about 44,347 Florida votes for Buchanan. (He actually got about 17,000 votes in Florida.) Moreover, the number of registered voters for Buchanan’s Reform Party in Palm Beach County was a mere 304 voters! What might be some of the reasons for this discrepancy? ᭤ There are two fundamentally different types of voting methods: preferential and nonpreferential. As the name suggests, a preferential voting system asks a voter to state a preference by ranking alternatives. This is usually done using a pref- erence table or preference schedule. For example, suppose the Math Club wants to order some pizzas for the end-of-year party. The Pizza House offers a special: three different one-topping pizzas—one jumbo, one large, and one medium—for only $20. The question is, Which topping to order on which pizza? The club members decide that the most popular topping should go on the jumbo pizza, the second-choice topping on the large pizza, and the third choice on the medium pizza; the topping choices are pepperoni, sausage, mushrooms, or anchovies. Each club member could fill out a preference ballot, and the results for the ballots might be sum- marized as in Table 14.1. PhotoDisc/Getty Images TABLE 14.1 Choice Joan Richard Suzanne First Sausage Sausage Pepperoni Second Pepperoni Pepperoni Mushrooms Third Mushrooms Anchovies Sausage Fourth Anchovies Mushrooms Anchovies If you were only considering each person’s first choice, and Joan, Richard, and Suzanne were the only voters, sausage would win 2 votes to 1. We say that sausage received a majority (2 out of 3) of the first-place votes. A candidate with a major- ity of the votes is the one with more than half, or 50%, of the vote. Looking at the table, you might argue that pepperoni is a better choice because each voter has it listed as first or second choice. If all the Math Club members were voting, listing all the ballots would take a lot of space because with only four toppings, there would be 4! ϭ 24 different ballots to consider. If you had five toppings, there would be 5! ϭ 120 ballots. You can summarize the results of an election by showing how often a particular outcome was selected with a preference table. Do it in steps. 1. Replace the word sausage with the letter S, pepperoni with the letter P, and TABLE 14.2 so on. 5 2. If several people have exactly the same list of preferences, list them together. Suppose five people all vote S, P, M, A. This fact is shown by using a table of S votes like Table 14.2. The number 5 at the top indicates that five people had P the exact results S, P, M, A on their ballots. Note that the first choice S appears at the top, the second choice P is next, and so on. M 3. Assume that all the club members chose one of 3 or 4 different rankings. The A voting methods we will study will work the same way no matter how many of the 4! ϭ 24 possible rankings were chosen. 304470-ch14_pV1-V59 11/7/06 10:01 AM Page 4 V4 14 Voting and Apportionment Now we are ready to analyze the results of elections using different voting systems: plurality, plurality with runoff, plurality with elimination, Borda count, and pairwise comparison. A. The Plurality Method As you can see from the results of the 2000 presidential election in Table 14.3, if there are three or more candidates it is possible that no candidate receives a majority (more than 50%) of the votes. In this case, one method of selecting the winner is to select the candidate with the most votes. This method is called the plurality method. In a U.S. presidential election, the candidate with the most popular votes does not necessarily win! TABLE 14.3 Results as of 6:00 P.M., EST, 11/17/2000 Candidates Votes Vote (%) D Gore 49,921,267 49 R Bush 49,658,276 48 G Nader 2,756,008 3 RF Buchanan 447,927 0 No winner declared Plurality Method Each voter votes for one candidate. The candidate with the most first-place votes is the winner.
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