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Mathematics and Democracy

Mathematics and Democracy: Designing Better Voting and Fair-Division Procedures

Steven J. Brams
Copyright Date: 2008
Pages: 390
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  • Book Info
    Mathematics and Democracy
    Book Description:

    Voters today often desert a preferred candidate for a more viable second choice to avoid wasting their vote. Likewise, parties to a dispute often find themselves unable to agree on a fair division of contested goods. InMathematics and Democracy, Steven Brams, a leading authority in the use of mathematics to design decision-making processes, shows how social-choice and game theory could make political and social institutions more democratic. Using mathematical analysis, he develops rigorous new procedures that enable voters to better express themselves and that allow disputants to divide goods more fairly.

    One of the procedures that Brams proposes is "approval voting," which allows voters to vote for as many candidates as they like or consider acceptable. There is no ranking, and the candidate with the most votes wins. The voter no longer has to consider whether a vote for a preferred but less popular candidate might be wasted. In the same vein, Brams puts forward new, more equitable procedures for resolving disputes over divisible and indivisible goods.

    eISBN: 978-1-4008-3559-1
    Subjects: Mathematics, Political Science

Table of Contents

  1. Front Matter
    (pp. i-vi)
  2. Table of Contents
    (pp. vii-xii)
  3. Preface
    (pp. xiii-xvi)

    • 1 Electing a Single Winner: Approval Voting in Practice
      (pp. 3-22)

      It may come as a surprise to some that there is a science of elections, whose provenance can be traced back to the Marquis de Condorcet in eighteenth-century France, Charles Dodgson (Lewis Carroll) in nineteenth-century England, and Kenneth Arrow in twentieth-century America. Since Arrow published his seminal book,Social Choice and Individual Values, more than fifty years ago (Arrow, 1951, 1963)—for which in large part he received the Nobel Memorial Prize in Economics in 1972—there have been thousands of articles and hundreds of books published on everything from the mathematical properties of voting systems to elect centrist candidates.¹...

    • 2 Electing a Single Winner: Approval Voting in Theory
      (pp. 23-45)

      In single-winner elections that attract multiple candidates, a voter may consider more than one candidate acceptable if there is no obviously “best” candidate. Even if there is, the voter may wish to approve of a more viable second choice if his or her first choice has little chance of winning.

      To make “acceptability” more precise, in this chapter I extend the usual social-choice framework to include information not only on how voters rank candidates but also on where they draw the line between those they consider acceptable and those they consider unacceptable. This new information, of course, is precisely that...

    • 3 Electing a Single Winner: Combining Approval and Preference
      (pp. 46-68)

      As compelling as approval voting (AV) is in theory, and as well as it has worked in practice, other single-winner voting systems may be preferable under certain circumstances. For example, if voters are willing and able to rate candidates on a 3-point scale (good, medium, bad), this system could help voters distinguish acceptable candidates who are good from those who are just medium. In fact, such a refinement of AV has been proposed by Felsenthal (1989), Yilmaz (1999), and Hillinger (2005).

      Of course, one could ask voters to make still finer distinctions—say, on a 5-point rating scale—or give...

    • 4 Electing Multiple Winners: Constrained Approval Voting
      (pp. 69-88)

      In electing a committee, a council, or a legislature comprising more than one person, the usual rationale is to afford different factions or interests the opportunity to gain representation in proportion to their numbers, which is referred to asproportional representation(PR).¹ In this chapter, I focus on one procedure for doing so that uses an approval ballot. As with AV, voters can approve or disapprove of each candidate, but who is elected will be constrained in certain ways, which is why I call the procedureconstrained approval voting(CAV).²

      CAV was inspired by the request of a professional association...

    • 5 Electing Multiple Winners: The Minimax Procedure
      (pp. 89-111)

      In this chapter I analyze a voting procedure, called theminimax procedure, which is designed to elect committees whose members are representative of the electorate as a whole. It is based on approval balloting—whereby voters approve of as many candidates as they like, as under approval voting (AV) and constrained approval voting (CAV)—but votes are not aggregated in the usual manner.¹

      Instead of selecting the candidates that receive the most votes, the minimax procedure selects the set of candidates that minimizes the maximum “Hamming distance” to voters’ ballots, where these ballots are weighted by their proximity to other...

    • 6 Electing Multiple Winners: Minimizing Misrepresentation
      (pp. 112-142)

      In this chapter, I extend and generalize proportional-representation (PR) systems that were first proposed by Monroe (1995). Monroe’s systems select the winning candidates for a representative body that minimize the sum of “misrepresentation” values of voters.¹ These values are based on information that the voters provide on their ballots, such as candidate rankings or approval votes.

      Suppose, for example, that one wishes to choose a set of candidates so that as many voters as possible approve of at least one candidate who is elected. Or if the voters rank candidates, suppose that one wishes to choose candidates who are as...

    • 7 Selecting Winners in Multiple Elections
      (pp. 143-170)

      In this chapter, I take a tack that is different from previous chapters, wherein I described and analyzed voting procedures that seem well suited for different kinds of elections. Here, by contrast, I begin by analyzing so-called aggregation paradoxes, which arise when voters are asked to make choices in multiple elections, and ask how these paradoxes might best be resolved.

      The elections, for example, might involve voting on a bill and its amendments in a legislature, or voting on several propositions in a referendum. I show not only that different aggregation procedures may produce radically different outcomes but also that...


    • 8 Selecting a Governing Coalition in a Parliament
      (pp. 173-198)

      In most parliamentary systems, it is rare for a single party to win a majority of seats and thereby be able to govern by itself. Typically, two or more parties that together hold a majority of seats will form a governing coalition, which cannot be overthrown unless some of its members defect.

      This coalition will usually be led by the largest party in parliament, whose leader becomes prime minister. But this may not be desirable if the parliament is ideologically fractured and this party is relatively extreme (on the left or the right).

      In this chapter, I propose a procedure...

    • 9 Allocating Cabinet Ministries in a Parliament
      (pp. 199-223)

      How coalition governments in parliamentary democracies form and allocate cabinet ministries to political parties is the subject of a substantial theoretical and empirical literature. By and large, a rule of proportionality, whereby parties are given more ministries or more prestigious ministries (e.g., finance, foreign affairs, or defense) in proportion to their size, is followed. However, small centrist parties that are pivotal in coalitions (e.g., the Free Democrats in Germany) have successfully bargained for larger-than-proportional allocations (Browne and Dreijmanis, 1982; Budge and Keman, 1990; Warwick, 2001; Warwick and Druckman, 2001).

      The degree to which party leaders are satisfied with their allocations...

    • 10 Allocating Indivisible Goods: Help the Worst-Off or Avoid Envy?
      (pp. 224-251)

      Probably the most-discussed trade-off in economics is between efficiency and equity (LeGrand, 1991). While efficiency is arguably best achieved by unfettered competition in the marketplace that maximizes total wealth, the marketplace may be extremely unfair in distributing this wealth, especially to people who lack certain skills or social support (Roemer, 2000).¹

      The efficiency versus equity trade-off has analogues in other fields. In politics, centralized government may be less democratic, but decentralized government may be more wasteful and corrupt (though some analysts would claim just the opposite). In law, economic justice may best be served by giving people rights to a...

    • 11 Allocating a Single Homogeneous Divisible Good: Divide-the-Dollar
      (pp. 252-270)

      Much work in the mathematical social sciences is devoted to showing the conditions under which individually rational actions can lead to collectively inferior outcomes. This problem is epitomized by the game of Prisoners’ Dilemma, in which each player has a dominant, or unconditionally best, strategy of not cooperating, but the resulting outcome, and the unique Nash equilibrium, is worse for both players than if they had both cooperated. The game is nonzero-sum in that both players may win (by cooperating) or lose (by not cooperating) simultaneously; what one wins is not necessarily equal to what the other loses, making the...

    • 12 Allocating Multiple Homogeneous Divisible Goods: Adjusted Winner
      (pp. 271-288)

      Most disputes—divorce, labor-management, merger-acquisition, and international—involve only two parties, but they frequently involve several homogeneous goods that must be divided, or several issues that must be resolved. In this chapter I describe a procedure calledadjusted winner(AW) that has been applied to disputes ranging from interpersonal to international (Brams and Taylor, 1996, 1999).

      To introduce AW, I discuss in section 12.2 two properties of fair division, proportionality and envy-freeness (the latter I discussed in a different context in chapter 10). The well-known cake-cutting procedure of “I cut, you choose” satisfies these properties, but I argue that it...

    • 13 Allocating a Single Heterogeneous Good: Cutting a Cake
      (pp. 289-304)

      In chapter 11 I proposed several procedures (DD1, DD2, and DD3) for dividing a single homogeneous good (money), and in chapter 12 I analyzed a procedure (Adjusted Winner, or AW) for dividing multiple homogeneous goods. In this chapter I describe three new procedures for dividing a single heterogeneous good, such as a cake with different flavors or toppings whose parts players may value differently.

      If there arenplayers, the procedures assume that the players make the minimal number of cuts (i.e.,n− 1) required to divide the cake. In addition, they provide insight into the difficulties underlying the...

    • 14 Allocating Divisible and Indivisible Goods
      (pp. 305-328)

      So far I have analyzed the fair division of indivisible goods (chapters 9 and 10) and three kinds of divisible goods—homogeneous goods like money (chapter 11), multiple homogeneous goods (chapter 12), and a heterogeneous good like a cake (chapter 13). In this chapter I assume that there are multiple indivisible goods but also a single divisible homogeneous good—namely, money.

      The money is a “bad” in the sense that it is used by the players to pay for the indivisible goods they receive. The fair-division procedure that I propose for allocating the indivisible goods and the divisible bad assumes...

  6. 15 Summary and Conclusions
    (pp. 329-336)

    In this brief chapter, I offer a capsule summary of findings in each chapter and then draw some more general conclusions.

    Since the mid-1980s, approval voting (AV) has been successfully used to choose presidents and other officers in several major professional societies, some with tens of thousands of members. AV almost always elects Condorcet winners if they exist and, with only one exception (the Institute of Electrical and Electronic Engineers, or IEEE), has proved uncontroversial. Contrary to the fears of some, the winners in these societies are not “lowest common denominators” but tend to be supported by all classes of...

  7. Glossary
    (pp. 337-342)
  8. References
    (pp. 343-362)
  9. Index
    (pp. 363-373)