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*Note: This letter was originally slated to appear in the September, 2001 issue of PS: Political Science and Politics, but due to editorial mishap the publication was delayed. PS apologizes for the delay.
On February 9, 1996, after a seventeen-month cease-fire, the Irish Republican Army (IRA) set off a bomb in East London. Less than a week later, the London police found and destroyed a bomb that the IRA had left in a telephone booth in the West End. A few days later, another IRA bomb went off on a doubledecker bus. The British government, under the leadership of John Major, asserted that it would have no official contact with Sinn Féin, the political arm of the IRA, until the paramilitary activities stopped. The government also deployed 500 additional troops in Northern Ireland. In October 1996, the IRA detonated two bombs at the British Army's headquarters in Lisburn, bringing the violent conflict back to Northern Ireland for the first time since the cease-fire.
On May 1, 1997, Tony Blair was elected Great Britain's prime minister. His election brought a renewed sense of optimism and good will to the conflict in Northern Ireland. Responding to Blair's overtures, including his warning that “the settlement train is leaving, with or without them” (Hoge 1997a, A7), the IRA announced on July 19, 1997, that they had “ordered the unequivocal restoration of the cease-fire” (Clarity 1997c, 1). Soon thereafter, Sinn Féin was invited to participate in the peace talks, which reconvened on September 15, 1997.
Our thesis in this chapter is that AW is not simply a fair-division procedure with desirable theoretical properties. It is also one that is eminently practicable in a variety of situations, ranging from international disputes to divorce settlements.
We begin section 5.2 by showing how AW might have been used in facilitating the resolution of the dispute between the United States and Panama over the Panama Canal treaty, which was negotiated in the 1970s and involved ten issues. In sections 5.3 and 5.4 we illustrate the application of AW to two divorce cases, one hypothetical and one real (from New York state). In the real case, AW probably would have produced a different resolution, which the dissenting judge in this case, when heard on appeal, supported. As a final application of AW, we revisit in section 5.5 the problem of dividing up the Vermont estate, which was first discussed in section 1.2, and compare this division with the division obtained using Knaster's procedure, which was described in section 3.2.
Knaster's procedure, when applied to two-party disputes involving multiple issues, bears some resemblance to AW insofar as the bids that the players make may be thought of as their placing points on the items to be divided.
The problem of fair division is as old as the hills, but our approach to this problem is new. It involves
setting forth explicit criteria, or properties, that characterize different notions of fairness;
providing step-by-step procedures, or algorithms, for obtaining a fair division of goods or, alternatively, preferred positions on a set of issues in negotiations; and
illustrating these algorithms with applications to real-life situations.
Our three-pronged focus on properties, algorithms, and applications pulls together work scattered in different disciplines. Thus, philosophers have devoted major attention to explicating the meaning of fairness, and related concepts like justice and equity, and spelling out some of their implications. Theoretical economists have postulated abstract properties that any fair-division scheme should satisfy, but their demonstration of the existence of such a scheme usually says little about how, constructively (i.e., by the use of an algorithm or step-by-step procedure), to produce a fair division. Mathematicians have proved nonconstructive existence results as well, but they have been more interested than economists in also developing algorithms that actually produce a fair division.
Political scientists, sociologists, and applied economists have taken a more empirical tack, seeking to determine conditions under which fairness, or departures from it, occur in the world and what consequences they have for people and institutions. Psychologists have especially paid heed to how perceptions of fairness impinge on people's attitudes and affect their behavior.
There are a host of voting procedures under which voters either can rank candidates in order of their preferences or allocate different numbers of votes to them, which we call preferential voting systems because they enable voters to distinguish more preferred from less preferred candidates. We shall describe four of the most common systems, discuss some properties that they satisfy, and illustrate paradoxes to which they are vulnerable. They are:
The Hare system of single transferable vote (STV)
The Borda count
The rationale underlying all these systems is that of affording different factions or interests in the electorate the opportunity to gain representation in a legislature or council proportional to their numbers, which we call proportional representation (PR). We shall briefly analyze each of these systems here and then make some comparisons, based on different criteria, at the end of the chapter. What is worth noting here, however, is that each of these systems offers a different approach to the problem of achieving PR, especially of minorities, which is the notion of fairness we take as our starting point in the study of elections with multiple winners.
The remainder of the chapter takes an unusual turn in that it was inspired by the request of a professional association to advise it on a voting procedure to use in electing its governing board.
In this brief concluding chapter, we draw attention to a few general themes that link the more specific findings of the book:
Fair division is an old subject with a rich past.
We traced concerns about fair division back to the Bible and ancient Greece, but probably in every culture issues of fair division have arisen and rippled through its institutions. Specific procedures, as well, have a venerable history, which have taken on different forms at different times. For example, divide-and-choose emerged as a legislative procedure in which one house divides and the other chooses in James Harrington's The Commonwealth of Oceana in the seventeenth century; nearly 350 years later it is the basis for allocating mining rights under the Convention of the Law of the Sea.
The pervasiveness of a fair-division ethic seems driven in part by its survival value. From a sociobiological perspective, the idea of giving people what they deserve – by reciprocating their help or kindness or, conversely, by not being responsive when they are not – leads to the development of conventions, customs, or even laws that institutionalize the allocation of goods or bads (like chores) with some modicum of fairness. It would be fascinating to study these from the perspective of the theory developed in this book, but we have chosen instead to concentrate on the theoretical analysis of procedures while offering some empirical illustrations.
In the Hebrew Bible, the issue of fairness is raised in some of the best-known narratives. Cain's raging jealousy and eventual murder of Abel is provoked by what he considered unfair treatment by God, who “paid heed” to Abel's offering but ignored his (Gen. 4:4). Jacob, after doing seven years of service in return for Laban's beautiful daughter, Rachel, was told that his sacrifice was not sufficient and that he must instead marry Laban's older and plainer daughter, Leah, unless he did seven more years' service, which he regarded as not only breaking a contract but also flagrantly unfair. Fairness triumphed, however, when King Solomon proposed to divide a baby, claimed by two mothers, in two. When the true mother protested and offered the baby to the other mother (whose baby had died), the truth about the baby's maternity became apparent, and “all Israel … stood in awe of the king; for they saw that he possessed divine wisdom to execute justice” (1 Kings 3:28).
Solomon's proposed solution is the first explicit mention of fair division that we know of in recorded history. But it is, of course, no solution at all: Solomon had no intention of dividing the baby in two. Instead, his purpose was to set up a game between the two women, described in Brams (1980, 1990b), that would distinguish the mother from the impostor.
In this chapter we present the solution to the problem first raised by Gamow and Stern (1958) of finding a finite algorithm for providing an envy-free division among any number of players. Before presenting this solution, however, we will describe two algorithms in section 7.2 – one continuous, involving a moving knife, and the other discrete – that provide allocations that are approximately envy-free, with the degree of error within any preset tolerance level.
We describe in section 7.3 an exact but “infinite” solution to the n-person envy-freeness problem, using what we call the “trimming procedure.” Because this procedure requires an infinite number of stages to accomplish, however, it is not, technically, an algorithm.
Fortunately, the trimming procedure can be rendered finite, but only by complicating it considerably. Without giving full details, we will describe enough of the finite procedure in section 7.4 to give the reader a good idea of how it works, and why it is unbounded – the number of stages it may require cannot be specified without knowing the preferences of the players. We will also illustrate this procedure with an example involving four players.
In section 7.5 we apply the trimming procedure to an inheritance example that involves indivisible as well as divisible goods, showing that it may be necessary to sell some of the indivisible goods to complete the procedure.
Divide-and-choose, as we showed in section 1.2, assures each player of a piece of cake he or she perceives to be at least 1/2 the total (proportionality), no matter what the other player does. Because there are only two players, this means that each player can get what he or she considers to be a piece at least tied for largest. To provide this guarantee, however, the cutter must play “conservatively” by dividing the cake exactly in two, according to his or her valuation of it. That way, whatever piece the chooser selects, the cutter is assured of getting 1/2.
Although proportionality and envy-freeness are equivalent when there are only two players, this is not the case when there are more than two players. In extending envy-freeness to more than two players, we seek procedures wherein each player has a strategy that guarantees him or her a piece that is at least tied for largest, no matter what the other players do.
We showed in chapters 2 and 3 that none of the n-person proportional procedures is envy-free: while these procedures guarantee a player a piece of size at least 1/n, one or more of the players might think that another player received a larger piece. This was not the case for the two-person pointallocation procedures we analyzed in chapter 4 and applied in chapter 5, which are not only proportional – and, therefore, envy-free – but also equitable.
Although a player like the president may propose legislation, almost invariably provisions are added to and subtracted from such a proposal, votes are taken, new changes made, and so on, as a bill wends its way through the legislative process. Likewise, the implementation of this legislation, once enacted, is also subject to significant modification through executive orders, court cases, citizen response, and the like.
Both divide-and-choose (section 1.2) and filter-and-choose (section 1.3) are strictly applicable only to two players. Although we suggested in section 1.4 that filter-and-choose is, effectively, played again and again in the legislative process, this two-person procedure provides an incomplete model if there are different players at each stage. Consequently, we ask whether there is an extension of divide-and-choose to more than two players.
The answer to this question, discovered by Hugo Steinhaus but discussed for the first time in Knaster (1946), extends divide-and-choose to three people. This result and others we shall describe in this chapter initiated the modern era of research on fair division, which focused on cake cutting but involved procedures for allocating indivisible goods as well (see chapter 3).
Hugo Steinhaus was a mathematician who, together with his colleagues, Bronislaw Knaster and Stefan Banach, began their research on fair-division procedures in Poland during World War II (Steinhaus, 1948).
The proportional procedures described in chapter 2 are not generally applicable to real-world situations involving indivisible goods, whose value is destroyed if they are divided. They were, in fact, designed for the case where the good being divided, such as cake or land, is divisible – that is, a cut can be made at any point.
In this chapter, by contrast, we assume that there are k different indivisible goods, which cannot be partially allocated. The fair division problem is to assign each such good to one and only one player so that each player thinks he or she is getting at least 1/n of the total.
We focus on two procedures for producing a proportional division of indivisible goods. The first is the “procedure of sealed bids” proposed by Knaster some fifty years ago (Steinhaus, 1948). This procedure seems quite practicable if the players all have an adequate reserve of money. The second procedure is the “method of markers” developed by William F. Lucas about twenty years ago (Lucas, 1994), which builds on ideas behind the lastdiminisher procedure and applies them to indivisible goods. These procedures are discussed in sections 3.2 and 3.3.
In section 3.4, we illustrate how these procedures might also be used to allocate goods when players have different entitlements, as in the case of estate division.