Fair Division

Steven J. Brams Department of Politics New York University

New York, NY 10003 steven.brams@nyu.edu

To appear in Barry R. Weingast and Donald Wittman (eds.), Oxford Handbook of Political Economy (Oxford University Press, 2005).

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Fair Division

1. Introduction

The literature on fair division has burgeoned in recent years, with five academic

books (Young, 1994; Brams and Taylor, 1996; Robertson and Webb, 1998; Moulin,

2003; Barbanel, 2004) and one popular book (Brams and Taylor, 1999b) providing

overviews. In this review, I will give a brief survey of three different literatures: (i)

division of a single heterogeneous good (e.g., a cake with different flavors or toppings);

(ii) division, in whole or part, of several divisible goods; and (iii) allocation of several

indivisible goods. In each case, I assume the different people, called players, may have

different preferences for the items being divided.

For (i) and (ii), I will describe and illustrate procedures for dividing divisible

goods fairly, based on different criteria of fairness. For (iii), I will discuss problems that

arise in allocating indivisible goods, illustrating trade-offs that must be made when not all

criteria of fairness can be satisfied simultaneously.

2. Single Heterogeneous Good

The metaphor I use for a single heterogeneous good is a cake, with different

flavors or toppings, that cannot be cut into pieces that have exactly the same composition.

Some of the cake-cutting procedures that have been proposed are discrete, whereby

players make cuts with a knife—usually in a sequence of steps—but the knife is not

allowed to move continuously over the cake. Moving-knife procedures, on the other

hand, permit such continuous movement and allow players to call “stop” at any point at

which they want to make a cut or mark. There are now about a dozen procedures for

dividing a cake among three players, and two procedures for dividing a cake among four

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players, such that each player is assured of getting a largest or tied-for-largest piece

(Brams, Taylor, and Zwicker, 1995) and so does not envy another player (resulting in an

envy-free division).

Only two 3-person procedures (Stromquist, 1980; Barbanel and Brams, 2004), and

no 4-person procedure, make an envy-free division with the minimal number of cuts (n -

1 cuts if there are n players). A cake so cut ensures that each player gets a single

connected piece, which is especially desirable in certain applications (e.g., land division).

For two players, the well-known procedure of “I cut, you choose” leads to an

envy-free division if the cutter divides the cake 50-50 in terms of his or her preferences.

By taking the piece he or she considers larger and leaving the other piece for the cutter

(or choosing randomly if the two pieces are tied in his or her view), the chooser ensures

that the division is envy-free. However, this procedure does not satisfy certain other

desirable properties (Jones, 2002; Brams, Jones, and Klamler, 2004).

The moving-knife equivalent of “I cut, you choose” is for a knife to move

continuously across the cake, say from left to right. Assume that the cake is cut when

one player calls “stop.” If each of the players calls “stop” when he or she perceives the

knife to be at a 50-50 point, then the first player to call “stop” will produce an envy-free

division if he or she gets the left piece and the other player gets the right piece. (If both

players call “stop” at the same time, the pieces can be randomly assigned to the two

players.)

Surprisingly, to go from two players making one cut to three players making two

cuts cannot be done by a discrete procedure if the division is to be envy-free (Robertson

and Webb, 1998, pp. 28-29; additional information on the minimum numbers of cuts

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required to give envy-freeness is given in Even and Paz, 1984, and Shishido and Zeng,

1999). The 3-person discrete procedure that uses the fewest cuts is one discovered

independently by John L. Selfridge and John H. Conway about 1960; it is described in,

among other places, Brams and Taylor (1996) and Robertson and Webb (1998) and

requires up to five cuts.

Although there is no discrete 4-person envy-free procedure that uses a bounded

number of cuts, Brams, Taylor, and Zwicker (1997) and Barbanel and Brams (2004) give

moving-knife procedures that require up to 11 and 5 cuts, respectively. The Brams-

Taylor-Zwicker (1997) procedure is arguably simpler because it requires fewer

simultaneously moving knives. Peterson and Su (2002) give a 4-person envy-free

moving-knife procedure for chore division, whereby each player thinks he or she receives

a smallest or tied-for-smallest piece of an undesirable item, that requires up to 16 cuts.

To illustrate ideas, I will describe the Barbanel-Brams (2004) 3-person, 2-cut

envy-free procedure, which is based on the idea of squeezing a piece by moving two

knives simultaneously. The Barbanel-Brams (2004) 4-person, 5-cut envy-free procedure

also uses this idea, but it is considerably more complicated and will not be described

here.

The latter procedure, however, is not as complex as Brams and Taylor’s (1995)

general n-person discrete procedure. Their procedure illustrates the price one must pay

for an envy-free procedure that works for all n, because it places no upper bound on the

number of cuts that are required to produce an envy-free division; this is also true of other

n-person envy-free procedures (Robertson and Webb, 1997; Pikhurko, 2000). While the

number of cuts needed depends on the players’ preferences over the cake, it is worth

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noting that Su’s (1999) approximate envy-free procedure uses the minimal number of

cuts at the cost of little error.1

I make two assumptions about cake-cutting:

1. The goal of each player is to maximize the minimum-size piece (maximin

piece) he or she can guarantee for himself or herself, regardless of what the other players

do. To be sure, a player might do better by not following such a maximin strategy; this

will depend on the strategy choices of the other players. However, all players are

assumed to be risk-averse: They never choose strategies that might yield them larger

pieces if they entail the possibility of giving them less than their maximin pieces.

2. The preferences of the players over the cake are continuous, enabling one to use

the intermediate-value theorem. To illustrate, suppose that a knife moves across a cake

from left to right and, at any moment, the piece of the cake to the left of the knife is A

and the piece to the right is B. If, for some position of the knife, a player views piece A

as being larger than piece B, and for some other position he or she views piece B as being

larger than piece A, then there must be some intermediate position such that the player

values the two pieces exactly the same.

I will refer to players by number— player 1, player 2, and so on—calling even-

numbered players “he” and odd-numbered players “she.” In the case of the 3-person, 2-

cut “squeezing procedure” to be described next, cuts are made by two knives in the end.

Initially, however, one player makes “marks,” or virtual cuts, on the line segment

1 See Brams and Kilgour (2001), Haake, Raith, and Su (2002), and Potthoff (2002) for other approaches, based on bidding, to the housemates problem discussed in Su (1999). On approximate solutions to envy-freeness, see Zeng (2000).

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defining the cake; these marks may subsequently be changed by another player before the

real cuts are made.

Squeezing procedure. A referee moves a knife from left to right across a cake.

The players are instructed to call “stop” when the knife reaches the 1/3 point for each.

Let the first player to call “stop” be player 1. (If two or three players call “stop” at the

same time, randomly choose one.) Have player 1 place a mark at the point where she

calls “stop” (the right boundary of piece A in the diagram below), and a second mark to

the right that bisects the remainder of the cake (the right boundary of piece B below).

Thereby player 1 indicates the two points that, for her, trisect the cake into pieces A, B,

and C:

A B C /-----------|-----------|-----------/ 1 1

Because neither player 2 nor player 3 called “stop” before player 1 did, each of players 2

and 3 thinks that piece A is at most 1/3. They are then asked whether they prefer piece B

or piece C. There are three cases to consider:

1. If players 2 and 3 each prefer a different piece—one player prefers piece B and

the other piece C—we are done: Players 1, 2, and 3 can each be assigned a piece that

they consider to be at least tied-for-largest.

2. Assume players 2 and 3 both prefer piece B. A referee places a knife at the

right boundary of B and moves it to the left. At the same time, player 1 places a knife at

the left boundary of B and moves it to the right in such a way that the amounts of cake

traversed on the left and right of B are equal for player 1. Thereby pieces A and C

increase equally in player 1’s eyes. At some point, piece B will be diminished

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sufficiently to B'—in either player 2 of player 3’s eyes—to tie with either piece A' or C',

the enlarged A and C pieces. Assume player 2 is the first, or tied for the first, to call

“stop” when this happens; then give player 3 piece B', which she still thinks is the largest

or the tied-for-largest piece. Give player 2 the piece he thinks ties for largest with piece

B' (say, piece A'), and give player 1 the remaining piece (piece C'), which she thinks ties

for largest with the other enlarged piece (A'). Clearly, each player will think he or she

received at least a tied-for-largest piece.

3. Assume players 2 and 3 both prefer piece C. A referee places a knife at the

right boundary of B and moves it to the right. Meanwhile, player 1 places a knife at the

left boundary of B and moves it to the right in such a way as to maintain the equality, in

her view, of pieces A and B as they increase. At some point, piece C will be diminished

sufficiently to C'—in either player 2 or player 3’s eyes—to tie with either piece A' or B',

the enlarged A and B pieces. Assume player 2 is the first, or the tied for first, to call

“stop” when this happens; then give player 3 piece C', which she still thinks is the largest

or the tied-for-largest piece. Give player 2 the piece he thinks ties for largest with piece

C' (say, piece A'), and give player 1 the remaining piece (piece B'), which she thinks ties

for largest with the other enlarged piece (A'). Clearly, each player will think he or she

received at least a tied-for-largest piece.

Note that who moves a knife or knives varies, depending on what stage is reached

in the procedure. In the beginning, I assume a referee moves a single knife, and the first

player to call “stop” (player 1) then trisects the cake. But, at the next stage of the

procedure, in cases (2) and (3), it is a referee and player 1 that move two knives

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simultaneously, “squeezing” what players 2 and 3 consider to be the largest piece until it

eventually ties, for one of them, with one of the two other pieces.

3. Several Divisible Goods

Most disputes—divorce, labor-management, merger-acquisition, and

international—involve only two parties, but they frequently involve several goods that

must be divided, or several issues that must be resolved. As an example of the latter,

consider an executive negotiating an employment contract with a company. The issues

before them are (1) bonus on signing, (2) salary, (3) stock options, (4) title and

responsibilities, (5) performance incentives, and (6) severance pay (Brams and Taylor,

1999a).

The procedure I describe next, called adjusted winner (AW), is a 2-player

procedure that has been applied to disputes ranging from interpersonal to international

(Brams and Taylor, 1996, 1999b).2 It works as follows. Two parties in a dispute, after

perhaps long and arduous bargaining, reach agreement on (i) what issues need to be

settled and (ii) what winning and losing means for each side on each issue. For example,

if the executive wins on the bonus, it will presumably be some amount that the company

considers too high but, nonetheless, is willing to pay. On the other hand, if the executive

loses on the bonus, the reverse will hold.

Thus, instead of trying to negotiate a specific compromise on the bonus, the

company and the executive negotiate upper and lower bounds, the lower one favoring the

2 Procedures applicable to more than two players are discussed in Young (1994), Brams and Taylor (1996, 1999b), and Moulin (2003).

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company and the upper one favoring the executive. The same holds true on other issues,

including non-monetary ones like title and responsibilities.

Under AW, each side will always win on some issues. Moreover, the procedure

guarantees that both the company and the executive will get at least 50% of what they

desire, and often considerably more.

To implement AW, each side secretly distributes 100 points across the issues in the

dispute according to the importance it attaches to winning on each. For example, suppose

the company and the executive distribute their points as follows, illustrating that the

company cares more about the bonus (it would be a bad precedent for it to go too high)

and the executive cares more about severance pay (he or she wants to have a cushion in

the event of being fired):

Issues Company Executive

1. Bonus 10 5 2. Salary 35 40 3. Stock Options 15 20 4. Title and Responsibilities 15 10 5. Performance Incentives 15 5 6. Severance Pay 10 20 Total 100 100

The underscored figures show the side that wins initially on each issue by placing

more points on it. Notice that whereas the company wins a total of 10 + 15 + 15 = 40 of

its points, the executive wins a whopping 40 + 20 + 20 = 80 of its points. This outcome is obviously unfair to the company. Hence, a so-called equitability

adjustment is necessary to equalize the points of the two sides. This adjustment transfers

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points from the winner (the executive) to the loser (the company). The key to the success

of AW—in terms of a mathematical guarantee that no win-win potential is lost—is to

make the transfer in a certain order. That is, of the issues initially won by the executive, look for the one on which the

two sides are in closest agreement, as measured by the quotient of the winner’s points to

the loser’s points. Because the winner-to-loser quotient on the issue of salary is 40/35 =

1.14, and this is smaller than on any other issue on which the executive wins (the next-

smallest quotient is 20/15 = 1.33 on stock options), some of this issue must be transferred

to the company. But how much? The point totals of the company and the executive will be equal

when the company’s winning points on issues 1, 4, and 5, plus x percent of its points on

salary (left side of equation below), equal the executive’s winning points on issues 2, 3,

and 6, minus x percent of its points on salary (right side of equation):

40 + 35x = 80 - 40x 75x = 40.

Solving for x gives x = 8/15 = 0.533. This means that the executive will win about 53%

on salary, and the company will lose about 53% (i.e., win about 47%), which is almost a

50-50 compromise between the low and high figures they negotiated earlier, only slightly

favoring the executive. This compromise ensures that both the company and the executive will end up with

exactly the same total number of points after the equitability adjustment:

40 + 35(.533) = 80 - 40(.533) = 58.7.

On all other issues, either the company or the executive gets its way completely (and its

winning points), as it should since it valued these issues more than the other side.

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Thus, AW is essentially a winner-take-all procedure, except on the one issue on

which the two sides are closest and, therefore, is the one subject to the equitability

adjustment. On this issue a split will be necessary, which will be easier if the issue is a

quantitative one, like salary, than a more qualitative one like title and responsibilities.3

Still, it should be possible to reach a compromise on an issue like title and

responsibilities that reflects the percentages the relative winner and relative loser receive

(53% and 47% in the example). This is certainly easier than trying to reach a

compromise on each and every issue, which is also less efficient than resolving them all

at once according to AW.

In the example, each side ends up with, in toto, almost 59% of what it desires,

which will surely foster greater satisfaction than would a 50-50 split down the middle on

each issue. In fact, assuming the two sides are truthful, there is no better split for both,

which makes the AW settlement efficient.

In addition, it is equitable, because each side gets exactly the same amount above

50%, with this figure increasing the greater the differences in the two sides’ valuations of

the issues. In effect, AW makes optimal trade-offs by awarding issues to the side that

most values them, except as modified by the equitability adjustment that ensures that both

sides do equally well (in their own subjective terms, which may not be monetary). On

the other hand, if the two sides have unequal claims or entitlements, as specified, for

example, in a contract, AW can be modified to give each side shares of the total

proportional to its specified claims.

3 AW may require the transfer of more than one issue, but at most one issue must be divided in the end.

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Can AW be manipulated to benefit one side? It turns out that exploitation of the

procedure by one side is practically impossible unless that side knows exactly how the

other side will allocate its points. In the absence of such information, attempts at

manipulation can backfire miserably, with the manipulator ending up with less than the

minimum 50 points its honesty guarantees it (Brams and Taylor, 1996, 1999b).

While AW offers a compelling resolution to a multi-issue dispute, it requires

careful thought to delineate what the issues being divided are, and tough bargaining to

determine what winning and losing means on each. More specifically, because the

procedure is an additive point scheme, the issues need to be made as independent as

possible, so that winning or losing on one does not substantially affect how much one

wins or loses on others. To the degree that this is not the case, it becomes less

meaningful to use the point totals to indicate how well each side does.

The half dozen issues identified in the executive-compensation example overlap to

an extent and hence may not be viewed as independent (after all, might not the bonus be

considered part of salary?). On the other hand, they might be reasonably thought of as

different parts of a compensation package, over which the disputants have different

preferences that they express with points. In such a situation, losing on the issues you

care less about them than the other side will be tolerable if it is balanced by winning on

the issues you care more about.

4. Indivisible Goods

The challenge of dividing up indivisible goods is daunting. The main criteria I

invoke are efficiency (there is no other division better for everybody, or better for some

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players and not worse for the others) and envy-freeness (each player likes its allocation at

least as much as those that the other players receive, so it does not envy anybody else).

But because efficiency, by itself, is not a criterion of fairness (an efficient

allocation could be one in which one player gets everything and the others nothing), I

also consider other criteria of fairness besides envy-freeness, including Rawlsian and

utilitarian measures of welfare.

I present two paradoxes, from a longer list of eight in Brams, Edelman, and

Fishburn (2001),4 that highlight difficulties in creating “fair shares” for everybody. But

they by no means render the task impossible. Rather, they show how dependent fair

division is on the fairness criteria one deems important and the trade-offs one considers

acceptable. Put another way, achieving fairness requires some consensus on the ground

rules (i.e., criteria), and some delicacy in applying them (to facilitate trade-offs when the

criteria conflict).

I make five assumptions. First, players rank indivisible items but do not attach

cardinal utilities to them. Second, players cannot compensate each other with side

payments—the division is only of the indivisible items. Third, players cannot randomize

among different allocations, which is a way that has been proposed for “smoothing out”

inequalities caused by the indivisibility of items. Fourth, all players have positive values

for every item. Fifth, a player prefers one set S of items to a different set T if (i) S has as

many items as T and (ii) for every item t in T and not in S, there is a distinct item s in S

and not T that the player prefers to t. For example, if a player ranks items 1 through 4 in

order of decreasing preference 1 2 3 4, I assume that it prefers

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• the set {1,2} to {2,3}, because {1} is preferred to {3}; and

• the set {1,3} to {2,4}, because {1} is preferred to {2} and {3} is preferred to {4},

whereas the comparison between sets {1,4} and {2,3} could go either way.

Paradox 1. A unique envy-free division may be inefficient.

Suppose there is a set of three players, {A, B, C}, who must divide a set of six

indivisible items, {1, 2, 3, 4, 5, 6}. Assume the players rank the items from best to worst

as follows:

A: 1 2 3 4 5 6

B: 4 3 2 1 5 6

C: 5 1 2 6 3 4

The unique envy-free allocation to (A, B, C) is ({1,3}, {2,4}, {5,6}), or for simplicity

(13, 24, 56), whereby A and B get their best and 3rd-best items, and C gets its best and 4th-

best items. Clearly, A prefers its allocation to that of B (which are A’s 2nd-best and 4th-

best items) and that of C (which are A’s two worst items). Likewise, B and C prefer their

allocations to those of the other two players. Consequently, the division (13, 24, 56) is

envy-free: All players prefer their allocations to those of the other two players, so no

player is envious of any other.

Compare this division with (12, 34, 56), whereby A and B receive their two best

items, and C receives, as before, its best and 4th-best items. This division Pareto-

4 For a more systematic treatment of conflicts in fairness criteria and trade-offs that are possible, see Brams and Fishburn (2000), Edelman and Fishburn (2001), Herreiner and Puppe (2002), Brams, Edelman, and Fishburn (2004), Brams and Kaplan (2004), and Brams and King (2004).

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dominates (13, 24, 56), because two of the three players (A and B) prefer the former

allocation, whereas both allocations give player C the same two items (56).

It is easy to see that (12, 34, 56) is Pareto-optimal or efficient: No player can do

better with some other division without some other player or players doing worse, or at

least not better. This is apparent from the fact that the only way A or B, which get their

two best items, can do better is to receive an additional item from one of the two other

players, but this will necessarily hurt the player who then receives fewer than its present

two items. Whereas C can do better without receiving a third item if it receives item 1 or

2 in place of item 6, this substitution would necessarily hurt A, which will do worse if it

receives item 6 for item 1 or 2.

The problem with efficient allocation (12, 34, 56) is that it is not assuredly envy-

free. In particular, C will envy A’s allocation of 12 (2nd-best and 3rd-best items for C) if it

prefers these two items to its present allocation of 56 (best and 4th-best items for C). In

the absence of information about C’s preferences for subsets of items, therefore, we

cannot say that efficient allocation (12, 34, 56) is envy-free.5

But the real bite of this paradox stems from the fact that not only is inefficient

division (13, 24, 56) envy-free, but it is uniquely so—there is no other division, including

an efficient one, that guarantees envy-freeness. To show this in the example, note first

that an envy-free division must give each player its best item; if not, then a player might

prefer a division, like envy-free division (13, 24, 56) or efficient division (12, 34, 56),

5 Recall that an envy-free division of indivisible items is one that, no matter how the players value subsets of items consistent with their rankings, no player prefers any other player’s allocation to its own. If a division is not envy-free, it is envy-possible if a player’s allocation may make it envious of another player, depending on how it values subsets of items, as illustrated for player C by division (12, 34, 56). It is envy-ensuring if it causes envy, independent of how the players value subsets of items. In effect, a division that

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that does give each player its best item, rendering the division that does not envy-possible

or envy-ensuring. Second, even if each player receives its best item, this allocation

cannot be the only item it receives, because then the player might envy any player that

receives two or more items, whatever these items are.

By this reasoning, then, the only possible envy-free divisions in the example are

those in which each player receives two items, including its top choice. It is easy to

check that no efficient division is envy-free. Similarly, one can check that no inefficient

division, except (13, 24, 56), is envy-free, making this division uniquely envy-free.

Paradox 2. Neither the Rawlsian maximin criterion nor the utilitarian Borda-score

criterion may choose a unique efficient and envy-free division.

Unlike the example illustrating paradox 1, efficiency and envy-freeness are

compatible in the following example:

A: 1 2 3 4 5 6

B: 5 6 2 1 4 3

C: 3 6 5 4 1 2

There are three efficient divisions in which (A, B, C) each get two items: (i) (12, 56, 34);

(12, 45, 36); (iii) (14, 25, 36). Only (iii) is envy-free: Whereas C might prefer B’s 56

allocation in (i), and B might prefer A’s 12 allocation in (ii), no player prefers another

player’s allocation in (iii).

Now consider the following Rawlsian maximin criterion to distinguish among the

efficient divisions: Choose a division that maximizes the minimum rank of items that

is envy-possible has the potential to cause envy. By comparison, an envy-ensuring division always causes

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players receive, making a worst-off player as well off as possible. Because (ii) gives a

5th-best item to B, whereas (i) and (iii) give players, at worst, a 4th-best item, the latter two

divisions satisfy the Rawlsian maximin criterion.

Between these two, (i), which is envy-possible, is arguably better than (iii), which

is envy-free: (i) gives the two players that do not get a 4th-best item their two best items,

whereas (iii) does not give B its two best items.6

A modified Borda count would also give the nod to envy-possible division (i).

Awarding 6 points for obtaining a best item, 5 points for obtaining a 2nd-best item, . . ., 1

point for obtaining a worst item in the example, (ii) and (iii) give the players a total of 30

points, whereas (i) gives the players a total of 31 points.7 This criterion might be

considered a measure of the overall utility or welfare of the players. Thus, neither the

Rawlsian maximin criterion nor the utilitarian Borda-score criterion guarantees the

selection of the unique efficient and envy-free division.

5. Conclusions

The two foregoing paradoxes pinpoint difficulties in dividing up indivisible items

so that criteria like efficiency, envy-freeness, and Rawlsian and utilitarian notions of

welfare are all satisfied. These conflicts are independent of the procedure of fair division

envy, and an envy-free division never causes envy. 6 This might be considered a second-order application of the maximin criterion: If, for two divisions, players rank the worst item any player receives the same, consider the player that receives a next-worst item in each, and choose the division in which this item is ranked higher. This is an example of a lexicographic decision rule, whereby alternatives are ordered on the basis of a most important criterion; if that is not determinative, a next-most important criterion is invoked, and so on, to narrow down the set of feasible alternatives. 7 The standard scoring rules for the Borda count in this 6-item example would give 5 points to a best item, 4 points to a 2nd-best item, . . ., 0 points to a worst item. I depart slightly from this standard scoring rule to ensure that each player obtains some positive value for all items, including its worst choice, as assumed earlier.

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that is used. While the procedures I illustrated for dividing up cake among three players,

or several divisible items between two players, ensure efficiency and envy-freeness,

equitability is not satisfied for cake-cutting, and efficiency, envy-freeness, and

equitability cannot all be guaranteed if there are more than two players.

Patently, fair division is a hard problem, whatever the things being divided are.

While some conflicts are ineradicable, as the paradoxes demonstrate, the trade-offs that

best resolve these conflicts are by no means evident. Understanding these may help to

ameliorate, if not solve, practical problems of fair division, ranging from the splitting of

the marital property in a divorce to determining who gets what in an international dispute.

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