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).
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
2
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
3
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 BramsTaylor-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
4
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 evennumbered players “he” and odd-numbered players “she.” In the case of the 3-person, 2cut “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 envyfreeness, see Zeng (2000).
5
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
6
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
7
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).
8
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
100
100
Total
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
9
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 nextsmallest 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.
10
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.
11
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
12
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
13
• 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 4thbest items. Clearly, A prefers its allocation to that of B (which are A’s 2nd-best and 4thbest 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).
14
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 envyfree. 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 envyensuring if it causes envy, independent of how the players value subsets of items. In effect, a division that
15
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
16
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.
17
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.
18
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