Weingast B.R., Wittman D. The Oxford Handbook of Political Economy
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432 fair division
determine what winning and losing means on each. More specifically, because the
procedure uses 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 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, such as a car, a boat, or a house in a
divorce, is daunting, though sometimes such goods can be shared (usually at different
times). The main criteria I invoke are efficiency (there is no other division better for
everybody, or better for some 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 alloca-
tion 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 (to be defined).
I present two paradoxes, from a longer list of eight in Brams, Edelman, and
Fishburn (2001),
9
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-offsone
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-offswhenthecriteriaconflict).
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 (e.g. money)—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
⁹ 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 2003; Brams and Kaplan 2004; Brams and King 2005.
steven j. brams 433
adifferent set T if (i) S has as many items as T and (ii) for every item t in T and not in
S,thereisadistinctitems in S and not in T that the player prefers to t. For example,
if a player ranks four items in order of decreasing preference, 1234, I assume that it
prefers
r
the set {1, 2}to{2, 3}, because {1}ispreferredto{3}; and
r
the set {1, 3}to{2, 4}, because {1}ispreferredto{2}and{3}ispreferredto{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: 123456
B: 432156
C: 512634
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 third-best items, and C gets its best and
fourth-best items. Clearly, A prefers its allocation to that of B (which are A’s second-
best and fourth-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
twoplayers,sonoplayerisenviousofanyother.
Compare this division with (12, 34, 56), whereby A and B receive their two best
items, and C receives, as before, its best and fourth-best items. This division Pareto-
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
betterwithsomeotherdivisionwithoutsomeotherplayerorplayersdoingworse,
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 item 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)isthatitisnotassuredly envy-
free. In particular, C will envy A’s allocation of 12 (second-best and third-best items
for C) if it prefers these two items to its present allocation of 56 (best and fourth-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)isenvy-free.
10
¹⁰ Recall that an envy-free division of indivisible items is one in which, 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
434 fair division
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)orefficient division
(12, 34, 56), that does give each player its best item, rendering the division that does
not do so 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
envyanyplayerthatreceivestwoormoreitems,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 compat-
ible in the following example:
A: 123456
B: 562143
C: 365412
There are three efficient divisions in which (A, B, C) each get two items: (i) (12, 56,
34); (ii) (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
players receive, making a worst-off player as well o
ff as possible.
11
Because (ii) gives a
fifth-best item to B, whereas (i) and (iii) give players, at worst, a fourth-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 fourth-best item their two best
items, whereas (iii) does not give B its two best items.
12
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 is envy possible has the potential to cause envy. By comparison, an envy-ensuring division
always causes envy, and an envy-free division never causes envy.
¹¹ This is somewhat different from Rawls’s (1971) proposal to maximize the utility of the player with
minimum utility, so it might be considered a modified Rawlsian criterion. We introduce a rough
measure of utility next with a modified Borda Count.
¹² 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
steven j. brams 435
Next consider a modified Borda Count, which also gives the nod to envy-possible
division (i). Awarding 6 points for obtaining a best item, 5 points for obtaining a
second-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.
13
This criterion might be thought of as 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 a unique efficient and envy-free division.
5 Conclusions
.............................................................................
The squeezing procedure I illustrated for dividing up cake among three players
ensures efficiency and envy-freeness, but it does not satisfy equitability. Whereas
adjusted winner satisfies efficiency, envy-freeness, and equitability for two players
dividing up several divisible goods, all these properties cannot be guaranteed if there
are more than two players. Finally, the two paradoxes relating to the fair division
of indivisible goods, which are independent of the procedure used, illustrate new
difficulties—that no division may satisfy either maximin or utilitarian notions of
welfare and, at the same time, be efficient and envy free.
Patently, fair division is a hard problem, whatever the things being divided are.
While some conflicts are ineradicable, 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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part vii
..............................................................
PUBLIC FINANCE
AND PUBLIC
ECONOMICS
..............................................................
chapter 25
..............................................................
STRUCTURE AND
COHERENCE IN
THE POLITICAL
ECONOMY OF
PUBLIC FINANCE
..............................................................
stanley l. winer
walter hettich
No one can quarrel with the requirement that the budget plan should be
designed to maximize welfare. . .
(Musgrave and Peacock 1958,xi)
At first sight it might be argued that anyone who set himself up as a judge
and wished to establish standards for the distribution of expenditure or
amounts of revenue different from those approved by Parliament, must
belongtooneofthreecategories:eitherhemustbementallytheequal
of that average intelligence [in Parliament], in which case he could not
∗
For helpful comments we thank Timothy Besley, Tom Borcherding, Roger Congleton, Amihai
Glazer, Randall Holcombe, Kai Konrad, Gary Miller, Paul Rothstein, Dan Usher, and editors Barry
Weingast and Don Wittman, as well as participants in seminars at Carleton University, the Claremont
Graduate School, and the University of Western Ontario. Winer’s research was supported by a
Canada-United States Fulbright Scholarship during the 2003–4 academic year and by the Canada
Research Chair program. The hospitality extended to Winer by the Department of Economics and the
Center for the Study of Democracy at UC Irvine is also gratefully acknowledged. Errors and omissions
remain the responsibility of the authors.