Priority Rules and Other Asymmetric Rationing Methods
Hervé Moulin
Econometrica, Vol. 68, No. 3. (May, 2000), pp. 643-684.
Stable URL:
http://links.jstor.org/sici?sici=0012-9682%28200005%2968%3A3%3C643%3APRAOAR%3E2.0.CO%3B2-O
Econometrica is currently published by The Econometric Society.
Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at
http://www.jstor.org/about/terms.html. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained
prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in
the JSTOR archive only for your personal, non-commercial use.
Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at
http://www.jstor.org/journals/econosoc.html.
Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed
page of such transmission.
The JSTOR Archive is a trusted digital repository providing for long-term preservation and access to leading academic
journals and scholarly literature from around the world. The Archive is supported by libraries, scholarly societies, publishers,
and foundations. It is an initiative of JSTOR, a not-for-profit organization with a mission to help the scholarly community take
advantage of advances in technology. For more information regarding JSTOR, please contact [email protected].
http://www.jstor.org
Mon Mar 31 06:18:12 2008
Econon~etrica,V ol. 65, No. 3 (May. 20001, 643 -654
PRIORITY RULES AND OTHER ASYMMETRIC RATIONING METHODS In a rationing problem, each agent demands a quantity of a certain commodity and the
available resources fall short of total demand. A rationing method solves this problein at
every level of resources and individual demands. We impose three axioms:
Consistency-with
respect to variations of the set of agents-Upper Composition and
Lower Composition-with respect to variations of the available resources.
In the model where the commodity comes in indivisible units, the three axioms
characterize the family of prioriy rz~les,where individual demands are met lexicographically according to an exogeneous ordering of the agents.
In the (more familiar) model where the commodity is divisible. these three axioms plus
Scale Invariance-independence of the measurement unit-characterize a rich family of
methods. It contains exactly three winmetric methods, giving equal shares to equal
demands: these are the familiar proportional. uniform gains, and uniform losses methods.
The asymmetric methods in the family partition the agents into priority classes; within
each class, they use either the proportional method or a weighted version of the uniform
gains or uniform losses methods.
KEYWORDS:Rationing, priority rule, consistency, composition, scale invariance
1.
RATIONING: ENDOGENEOUS DEMANDS, EXOGENEOUS ASYMPvlETRIES
THE SIMPLE MODEL OF RATIONING DISCUSSED in this paper is perhaps the oldest (O'Neill (1982), Banker (1981), Rabinovitch (1973)) and surely the simplest
formal model of distributive justice. A rationing problem among the agents from
N = {I,2,. . . , n} is a list of "demands7'x,, a nonnegative number for each agent i
in N, and an amount t , also a nonnegative number, to be divided in nonnegative
shares among the agents in N. We speak of rationing because the available
resources cannot satisfy all demands t < C , x , .
The same formal model arises in a variety of contexts. In the "inheritance"
context (07Neill(1982)) the demands are documented by legal deeds signed by
the deceased, so that the number xi represents a legitimate claim on the
resources; the "bankruptcy" interpretation (Aumann and Maschler (1985)) is
similar, with each creditor producing a verifiable debt to support his demand,
and the total debt exceeding the liquidation value of the bankrupt firm. In the
"taxation" interpretation (Young (1988, 1990)), xi represents agent i's tax
liability (her taxable income) and C , x , - t is the total tax to be levied; a related
interpretation is the cost-sharing of a public good (Moulin (1987)) where xi is
agent i's benefit from the public good and C,\,X, - t is its cost, to be shared
among beneficiaries.
'
Many stimulating conversations wit11 Scott Sllenker have been vei-jr helpful. The constructive
comments of three referees and of the Editor have greatly improved the paper. This paper was born
during a visit at the Universitat Autonoma de Barcelona, May-June, 1997; financial support by the
Foundation BBV is gratefully acknowledged.
Rationing is by far the richest interpretation of the model: the examples range
Lrorn the distribution of medical assistance in war or disaster situations (the
"'medical triage" problem: Winslow (1982)), of food supplies to refugees, organs
f o ~transplant (Elstel (1992)), scats in theaters, overbooked planes or colleges
(Hofstee (1990)), and visas to potential immigrants. Elster (1992, especially
Chapters 2, 3) offers a good survey of the empirical literature on rationing. In
tlic microeconomic literature, rationing is a consequence of the rigidity of prices
and inspires a whole literature on "disequilibrium" reviewed in Benassy (1982)
(see also Drkze (1975)). In the network literature, queuing is almost synonymous
with rationi~ig:each user of tlie network sends a certain number of "jobs," and
the serving algorithm decides which job is handled first and so on. See Shenker
(1995), Demers, I<esliav, and Shenker (1990), as well as Gelenbe (19831, Gelenbe
and Mitrani (1980).
Naturally, tlie actual rationing method used depends much on the specific
context. For instance, egalitarian methods are likely to prevail when distributing
limited food supplies, because the "demands" represent an objective need for
nourishment, whereas proportional rationing is compelling when sharing the
joint cost of a public good among business partners. In the case of taxation, the
tax schedule is designed to strike a compromise between the redistribution goal
(progressivity of taxation), and the incentives considerations pulling toward
proportional taxation, or even head tax (Young (1988)). And so on.
Despite the heterogeneity of problems to which the formal model is being
applied, we can gain deep insights into the logic of rationing by looking at
several structural properties that transcend the particular application. Typically
these properties express an invariance of the solution to certain changes in the
parameters of the problem, such as a change in the measurement unit of the
resources being distributed (the Scale Invariance Axiom; see below), or a change
in the set of agents among whom the distribution must take place (the Consistency Axiom; see below).
This paper follows that tradition, by offering a complete characterization of
in
all rationing methods satisfying three or four such invariance axioms."et
one important aspect this paper departs from the existing literature on rationing
nlethods. The latter, with very few exceptions, focuses on g)s?zinetric methods,
that is to say, imposes the Equal Treatrnerzt of Equals Axiom: if two participants
in the allocation problem make equal demands, then they should receive the
same shares of resources. In other words, the agents in society N have a pnori
equal rights, and the differences in their individual shares is motivated exclusively by the pattern of differences in individual demands. The prominent
symmetric methods are the proportional method, the uniform gains methods
and a few others.'
' Three in the discrete model, and four in the co~ltinuousmodel, as explained below.
,'Such as the Talmudic method of Aumann and Maschler (19851, the equal sacrifice methods and
the more general parametric methods: see Section 3 and Example 3 in Section 6.
PRIORITY RULES
645
E q ~ a Treatment
l
of Equals is conlpelling in the manly contexts where 110 a
priori discrimination can be allowed (e.g. when designing a direct income tax
schedule). In other cases, we (lo want to discrimiaate anlong the recipients of
the resources independently of the size of their demand. Creditors in a
bankruptcy situation are cornnlonly partitioned in priority classes, so that two
creditors with equal debts but different priority status receive, in general,
unequal shares: an example is the American bankruptcy law (Kaminski (1997)),
discussed at the end of Section 2. A paradigmatic example of priority ordering is
the sharing of a kill among a pride of lions, with the dominant male lion getting
a full meal, before the lion next in order can approach the kill and so on.
The goal of this paper is to study axiomatically the rationing problem without
imposing Equal Treatment of Equals. There are often differences between
agents that are not captured by the size of the individual demands: these
differences may reflect exogeneous rights-as
in the bankruptcy example
above-or some less precise notion of need, merit, and so on. In order to
accommodate these differences, it is necessary to drop the symmetry recluirement and let the arbitrator choose a method that fairly reflects the exogeneous
(i.e., independent in the differences in demands) asymmetries between agents.
Our axioms do not, by themselves, force any symmetry or asymmetry into the
method, they are typically satisfied by some symnletric rationing methods and by
some asymmetric ones. Anlong methods treating equal demands equally, the
proportional and uniform gains methods stand out by virtue of their multivarious axiomatic properties (see, for instance, the literature survey of Thomson
(1995)). Similarly the priority rules following a fixed priority ordering as rigorously as hungry lions, stand out as the most natural methods among asymmetric
ones: indeed they are the nlost asymmetric rationing methods. Our main result
(Theorem 2) characterizes a family of rationing methods containing the priority
rules, the proportional and uniform gains methods, as well as a family of
intermediate methods capturing arbitrary degrees of asymmetry.
2.
OVERVIEW OF THE RESULTS
The first result of the paper explains the prominence of priority rules in the
rationing model where demands as well as the resources to be distributed come
in indivisible units: the numbers .v,, t , as well as the individual shares j; are all
no~lnegative integers. This lllodel encompasses the allocation of organs for
transplants, college adn~issions,and in general queuing problems where indiviclual demands consist of a finite number of "jobs" (e.g., packets processed by the
Internet) and so on. We call it the discrete rationing model.
Despite its empirical relevance, the discrete inodel has been all but ignored by
the axiomatic literature on distributive justice: the only exception is in the
Appendix of Young (1994) discussed in Section 3; the axiomatic survey on
rationing by Thomson (1995) does not even mention the discrete model.
In the discrete model, Equal Treatment of Equals is structurally impo~sible,~
and so the priority rules are the only "natural" rationing methods; the proportional method, for instance, can only be approximated in the discrete model.
Theorem 1 offers a characterization of priority rules by means of three axioms
discussed below.
The continuous rationing model is the one discussed by virtually all the
literature. The individual demands xi, the resources t to be divided, and the
individual shares y, are now nonnegative real numbers.'
The three axioms that are the subject of this paper are called Consistency,
Upper Composition, and Lower Composition. Their definition is the same in the
discrete and continuous models. Consistency expresses the invariance of our
rationing method to certain changes in the jurisdiction of the problem, namely
the set N of agents among which the division takes place. Specifically, Consistency says that if y =
is the division of t selected by our rationing
method at the demand profile x, then y _ i = (y,), N,i should be the division of
t -yi selected at the demand profile x - ~ M
. ore generally, if we restrict the focus
of the distribution problem to a subset M of N, among whom is to be
distributed what is left over after the members of N \ M have taken their
shares, then the shares for the members of M remain the same as in the
original problem.
The consistency property has been studied in a variety of models of distributive justice, such as axiomatic bargaining, values of cooperative games, public
decision-making with transfers, matching, fair division of unproduced commodities, cooperative production, and more. Thomson (1990) and Maschler (1990)
give good surveys of the abundant literature. Without a doubt, consistency has
been the most important subject of research within the area of axiomatic
allocation of resources in the last fifteen years or so.
The Upper and Lower Compositions properties are invariance properties with
respect to changes in the amount of resources t being distributed. Given an
initial profile of demands x, suppose that the arbitrator is told first that at most
the amount t will be available and computes the corresponding shares y. Next
an additional cut is announced, and she learns that only t ' units, t ' < t , are
available after all; she should compute the correct shares y ' from the initial
demand x; however the axiom Upper Composition allows her to use the
"optimistic" shares y as the demand profile, that is to say, Upper Composition
requires that y ' be the correct shares for that demand profile as well. In other
words, when Upper Composition holds, she can use the optimistic assessment t
to update the demand profile from x to y and forget about the initial demands
entirely.
.
Think of two agents asking one candy each, when there is only one candy to give away.
'A third model, relevant for the apportionment of seats in a legislative body, has real numbers x,
and integers t , y,. Think of x, as the fraction of the total population in state i, and of t as the
number of seats in Congress; the problem is to allocate yi seats to state i so as to approximate the
ideal of "one man one vote": see Balinski and Young (1982).
PRIORITY RULES
647
The axiom Lower Composition expresses a "dual" invariance property pertaining to a pessimistic assessment of the available resources6 Now our arbitrator is given a demand profile x and learns that the available resources will be at
least t. The rationing method recommends to allocate t as J, among N. Once this
is done, agent i's demand is reduced from xi to xi -yi, and any additional
resources t ' that become available later will be distributed according to the
demand profile x - y. Now Lower Composition requires that this allocation of
the resources t t' in two steps, yields precisely the same result as the direct
allocation of t + t ' according to the initial demands x.
The Upper and Lower Compositions properties have been introduced in the
rationing model by Moulin (1987) and Young (1988). They have fewer relatives
than Consistency in other models of distributive justice. Still Lower Composition
is related to the property "Step by Step Negotiation" introduced in the model of
axiomatic bargaining by Kalai (1977).' It is also reminiscent of the "Path
Independence" Axiom for choice functions (Plott (1973)). Finally in Section 7 we
discuss the close link of Upper and Lower Compositions with the Distributivity
Axiom in the cost sharing model of Moulin and Shenker (1999).
This paper offers two characterizations. The first one, in the discrete model,
says that the priority rules (with a priority ordering independent of demands) are
the only rationing methods satisfying the three properties Consistency, Upper
and Lower Composition (Theorem 1 in Section 5). Contrast this sharply negative
result with the rich gamut of rationing methods meeting the same three
properties in the continuous model (described in the next paragraphs). 111 the
discrete model the three properties together rule out any compromise for the
sake of equity. Section 5 offers a couple of approximately equitable methods
meeting tsvo of the three axioms.
111 the continuous model, the priority rules satisfy Consistency, Upper and
Lower Compositions but so do many other methods. Two important examples
are the proportional method (shares yi are proportional to demands xi) and the
uniform gains method (equalizing shares as much as possible under the constraint yi Ix,;see Section 6). On the other hand, some often discussed symmetric methods fail Upper and Lower Compositions: an example is the contested
garment method (defined for the case n = 2) and its consistent extension as the
Talmudic method of Aumann and Maschler (1985). The random priority method
(O'Neill(1982)) even fails all three axioms Upper Composition, Lower Composition, and Consistency.
Our second characterization (Theorem 2 in Section 6) describes in the
continuous model all rationing methods satisfying Consistency, Upper and
Lower Compositions, as well as a fourth axiom called Scale Invariance. The
+
In Section 4 we define formally the duality operation on rationing methods. and show that
Lower Composition is indeed the dual property of Upper Composition.
S t e p by Step Negotiation considers two nested bargaining sets. and uses the bargaining solution
for the smaller bargaining set as the disagreement point for the larger bargaining set. It is the basis
for the characterization of the egalitarian solution.
648
H E R V ~MOULIN
latter is the familiar requirement that a simultaneous increase, in the same
proportions, of all demands x, and the resources t must result in the same
proportional increase of the shares y,. The usual interpretation of this property
is that a change in the measurement unit of the commodity in which the
demands, resources and shares are all written, is irrelevant; equivalently the size
of the resources and individual denlands does not matter: the same rules govern
the allocation of $1 or of $10,000.
The family of (continuous) rationing methods meeting Consistency, Upper
and Lower Compositions, and Scale Invariance is quite interesting. It contains a
continuum of methods, striking a compromise between maximal asymmetry, i.e.,
the priority rules, and full symmetry, i.e., Equal Treatment of Equals. But the
path along which the arbitrator can adjust the degree of asymmetry is quite
narrow. If Equal Treatment of Equals is required, exactly flzree methods are
possible: proportional, uniform gains, and its "dual" method uniform loses
(equalizing losses x,-y, as much as permitted by the constraint y, 2 0): Corollary to Theorem 2. If the arbitrator aims at some asymmetry but rules out any
absolute priority of one agent over another, then her choice is limitedqo the
proportional method, and to asymmetric generalizations of uniform gains and
uniform losses in which the gains (or losses) follow some exogeneous weights
among agents (again, adjustments must be made for the constraints y,sx,and
y,2 0 respectively: see Section 6). We call these methods weighted gains and
weighted losses methods respectively.
Finally, the full family of methods described in Theorem 2 results from the
combination of a priority yreordering, and the basic methods proportional,
weighted gains, and weighted losses. For instance, the American bankruptcy law
partitions the creditors in four groups, federal government, secured claims,
trustees, unsecured claims; within each group of creditors proportional rationing
prevails; across groups, a rigorous priority is enforced with the federal government having the highest priority followed by the secured claims and so on (see
Kaminski (1997) for more details on this and other examples of asymmetric
rationing). The methods uncovered in Theorem 2 allow other variants where
among some groups of creditors, proportional rationing is replaced by uniform
(or weighted) gains (or losses).
3.
RELATION TO THE LITERATURE
As mentioned above, the only previous results in the discrete rationing model
of which I am aware are two theorems in the Appendix of Young (1994).
Theorem 1 there is (almost) a particular case of our Theorem 1 in the case
where all agents demand at most one unit of the good. Remark 1 in Section 5
explains the connection in detail. The other relevant result in Young (1994) is
discussed below in this section.
This claim holds only if thrce or morc agents are involved: the case n
methods: sce Lemma 4 in Section 6.
=2
allows for more
.
.
The main rcsult in tllc literature on continuous rationing mcthods 1s Young
(1983, cl~aracterizingthc family of pnizr171~ft.i~
ii~ethodsby thc combination of
Consistency. a stronger version of Equal Treatment of Equals. ancl a contin~~ity
requirement. A parametric method is constructed by choosing a real valued
function lr( A, s,) that is defined o\.cr R _,continuous and strictly increasing in A,
and such that /do, .Y,)= 0 and / I ( -;
x.s,)= .Y!. Then for all s , . . . . , .)K,! and all t
such that t < C , .Y,.the system
has a unique solutioil. defining the parametric rationing method in cluestion.
All three symmetric methods allov.ecl by our Theorem 3--proportional.
uniform gains, and uniform losses-are parametric, although the corresponding
function 11 is not strictly increasing in h in the case of the latter two mcthods:
IT( A, s, 1 = inill{ A,s , } for thc uniform gains method,
'l'f-:' \
h ( A . , ~ , ) = m a x1
O \ for the uniiom~li>iiei oiethotl
Yet another result by Young (1988) bears an even closer relation to our
Theorem 2. That result explores the impact of the Upper Composition and Scale
l~lvariancei-4.xioms on the set of parametric methocls. Young charucterizes the
fa~lliiyof equal sacrifice methods (described in Example 3, Section 6) by the
combination of Upper Compositio~l \\it11 Consistency, Equal Treatment of
Ecluals, and Strict Resource Monotonicity (a strengthening of our Resource
Monotonicit): (2) in Sectiori 4); he also describes the subset of those meeting
Scale Iitvaria~lceas well.
Because of the strict resource monotonicity recluireinent. neither the uniform
gains nor the utlifornl losses method is an cqual sacrifice methotl; tl-rerefore the
Coro1lal-y to our Theorem 2 (characterizing the three sq-mmetric neth hods within
our larger family) cannot be deduced from Young's result. Hon-ever. the
co~lncctionis stro~lg,anci leads to a couple ol' open questions described
int~~itive
in tlie co~lcludingSection 7.
From the point of view of the literature on Consistency. the contribution of
this paper is to explore a subset of a.s~~i~zr71ctr.i~
consiste~ltrationing methods,
\\here the previous literature only considers symn~etricones. Thc general model
of division according to "~ipes"(introduced in the Appendix of Young (lCj94))is
no exception. There individual demands are replaceci by the Illore general
notion of "type," and the method treats different types differently, itlcludirlg the
possilbility that one type has absolute priority over another type. In particular
Theorem 7 , page IS6 of Young (1994) is cast in the contest of the discrete
rationing ~llodel(with individual denlallds of arbitrary size, unlilce llis Thcorem
1) and offers a characterization of tlie priority rules based on a weaker version
of Consistency; this is quite siitiilar to our Theorem 1 in Scction 5 hut the
analogy is more superficial than it sounds. Indeed a key axiom for Young's
Theorem 7 is a version of Equal Treatment of Equals (adapted for the discrete
context by only requiring that the shares of two agents of the same type differ by
at most one unit), and the results depend crucially on the possibility of
replicating agents of the same type. Therefore the type model can not accommodate the finite societies, where each agent is of a different type, that are the
subject of this paper.
A close relative of the literature on rationing is the recent stream of papers
addressing the fair division of a single commodity under single peaked preferences. There a given amount of the commodity must be distributed among N
and agent i's preferences over his share y, of the commodity are single peaked,
with their peak at x,. The central axiom is the property of strategyprooj%zess,
namely the fact that truthful report of one's peak is a dominant strategy for all
agents at all profiles. The main result is that the uniform gains method is the
only symmetric and strategyproof method: Sprumont (1990, Ching (1994L9 But
the set of all strategyproof methods contains many asymmetric ones, and this
complicated family is described in Barber&, Jackson, and Neme (1997).
A rationing problem can be viewed as a "one sided" fair division problem
where the resources are always in short supply. If we interpret agent i's demand
x, as her most preferred consumption it makes sense to assume that agent i's
preferences are strictly increasing on [O, x,] and decrease, however slightly,
afterward. Then we can speak of a strategyproof rationing method in the same
way as for the fair division problem.
It is easy to show that the priority rules are all strategyproof, and so are the
weighted gains methods for any choice of weights. Thus the family uncovered in
our Theorem 2 contains many strategyproof methods, and all methods characterized in Theorem 1 are strategyproof. In a companion paper (Moulin (1999))
the two properties of Strategyproofness and Consistency are jointly applied to
rationing methods: in the discrete as well as continuous models they characterize the family of fixed path methods (such as Example 1, Section 5). In the
discrete model this family contains much more than the priority rules. In the
continuous model it is different from-but related to-the family uncovered in
Theorem 2.
The strategyproof allocation of indivisible goods is the subject of Papai (1999),
where the goods are heterogeneous (whereas they are homogeneous in our
discrete model): under the additional requirement of Pareto Optimality and
Non-Bossiness, that paper characterizes, essentially, the priority rules.
One last paper closely related to this one is Moulin and Shenker (1999). That
paper looks at the cost-sharing problem with variable demands of a homogeneous good, when the cost-sharing method can take into account the whole cost
function. The two axioms of Additivity (of cost shares with respect to the cost
Symmetry is interpreted as Anonymity, a stronger form of Equal Treatment of Equals, or as No
Envy in Sprumont (1991) and as Equal Treatment of Equals in Ching (1994).
65 1
PRIORITY RULES
function) and of Distributivity (nit11 respect to the composition of cost functions)
are combined to characterize a certain family of cost-sharing methods. That
result has deep connections with the present Theorem 2; see the last concludi~lg
comment in Section 7.
4.
THE TWO MODELS AND THE THREE MAIN AXIOMS
The following notdtion is used. We denote by N the set of nonnegative
integers and by R thdt of nonnegdtive red1 numbers. For any f~nzteset N we
denote by 8' the zth coordinate vector of R' dnd by ,i",the unit simplex of R'.
For any x in R' (or N \ ) and any subset M of N, we write
+
x,,,
C x, ;
=
1
x " is the projection of x on R" (or N "1.
t '$1
Throughout the paper we fix a set .,Pi,of (potential) agents: this set can be
finite or countably infinite. Yet we only consider rationing problems involving a
finite set N of agents.
A mtioni~zgproblemzis given by a finite subset N of .dl", a profile of individual
demands x,,one for each i E N , and a quantity t of "resources" to be divided
among N. We always assume: 0 I r IX,. A discrete rationing problem is one
where each demand xi and the resource t are ~lonnegativeintegers, namely, the
good to be divided comes in individual units. A continrio~isrationing problem is
one where each xi as well as t are nonnegative real numbers, namely the good
in question is divisible. As no confusion may arise, we use the same notation for
rationing problems and rationing methods in the discrete and continuous
models.
Each agent i in .4 has a maximal demand Xi. We assume Xi r + m , so
Xi = + x means that agent i's demand is not bounded above. Thus a demand
,v[O,XI], where the interval
profile x for the society N varies in X( N ) =
[0, Xi] is taken in N or R +.
n,,
DEFINITION
1: Given throughout the paper are .A/,, finite or countably infinite,
and XI, O < X I I + x,for each agent i in ./l;.
A rationing method associates with every (discrete or continuous) rationing
problem ( N ; t; x), where x E X ( N ) and O 5 t r x , , a profile of individual shares
y denoted y = r(N; t; x ) and such that
(1)
y
E
N'" (resp. y E R ? ) ;
O<yl I X , for all EN; 41,' = t
Throughout the paper, we only consider rationing methods meeting the
following mild property:
INDEPENDENCE
OF NULLD E ~ N D S
For
: all N, M such that M cN
x E X( N), all t
(2)
{x, = O for all i E N \ M } * { [ r ( ~t ;; x)]"
=r(~;t:x')}.
c . 4 ,all
This establishes a minimal link between the rationing method applied to society
N and to the subsociety M . If agents in N \ M demand nothing, they receive
nothing (by the feasibility condition (I)), and Independence of Null Demands
says that the di\ision of f among M is the same whether or not these null agents
are present. Independence of Null Demands is a consequence of-but a much
weaker requirement than-the powerful axiom of Consistency (see below).
For every subset ./ff of 14,we denote by r ' the natural projection of the
rationing method r on ./F:
for all Nc.A/, all x ~ X ( N ) , a l tl :
r '(N;t;x)=r(N;t;x).
For any finite subset N of ./& the Independence of Null Demands property
implies that r"' is entirely determined by the restriction r ( N ) of I. to the
rationing problems for society N, namely r(N)(t; x) = r(N; t ; x). Therefore
\\hen "/Pi,is finite, the entire rationing method is determined by 4 4 ) .
We denote by !Il(/"J;) the set of rationing methods (Definition 1) satisfying
Independence of Null Demands.
Note that, in the continuous model, a rationing method r(N; x; t) is not
necessarily a continuous function of either the demand profile x, or the
resources t. However, all methods characterized in Theorem 2 are globally
continuous with respect to x and t.
There is a natural duality operation on rationing methods that plays a key role
below. Given r in ML4'"), its dual r": is the following rationing method:
The rationing method r allocates a total loss t (namely x,%,- t units of resources) by deducing xi - r,(N; x , - t; x) from agent i's demand. Thus the dual
rationing method I.:" splits t units of resources as the method r would split t
units of losses. We let the reader check that r" meets properties (1) and (2), and
that we have: (I-")" = r.
We now define the three main invariance axioms; their definition is identical
in the discrete and continuous models.
CONSISTENCY:
For all N, all i, j E N, i # j, all t and all x,
UPPERCOMPOSITION:
For all N fixed (omitted in formula) all t, t ' and x,
For all N fixed (omitted in formula) all t, t ' and x,
LOWERCOMPOSITION:
0s t ' s t s x , * r ( t ; x ) = l . ( t l ; x ) + r ( t - t r ; x - r ( t J , x ) ) .
653
PRIORITY RULES
The Consistency Axiom is a well known and powerful requirement, linking
rationing methods for a "society" N and its subsocieties (see Section 2). If a
certain distribution of shares J among N is recommended by the rationing
method for a certain profile x of demands and quantity t, Consistency insists
that the restriction of y to any subset N ' of N (simply ignoring the shares of
agents in N \ N 1 ) be recommended for the restriction of x to N ' and the
quantity t ' = y , allocated to N ' in the initial problem: shrinking the jurisdiction
of the problem does not alter the correct decision.
Upper Composition (called Path Independence in Moulin (1987)) allows to
carry out a "partial" rationing when we know an Lipper bound t ' on total
;
resources: we may lower agent i's demand from its initial value x, to xi = ~ ; ( t 'x),
and consider xi as his new demand whenever further rationing of the resources
to distribute (from t ' to t) occurs.
Lower Composition (called Composition Principle in Young (1988) allows a
partial distribution of the resources when we know a lo11,er bound t ' of the
available resources: we distribute the shares d t ' , x) among N, and use agent 1's
residual demand x, - r,(tl; x) as the basis for distributing the additional resources t - t ' that become available later on.
An immediate consequence of Upper Composition and of feasibility (1) is the
familiar solidarity property known as Resource Monotonicity.
-
RESOURCE
MONOTONICITY:
For all N G-45,all x E X(N), all t, t ':
(3)
{0 t I t ' ~ x , } { r ( N ; t ; x ) 5 r ( N ; t ' ; x)}.
If we have more resources to distribute, no one ends up with a smaller
share.''
Note that Upper and Lower Compositions are "dual" axioms, that is to say
the rationing method I- meets Lower Composition if and only if its dual method
r meets Upper Composition. By contrast, Consistency is a self-dual axiom,
namely r is consistent if and only if I . - is consistent (we omit the straightforward
proof of these facts).
We make a final comment on the combination of the Upper and Lower
Compositions properties. Suppose the society N and demand profiles x are
known, but the actual amount of resources t to be divided is only known to be in
the interval [t,, tz], where 0 5 t, 5 t2 IX,~,.Then the combination of Upper and
Lower Compositions reduces the rationing problem to one with a reduced
demand profile 2 and total demand t - t,:
'"
set
,
then
d t , x)
-t
=d
,
t , ,x)
)
and
;=
t - t,;
+ r(F, 2 ) .
Some meaningful rationing methods fail Resource Monotonicity (hence Upper Composition as
well). For instance we may choose the shares p so as to maximize C iil,(x,-y,), where the fixed
functions 11, are increasing: if these fiinctions are not concave, the resulting rationing method fails
Resource Monotonicity (thanks to the co-editor for this remark).
Hence the rationing process can be more closely approximated as the bounds of
t become tighter. To check the above equality, set x ' = r(t2, x) and invoke
Upper and Lower Compositions several times:
=r(t,, x)
5.
+ r(t - t,, r ( t 2 ,x ) - r ( t l , x)).
PRIORITY RULES IN THE DISCRETE MODEL
First we define formally the priority rules in both the discrete and continuous
models. Then we show that in the discrete model, priority rules are characterized by the combination of Consistency, Upper and Lower Compositions.
We denote by a an ordering of No, namely a complete, transitive, and
antisymmetric binary relation. If a ranks i above j, we say that i has priority
over j. The restriction of a to a finite subset N of No, with cardinality n, is
represented by a bijection from {1,2,. . .,n} into N, also denoted a , with the
interpretation that a(1) = i means "agent i has the highest priority in N,"
a(2) =j means "agent j has the second highest priority," and so on."
The priority method r U ( N ) associated with the priority ordering a of N is
now defined. For all t, x the vector y = rU(N)(t;x) is the unique vector of
shares meeting property (1) and such that
Equivalently, we can compute y from the unique integer i*,O s i* ~n
such that
= IN1
by setting
i* yu(i)=
for
i = 1,.. . ,i*,
yu(ir+
=t -
xu(i),
i= l
y,,j,=O
for j = i h + 2 ,..., n.
Checking that r " meets property (2) is straightforward.
THEOREM
1: For any ordering a of Sv',, the priority rationing method r " satisfies
the three properties Consistency, Upper and Lower Compositions.
Conuersely, in the discrete model, a rationing method satisfiing the three properties Consistency, Upper and Lower Compositions, is the priority method r " for some
ordering a of No.
11
Note that if Mo is infinite, the initial ordering u is not necessarily representable as an
enumeration of No,namely a bijection from N into No",.
655
PRIORITY RULES
All proofs are gathered in the Appendix. Note that the first statement (r" is
consistent and meets Upper and Lower Compositions) is easy to check, whether
in the discrete or continuous model.
Theorem 1 is a negative result. The combination of our three requirements
leaves no room for even the slightest degree of compromise. A good example of
an approximately equitable method is now given. It shows why the combination
of the two composition axioms is so demanding in the discrete model.
EXAMPLE
1: Approximately Uniform Gains." The uniform gains method in
the continuous model gives the same share to each agent whose demand is not
met, and this common share is not smaller than any demand that is met; see
Section 6.
In the discrete model, we also need to choose a tie-breaking rule: the easiest
way is to fix an ordering of N, say N = {1,2,. . . , n}. Then we distribute the units
one at a time, in successive rounds where the agents are served in the given
ordering, deleting agents who have received their full demand. For instance,
with n = 3 and x = (3,1,4) we distribute 5 units as y = (2,1,2) and 6 units as
(3,1,2).
We define formally the solution y to the rationing problem (t, x). For any
integer A, we denote N(A) = {i E Nlx, 5 A} and M(A) = {i E Nlx, > A} with
cardinality m(A). Let A* be the largest integer satisfying
equivalently A* is the unique integer such that
h*.m(A*) +x,(,., 5 t < (A"
+ l).m(A*) +x,(,-,.
Set 6 = t - A" am( A*) - x,(,+, and define
= A*
- A*
+1
if i is ranked at most 6 within M(A"),
if i E M ( A"), yet is ranked strictly above 6.
A straightforward argument (omitted) shows that the rationing method just
defined is consistent and meets Upper Composition. On the other hand, it fails
Lower Composition. To see this, it is enough to choose n = 2 and to compute
t
= 4,
x = (3,2) =, r(4; (3,211 = (2,2)
1 3 , 2=1 0
3
2
=
2 ,1
with
A*
with
A* = 0,
with
A*
which is in contradiction of Lower Composition:
12
Thanks to an anonymous referee who suggested this example.
= 2,
=
1,
The lesson of Example 1 is this: if we want to equalize individual shares y, as
much as possible in spite of the indivisibilities, we are forced to keep track of
who has just been served; the distribution can't be history-independent as the
Lower Composition Axiom would require.
Note that the LC property is not, per se, incompatible with approximate
equity, namely the requirement that the shares of two agents with identical
demands differ by at most one unit. Indeed consider the dual of the method just
defined (approximately uniform losses): it satisfies Lower Composition and
Consistency, and fails Upper Composition. It is only the combination of Upper
and Lower Compositions that is hard to meet. For instance, if C"S,/= 2, Consistency is vacuously satisfied and Theorem 1 states the incompatibility of UC and
LC.
To conclude this section, we check that Theorem 1 is a tight result. Drop
Lower Composition and the method of Example 1 emerges; drop Upper
Composition and its dual emerges. Finally, we drop Consistency and assume that
Sv', contains at least three agents.
EXAMPLE
2: A Method Meeting Upper and Lo11,er Compositions but not Consistency. Assume .4= {1,2,3} and X , = 1 for i = 1,2,3. Consider the following
method r(Jy,):
d l ; (1, 171))= ( l , 0 7 0 ) ;
r(2; ( l , l , 1)) = ( l , l , O ) ;
r(l;(l,l,O~)=(1,O,O); r(l;~O,l,l))=~O,l,O);
r ( l ; ( l , O , l ) ) = (O,O, 1).
The method r and its dual are represented on Figure 1. On all figures we
describe a rationing method by highlighting some of the rationingpaths t -,r(t, x),
657
PRIORITY RULES
from 0 to x when t varies from 0 to n,. All methods are resource monotonic
(property (3)); therefore the paths are increasing.
In the discrete model with X, = 1 for all 1 . the set of demand profiles x is the
unit cube in N Upper Composition means that if n ' is on the path from 0 to x,
then the path from 0 to x ' obtains from the above by simple truncation.
Therefore the method is entirely described by the set of its maximal rationing
paths (paths that cannot be further extended), reaching every vertex of the unit
cube.
In the case J<, = {1,2,3) and the above method r , there are three maximal
paths depicted by a thick line on Figure 1: their endpoints are (1, I, I), (1,0, I),
and (0,1,1). For the dual method 1. , the path from 0 to x obtains from the
corresponding path in r by symmetry around x/2; see Figure 1, establishing
that both r and r meet Upper Composition. Hence, by duality. they both meet
Lower Composition. To see that they are not consistent, observe that
'.
( { 1 2 ;2
pet
1
=
1 and
r2({1,2,3);2;( I >1 , l ) ) = 1;
r,({1>3};2 - 1; ( 1 , l ) ) = 0.
RE\L~RK
1: Theorem 1, page 175 111 Yoctrzg (1994) rs closeb related to our
I . Corlslder tlze case ]there each agerlt dernarlds at most one ztnzt of
Tl~eoren~
zr~dz~
zszble good: XI = 1 for all r E /Po. Then It tc1r71~o ~ l tthat Conszrtency zn~plzes
both Upper and Lol~erColnposltzons (the proof of thzs fact u presented at the end of
that of Theor.enz 1 irz the Appenclrx). Tlzereforz our. Theorem rectds: a ratzolzirzg
method w a prrorzty nzethod z f aid orzk zf zt 1s conszstent. Yoe~r~g's
rzseilt is a
geneizrlzzatiorz of tlle latter statenlent to t l case
~
where the r.atzorzzrzg nzetlzocl a
nzultir~aluedarzd tlze prrouty order?rlg allolts for. rrzdrffererzces.
6.
CONTISUOUS
MODEL:
THE MAIS THEOREM
In the continuous model the priority rules satisfy. naturally, the same three
properties: Consistency, Upper Composition, and Lower Composition, but so do
many other methods. In particular. the three rationing methods most often
discussed in the literature meet these three axioms. They are defined as follows,
for any N , t , and x:
t
PROPORTION~L
METHOD:PRO. pro(N; t ; X) = - .X.
xv
U\IFOKMGAINSMETHOD: UG. For all i, ug,(N;t ; x-)
= min{h, x,)
where
C, min{h, x,) = t.
USIFORMLOSSESMETHOD:UL. For all i, ul1(N;t : x) = max{x, - p, 0) where
C, max{x, - p,O) = t.
The uniform gains and uniform losses methods are often called "constrained
equal gains" and "constrained equal losses" in the literature. Note that ug and
ul are dual of each other, whereas pro is self-dual.13 For our main characterization we need a fourth invariance property (a familiar requirement: see, e.g.,
Young (1988), Moulin (1987)), ruling out any influence of the measurement unit
of the resources being distributed:
SCALEINVARIANCE:
For all N, t, x, all A, 0 < A < 1,
r ( N ; At; Ax)
=
A.r(N; t; x ) .
Observe that pro; ug, ul, as well as all priority methods are scale invariant.
We now define two asymmetric generalizations of the uniform gains and
uniform losses methods respectively, that meet the four invariance properties
Consistency, Upper and Lower Compositions, and Scale Invariance. These
methods play a key role in Theorem 2.
DEFINITION
2: Given a set of positive weights wi, i €4,
t he Weighted Gains
method g" is given by:
forallN,t,xandalli~N:
g r ( N ; t ; x ) = m i n { A w i , x i } where
The Weighted Losses method lw,its dual, is given by:
for all N, t, x and all i E N :
l;(N; t; x )
= max{xi -
Aw,, O} where
LEMMA1: The Weighted Gains and Weighted Losses Methods meet the four
axioms Consistency, Upper and Lower Compositions, and Scale Invariance.
The straightforward proof is omitted. Figure 2 illustrates Definition 2 -in the
two agents case. The dashed line with slope w,/w, = 1/2 plays the role of the
diagonal in the Uniform Gains and Uniform Losses method. On Figure 2 the
demands (x,, x,) are fixed (to about (3,311 and the two rationing paths t -t
gW(t;x) and t -t lw(t;x) are depicted.
Note that a given priority method can be viewed as the limit of a sequence of
weighted gains (or losses) methods, where the weight wi becomes infinitely
larger than wj whenever i has priority over j (we omit the straightforward
details).
l 3 Young (1988) shows (Theorems 3, page 334) that pro is the only self-dual rationing method
satisfying Upper Composition (or, equivalently, Lower Composition). As Young himself remarks, self
duality is a strong property with no clear ethical meaning.
PRIORITY RULES
Next we define the operation of composition of rationing methods by a
priority preordering and note that this operation respects our four invariance
properties: hence we construct a rich family of methods with these four
properties.
DEFINITION
3: Given a rationing method r in M(SS,), and two agents i, j, we
say that r gives priority to i ouer j if j is never allocated any resource until i's
demand is met in full:
for all N such that i , j
E
N , for all t, x such that t I
Exi:
i
y, > O - y i = x , . In the next Lemma, we are given a preordering 2 of .Hi, namely a complete
and transitive binary relation onXo. We interpret the strict relation > associated
with 2 (i +j iff i 2 j but not j 2 i) as a strict priority relation. We denote by.&'"
an indifference class of 2 .
LEMMA2: Given are 2 , a preordering of A(,,and for each indifference class .Af
of 2 , a rationing method r , M
~ ( 4 . There exists a unique rationing method r,
r E %(Mi) such that:
.
.
r projects onto r,. for eveiy equiualer?ceclass Mi ry= r,;
r giues priority to i ouer j if i +j, for any i, j in 4.
We call r the 2 -priority composition of the methods r,J,.
660
HERV~.MOULIN
To prove Lemma 2, we derive the explicit formula for r as follows. Any finite
subset N of A/;is partitioned by the equivalence classes of 2 as N,, .. ., NK,
for all k. Given a demand profile x, denote t h = Cf=,x , , so that
with N, > N,
0 t' 5 ... 5 t = x , ~ a, nd compute the shares allocated by r as follows. If
t k I t 5 t L+ 1 :
+
(where
,
is the equivalence class of 2 containing N,).
LEMMA3: Notatiolzs as irz Lenznza 2. If each nzethod I;, rneets any one of the
four axioms Consistency, Upper and Lower Compositions, and Scale Inr>aliance,so
does their 2 -priority conzposition.
The proof follows by inspection of formula (4). Lemma 3 implies that any
2 -priority composition of methods taken among Proportional, Weighted Gains,
and Weighted Losses, does meet our four invariance axioms. Hence there is a
fairly large family on offer to the mechanism designer.
One common feature of the methods pro, g ' b n d I1+is that they do not
involve any strict prioritizing between agents: if the weights w are very unequal,
the methods g"' and I" are serving individual demands at a very asymmetric
rate, but it is not the case that an agent i with a large weight has priority over an
agent j with a small weight in the sense of Definition 3. On the contrary, in g'+
every agent with a positive demand receives a positive amount of resources (if t
is positive). Similarly in l", no agent receives her full demand unless t = x , .
Before stating Theorem 2, it remains to define a set of irreducible rationing
methods where no agent has priority over any other agent (Definition 3).
Theorem 2 states that any method meeting our four invariance axioms must be
the 2 -priority composition of such irreducible methods. For the case of three
agents or more, these irreducible methods consist exactly of the Proportional,
Weighted Gains, and Weighted Losses methods. But in the case of (an equivalence class .N with) exactly two agents, the Consistency Axiom is vacuous and, in
turn, the family of irreducible methods contains (infinitely) more elements than
pro, g'" and 1'". Our last preliminary result gives the precise meaning of this
claim.
, unit simplex of R ~ ' i,s borrowed
The notion of ordered covering of Y Nthe
from Moulin and Shenker (1999). It is useful in the proof of Theorem 2 (see
Appendix). For the sake of stating Theorem 2, however, we need only to define
this notion for the case N = {I,2).
661
PRIORITY RULES
is a partition (not necessarily finite) of 9<1,2)
An ordered coaering of q,,,)
by singletons {e} and oriented open intervals ]el, e2[. Note that if i
F contains an
interval ]el, e2[, it also contains the singletons {el} and {e,}. Given the ordered
covcring i? and some x E R(:~)\{O},we denote by C0(x) the smallest element of
527' containing ?, = (l/x,%,).x:it can be 2 itself or an interval lei, e2[ containing 2.
LEMMA4: Assume .A6 = {1,2}. To each ordered coaering (G of
associate a rationing method in !%(-I/b) as follows: for all x E R(:'}\ (0)
(5) q,,2,
we
if C O ( x )= ] e l , e2[, then x = h , e 1 i h 2 e ' for some A , , A, > 0, and
r ( t ; x ) = t . e l for O i t < A , ,
r(t:x)=A,.e1+(t-h1).e2
for
h,itIxjlzY
This rationing method r meets the properties Upper and Lower Compositions, and
Scale Inaariance. Con~erselyany rationing vrzetlzod in /@meeting
Upper and Lower
Compositions, and Scale In~~ariance,
is associated by ( 5 ) with an ordered cor%ering
of .?,,,).We write Z;C/1',) for the set of rationing methods thus constructed.
The proof of Lemma 4 is in the Appendix. We illustrate the family Z; by
some examples.'"f
B' consists of all singletons {el of 9<1,2,,its rationing
method is the proportional one. If W = ({s'}, {s2},Is',: &I[) (recall that E' is the
ith coordinate vector), it yields the {2,l}-priority rule. The uniform gains method
is derived from the covering ({~'),{s'},{e},]e,sl[,]e, s2[), where e = ;(s1 + s 2 ) .
Its dual, uniform losses, obtains by exchanging the orientation of the two
intervals.
Obviously a?* contains (infinitely) many more methods. For instance, the
ordered covering may contain all the singletons {e}between s1and e as well as
{E'}, and the oriented interval ]c, s2[, SO the associated rationing method is a
hybrid of the proportional method "to the right of e mand of uniform gains "to
the left of e"; see Figure 3, showing the shape of the rationing path t + r(t. .XI
for four choices of x.
DEFINITION
4: Given a (finite or infinite) subset JV' of Mi,we define an
irredz~ciblemethod on JV as follows:
.
if
methods;
=
2, a method in 2Y;C,l') with the exception of the two priority
'' For a more detailed discussion, see Moulin and Shenker (1099).
if IMl2 3, one of the following methods:
proportional (restricted to M);
weighted gains gw,w E R<+;
weighted losses I", w E R$+ .
We denote b y Z * W ) the set of irreducible methods on M.
THEOREM2: Given are No (finite or infinite) and the maximal demand Xi,
0 < X i 5 + m, for each i €4.
The rationing method r E %(No)satisfies the four
axioms: Consistency, Upper and Lower Compositions, and Scale Invariance i f and
only if there exists a priority preordering 2 of 4,and, for each indifference classM
such that r is the t -priority composition
of 2 , an irreducible method r, EX*W),
of the methods r,(Lemma 2). We denote by X (NOthe set of methods thus defined.
Equal Treatment of Equals is the basic symmetry requirement (discussed in
the introduction) that two equal demands receive the same share:
for all N , t , x , all i , j :
xi=xj*ri(N;t;x)=r,(N;t;x).
Within the family X (No),Equal Treatment of Equals is only satisfied by three
methods.
COROLLARY
OF THEOREM2: Assume 1
4
1r 3 and to the above four axioms,
add Equal Treatment of Equals. Then there are exactly three rationing methods
meeting these five properties: the proportional, uniform gains and uniform losses
methods.
Theorem 2 and its Corollary are tight results, as the following examples
demonstrate.
663
PRIORITY RULES
EXAMPLE3: Equal SacriJice Methods (Young (1988)). In this interesting class
of methods, all axioms but Lower Composition-or Upper Composition-are
satisfied, and the methods are outside the set A? (4).
These methods use a
reference "utility function" u, common to all agents, and compute the cost
shares y by solving the system:
u(xi) - u(yi) = u(xj) - u(yj)
for all i , j.
Thus u(xl) - u(y,) measures the sacrifice inflicted upon agent i by the rationing
method. Of course, the function u must be chosen carefully so that the above
system, combined with y , = t, has a unique solution for all x, t. For any such
choice of u, the rationing method thus defined meets Consistency, Upper
Composition, and Equal Treatment of Equals. If one chooses u as a power
function, the method is also Scale Invariant. For instance, 4 x 1 = -(l/x) yields
the method
Xl
y1 =
l+hx.
where h solves: h 2 0
and
=t
Figure 4 illustrates the rationing paths of this method. It is easy to see on this
figure that the method violates Lower Composition. Naturally, the dual of the
above method meets all the requirements of the corollary with the exception of
Upper Composition.
Young (1988) characterizes equal sacrifice methods by the combination of
Consistency, Upper Composition, and Equal Treatment of Equals, together with
Strict Resource Monotonicity (property (3) with strict inequalities everywhere),
and Strict Ranking (x, < xj *yl <y,); see his Theorem 1 as well as Theorem 2
for the case where Scale Invariance is added to the list of requirements. Those
results are the closest to our Theorem 2 in the literature.
EXAMPLE
4: A Method Meeting Upper and Lower. Conzpositioiu, and Consistency, but Failing Scale Incariance. Fix .&'= {1,2} so that Consistency is vacuously
satisfied and define a rationing method as follows. For all t, x,, x2 such that
0 I t s x , + x 2 and x, > 0, the shares (y,, y,) arc the unique solution of the
system
Clearly Scale Invariance is violated. To check Upper Composition is straightforward. Lower Composition can bc checked directly or deduced from a duality
argument: the dual of the above method is defined by the similar system:
EXAMPLE
5: A Method Meeting Upper and Lower Compositiorzs, Scale Itzi*arirrnce, but Failing Corzsistency. Here we use the rich space of rationing methods
described in Moulin and Shenker (1999) (in the closely related context of
additive cost-sharing methods). All such methods meet Upper and Lower
Compositions, and Scale Invariance, and they are derived from arbitrary ordered
coverings of the simplex 3,(see proof of Theorem 2 in Appendix, or Definitions 1,2 in Moulin and Shenker (1999).
For instance, with ,^S, = {l,2,3} we can combine the 2-persons uniform gains
method between (1) and (23). with a proportional method among {2.3}. This
gives the following shares y = r(t; x):
This method is not consistent because the projection on {12} of the rationing
path to a demand profile ( x , , x,, x,) depends on x,. We let the reader check
that it meets Upper and Lower Compositions and Scale Invariance.
It is equally easy to find a method meeting Upper and Lower Composition,
Scale Invariance, and Equal Treatment of Equals, but failing Consistency. For
instance consider the following method:
PRIORITY RULES
Assume x, < x 2 , x,.
I f O ~ t ~ 3 x ~y L: =
t
7,for i = 1,2,3;
I f 3 x , < t < x 1 2 , : y, = x , ,
y,=x,+
(x,-x1)
(x2,-2x1)
.(t-3x,)
for i=2,3;
all other cases by exchanging the roles of agents.
7.
CONCLUDING COMMENTS
In the discrete rationing model, the analog of the Scale Invariance Axiom is
Replication Incariance:
(strictly speaking we must restrict the property to those numbers h such that
Ax EX(N)). The priority methods 7" are replication invariant. Yet Replication
Invariance cannot be interpreted as an invariance with respect to a change in
the measurement unit. In fact, it is not a compelling property. To see this,
consider the case N = {1,2) and assume r(1; (1,l)) = (1,O): the method favors
agent 1 when only one unit is available. Replication Invariance requires
r(2q; (2q, 2q)) = (2q, 0) for all q. However it makes good sense to distribute 2q
units fairly r(2q; (2q, 2q)) = (q, q): the fact that one unit had to be allocated
unfairly follows simply from the indivisibility, and does not imply that the
method has to favor the same agent at every level of resources.
Theorem 1 suggests the following open questions in the discrete model: what
is the set of rationing methods satisfying Consistency and Upper Composition?
Lower and Upper Compositions? What if we add the requirement of approximate equity (two agents with identical demands receive identical shares up to at
most one unit)?
Finally, we state three open problems in the continuous model directly
inspired by Theorem 2. Consider first the combination of the three axioms
Consistency, Upper Composition, and Scale Invariance. Recall that Young
(1988) adds Equal Treatment of Equals, Strict Resource Monotonicity, and
Strict Ranking to these three and characterizes a one-dimensional family of
"equal sacrifice" methods (see Example 3 in Section 6). If we drop Strict
Resource Monotonicity and Strict Ranking from the list, we capture at least the
Uniform Gains and Uniform Losses methods. What other methods can be
added? More difficult is to drop Equal Treatment of Equals as well: the
combination Consistency, Upper Composition, Scale Invariance, allows our
entire set X(N),as well as the generalized equal sacrifice methods where each
individual sacrifice is measured along a different utility function. What is the
general form of the methods meeting Consistency, Upper Composition, and
Scale Invariance?
Similarly, consider the combination Consistency, Upper and Lower Compositions. Example 4 lives here, but it is not at all clear what is the most general
form of a method meeting these three requirements.
The last relevant triple of axioms for which the corresponding family of
rationing methods is not known is Upper and Lower Compositions, and Scale
Invariance. Here some clues toward an answer are given by Theorem 1 in
Moulin and Shenker (1999). That paper looks at cost-sharing methods with
variable demands of a homogeneous good. That is to say, a cost-sharing problem
is given by a demand profile x and a cost function C, from R + into itself. The
cost-sharing method must select a profile y of cost shares so that y, = C(x,).
The familiar assumption of Additivity of cost shares y with respect to the cost
function C essentially implies that our cost-sharing method is entirely determined by a certain rationing method, via the integral formula
The result in question (Theorem 1 in Moulin and Shenker (1999)) explores the
consequences of the property of Distributivity of cost shares with respect to the
composition of cost functions. It turns out that this property implies that the
associated rationing method meets Upper and Lower Compositions, and Scale
Invariance. Therefore the entire family of cost-sharing methods characterized
there gives us new rationing methods meeting Upper and Lower Compositions,
and Scale Invariance. They are built, just like the methods in X(N),with the
help of an ordered covering of the simplex PN,a nd generalize to an arbitrary n
the methods described in Lemma 4 above. I conjecture that these methods
exhaust the possibilities under the triple requirement Upper and Lower Compositions, and Scale Invariance.
Dept. of Economics, Rice Unicersity, 6100 Main St., Houston, TX 77005-1892,
U.S.A.
Manuscript receil'ed Nocember, 1997;final recision receil'ed May, 1999.
APPENDIX: PROOFS
1. Proof of Theorem I
We omit the easy verification of the direct statement: a priority method meets all three invariance
properties. To prove the converse statement, we proceed in four steps.
In Step 1 we describe a rationing method by a family of rationing paths, when x is lixed and t
varies from 0 to x,. These paths are derived from the successive derivatives of r with respect to t .
In Step 2 we look at the 2-agent case, and show that Upper and Lower Composition together force a
priority rule. In Step 3 we reformulate Consistency as the property that "the projection of the
rationing path to x is the rationing path to the projection of x." Step 4 concludes the proof for 3 or
more agents: a consistent rationing method projecting onto a priority rule on every two-dimensional
subspace must be a priority rule.
667
PRIORITY RULES
We ~ I X throughout the proof a rationing method r meeting the three properties Consistency,
Upper and Lower Compositions.
Step 1: Preliminaly notation and obsen~ations
Because the method r is resource monotonic ((311, the feasibility condition ( 1 )implies that, for all
N and all x the rationing path t + r ( N ;t ; x ) moves by increments equal to a coordinate vector:
for all t,O s t g x ,
1:
-
there exists i E N such that r ( N ;t
+ 1;x ) - r ( N ;t ; x ) =
ci.
Therefore, the path t -t r ( N , t ; x ) can be described by the sequence p(N, x ) of its derivatives. This
and its tth element is i if and only if r ( N ;t ; x ) - r ( N ;t - 1; x ) = E ' . Note
sequence is of length
that, for all i , agent i appears exactly x, times.
Clearly a rationing method is entirely described by the family of sequences p(N, x), for all N and
all x. For instance, consider the priority method with ordering u . For any N with cardinality n , and
any x, the sequence p(x, N ) is
( ~ ( 1,...,
) u ( l ) , u ( 2 ),..., u ( 2 ) ,..., u ( n ),...,u ( n ) ) .
xr (1)
%(2)
~ S ( , Z )
Notation: For any N and any subset M of N we denote by [ p(N, x)lM the "projection" of the
sequence p ( N ; x ) on M , namely the sequence of length x , in M obtained by deleting all terms in
N\M.
Step 2: The case of two agents: n
=
2
Assume N = {1,2).We show that r ( N ) must be a priority method. As N remains lixed in this
step, we simply write r(t; x ) in lieu of r ( N ;t ; x). Denote by p the "predecessor" mapping, namely
p ( x ) = r ( x , - 1; x). Upper Composition is equivalent to the property:
r ( t , x ) = p(*,"-')(x)
for all x , all t ,
0 5 t 5x,
where p(") denotes the ath "power" of p. Next Lower Composition implies the following:
(6)
~ ( x=)r ( 1 ;x ) + r ( x ,
-
1; x
- r ( l ; x ) ) * p ( x ) = r(1; x
) + p ( x - r ( 1 ;x ) ) .
Now we fix x and assume that the sequence p(N, x ) starts and finishes with the same agent i ,
namely
(7)
r ( 1 ;x ) = E '
and
x -p(x)
= 8'.
Setting x' = p ( x ) and invoking Upper Composition, we have
Moreover, (6) and (7) imply
Therefore x' satisfies the property (7):the sequence p ( N ; x ' ) starts and finishes by i. Now we repeat
)
(7) as well, and so on. Hence the sequence
the argument to show that xu = p ( x l )= p ( 2 ) ( x satisfies
p(N; x ) contains agent i only. Thus the only vectors x satisfying (7) are coordinate vectors x = x i . E ' .
Next we pick an arbitrary x and t , 2 1 t gx,,., and assume that the sequence p ( N , x ) has i for
first and tth element:
r ( 1 ;x ) = c i
Set y
= r ( t ;x )
and
r ( t ; x ) - r ( t - 1; x ) = c i
and deduce from Upper Composition that
By the above argument, the whole sequence p(N; y ) is constant and equal to i, which in turn says
that the first t elements of p(N; x) are equal to i.
We have just proved that for all x, the sequence p(N; x) must be one of the two priority
sequences:
- -I
(- 1,..,, 1 , 2,...,2 I or
2 ,..., 2 , 1 , , . . ,
11.
XI
X2
x2
XI
It remains to prove that the priority ordering in p(N; x) is independent of x, for any x such that
x, > 0 and x2 > 0 (note that if one of x, is zero, there is nothing to prove). Assume that p(N; x)
gives priority to agent 1, p(N, x) = (1,. . . ,1,2,. . . ,2). Note that p(x) = x - 8' (because x, > 0) and
that the sequence p(N,p(x)) obtains from p(N,x) by deleting the last term (by Upper Composition). Therefore p(N, x E ' ) also gives priority to agent 1. Next we show that p(N, x - 8') gives
priority to agent 1, if x, 2 2. Compute, with the help of (6):
-
Thus the last term of p(N, x - 8') is 2; hence p(N, x - 8 ' ) must give priority to 1.
We have shown that if p(N,x) gives priority to 1 for some x in X ( N ) with both coordinates
positive, the same holds for all x ' bounded above by x. This implies at once that the priority
ordering is constant over X(N).
In Step 2 we have shown that when N l = 2 the rationing method r ( N ) is a priority method.
Step 3: Aiz equii'nlent fornl~~lflti011
of Consisrerlc)j
We claim that the rationing method r defined on J9h satisfies consistency if and only if
(8)
for all T c N c/Pb,all x in X ( N ) :
p ( ~ ; x ' )= [ p ( ~ , x ) ] ~
In other words, Consistency amounts to the commutativity of the mapping x + p(N, x) with the
projection over any subset of N. First we observe that property (8) holds true if it holds whenever T
takes the form N \ i for some i in N. This is because the projection satisfies [[ p(N,x)lT]" =
[ p(N, x)]%henever S c T c N.
Next we introduce a notation. Given a sequence w in N of length at least t, we write n ( t ; w) for
the "t-head of w," namely the sequence (of length t) made from the first t elements of w. We also
write O(i; w) for the number of times agent i appears in w. Therefore we have for all N, x, and t:
The followillg equivalence follows from (9) as well as from the definition of
operator:
-
p ( N \ j,.xzy\') = [ p ( ~ , x ) ] ~ ' ' for all r ,
(10)
[ ~ ( tp; ( ~x))]'"'
,
T,
p, and the projection
1 S t Sx,.;
=
.ii(t - q ( N ; t ; x); p ( ~ \ j x, N \ ' ) ) .
We are now ready to prove the equivalence of Consistency and of property (8). Assume the latter.
Then compute from (81, (91, and (101,
Conversely, assume Consistency and use the same composition as above to derive
669
PRIORITY RULES
As the dbove equallty holds for all
shows thc des~redequallty, namely
1
ln N \
J
and all r , 1 s t
I
1,
, an easy lnduct~onalgun~enton r
Indeed for t = 1, if p ( N , x ) starts by j, then the two above sequences are empty, whereas if
p(Ar, x) = i for some i different from j, then (11) shows that both sequences start by i. Next for
t = 2, if the second term in p( N. n) is j, both sequences do not change (because 2 5 ( N ; 2; x) = 1 r;(N; 1; XI), whereas if the second term of p i N , x) is i, i + j , this same i is added to both sequences.
And so on.
-
Step 4: Etzd of proof
By Stcp 2 wc can define a complcrc binall. relation in i , as follows:
{i > j )
iff
{r({iJ}) gives priority to 1 over j } .
This relation is clearly antisymmetric. We show by contradiction that it is transitive If 1 > 2, 2 > 3,
and 3 > 1 consider p(N, x) for N = {123) and x = ( l , l , 1). By (2) and the definition of > we have
a contradiction. Therefore > is an ordering of . ~ iIt" .remains to check that r. is thc corresponding
priority mcthod. Fix N in ./l/l and r in X ( N ) and consider the sequence p(N,,v). By property (8)
applied to N and T = { i j ) we know that all occurrences of i in p(A1, s) must precede those of j. The
Q.E.D.
desired col~clusionfollows at once.
Proof of' Renzalk I
Assume X, = 1 for all i. Then the sequence p(N, s)(see Step I ) defines an ordering of the
support of .v, namely N, = {i E N//c, = 1). Assume A!, I = ti1 and denote
By (8) applied to T = AT\ 1: P ( N \ 1, x'"',')
for a E N , i
1:
=
{ m ,m
1,. .. ,2). Hence for all t, t 5
-
I , we have
r.,(N;t;.x)=~;(~\l;t;x~\,')
Denote by y the following vector in {0,1)": y, = 0,
again and the fact that r , ( N ;t : y ) = 0, we get:
for all EN, i # 1:
-
v, = 1, for all I EAT\ 1. Then. by
Consistency
ri(N;t;y)=r.i(~\l,t;,t~v").
Combining the above two equalities and the fact that r,(N; r ; x ) = 0 (because r s
-
I). we have:
Note that y = r(,V: x, - 1; x ) so that the above equation expresses Upper Co~npositionfor the pair
t, t' = x , - 1. The full Upper Conlposition property follows from an obvious induction argument.
Finally, we show that r satisfies Lower Composition. Because r." is consistent (Consistency is
self-dual: see Section 4), the ahove argument shows that it meets Upper Composition; Lower
Composition is the dual of Upper Composition, which implies the desired conclusion.
2. Proof of Lenzma 4
After checking the direct statement-a method i n x ( , N ; ) meets the three invariance axioms-we
prove the converse statement in four steps. In the first three steps, we fix a strictly positive profile of
demands x and analyze the geometric structure of the rationing path to x. In Step 1 we show that
this path is everywhere strictly above the segment [O, x], or everywhere strictly below it, or coincides
with this segment; in Step 2 we show that if it is strictly above [0, XI,then it is strictly above any of its
chords; in Step 3 we deduce that the path consists of at most two linear pieces. Step 4 concludes the
proof.
The direct statement is straightforward: the rationing method associated with an ordered covering
of ,5'iI2, meets Upper and Lower Compositions and Scale Invariance. Scale Invariance is
immediate. As for Upper Composition, we fix an arbitrary x in R(:"\{o) and show r(t, x) =
r(t, r ( t l , x)) for all t, t',O s r s r ' <x,. If c O ( x )= (l/x,).x, the claim is obvious. Assume next
c 0 ( x )= l e i , ea[ and set x = h,el + h2e2. Recall that, by definition of an ordered covering, we have
C O (p,el) = e l for all p, > 0. We distinguish two cases:
If r ' 5 A,, then (5) implies r.(rt,x) = t'e' and r(r,r(t1, x)) = tel as desired.
If A, < t ' , then r ( t l , x ) = h,el
~ ( tr ;( t f ;x ) ) =re1
=
+(r'
for t
Ale1 + (t
-
-
h1)e2;therefore by (5) again
5 A,,
hl)e2
for hl < t 5 t ' ,
and the claim is proved. The similar proof of Lower Composition is omitted.
Conversely, we fix a rationing method r satisfying Upper and Lower Composition and Scale
Invariance and we show that r is associated with an ordered covering of q,,,as stated in Lemma 4.
For simplicity, here and in the proof of Theorem 2, we assume X , = + x so that individual
demands valy in R + . The careful reader will check that all Steps of the proof are unchanged when
some X, are finite.
In the current proof, we write r(t; x ) instead of ~((121;t; x ) as no confusion will arise. Two more
pieces of notation: the straight line connecting O and a (nonzero) vector x is called the x-line and
the rationing path r -t r(t; x) is called the x-path.
Step 1:
We fix an arbitrary vector 2 in R(:'). The i-path is monotonic: property (3). Because r,(t, 2 ) = t
for all r, this implies that r is continuous with respect to t.
We show in Step 1 that the 2-path either coincides with the i-line, or is everywhere strictly above
this line (except for r = O and t = i N ) , or is everywhere strictly below it. If 2, or 2, is zero, the claim
is obvious, so we can assume i,> O for i = 1,2. For all t, O 5 t s 1,, we define
Note that the indifference contours of the function f ( y ) = i,y2 - Z2y, are the straight lines parallel
to the .Cline. Moreover, f is positive above this line and negative below it.
In particular d ( t ) = O if and only if r(t; 2 ) is proportional to 2, therefore if A(t) = O for all t,
the i-path follows the segment [O, 21. Assume next that A(t) > O for some t. We show
0 < r s,?,.,
that for all t', O < t ' <2,\,, we have d ( t ' ) > O as well. The proof is by contradiction. Suppose that
(12)
d(t)>O
and
d(rf)50
forsomet,t't]O,2~[.
Let t + be the largest number achieving the maximum of d on [O, I, 1: this is well defined because r ,
hence d as well, is continuous in r. As A(0) = d(2,) = 0, we know that O < t + < 2,. By continuity of
d and (12) there exlsts to, O < r 0 < i N , such that d ( t o ) = 0. Denote .I+= r ( t + ; i )and x 0 = r(ro;1).
We can write xo = A.2 for some A, O < A < 1. Assume first t + < to, so that Upper Composition
lmplies x + = r ( t + ; xO).Then invoke Scale Invariance:
671
PRIORITY RULES
where (t+/A) < I , because t + < t o and t o = A,,?,. The above equality implies A(t+/h) > A([+), a
contradiction of the definition of t + .
The second case to consider is t o < t + . In this case we define the function A':
and check that A:' is negative and rniriimized over [O,.?,v] at .?, - t + : this is because the path
t + r"(t; i ) obtains from the I-path by symmetry around 2/2. Next we invoke ,?, - t + <.?, - t o and
Upper Composition of the dual method r.' :
Compute r W ( i ,- t o ;2 ) = i- x o
r +imply
= (1 -
All. Therefore the above equation and Scale Invariance of
(1 - A ) r (t ; I ) = " ( i - t ; i )
where
t
=
2,. - t -
------
I-A
Therefore,
Note that t < I , follows from r 0 < t + and r o = h .I,..
(l-A)d"(r)=d.:.(i,y-t+)
where
O<(l-~)<1,
(recall that A" is negative at
a contradiction of the fact that I,. - r - minimizes A" on [O, i].,
2,. - t + ) .
We have shown that if d is positive somewhere on 10, I,.[ it must be positive everywhere on this
interval. A similar argument shows that if d is negative somewhere on lo,.?,.[, it is negative
everywhere. Therefore the ,?-path is either the segment [O,.?], or is everywhere above the corresponding line, or is everywhere below.
Step 2: A ratiortirigpatlt is nei,er. below ariy of its c/~or.ds,or. is nei3er.0boi.e ariy of its chords
; r.(t2;s ) ] joining two points on the path. In
Given s,a chord of the x-path is the segment [ ~ ( r 'x),
Step 2 we show that for any x in R? , one (and only one) of the following statements l~olds:
,.(t; x )
oraboi,e the cliorrl [r.(rl;x ) , r ( t 2 ;x)l
r ( t ; x ) is otl or. below the chor.rl [r.(rl;x ) , r.(t2;x ) ] .
If x, = 0 for some i = 1,2 the claim is obvious. We fix now 2 such that I, > O for i = 1,2. If the
I-path is borne by the I-line there is nothing to prove so by Step 1 we can assume, without loss of
generality, that the 2-path is everywhere above the ,?-line. Let a' and a 2 be two arbitrary distinct
points on the I-path: we must prove that the path between ( I ' and a' is never below the chord
[a1,a']. See Figure 5.
Set a' = r(ti; I ) ; i = 1 , 2 with t 1 < r' and invoke Upper and Lower Composition: the rationing
path t + r(t; a' a'), for 0 s t < t' r1 coincides, up to the translation by -a1, with the portion of
the i-path for r such that t 1 5 r i t 2 . Applying Step 1 to s = a' a ' , we deduce that; between t 1 and
t', the i-path must be everywhere above the chord, or e v e l y h e r e below it, o r must coincide with
the chord. We now assume that it is everywhere below and derive a contradiction. Distinguish two
cases.
-
-
-
Case 1: T11e slope of a' rs larger tho17 tlzat of a'
- a':
see Figure 5.A.
Because the i-path between a ' and a' is continuous and evenwhere below [ a 1 , a 2 ] ,we can
choose a point b on this part of the path such that the slope of a' b is smaller than that of n2
and larger than that of a' - a 1 ; see Figure 5.A. By continuity again, this time applied to the ipath between 0 and t l , we can find 6 ' on the path below a', and aligned with a 2 and b (i.e.,
slope ( a 2 b') = slope(n2 - b)). Now consider the path between 0 ' and a 2 : it contains a point
strictly above the chord a 2 b ' (namely a') and one on the chord (namely b), which is a
contradiction of Step 1 (applied to .x = a 2 - b').
-
-
Case 2: The slope of 2
-
a'
IS
srnallel tlzar7 that of a'
-
a', see Flgule 5.B
The argument is similar: we choose first a point b on the path between a' and a 2 such that the
slope of b - a ' is smaller than that of a 2 - a 1 and larger than that of s - a'. Then we choose b ' on
the path between (I' and i aligned with a ' and b (see Figure 5.B) and obtain a contradiction of Step
1 applied to x = b' a ' .
-
T o conclude the proof of Step 2, one checks that Cases 1 and 2 are exhaustive (and nonexclusive):
the easy argument is omitted.
Step 3: A ratior~ir~gpath
is made of orze or two lir~enrpzeces
Fix ,x such that x, > 0 for i = 1,2, and assume that the s-path is evelyvhere above the x-line. By
Step 2; it is never below any of its chords.
We denote, as in Step 1, x + = r ( t + ; s ) the highest point on this path maximizing the distance to
[O; x], i.e., t + is the highest maximizer of A on [O, x, 1. We show first that the s-path follows [x+,x ]
between t + and x,.. Next we prove that the x-path follows [0, x + ] between 0 and t + .
We prove the first claim by contradiction: if it is false, we can choose a point x' on the x-path
between x+ and x and strictly above the chord [ s + , s ] . We choose s1 such that its distance to
[ x + ; s ] is maximal (on this portion of the path), and we set x' = r ( t l ; x), t + < t' <x,.. See Figure 6.
The definitions of s + and s1 imply the following inequalities on slopes:
PRIORITY RULES
The right-hand inequality is strict because x + is the highcst maximizer of A. Moreover. for any
point x L on the ,I.-path before x'. thc definition of .I.' implies
Below wc construct a point x ' o n the x-path before .x' and such that I, x ' , and I ' are aligned: thus
slope(xl - s ' : ) = slope(x n ' ) in contradiction of (13) and (14). This contradiction cstablishes the
claim that thc x-path follows [s'. s ] hetween t+ and I,.
Denote .v2 =s' - x + : the right-ha~idinequality in (13) implies that x' is helow [O;x]. Note that
x 2 is on thc d~lcrlpath to x', namcly
x 2 =y9(t2;.x1)
where
t'
=
t1 - f t
See Figure 6. So the dual path to s 1colltaills xi, a point strictly below [O, x ] and ends at x ' , strictly
above [O, .XI:by continuity there exists t h s u c h that
t % i 3 < t l and for s o ~ n cA,O < A < 1:
x 3 =r.-"(t", -T I ) = A . X
Now the dual path to x contains .r . r ' , because the x-path contains x'. By Scale Invariance of r.',
the dual path to s3must contain thc point A.(x -,TI). Thc dual path to x ' colltai~~s
.r" so by Upper
Composition (applied to r.':'), it contains A,(.\. - x ' ) as well. This in turn means that the xl-path
contains the point I" =x' A(.x-XI); see Figure 6. Thus I". x, and I' are aligned and we conclude
to a contradiction as announced above.
We have shown so far that thc x-path equals the segment [,I.+,r ] above x + . Define x - = r.(t-; s )
to be the lowest point on this path luaxirnizing the distance to [O, x]. We assume ,I. # s- and derive
a contradiction. The path to x must fc>llow[x-;x + ] betwcen t and t + , because it is never below
any of its chords and a point abovc [r-,,I.+] has a higher distance to [0, x]. Therefore the dual path
to .I.+ follows [O,xo], where .I.'= x + - x - , up t o t o = t + - r-; sce Figure 7. By Lowcr Composition,
h
[O. x 0 ] as well. But sois proportional to
the dual path t o so equals [O; .vo], hence the ~ ' - ~ a tequals
.Y so by Scalc Invariance, the x-path must be [0, x]; that was ruled out in the first place. This proves
-
+
X+ =X-. Finally, we consider the dual path to .v: because r*' satistics all three properties Consisteilcy,
Gpper and Lower Compositions, the above arguments apply to the path r . " ( l , x ) as well. In
particular, the dual path is always below [O; x ] because it contains the point .x -x+: hclow [O, x ] by
(13). Morcover the dual path is lincar above the highest point with maximal distance to [O, s ] ,
namely above x x = x -I+,and so the dual path is lincar beyond x - x + ; thus the x-path is linear
below x + .
1 - 1 ~ MOULIN
~~6
Step 4: Erul of Proof
W e have shown in Step 3 that the s-path consists of the two segments [O; x + ] and [ x + , x ] .
Denoting by e' and e2 respectively the directions of these two segments, normalized to be in q 1 2 ) ,
we conclude that either the path to .r follows [0, XI. or it is given by formula (5).
Now we define for all nonzero x:
C"(s
=
{ ( l / . ~ , ~ ) . xif )the path to x follows [O, XI:
c n ( x )= ] e l , e'[
if the path to .x is given by (5).
We check now that the sets CO(x),w hen .r varies, constitute an ordered covering of .q12,
and that r
is the associated rationing method.
and must show that they are either equal or mutually
We already know that these sets cover
= ~ , e+
' h,e2 and C@(.?)=]el. e2[, and we
disjoint. To see this. we take any i.e l , e' such that .i
prove that for all s such that .r = ,u,el + ,u2e2,pi > O for i = 1,2; a e must have ~ " ( s )=]el. e2[ as
well, namely the x-path is given by formula (5) (where p, replaces A!).
Consider a point x o n the half-line borne by e' at r+=hlel; excluding s+. See Figure 8.
(15)
x- = s + + ,u2e2= h l e l
+ p2e2
where
,u2 > 0.
If x is on [x+,,?I, Upper Composition implies that the s-path follows [O; x + ] at first. then [x+,x ] as
r e q ~ ~ i r eby
d (5). If s is beyond 2 on the halfline (i.e.. ,u2 > A,), there is a number A,O < A < 1, such
675
PRIORITY RULES
that A.x is on the segment [is';.?]; see Figure 8. Thus by Upper Composition of r.", the dual
~ s - x + ] first, then [ i x ' , Ax]. Hence the path to A.s is as required by (5);
path to h . s f o l l o ~ ~[O,i
by Scale Invariance, the same holds for x. So the path to any point on the haltline L give11 by (15)
takes the required form. By Scale Invariance the same holds true for any poilit oil a halfline AL, for
any A > 0. Such points cover the cone
{PLL,el + p,e2/p, > 0. i = 1 , 2 )
and the proof that all points in this cone have C?(s) =]e',e2[ is now complete. Tlle desired
conclusion, that two sets d1(x) and C (s') are either identical or disjoint, follows at once.
3. Proof of 711corenz 2
We already know that evely method in %No) meets Consistency, Upper and Lower Compositions, and Scale Invariance. Conversely, we fix a method meeting thcse four axioms and show that it
belongs to z I C ~ l ) .
Step 1 offers some prelimina~ymathcnlatical results including a reformulation of Consistency as:
the projection of an x-path is the rationing path to thc projection of x. In Step 2 we use Consistency
to show that if a rationing method projects onto every two dimensional subspace as onc of the
methods described in Lemma 4, this method partitions the positive orthant into relatively open
cones 011 which it is "piecewise linear.'' 111 Step 3 wc offer yet another equivalent formulation of
Consistency, that is used in Step 4 to derive the general shape of our rationing method in the three
agents case. This is the longest step requiring numerous geometric arguments in the simplex of R .
In Steps 5 and 0 we use an induction argulnent to extend the results to any n , 1 1 2 4.
Step 1: Prelinlinar?, notatiorz and prelimirziziy res~ilts
We fix N. finite, and we say that a finite set of vectors inv;/:
is of full rank if these vcctors are
linearly independent. Given a sequence {e', .... e") in Ly,-and of full rank, we denote by
T ( e 1 , .. . , e") the relatively open cone
K
x E ~ ( e ' . ... ,e")
iff
3A,, Ak
> 0. k = 1.. . . ,K such that x =
h,.ei.
1
Notc that the decomposition is unique.
R \{O), a path y ( N ; s) is a subset of R: connecting 0 to
Given the finite set N , arid a point x in :
x, continuously and monotonically. Formally, we say that y ( N . x) is a path if there exists a
nondccreasing mapping S from [0, .Y,\~] into R! such that
S(0) = O ;
S ( x , )=x,
i3,.(t) = t
for all t , O < t < s Y
and if y ( N , s) is the range of 6 (the usual terminology is "monotone path" but we will not consider
nonnlonotone paths). Note that the mapping 8 must be continuous. 4 path y ( N . .r) has a canonical
projection [y(N, x)]"' on R?', for every subset M of N: it is an easy mattcr to check that the
projection of an N-path is all M-path. Simply define the mapping 6 ' I by:
for all t , O i t s x , , :
S."(t) = [ 8 ( t t ) ] "
for all t ' such that O<tr<x,. and S,,,(tf) = t ,
and check that 6"' is well defined and is nondecreasing from O to xZ'. We omit the details.
W e state next a mathematical property:
CKIQUENESS
LF.MMA:Gi~'e11a finite !V. with j:Vl z3, a poiilt x in R: \ (0) and for each i E N . a
path n(iV\i; x,'\'). there e.xi.7t.r at most oizepatl~y ( N ; x) sucfz that
Moxocer, if ear11path yi is pieceitise linear (rraith finitely nlarzy pieces). so is y
T o prove the lemma, denote by 6, a monotone mapping representing y, and assume that we have
two distinct paths y '( N; x) and y '(N; x) projecting onto y, for all i. Because y' # y there exists
two points z' in yi(iV; I), i = 1 , 2 such that z\. = 2.: and z ' # z'. Thus we can find two coordillates i
and j such that zj <z: and zf <z,!. Let k be a third coordinate (liV r 3): by assumptioll we have
[zilh"
=
ak(t')
for somc t r , o s t '
IX,:,~
As 6, is nondecreasing. the two vectors [z'lr\%re Pareto comparable: contradiction.
The straightfo~wardproof of the second statement-y
is piecewise linear if each y, is-is
omitted. In order to state the second mathematical fact. me introduce some more notation. For N ,
M finite and 1V c N we define an operator p,, from the simplex 9,
into
. y', U {O):
for all e r,~Y',:
p,,,(e)
if e A'= 0.
=0
1
(16)
pM ( e ) = -. e W
edf
if e"' i0.
Next for any sequence {el,. .. , e") in .yv,u-e denote by p,,,(el,. . . ,e") the sequence in y,,obtained
from {p,,,(el),. ..,p,,(eK)} by removing zero vectors and merging collsecutive elements if they are
equal. Thus the sequence p,,(e1, . . . ,e h ) might have fewer than I< elements (e.g., a single clcment if
0); it might even be empty, if evely e k , k = 1.. .. . K. projects to 0 on RC'.
el = ... = e" and (e')." i
FULLRANI<LEMMA:Gir.ei1 iVfitzire, with 1 Ni r 3, rrtld a seqcletlce {e', . . . , e") in PA
, q/'~vhichtwo
coi1sec1itii.eelerilet~tsare not eq~ral.s~lpposethat for all i r N, the sequence p,v,,(e'. . . . , e K ) is of,fi~Il
the seq~rei~ce
{el,. . . . e K ) is offirll rank in R". In particular.. K 5 n.
r.cri11cin R" ','. T~ZCII
In the above statenlent we adopt the collvelltion that the empty set is of full rank. The proof of
this fact is rclcgatcd to Step 6 below.
Finally, we rcforrnulate the Consistency Axiom. We denote hy p(iV, x) the path associated with
our rationing method r. (namely the image of r ( N ; t : s ) when r varies in [0, x,]). Co~isistellcyis
equivalent to thc following property (analogous to property (8) in the discrete model):
(17)
for all finite A l .
N.with M c N and for all x E R: \{O}
Indeed, by definition of Consistency, we have for all :'\I and all j in
p ( ~s.")
, = [ P ( N .x)]'".
N
The two sets on cach side of the inclusioli are (monotone) paths from 0 to ~ ' (the
l " projection
~
of a
path is a path). hence they must he equal. Repeated applications of this argument yield (17).
Conversely. property (17) applied to A4 = N \ j implies for all t, O 5 t I X ~ ~ :
there exists t ' , 0 i r ' < s , , , :
The equality t ' = r
-
(r.(N; t : x ) ) " \ ' = r . ( ~ \ j ; t ' ; x Y \ j ) .
r;.(lV; t; x) follou~sat once.
Step 2: The ordeicd t oi3euilgof 9,
In Step 2 we prove the following claii11: for all 2 in R': \{O}. there exists an ordered sequence, of
full rank, in .:/Isuch that T ( e l , . .. , e") contains 2 and for all .r in T ( e l , . . . , e"), thc rationing
method is computed as follows:
K
if x
(18)
=
t:h,.r\
i;
write llis =
1
1A,,
for k
=
I , . . . . K. and it,
= 0;
the11
1
k
r ( h r ; t ; x ) = E h k , e h ' +(
t /li).ehL'
forall t such that i l , s r s 1 t , + ,
677
PRIORITY RULES
Clearly, the claim implies that when i varies, the (relatively open) cones T ( e l , .. . , e " ) form a
partition of R ~ \ { O ) that we call the ordered covering of .Y;, induced by 1.. This terminology
generalizes that of Lemma 4, from the case n = 2 to an arbitrary rl.
We prove the claim by induction on IN(. Lemma 4 establishes the claim when IN1 = 2 so we now
assume n 3 and that it holds for all M such that M I I r l - 1. We fix arbitrarily s in R: \{O}. By
the induction assumption applied to x R\', for any i E N , we know that the path p(iV\i, x.'\') is
either trivial (if x "'\' = 0) or is a piecewise linear path with s~iccessir,egradients {el(i),. . . ,eK(i)}; a
full rank sequence of K , vectors in .Y;,,,,, with K, I 11 - 1. By (17) this implies that the projection of
the path p ( N , x ) on every subspace N \ i is either trivial or is piecewise linear. Therefore p(N, x) is
piecewise linear (Uniqueness Lemma). We write its successive gradients as {el,. . . . eK).
We .check next that the sequence {el,.. . , e K ) is of full rank. The sequence of gradients
. . . ,(eK?.' \'I by deleting zero vectors and
{el(i),. . . . eK(i)) obtains from the sequence
merging consecutive elements if they are equal: that is; {el(i),. . . , e K ~ ( i )=) p , \ , ( e l , . . . e" ). Thus
the sequence {el,. . . , e K }satisfies the assumptions of the Full Rank Lemma; hence it is of full rank
(and K s 1 1 ) . By definition of this sequence, the vector x is a strictly positive linear combination
Cf A, .e%nd the rationing method r.(N; t; s ) is given by (18) for all f in [O; x "1 (recall that e q s the
sequence of successic'e gradients starting from r = 0).
It remains to show that for any other element i in T ( e l , . . . , e"?, with decomposition i = XA,.ek,
the path r.(iV; t; i ) is given by (18). Fix such an T
. and observe that'\'.).I?.(
is in T(e'(i), . . .; eK8(i?)
for all i E N. By the induction assumption; this means that the method r.(N\i; r;[,T]"\') is
computed by (18):
;
follows the direction e v i l on the interval [ :l,\(i), A,+,(i)], for all k
=
1;.. .; K , .
Now denote by ?(N; i?the path that follows successively the direction e L + ' o n the interval
[A,, A,+ ,], for k = 0,.. .; K 1 (as in (18)). Its projection on iV\i is precisely the path t + T ( N \
i; t; (.T)'"\')
just described above, because the projection is linear, and p , , , ( e l , . . . , e") =
{el(i?,. . . ,eKz(i?).On the other hand, (17) implies that the path p(N; i ) associated with r. projects
on N \ i as p(iV\i; [I]""?, namely the same path r + r ( N \ i ; t;[.T]"'\'). By the uniqueness Lemma,
these two paths coincide, and the proof of the claim is con~plete.
-
Step 3: A reform~ilntionof Corisisteriq
In Step 2 we showed that to each point x in R: \{O) we can associate a relatively open cone
r ( e l . . . . , e K ) containing x and such that the rationing method r. is given by (18) in this cone. The
ordered sequence {el,.. . , e K } in .PPYis uniquely defined for a given s, and we denote C:(s) =
CO(el,. .. , e") the ordered polytope with ordered vertices e l , . . . , ex. Step 2 implies that when x
varies, the relative interiors of the polytopes C(x) form a partition of .Y', . Note that in the
statement and proof of Lemma 4, we used these relative interior sets CO(x?.In the current proof, it
is more convenient to deal with their closure C(.x). We call cliinerzsiori of x the dimension of its
ordered polytope, namely K.
T o complete the proof of Theorem 2, it remains to apply the full force of Consistency on these
ordered coverings of Y<v.a nd to show that they are indeed generating a method in .;7/(./1V0). The key
observation is that Consistency can be expressed in terms of the ordered polytopes just defined as
follows:
(19)
for all ,YER.? \{o}. all M c N :
C(.Y) = c O ( e l , . . ., e") -C(,Y."? = c o ( ~ , ~ ~ .(..~. 'e")).
,
Recall that p,if(e' ,....e K ) stands for the sequence in .Y;I obtained from {p,,,(el),....p,,,(eK)} by
removing zero vectors and merging consecutive elements if they are equal. As the path P(N, x ) is
piecewise linear with successive gradients {el,... , e"}, its projection [ p(N, x)].'' is piecewise linear
with successive gradients p,,4(e1,.. . ,e"). The equivalence of (19) and (17) follows at once.
A useful consequence of (19) is that if x is of dimension 1, namely C(x)= {(l/sN).x) and
p ( N . x) = [0, XI, then x." is zero or is of dimension 1 as well.
~1 =3
Step 4: End of the Proof w h e ~1 N
We fix N = {1,2,3) and write simply 9
'instead of 9%.
Throughout Step 4 we interpret the
restriction of p,, to the simplex as a "projection." For any x in 9,
x # E', p(,,)(x) is the intersection
of the line E ~x ,and of the (12)-face; and p ( 1 Z J ( ~=3 0.
) See Figure 10. where y =p(,,)(z) and
x =p(lZ)(z).
If x is in the relative interior of 9; the polytope C(x) is of dimension 1; 2, or 3 (it is a point, an
'
;
C(x) is of dimension 1 or 2.
interval, or a true triangle). If x is in a 2-face of 9
C(x) is a true triangle C(x) = cO(el, e2,e3); and
We first assume that for some x interior to 9,
we show that the shape of this triangle is very particular. As C(x(',')) is of dimension 1 or 2, (19)
implies that the sequence p(,,)(el, e2,e3) is 'of rank 1 or 2. It cannot be of rank 1 because (el, e2,e3)
contains either one zero element
is of rank 3. Therefore the sequence (p(,2)(e'),p(12)(e2),p(,2)(e3))
or two equal consecutive (nonzero) elements.
Applying the above property for all three projections on the 2-faces of 9(i.e., the faces [E', d l ) ,
an easy argument shows that the triangle cO(el, e 2 ,e" must have one of the following two forms:
(20) el, e2 on a face {i,j) and e 2 ,ehligned with the vertex
(i.e., e2 is the projection of e3 on the face {i,j));
E"
or the symmetrical configuration from exchanging e' and e3.
Figure 9 describes these configurations; the formal argument is omitted. Next we distinguish three
cases, depending on the dimension of the ordered polytopes covering 9.
Case 1: All points on all 2-faces of Y are of dimension 1.
Cose 2: At least one point on a 2-face of 9' is of dimension 2 and all interior points of Y are of
dimension 2 or 3.
Yes Yes
PRIORITY RULES
Case 3: At least one point on a 2-face of Y is of dimension 2 atzd at least one point interior to 2%
is of dimension 1.
If C(x) is of dimension 2 or more, the
Assume we are in Case 1. Pick any x interior to 9.
projection of C(x) on at least one 2-face is of dimension 2, a contradiction. Thus all points in Y a r e
of dimension 1 and we have the proportional method.
Next consider Case 2. We claim that the following configuration is impossible: x is interior to the
(12)-face, y is interior to the (13)-face and .x, y are both of dimension 1. We prove the clam by
contradiction: the intersection z of [E" x] and [ c 2 ,y] would be interior to 9and the polytope C(Z)
would project onto the (12)-face as x and on the (13)-face as y; therefore C(z) would be of
dimension 1, which is ruled out in Case 2. See Figure 10.
The claim implies that there are at least two 2-faces, say (12) and (131, such that all their interior
points are of dimension 2: this implies (by Lemma 4) that the methods r(12) and r(13) are priority
rules, which leaves only two possible methods for 412) and two for r(13).
Case 2.a: Suppose first that r(12) gives priority to 1 over 2 whereas r(13) gives priority to 3 over 1.
We claim that there cannot exist a point x interior to the (23)-face and of dimension 1. If such a
point exists, consider y interior to [E', x] (say the midpoint of this interval). The polytope C(y) must
be within [E',x] (by (19) applied to (23)) and must project on (12) as {E', E ' ) (by (19) applied to
(12)). Hence C(y) = [ s l , x ] . But its projection on (13) is not [E', s'] as required by (19) and our
assumption that r(13) gives priority to 3 over 1. The claim is proven, and implies that r(23) is a
a d check that the only polytope C(x)
priority rule as well. Take now an arbitrary x, interior to 9,
projecting on each face as the full face is the whole simples .Y, namely C(x) = CO(c', s J ,c k ) . Thus
our method is the priority rule with priority ordering 3, 1, 2.
Case 2.h: Suppose next that r(12) and r(13) both give priority to 1. Pick any x interior to 9and
assume C(.X(~'))= CO(e',e2). As the projections of C(x) on the other two faces are [E', s 2 ] and
[ c l , c3], it follows that C(x) = CO(E', el, e?). Similarly, if C(X(")) = {e), then C(x) = [ e l , el. Thus in
Case 2.b the method r(123) is the composition of the method r(23) in X2((23)) by the priority
ordering giving 1 priority over (23).
Case 2.c: Suppose finally that r(12) gives priority to 2 and r(13) gives priority to 3. An argument
similar to that of Case 2.b shows that r is the compositio~lof r(23) with the priority ordering giving
(23) priority over 1.
Finally we consider Case 3. We pick a 2-face, say (2.3) and two points el, e2 in this face (in
[ c 2 ,c3I) such that ]el. e" or ]e2,el[ is in the covering of r(23). We also pick z, interior to Y a n d of
of dimension
3
1, hence cannot be in ]el, e2[; therefore [el, e2] is a strict
dimension 1. Clearly, ~ ( is ~
subset of [ s 2 ,s3]. See Figure 11. Assuming without loss of generality that z(2" is between c 2 and
[el, e2], we construct a point a at the intersection of the line borne by [ c 2 ,Z] and of [ E ' ,el], where
e' is the vertex of [e',e2] closest to c 2 . As the projection of C(a) onto (23) is {e') (because it
contains e' and (e') is in the covering of r(23)), and its projection onto {I31is (21")) (because ~ ( ' ' 1 is
in the covering of 1.(13)), it follows that a is of dimension 1 as well; see Figure 11.
Summarizing. we have now a point a interior to 9'
and of dimension 1, and a proper subinterval
[el. e 2 ] of [s'. s ' ] such that lei. e2[ or ]e2,el[ is in the covering of r.(23), and such that one of its
endpoints, say el, is the projection of a on (23).
We clr~in~
that the other endpoint e 2 must be one of s 2 or E ' . Wc prove the claim by
contradiction, assuming that e 2 is strictly between e1 and e 3 as shown on Figure 11. We construct
the points b and c as the intersection of [ c l , e ' ] with the lines e2a and 8% rcspectively: see Figure
12. The triangle [ahc] is of full dimension (a, 0. and c are not aligned) and we pick an interior point
x. If .x is of dimension 3;the triangle C(x) must bc contained in [abcl becausc all points on the faces
of [abc] are of dinlension 1 at most: because e' # s 3 , i = 1 , 2 such a triangle would not be of the type
(20); hence our point .x can be of dimension 2 at most. If x is of dimension 1, its projection x-(") is
of dimension 1 as well. which is impossible because it is a point in lei, r 2 [ .Thus x is of dimension 2.
C(x) is an interval contained in [abcl, and its projection on (23) is lei, eL[or ]e2,e'[. Denoting by x'
the intersection of the line m with s ' e 2 , we conclude that C(.x) is one of la, .x'[ or Ix'. a[; see Figure
12. The announced contradiction follows from considering two points x, y interior to [abc] and
aligned with E ' ; thc projection of the two illtcrvals ]a, .x'[ and ]a, y'[ on {1,3) do not coincide, a
violation of (19). See Figure 12.
PRIORITY RULES
Thus the claim e' = s3 is established. We now refer to Figure 13, where i ' denotes the
intersection of the lines sin and EJE'.The proof of the claim (in the previous paragraph) also shows
that any x interior to the triangle CO(a, s', if2) is of dimension 3. Now consider the ordered triangle
C(s): it is contained within the triangle {a, b, s'), its projection on (23) must be [el, E" or [s', e']
and its shapc is as described in property (20). Clearly the only two possibilities are CO(n, b, s') and
CO(E', b, a).
We show next that the situation is similar for the five other triangles cut by the lines sin; see
Figure 13. Namely we prove that for each point x interior to one such triangle {a, s', d!): the ordered
triangle C(.x) is either CO(a,il, 8') or CO(E', d J , i r ) . Pick x interior to (a, s 3 . i 2 ) ; we know that
) CO(P', 8 ' ) or CO(:', i3). Thus
C(.x) is CO(a, i 2 ,s 3 ) or CO(E', i<0). By (19) we have ~ ( x { " ) =
the covering of
contains ]g3,el[ or Is', i 3 [ and we can rcpeat the proof of the claim for the
triangle {ri, s l , i2}, upon exchanging the role of 1 and 3. The desired conclusion thus holtis for this
triangle as well. Next we pick .x interior to (0, E ' , i') and project it on the (13) face; by (19) we get
c(.Y{~")= C0(z2, ~ l or) C O ( s l , z2) and we repeat the proof of the claim, this time for the triangle
{n, s l , i 3 ) ,upon exchanging the roles of 2 and 3. Next we project an interior point of {{I,e ' , i'} on
the (23) face to show that the covering of ci/i231contains ]ifL,s 2 [ or ] s 2 , i ' [ and the desired
conclusion follows for the triangle (a, s 2 ,i'}. And so on.
We have identified the triangle C(.x) (up to one of two possible orientations) for all s interior to
any one of the 6 triangles. Next assume x is on ]n,il[. The projection of C(n) on the (13) face is
{i2,s3} (up to orientation) and its projection on the (12) face is {t" s 2 ) (up to orientation). Hence
C(x)is (a, i') (up to orientation). Similar arguments show that C ( s ) is {a,5') if s is in ]a, E'[ and is
{a, s ' ) if x is in ]a, s f [ .
Finally we check that the ordering of the six triangles must coincide on their common faces. For
instance, say CO(a, i 2 ,s 3 ) is in the covering of 5";(19) implies that CO(cz, e 2 ) is in the covering as
well; therefore C 0 ( t . ' ; e 2 ,a) cannot be in the covering, etc. In the end we are left with only two
possible coverings corresponding respectively to the weighted gain and weighted loss methods with
weight n.
Step 5: t.,'rzd of Proof wlzer~ is firlire arzri IN/ 2 3
We use an induction argument on the size of iV. Fix N,with ;iV1 = 11 r 4, and assume Theorem 2
holds for all N' of cardinality at most 11 1. For each i r N consider the restriction of our method r
to N\i. By induction it is a method in ,Y(N\i) and we denote hy R , its priority preordering of iV\i.
Any two such preorderings R , , R , coincide 011 i\T\{ij}-with the priority preordering of the
method on ,V\(ij). Therefore, therc exists a unique preordering 2 on N of which the restriction to
-
682
H E R V ~MOULIN
N\i
equals R,, for all i: this preordering compares i and j in the same way as R,, for any
k , k # i , j. It is transitive because I Nl r 4; for any triple i , j, k, the preordering coincides with R,,. for
any m different from i , j, k.
Assume first that 2 is not the overall indifference, and let hrlii N2 be a partition of N such that
R ranks all agents in Nl above all agents in N2. We check that r must give priority to N , over N,
(Definition 3). Suppose not: we can find x E R:, two agents 1,2 with 1 EN,, 2 EN^ and a point
y ~ p ( N , x such
)
that y, < x l and x , > 0. By (171, the path p(N\{3],xh'\(31) contains yN\(,l, a
contradiction of the fact that R , gives priority to 1 over 2. The claim is proven. Now the induction
argument shows that the restrictions of r to
is in
i = 1,2; hence r is in 2 ( N ) after all.
It remains to take care of the case where R is the overall indifference: in this case r(N\i) is
irreducible (Definition 4 ) for all i E N . AS 1N\il2 3, this leaves only the proportional, weighted
gains and weighted losses methods. We distinguish two c a m . If for some agent i the method
r(N\i) is proportional, then r(N\ij) is proportional as well. Therefore r(N\j) cannot be a
weighted gains or weighted losses method. Thus r(N\ j) is proportional for all j. Take any point x
interior to , Y N :for all j its (N\ j ) projection is of dimension 1; therefore x is of dimension 1 as well,
so that r( N ) is the proportional method.
The last case is when for all i, r(N\i) is either gW1(N\i) or l'"l(N\i). Assume that r(N\ 1) =
g'"l(N\ 1) and notice that r(N\{12]) is the weighted gains method with weight vector w?\l2 (a
vector in R!>(l2)).This; in turn, implies that r(N\2) must be a weighted gains method, namely
r(N\2) = gN'2(N\2) for some w2. Moreover wf\12 and wr\l2 are parallel.
Thus we have for all i a weight vector wi, wi E
such that w:\"
is parallel to w F i J for all
i , j. Because fz 2 4, this implies the existence of a weight vector w. w E R:,. of which thk projection
on each N\i is parallel to wi. Now we have identified the method g n ' ( N )and shown that it has the
same projection as our method r ( N )on every subspace N\i. The conclusion r ( N ) = g W ( N follows
)
by the Uniqueness Lemma. The case where r(N\ 1) is a weighted loss method implies, similarly,
that r ( N ) is a weighted loss method.
aN,),
Step 6: Proofof Theorem 2 when .No is itzjifzite
Once Theorem 2 is established for any finite subset N of .I&,its extension to a countably infinite
society.Nb is straightforward, hence omitted.
Step 7: Proof of Full Rank Lemma
The K-lemma says: if a sequence { e l , ... ,e K ]in PN meets assumption, P ( K ) , the sequence is of
full rank. Assumption P ( K ) is: two consecutive elements are not equal, and for all i . N,~
pN\i(e',. . . ,e K )is of full rank in R ~ \ ' .
We proceed by induction on the length K of the sequence. For K = 1 there is nothing to prove.
Assume the ( K - 1)-Lemma holds and consider a sequence {el,.. . ,e K }meeting P ( K ) . For simplicity, we denote eki =p,\,(ek), for k = 1
K. We suppose that {el,... , e K ]is not of full rank and
derive a contradiction. There is a nonzero vector h in R~ such that
$...,
Note that the sequence { e 2 , .. . ,e K ]meets the P ( K - 1) assumption: two consecutive elements are
different and for all z,p,,,(e2,. . ..e K )is a subsequence of pN\,(el,. .. ,e K ) ,so it is of full rank. The
induction assumption shows that { e 2 , ... ,e K }is of full rank, whence h, # 0.
PRIORITY RULES
683
We now compare e l , and e 2 i. Assume for some i, we have e' # 0, e 2 # 0, and e', # e 2 i.Then
the sequence PN\i(el, e 2 , .. . , e K ) has e L i and e l as its first two elements. Denoting this sequence
(a', a 2 ,a 3 , . . . ,aKg) (where K, I K), equation (21) yields, upon projecting on YN,,:
c
K
~ , [ e ' +] ~ ~~ ~ [ e "= ~0
\ ~
where p2 is given by
1
p2 = T ( A 2
eN\!
+ Ak)
if there is an integer k , k r 3 (necessarily unique) such that
and where k g , .. . , pK, are similarly constructed.
The above equation, given A, # 0, contradicts the full rank of { a 1 , .. . ,aKi). Thus we have shown
for all i E N :
(22)
{el # E' and e 2 # ss')= l e i , = e l , .
To end the proof, we look successively at the following cases:
Case 1: For all i , e 1 # E' and e 2 # s i . Then (22) implies e' = e 2 , a contradiction.
Case 2: For all i , e' # s1and e 2 = s j for some j. Here for all i # j , we get e k , = EL, = sJ so that
e' = ~j (as one checks easily), again a contradiction.
Case 3: el = ~j for some j and for all i , e 2 # zi: similar to Case 2.
# ( E ~ )=- e~2 for all
Case 4: e' = e i and e 2 = ~j for some i and some j. Note that E' =
k # i, j, hence a contradiction of property (22).
This completes the proof of Step 6, and of Theorem 2.
Step 7: Proof of Corollaly
If r satisfies Equal Treatment of Equals, no agent i can have priority over another agent j: this is
clear by comparing Definition 3 with the Equal Treatment of Equals Axiom. Therefore, a method in
X(4)meeting Equal Treatment of Equals must be irreducible. If I.Nnl
2 3, the irreducible methods
are the Proportional, Weighted Gains, and Weighted Losses. Clearly Equal Treatment of Equals
forces equal weights for every agent.
REFERENCES
AUMANN,
R. J.,
AND M. MASCHLER
(1985): "Game Theoretic Analysis of a Bankruptcy Problem from
the Talmud," Journal of Ecoizomic Tkeoly, 36, 195-213.
M., AND H. P. YOUNG(1982):
Fair Representatiorz: Meetzng the Ideal of One Man, One Vote.
BALINSKI,
New Haven: Yale University Press. BANKER,R. (1981): "Equity Considerations in Traditional Full Cost Allocation Practices: An
Axiomatic Perspective," in Joirzt Cost Allocatiorzs. ed. by S. Moriarity. Oklahoma City: University
of Oklahoma Press, 110-130.
BARBERA.S., M. JACKSON.A N D A. NEME (1997): "Strategy-Proof Allotment Rules," Ganzes and
Economic Behucior, 18. 1-21.
BENASSY.
J.-P. (1982): The Ecorzor?zics of Market Diseqz~ilibrium.New York: Academic Press.
CHING,S. (1994): "An Alternative Characterization of the Uniform Rule," Social Choice and W e b r e .
11, 131-136.
DEMERS,A,, S. KESHAV,A N D S. SHENKER(1990): "Analysis and Simulation of a Fair Queuing
Researclz ar~dE.vperience, 1, 3-26.
Algorithm," Ir~ternefit~orking:
DRRZE, J. (1975): "Existence of an Equilibrium under Price Rigidity and Quantity Rationing,"
Irlternntionul Ecorlonzic Rel,iew, 16, 301-320.
ELSTER,J. (1992): Local Jzlstice. New York: Russell Sage Foundation.
GELENBE,E . (1983): "Stationaly Deterministic Flows in Discrete Systems," Tlzeoleticul Computer
Sciences, 23, 107-127.
(1980): Arznlysis and Synthesis of Computer System Models. New York:
GELENBE.
E.. A N D I. MITRANI
Academic Press.
HOFSTEE,W. (1990): "Allocation by Lot: A Conceptual and Empirical Analysis," Social Science
hzformatiorz, 29, 745-763.
M. (1997): "A 'Hydraulic' Theory of Rationing," forthcoming in Matlzernaticnl Social
KAMINSKI,
Scierzces.
KALAY,E. (1977): "Proportional Solutions to Bargaining Situations: Interpersonal Utility Compar45, 1623--1630.
isons," Eco~ror~zetrica,
MASCHLER,M. (1990): "Consistency," in Ganze Tlzeogt rind Applications, ed. by T. Ichiishi, A.
Neyman, and Y. Tauman. New York: Academic Press, 183-186.
MOULIN,H. (1987): "Equal or Proportional Division of a Surplus. and Other Methods." Internntior~nl
Jourrzal of Garne Tlzeory. 16, 161-186. '
---- (1999): "Rationing a Commodity &long Fixed Paths," Jourrzal ofEcorzor?zic Tlleory, 84, 41-72.
(1999): "Disrributive and Additive Cost Sharing Methods." Ganzer and
MOULIN,H.. A N D S. SHENKER
Ecorlomic Behai'ior, 27, 299-330.
O'NEILL, B. (1982): "A Problem of Rights Arbitration from the Talmud," Matlzematical Social
Sciences, 2. 345-371.
PAPAI,S. (1999): "Strategyproof Assignments by Hierarchical Exchange," forthcoming in Econometrica.
PLOT, C. R. (1973): "Path Independence, Rationality and Social Choice." Ecorzorizetrica, 41,
1075-1091.
N. (1973): Probabilitj~and Statistical Iriference in ~Mediei~nl
Jewish Literature. Toronto:
RABINOVITCI-I,
University of Toronto Press.
SHENKER,
S. (1995): "Making Greed Work in Networks: A Game Theoretical Analysis of Switch
Selliice Disciplines," IEEE/ACM Trnrzsnctions or1 Neta:or,kirlg,3, 819-831.
Y. (1991): "The Division Problem with Single-Peaked Preferences: A Characterization of
SPRU*\IONT,
the Uniform Allocation Rule." Ecorlor?letrica.59, 509-519.
THOMSON,W. (1990): "The Consistency Principle." in Game Theory arzd Applicatiorzs. ed. by
T. Ichiishi, A. Neyman, and Y. Tauman. New York: Academic Press, 187-215.
----- (1995): "Axiomatic Analyses of Bankruptcy and Taxation Problems: A Sunjey," Mimeo.
University of Rochester.
WINS LO^, G. R. (1982): Triage and J~rstice.Berkeley: University of California Press.
YOUNG,H. P. (1987): "On Dividing an Amount According to Individual Claims or Liabilities."
Matl~eri~atic.~
of Operrttiorzs Rerearclz, 12, 398-414.
----- (1988): "Distributive Justice in Taxation," Jozlrrzal ofEcorzonzic Theoiy, 48, 321-335.
----- (1990): "Progressive Taxation and Equal Sacrifice," Alilericarl Econornic Recierv. 80, 1.
253-266.
---- (1994): Equirp, irz Tlreory arzd Anctice. Princeton: Princeton University Press.
http://www.jstor.org
LINKED CITATIONS
- Page 1 of 2 -
You have printed the following article:
Priority Rules and Other Asymmetric Rationing Methods
Hervé Moulin
Econometrica, Vol. 68, No. 3. (May, 2000), pp. 643-684.
Stable URL:
http://links.jstor.org/sici?sici=0012-9682%28200005%2968%3A3%3C643%3APRAOAR%3E2.0.CO%3B2-O
This article references the following linked citations. If you are trying to access articles from an
off-campus location, you may be required to first logon via your library web site to access JSTOR. Please
visit your library's website or contact a librarian to learn about options for remote access to JSTOR.
[Footnotes]
9
The Division Problem with Single-Peaked Preferences: A Characterization of the Uniform
Allocation Rule
Yves Sprumont
Econometrica, Vol. 59, No. 2. (Mar., 1991), pp. 509-519.
Stable URL:
http://links.jstor.org/sici?sici=0012-9682%28199103%2959%3A2%3C509%3ATDPWSP%3E2.0.CO%3B2-Q
References
Existence of an Exchange Equilibrium under Price Rigidities
Jacques H. Drèze
International Economic Review, Vol. 16, No. 2. (Jun., 1975), pp. 301-320.
Stable URL:
http://links.jstor.org/sici?sici=0020-6598%28197506%2916%3A2%3C301%3AEOAEEU%3E2.0.CO%3B2-E
Path Independence, Rationality, and Social Choice
Charles R. Plott
Econometrica, Vol. 41, No. 6. (Nov., 1973), pp. 1075-1091.
Stable URL:
http://links.jstor.org/sici?sici=0012-9682%28197311%2941%3A6%3C1075%3APIRASC%3E2.0.CO%3B2-K
NOTE: The reference numbering from the original has been maintained in this citation list.
http://www.jstor.org
LINKED CITATIONS
- Page 2 of 2 -
The Division Problem with Single-Peaked Preferences: A Characterization of the Uniform
Allocation Rule
Yves Sprumont
Econometrica, Vol. 59, No. 2. (Mar., 1991), pp. 509-519.
Stable URL:
http://links.jstor.org/sici?sici=0012-9682%28199103%2959%3A2%3C509%3ATDPWSP%3E2.0.CO%3B2-Q
Progressive Taxation and Equal Sacrifice
H. Peyton Young
The American Economic Review, Vol. 80, No. 1. (Mar., 1990), pp. 253-266.
Stable URL:
http://links.jstor.org/sici?sici=0002-8282%28199003%2980%3A1%3C253%3APTAES%3E2.0.CO%3B2-N
NOTE: The reference numbering from the original has been maintained in this citation list.