Priority Rules and Other Inequitable Rationing Methods
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1
Priority Rules and Other Inequitable Rationing Methods
1.
Rationing: endogeneous demands, exogeneous rights
The simple model of rationing discussed in this paper is perhaps the oldest (O’Neill
[1982], Rabinovitch [1973]) and surely the simplest formal model of distributive justice. A
rationing problem among the agents from N {1,2,, n} is a list of “demands” x i , 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 N xi .
The same formal model arises in a variety of contexts. In the “inheritance” context
(O’Neill [1982]) the demands are documented by legal deeds signed by the deceased, so that the
number x i 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]), x i represents agent i’s tax liability (her
taxable income) and ( N xi t ) is the total tax to be levied; a related interpretation is the
costsharing of a public good (Moulin [1987]) where x i is agent i’s benefit from the public good
and ( N xi t ) is its cost, to be shared among beneficiaries.
Rationing is by far the richest interpretation of the model: the examples range from the
distribution of medical assistance in war or disaster situations (the “medical triage” problem:
Winslow [1982]), of food supplies to refugees, organs for transplant (Elster [1992]), seats 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 the microeconomic literature, rationing is a consequence of the rigidity of prices
2
and inspires a whole literature on “disequilibrium” reviewed in Benassy [1982] (see also Drèze
[1975]). In the network literature, queuing is almost synonymous of rationing: each user of the
network sends a certain number of “jobs,” and the serving algorithm decides which job is
handled first and so on. See Shenker [1995], Demers, Keshaw and Shenker [1990], as well as
Gelenbe [1987], Gelenbe and Mitrani [1980].
Naturally, the 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 context of 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 all rationing
methods satisfying three or four such invariance axioms.1 Yet in one important aspect this paper
departs from the existing literature on rationing methods. The latter, with very few exceptions,
focuses on equitable methods, that is to say, imposes the Equal Treatment of Equals axiom (in
short ETE): 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 priori equal
rights, and the differences in their individual shares is motivated exclusively by the pattern of
differences in individual demands. The prominent equitable methods are the proportional
method, the uniform gains methods and a few others.2
1
Three in the discrete model, and four in the continuous model, as explained below.
Such as the Talmudic method of Aumann and Maschler [1985], the equal sacrifice methods and the more general
parametric methods: see Section 3 and Example 3 in Section 6.
2
3
Equal Treatment of Equals is compelling in the many contexts where no a priori
discrimination can be allowed (e.g. when designing a direct income tax schedule). In other cases,
we do want to discriminate among the recipients of the resources independently of the size of
their demand. Creditors in a bankruptcy situation are commonly 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.
One unambiguous conclusion emerges from the axiomatic analysis of rationing methods:
among methods treating equal demands equally, the proportional and uniform gains methods
stand out by virtue of their multifarious 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 inequitable ones: indeed
they are the most inequitable rationing methods.
The goal of this paper is to study axiomatically the rationing problem without imposing
Equal Treatment of Equals. As our axioms do not, by themselves, force any equity or inequity
into the method, they are typically satisfied by some equitable rationing methods and by some
inequitable ones. Thus if one accepts our axioms as compelling, the results below describe
precisely what freedom is left to the mechanism designer wishing to incorporate exogeneous
rights into the rationing of endogeneous demands.
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 x i , t, as well as the individual shares yi are all nonnegative integers. This model
encompasses the allocation of organs for transplants, college admissions, and in general queuing
problems where individual demands consist of a finite number of “jobs” (e.g., packets are
processed by the Internet) and so on. We call it the discrete rationing model.
4
Despite its empirical relevance, the discrete model 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 impossible,3 and so the
priority rules are the only ‘natural’ rationing methods; the proportional method, for instance,
could 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 x i , the resources t to be divided, and the individual shares yi are now
nonnegative real numbers.4
The three axioms that are the subject of this paper are called Consistency (in short C),
Distributivity (D), and Distributivity* (D*). 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, C says that if ( y1 ,, yn ) is the division of t selected by our rationing
method at the demand profile ( x1 ,, xn ) , then ( y2 ,, yn ) should be the division of ( t y1 )
selected at the demand profile ( x2 ,, xn ) . More generally, if we restrict the focus of the
distribution problem to a subset M of N, then their shares computed in the N-problem are still
“correct” in the M-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 [1996] 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.
3
4
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 i and integers
Think of x i as the fraction of the total population in state i, and of
t , yi .
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].
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The Distributivity and Distributivity* properties are invariance properties with respect to
changes in the amount of resources t being distributed. Given an initial profile of demands
( x1 ,, xn ) , suppose that the arbitrator is told first that at most the amount t will be available and
computes the corresponding shares ( y1 ,, yn ) . 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
( y1,, yn ) from the initial demand ( x1 ,, xn ) ; however the axiom D allows her to use the
“optimistic” shares ( y1 ,, yn ) as the demand profile, that is to say, D requires that ( y1,, yn )
be the correct shares for that demand profile as well. In other words, when D holds, she can use
the optimistic assessment t to update the demand profile from ( xi ) to ( yi ) and forget about the
initial demands entirely.
The axiom D* expresses a “dual” invariance property pertaining to a pessimistic
assessment of the available resources.5 Now our arbitrator is given a demand profile ( xi ) and
learns that the available resources will be at least t. The rationing method recommends to
allocate t as ( y1, y n ) among N. Once this is done, agent i’s demand is reduced from
xi to ( xi yi ) , and any additional resources t that becomes available later will be distributed
according to the demand profile (( xi yi )) . Now D* 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 ( xi ) .
The D and D* properties have been introduced in the rationing model by Moulin [1987]
and Young [1988]; unlike Consistency, they do not have relatives in many other problems of
distributive justice (see, however, the discussion of Moulin and Shenker [1997] in Section 7).
This paper offers two characterization results. 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, Distributivity and Distributivity* (Theorem
1 in Section 5).
In the continuous model, the priority rules satisfy C, D, and D* but so do many other
methods. Two important examples are the proportional method (shares yi are proportional to
5
In Section 4 we define formally the duality operation on rationing methods, and show that D* is indeed the dual
property of D.
6
demands x i ) and the uniform gains method (equalizing shares as much as possible under the
constraint yi xi ; see Section 6). On the other hand, some often discussed equitable methods
7
fail D and D*: an example if 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 D, D*, and C.
Our second characterization result (Theorem 2 in Section 6) describes in the continuous
model all rationing methods satisfying C, D, D*, as well as a fourth axiom called Scale
Invariance (SI). The latter is the familiar requirement that a simultaneous increase, in the same
proportions, of all demands x i and the resources t must result in the same proportional increase
of the shares yi . This property is commonly justified by the irrelevance of a change in the
measurement unit of the commodity in which the demands, resources and shares are all written.
The family of (continuous) rationing methods meeting C, D, D*, and SI is quite
interesting. It contains a continuum of methods, striking a compromise between maximal
inequity, i.e., the priority rules, and full equity, i.e., Equal Treatment of Equals. But the path
along which the arbitrator can adjust the degree of inequity is quite narrow. If Equal Treatment
of Equals is required, exactly three methods are possible: proportional, uniform gains , and its
“dual” method uniform losses (equalizing losses xi yi as much as permitted by the constraint
yi 0 ): Corollary to Theorem 2. If the arbitrator aims at some inequity but rules out any
absolute priority of one agent over another, then her choice is limited6 to 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 yi xi and yi 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 preordering, 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 rationing). The methods uncovered in
This claim holds only if three or more agents are involved; the case n 2 allows for more methods: see Lemma 4
in Section 6.
6
8
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 that I am
aware of 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.
The main result in the literature on continuous rationing methods is Young [1987],
characterizing the family of parametric methods by the combination of Consistency, a stronger
version of Equal Treatment of Equals, and an assumption of strict monotonicity of shares in the
resources (a strengthening of our Resource Monotonicity, (2) in Section 4). A parametric method
is constructed by choosing a function h( , xi ) continuous and strictly increasing in such that
h( 0, xi ) 0 and h( , xi ) xi . Then for all x1 ,, xn and all t the system
n
h( , x ) t
y i h( , x i )
i
i 1
has a unique solution, defining the parametric rationing method in question.
All three equitable methods allowed by our Theorem 2proportional, uniform gains and
uniform lossesare parametric, although the corresponding function h is not strictly increasing in
in the case of the latter two methods:
h( , xi ) min{ , xi }
1
h( , xi ) max{xi ,0}
for the uniform gains method
for the uniform losses method
Yet another result by Young (Young [1988]) bears an even closer relation to our Theorem
2. That result explores the impact of the Distributivity and Scale Invariance axioms on the set of
parametric methods. Young characterizes the family of equal sacrifice methods (described in
Example 3, Section 6) by the combination of D with C, Equal Treatment of Equals and Strict
Resource Monotonicity; he also describes the subset of those meeting Scale Invariance as well.
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Because of the strict resource monotonicity requirement, neither the uniform gains nor the
uniform losses method is an equal sacrifice method, therefore the Corollary to our Theorem 2
(characterizing the three equitable methods within our larger family) cannot be deduced from
Young’s result. However, the intuitive connection is strong, and leads to a couple of open
questions described in the concluding Section 7.
From the point of view of the literature on Consistency, an important contribution of this
paper is to explore a subset of inequitable consistent rationing methods, where the previous
literature only considers equitable ones. The general model of division according to “types”
(introduced in the Appendix of Young [1994]) is no exception. There individual demands are
replaced by the more general notion of “type,” and the method treats different types differently,
including the possibility that one type has absolute priority over another type. In particular
Theorem 7, page 186 of Young [1994] is cast in the context of the discrete rationing model (with
individual demands of arbitrary size, unlike his Theorem 1) and offers a characterization of the
priority rules based on a weaker version of consistency; this is quite similar to our Theorem 1 in
Section 5 but 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 yi of the
commodity are single peaked, with their peak at x i . The central axiom is the property of
strategyproofness, 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 equitable
and strategyproof method: Sprumont [1991], Ching [1994].7 But the set of all strategyproof
7
Equity is interpreted as Anonymity, s stronger form of ETE, or as No Envy in Sprumont [1991] and as Equal
Treatment of Equals in Ching [1994].
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methods contains many nonequitable ones, and this complicated family is described in Barbera,
Jackson and Neme [1995].
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 i as her most preferred
consumption it makes sense to assume that agent i’s preferences are strictly increasing on [0, xi ]
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 [1997]) 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 (described in Example 4, Section 6). In the
discrete model this family contains much more than the priority rules. In the continuous model it
is different frombut related tothe family uncovered in Theorem 2.
One last paper strongly related to this one is Moulin and Shenker [1997]. That paper
looks at the costsharing problem with variable demands of a homogeneous good, when the
costsharing method can take into account the whole cost function. The two axioms of Additivity
(of cost shares with respect to the cost function) and of Distributivity (with respect to the
composition of cost functions) are combined to characterize a certain family of costsharing
methods. That result has deep connections with the present Theorem 2: see the last concluding
comment in Section 7.
4.
The two models and the three main axioms
The following notations are used. We denote by N the set of nonnegative integers and
by R that of nonnegative real numbers. For any finite set N we denote by i the i-th coordinate
vector of R N and by
of N, we write:
SN
the unit simplex of R N . For any x in R N (or N N ) and any subset M
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x M xi ; x M is the projection of x on R M (or N M )
iM
Throughout the paper we fix a set
N0
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 rationing problem is given by a finite subset N of N0 , a profile of individual demands
x i , one for each i N , and a quantity t of “resources” to be divided among N. We always
assume: 0 t x N . A discrete rationing problem is one where each demand x i and the resource
t are nonnegative integers, namely, the good to be divided comes in individual units. A
continuous rationing problem is one where each x i as well as t are nonnegative real numbers,
namely the good in question is divisible.
Each agent i in
N0
has a maximal demand X i . We assume X i , so X i
means that agent i’s demand is not bounded above. Thus a demand profile x for the society N
varies in X ( N ) [0, X i ] , where the interval [0, X i ] is taken in N or R .
iN
Definition 1
Given throughout the paper are
for each agent i in
N0 , finite or countably infinite, and
X i , 0 X i ,
N0 .
A discrete (resp. continuous) rationing method associates with every discrete (resp.
continuous) rationing problem ( N ; t; x ) , where x X ( N ) and 0 t x N , a profile of individual
shares y denoted y r ( N ; t; x ) and such that
y N N ( resp. y R N ); 0 yi xi for all i N ;
yN t
(1)
Throughout the paper, we only consider rationing methods meeting two mild properties as
follows:
Resource Monotonicity
for all N N0 , all x X ( N ), all t , t : {0 t t x N } {r ( N ; t; x ) r ( N ; t ; x )}
(2)
If Resource Monotonicity fails, it is hard to interpret r as a rationing method, as more resources
to distribute may result in a smaller share for some agents.
Independence of Null Demands
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for all N , M s. t. M N N0 , all x X ( N ), all t
{xi 0 for all i N \ M } {[r ( N ; t; x )]M r ( M ; t: x M )}
(3)
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 (1)), and
Independence of Null Demands says that it makes no difference to ignore these agents entirely.
Independence of Null Demands is a consequence ofbut a much weaker requirement thanthe
powerful axiom of Consistency (see below).
For every subset
method r on
N
of
N0 , we denote by r N
the natural projection of the rationing
N:
for all N
For any finite subset N of
N, all x X ( N ), all t:
N0
r N ( N ; t; x ) r ( N ; t; x )
the Independence of Null Demands property implies that r N is
entirely determined by the restriction r ( N ) of r to the rationing problems for society N, namely
r ( N )( t; x ) r ( N ; t; x ) . Therefore when
by r(
N0
is finite, the entire rationing method is determined
N0 ) .
We denote by
( N0 )
the set of rationing methods (Definition 1) satisfying Resource
Monotonicity and Independence of Null Demands. Depending on the context, we speak of a
continuous method (when x, t, y are real numbers) or of a discrete method (when they are
integers). As no confusion will arise, we use the same notation in both contexts.
Note that a “continuous” rationing method r ( N ; x; t ) is not necessarily a continuous
function of the demand profile x, though properties (1) and (2) imply that it must be continuous
in the resources t. In fact, all methods characterized in Theorem 2 are continuous with respect to
x as well.
There is a natural duality operation on rationing methods that plays a key role below.
Given r in
( N0 ) , its dual r* is the following rationing method:
for all N , t , x: r *( N , t , x ) x r ( N ; x N t; x )
The rationing method r allocates a total loss t (namely ( x N t ) units of resources) by deducing
xi ri ( N ; x N t; x ) from agent i’s demand. Thus the dual rationing method r* splits t units of
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resources as the method r would split t units of losses. We let the reader check that r* meets
properties (1) (2) and (3), and that the duality operation is idempotent ( r*)* r .
We now define the three main invariance axioms C, D, and D*; their definition is
identical in the discrete and continuous models.
Consistency: for all N , all i, j N , i j, all t and all x:
ri ( N ; t; x ) ri ( N \ j; t rj ( N ; t; x ); x N \ j )
Distributivity: for all N fixed (omitted in formula) all t , t and x :
0 t t x N r ( t; r ( t ; x )) r ( t; x )
Distributivity*: for all N fixed (omitted in formula) all t , t and x :
0 t t x N r ( t; x ) r ( t ; x ) r ( t t ; x r ( t , x ))
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 y among N is
recommended by the rationing method for a certain profile x of demands, consistency insists that
the restriction of y to any subset N of N (simply ignoring the shares of agents in N \ N ) be
recommended by the method for the restriction of x to N : shrinking the jurisdiction of the
problem does not alter the correct decision.
Distributivity (also called Path Independence in Moulin [1987]) allows to carry out a
“partial” rationing when we know an upper bound t on total resources: we may lower agent i’s
demand from its initial value xi to xi ri ( t ; x ) , and consider x i as his new demand whenever
further rationing of the resources to distribute (from t to t ) occurs.
Distributivity* (also called Composition Principle in Young [1988]) allows a partial
distribution of the resources when we know a lower bound t of the available resources: we
distribute the shares r ( t , x ) among N, and use agent i’s residual demand xi ri ( t ; x ) as the basis
for distributing the additional resources t t that become available later on.
Note that Distributivity and Distributivity* are “dual” axioms, that is to say the rationing
method r meets D* if and only if its dual method r* meets D. By contrast, Consistency is a selfdual axiom, namely r meets C if and only if r* meets C (we omit the straightforward proof of
these facts).
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A final comment on the combination of the Distributivity and Distributivity* 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 [t1 , t2 ] , where 0 t1 t2 x N . Then the
combination of D and D* allows reduction of the rationing problem to one with a reduced
x and total demand t t :
demand profile ~
2
1
~
~
set ~
x r( t2 , x ) r( t1, x ) and t t t1, then r( t , x ) r( t1, x ) r( t , ~
x)
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 D and D* several times:
r ( t , x ) r ( t , x ) r ( t1 , x ) r ( t t1 , x r ( t1 , x )) r ( t1 , x ) r ( t t1 , r ( t2 , x ) r ( t1 , x ))
5.
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
C, D and D*.
We denote by an ordering of
N0 , namely a complete, transitive, and antisymmetric
binary relation. If ranks i above j, we say that i has priority over j. The restriction of to a
finite subset N of
N0 , with cardinality n, is represented by a bijection from {1,2,, n} into N,
also denoted , with the interpretation that (1) i means “agent i has the highest priority in
N,” ( 2) j means “agent j has the second highest priority” and so on.8
The priority method r ( N ) associated with the priority ordering of N is now defined.
For all t , x the vector y r ( N )( t; x ) is the unique vector of shares meeting property (1) and
such that:
for all i, j N:{ y j 0 and 1(i ) 1( j )} { yi xi }
Equivalently, we can compute y from the unique integer i*,0 i* n | N | such that:
i*
i*1
i 1
i 1
x ( i ) t x ( i )
by setting
8
Note that if N0 is infinite, the initial ordering
namely a bijection from N into N0 .
is not necessarily representable as an enumeration of N0 ,
15
i*
y ( i ) x ( i ) for i 1,, i*, y ( i*1) t x ( i ) , y ( j ) 0 for j i * 2,, n
i 1
Checking that r meets properties (2), (3) is straightforward.
Theorem 1
For any ordering of
N0 , the priority rationing method r
satisfies the three properties
Consistency, Distributivity, and Distributivity*.
Conversely, in the discrete model, a rationing method satisfying the three properties C,
D, D*, is the priority method r for some ordering of
N0 .
All proofs are gathered in the Appendix. Note that the first statement ( r meets C, D, and D*) is
easy to check, whether in the discrete or continuous model. The converse statement is harder to
prove.
Some intuition for the converse statement is provided by three examples showing that
Theorem 1 is a tight characterization result.
Example 1: A method meeting C and D but not D* (or C and D* but not D)
Assume
N0 {1,2}, X1 X 2
and consider the method r that gives priority to 2 for
all x such that x1 x2 , but gives priority to 1 for x such that x2 x1 :
x1 x2 : y r( t; x ) is s.t. y1 0 t x2
x2 x1 : y r( t , x ) is s.t. y2 0 t x1
The C property has no bite when |
N0 | 2 ; D holds because if x belongs to the area {x1 x2} ,
then the whole sequence r ( t; x ), t 1,, x12 stays in that area (and the complement area
{x2 x1} has the same stability property). The following shows that D* fails:
r (1;( 2,2)) ( 0,1)
yet
r (3;( 2,2)) ( 0,1) r ( 2;( 2,1))
Similarly the dual method r* meets C and D* but not D.
Example 2: A method meeting D and D* but not C
Assume
N0 {1,2,3} and
X i 1 for i 1,2,3 . Consider the following method r(
r ( 2;(111
, , )) (11
, ,0 ) ;
r (1;(11
, ,0)) (1,0,0) ;
N0 ) :
r (1;(111
, , )) (1,0,0)
r (1;( 0,11
, )) ( 0,1,0) ; r (1;(1,0,1)) ( 0,0,1)
The method r and its dual are represented on Figure 1.a and 1.b, thus establishing D and D*.
To see that C fails, observe that
16
r1 ({1,2,3};2;(111
, , )) 1 and r2 ({1,2,3};2;(111
, , )) 1 yet r1 ({1,3};2 1;(11
, )) 0
Remark 1
Theorem 1, page 175 in Young [1994] is closely related to our Theorem 1. Consider the
case where each agent demands at most one unit of indivisible good: X i 1 for all i
N0 .
Then
it turns out that the Consistency axiom implies both Distributivity and Distributivity* (the proof
of this fact is presented at the end of that of Theorem 1 in the Appendix). Therefore our
Theorem reads: a rationing method is a priority method if and only if it satisfies Consistency.
Young’s result is a generalization of the latter statement to the case where the rationing method is
multivalued and the priority ordering allows for indifferences.
6.
Continuous model: the main theorem
In the continuous model, the priority rules satisfy, naturally, the same three properties
Consistency, Distributivity and Distributivity*, but so do many other methods. In particular, the
three rationing methods most often discussed in the literature, meet C, D, and D*. They are
defined as follows, for any N, t and x:
Proportional method: pro
pro( N ; t; x )
t
x
xN
Uniform gains method: ug for all i ugi ( N ; t; x ) min{ , xi } where min{ , xi } t
N
Uniform losses method: ul for all i ul ( N ; t; x ) max{xi ,0} where max{xi ,0}
N
Note that ug and ul are dual of each other, whereas pro is self-dual.9 For our main
characterization result, 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:
Scale Invariance: for all N , t , x , all ,0 1: r ( N ; t; x ) r ( N ; t; x )
Observe that pro, ug, ul, as well as all priority methods are scale invariant.
9
Young [1988] shows (Theorems 3, page 334) that pro is the only self-dual rationing method satisfying D (or,
equivalently, D*). As Young himself remarks, self duality is a strong property with no clear ethical meaning.
17
We now define two inequitable (asymmetric) generalizations of the uniform gains and
uniform losses methods respectively that meet the four invariance properties C, D, D* and SI.
These methods play a key role in Theorem 2.
Definition 2
Given a set of positive weights wi , i
N0 , the Weighted Gains method
g w is given by:
for all N , t , x and all i N : g w ( N ; t; x ) min{wi , xi }where N min{wi , xi } t
The Weighted Losses method l w , its dual, is given by:
for all N , t , x and all i N : l w ( N ; t; x ) max{xi wi ,0}where N max{xi wi ,0} t
Lemma 1
The Weighted gains and Weighted losses methods meet the four axioms C, D, D* and SI.
The straightforward proof is omitted. Figure 2 illustrates Definition 2 in the 2 agents
case. 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 w j whenever i has
priority over j (we omit the straightforward details).
Next we define the operation of composition or rationing methods by a priority
preordering and note that this operation respects the four invariance properties C, D, D*, and SI:
this allows the construction of a rich family of methods with these four properties.
Definition 3
Given a rationing method r in
( N0 ) , and two agents i, j, we say that r gives priority to i over j
if j is never allocated any resource until i’s demand is met in full:
for all N such that i, j N , for all t , x:
y j 0 yi xi where y r( N ; t; x )
In the next Lemma, we are given a preordering
~ of
binary relation on
N0 .
N0 , namely a complete and transitive
We interpret the strict relation associated with
~ ( i j iff i
~ j but not j
~ i ) as a strict priority relation. We denote by
class of
~.
Lemma 2
N an indifference
18
Given are
~ , a preordering of
method r N (
N) .
N0 , and for each equivalence class N
of
~ , a rationing
There exists a unique rationing method r , r (
r projects onto r N for every equivalence class
r gives priority to i over j if and only if i j , for any i, j in
N0 )
such that:
N : r N rN
N0 .
We call r the
~ priority composition of the methods r N .
Once again, the proof of Lemma 2 is straightforward. We can give an explicit formula
for r as follows. Any finite subset N of
N0
is partitioned by the equivalence classes of
~ as
k
N1 ,, N K , with N k N k 1 for all k . Given a demand profile x, denote t k x N k so that
l 1
0 t1 t K xN , and compute the shares allocated by r as follows:
k
if t k t t k 1: yi xi for i N l
l 1
yi r Nk ( N k ; t t k , x N k 1 ) if i N k 1
yi 0 if
i
(4)
K
N
l
l k 2
(where
Nk
is the equivalence class of
~ containing N k .)
Lemma 3
Notations as in Lemma 2. If each method r N meets the four axioms C, D, D*, and SI, so does
their
~ -priority composition.
The proof follows by inspection of formula (4). Lemma 3 implies that any
~ -priority
composition of methods taken among Proportional, Weighted Gains, and Weighted Losses, does
meet the four axioms C, D, D*, and SI. Hence a fairly large family on offer to the mechanism
designer.
One common feature of the methods pro, g w and l w is that they do not involve any strict
prioritizing between agents: if the weights w are very unequal, the methods g w and l w are
serving individual demands at a very inequitable 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
19
contrary, in g w every agent with a positive demand receives a positive amount of resources (if t
is positive). Similarly in l w , no agent receives her full demand unless t x N .
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
~ 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 w and l w .
Our last preliminary result gives the precise meaning of this claim.
The notion of ordered covering of
SN , the unit simplex of
R N , is borrowed from
Moulin and Shenker [1997]. 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 {1,2} .
An ordered covering
ordered intervals [e1, e2 ] of
C of S{1,2} is a set (not necessarily finite) of singletons {e} and
S{1,2} , with nonempty interior, such that
their union covers
if [e1, e2 ] C then {ei } C as well, i = 1,2
the interiors ]e1, e2 [ and ]e1, e 2 [ of any two intervals in
S{1,2}
C
are disjoints, and they do not
contain any point {e} of C .
Given the ordered covering
element of
C
C
{12}
and some x R \ {0} , we denote by C( x ) the smallest
1
x itself or an interval [e1, e2 ] with ~
x in its
x
x : it can be ~
containing ~
xN
interior.
Lemma 4
Assume
N0 {1,2} .
( N0 )
{12}
as follows: for all x R \ {0}
To each ordered covering
C
of
S{1,2} we associate a rationing method in
20
if C( x )
1
t
x : r ( t; x )
x
xN
xN
if C( x ) [e1 , e2 ], then x 1e1 2e2 for some 1 , 2 0
r ( t; x ) t e1 for 0 t 1
r ( t; x ) 1 e1 ( t 1 ) e2 for 1 t x{12}
( 5)
The rationing method r meets the properties D, D*, and SI. Conversely any rationing method in
N0
meeting D, D*, and SI is associated with an ordered covering of
S2 .
We write
H2 ( N0 )
for the set of rationing methods thus constructed.
The proof of Lemma 4 is in the Appendix. We illustrate the family
C
consists of all singletons {e} of
H2
by some examples.10 If
SN , its rationing method is the proportional one.
C ({1},{ 2},[ 2 , 1 ]) (recall that i
If
is the i-th coordinate vector), it yields the {2,1} -priority
rule. The uniform gains method is derived from the covering ({ 1},{ 2},{e},[e, 1 ],[e, 2 ]) ,
where e
1 1
( 2 ) . Its dual, uniform losses, obtains by exchanging the orientation of the two
2
intervals.
Obviously
H2
contains (infinitely) many more methods. For instance, the ordered
covering may contain all the singletons {e} between 1 and e as well as {e2} , and the oriented
interval [e, 2 ] , so the associated rationing method is a hybrid of the proportional method “to the
right of e” and of uniform gains “to the left of e”: see Figure 3.
Definition 4
Given a (finite or infinite) subset
Nof N0 , an irreducible method on N
is one of the
following:
if |
N| 2
if |
N| 3 one of the following methods:
10
a method in
H2 ( N ) with the exception of the two priority methods.
proportional (restricted to
N
weighted gains g w , w R
N
weighted losses l w , w R
N)
For a more detailed discussion, see Moulin and Shenker [1997].
21
We denote by
H *( N)
the set of irreducible methods on
N.
Theorem 2
Given are
N0
(finite or infinite) and the maximal demand X i ,0 X i , for each i
The rationing method r (
N0 )
N0 .
satisfies the four axioms: Consistency, Distributivity,
Distributivity* and Scale Invariance if and only if there exists a priority preordering
~ of
and, for each indifference class
N
N0 ,
*
of
~ , an irreducible method r N H ( N) , such that r is
the
~ -priority composition of the methods r N (Lemma 2). We denote by
H ( N0 ) the set of
methods thus defined.
Equal Treatment of Equals is the basic equity requirement (discussed in the introduction)
that two equal demands receive the same share:
for all N , t , x, all i, j: xi x j ri ( N ; t; x ) rj ( N ; t; x )
Within the family
H ( N0 ) , Equal Treatment of Equals is only satisfied by three methods.
Corollary to Theorem 2
Assume |
N0| 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.
Example 3: Equal sacrifice methods (Young [1988])
In this interesting class of methods, all axioms but D*or Dare satisfied, and the
methods are outside the set
H ( N0 ) .
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( x j ) u( y j ) for all i, j
Thus u( xi ) u( yi ) 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 N t ,
has a unique solution for all x , t . For any such choice of u, the rationing method thus defined
meets C, D, and ETE. If one chooses u as a power function, the method is also Scale Invariant.
For instance, u( x )
1
yields the method
x
22
yi
xi
where solves: 0 and
1 xi
xi
1 x
N
t
i
Figure 4 illustrates the rationing paths of this method. It is easy to see on this Figure that the
method violates D*. Naturally, the dual of the above method meets all the requirements of the
corollary with the exception of D.
Young [1988] characterizes equal sacrifice methods by the combination of C, D, and
ETE, together with Strict Resource Monotonicity: see his Theorem 1 as well as Theorem 2 for
the case where SI is added to the list of requirements. Those results are the closest to our
Theorem 2 in the literature.
Example 4: Fixed path methods (Moulin [1997])
These methods meet C, D, and D* but fail SI and ETE. Consider the following
asymmetric version of uniform gains where the reference shares grow at unequal, nonlinear
paces. Choose a fixed positive weight i for each i
N0
and define:
yi min{ i , xi } where solves: 0 and N min{ i , xi } t
See Figure 5. To check the announced properties is straightforward.
The fixed path methods are characterized in Moulin [1997] by the two properties of
Consistency and Strategyproofness (discussed in Section 3). Finally, we can adapt a fixed path
method so as to obtain a method meeting C, D, D*, and ETE as well. Such a method is depicted
on Figure 5 in the case of 2 agents (we omit the straightforward formulas).
Example 5: A method meeting D, D*, SI, but failing C
Here we use the rich space of rationing methods described in Moulin and Shenker [1997]
(in the closely related context of additive costsharing methods). All such methods meet D, D*,
and SI, and they are derived from arbitrary ordered coverings of the simplex
SN
(see proof of
Theorem 2 in Appendix, or Definitions 1,2 in Moulin and Shenker [1997]).
For instance, with
N0 {1,2,3} 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 ) :
23
y1 ug1 ( t;( x1 , x23 ))
x
y2 2 ug2 ( t;( x1 , x23 ))
x23
x
y3 3 ug2 ( t;( x1 , x23 ))
x23
It is also easy to find a method meeting D, D*, SI, and ETE, but failing C. For instance define
the shares y by the formulas below for a demand profile x such that x1 x2 x3 and in the other
cases by similar formulas exchanging the roles of the agents:
if x1 x2 x3 and t 3x1:
if 3x1 t x123:
t
y 1 where 1 (111
,,)
3
y x1 1 is parallel to x x1 1
We let the reader check that the two above methods meet D, D*, and SI. They fail C because the
projection on {12} of the rationing path to a demand profile ( x1 , x2 , x3 ) depends on x3 .
7.
Concluding comments
In the discrete rationing model, the analog of the Scale Invariance axiom is Replication
Invariance:
for all x, t and all integer : r ( t; x ) r ( t; x )
(strictly speaking we must restrict the property to those numbers such that x X ( N ) ). The
priority methods r 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
, )) (1,0) : the method
property. To see this, consider the case N {1,2} and assume r(1;(11
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 equitably
r ( 2q;( 2q,2q )) ( q, q ) : the fact that one unit had to be allocated inequitably follows simply from
the indivisibility, and does not imply that the method has to favor the same agent at every level of
resources.
Finally, we state three open problems in the continuous model directly inspired by
Theorem 2. Consider first the combination of the three axioms C, D, and SI. Recall that Young
24
[1988] adds Equal Treatment of Equals and Strict Resource Monotonicity 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 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 ETE
as well: the combination C, D, SI, allows our entire set
H ( 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 C, D, and SI?
Similarly, consider the combination C, D, D*. We know that all fixed path methods
(Example 4) satisfy these three, but it is not at all clear what is the most general form of a method
meeting C, D and D*.
The last relevant triple of axioms for which the corresponding family of rationing
methods is not known is D, D*, SI. Here some clues toward an answer are given by Theorem 1
in Moulin and Shenker [1997]. That paper looks at costsharing methods with variable demands
of a homogeneous good. That is to say, a costsharing problem is given by a demand profile
( x1 ,, xn ) and a cost function C, from R into itself. The costsharing method must select a
profile ( y1 ,, yn ) of cost shares so that y N C( x N ) . The familiar assumption of Additivity of
cost shares ( yi ) with respect to the cost function C essentially implies that our costsharing
method is entirely determined by a certain rationing method, via the integral formula
yi
z
xN
0
C ( t )
dri
( t; x )dt
dt
The result in question (Theorem 1 in Moulin and Shenker [1997]) 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 D, D*, and SI.
Therefore the entire family of costsharing methods characterized in Theorem 1 gives us new
rationing methods in the family under investigation. They are built, just like the methods in
H ( N ) , with the help of an ordered covering of the simplex SN
, and generalize to an arbitrary
n the methods described in Lemma 4 above. I conjecture that these methods exhaust the
possibilities under the triple requirement D, D*, and SI.
25
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27
Appendix: Proofs
1.
Proof of Theorem 1
We fix throughout the proof a rationing method r meeting the three properties C, D, and D*.
Step 1 Preliminary notations and observations
Given N, a finite subset of for all T N
N0 , all x in X ( N ):
(T ; x T ) [( N , x )]T , and x
in, properties (1) and (2) imply that the path moves by increments equal to a coordinate vector:
(3)
Therefore the path t r ( N ; t; x ) is equivalently described by the sequence ( N , x ) of its
derivatives, namely the sequence in N with t-th element i given by (3). The sequence ( N , x )
has x N elements and agent i appears exactly x i times, for all i N .
Clearly a rationing method is entirely described by the family of sequences ( N , x ) , for
all N and all x. For instance, consider the priority method with ordering
. For any N with
cardinality n, and any x, the sequence ( x , N ) is
{
(1
),
, (
1),
2
,
(
2),,
n ),
( n
)}
(
,
(
,
x ( 1 )
x ( 2 )
x ( n )
Notation: for any N and any subset M of N we denote by [( N , x )]M the “projection” of
the sequence
( N ; x ) on M, namely the sequence 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 fixed 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 N 1; x ) . Distributivity is equivalent to the property:
r( t , x ) p( xN t ) ( x ) for all x, all t ,0 t xN
where p( a ) denotes the a-th “power” of p. Next Distributivity* implies the following:
p( x ) r (1; x ) r ( x N 2; x r (1; x )) p( x ) r (1; x ) p( x r (1 x ))
(6)
28
Now we fix x and assume that the sequence ( N , x ) starts and finishes by the same agent i,
namely:
r(1; x ) i
and
x p( x ) i
(7)
Setting x p( x ) and invoking Distributivity, we have
r (1; x ) r (1; r ( x N 1; x )) r (1; x )
Moreover, (6) and (7) imply
p( x ) p( x i ) p( x ) i x i x p( x ) i
Therefore x satisfies the property (7): the sequence ( N ; x ) starts and finishes by i. Now we
repeat the argument to show that x p( x ) p( 2 ) ( x ) satisfies (7) as well, and so on. Hence the
sequence ( N ; x ) contains agent i only. Thus the only vectors x satisfying (7) are coordinate
vectors x xi i .
Next we pick an arbitrary x and t ,2 t x N , and assume that the sequence ( N , x ) has i
for first and t-th element:
r(1; x ) i and r( t; x ) r( t 1; x ) i
Set y r ( t; x ) and invoke Distributivity to show that
r(1; y ) i and y p( y ) r( t; x ) r( t 1; y ) r( t; x ) r( t 1; x ) i
Hence by the above argument, the whole sequence ( N ; y ) is constant and equal to i, which in
turn says that the first t elements of ( N ; x ) are equal to i.
We have just proved that for all x, the sequence ( N ; x ) must be one of the two priority
sequences:
{1
,
,1, 2
,
,2} or {2,
,2,1
,
,1}
x1
x2
x2
x1
It remains to prove that the priority ordering in ( N ; x ) is independent of x, for any x such that
x1 0 and x2 0 (note that if one of x i is zero, there is nothing to prove). Assume that ( N ; x )
gives priority to agent 1, ( N , x ) {1,,1,2,,2} . Note that p( x ) x 2 (because x2 0 ) and
that the sequence ( N , p( x )) obtains from ( N , x ) by deleting the last term (by Distributivity).
Therefore ( N , x 2 ) also gives priority to agent 1. Next we show that ( N , x 1 ) gives
priority to agent 1, if x1 2 . Compute, with the help of (6):
29
p( x 1 ) p( x r(1; x )) p( x ) 1 ( x 1 ) 2
Thus the last term of ( N , x 1 ) is 2 hence ( N , x 1 ) must give priority to 1.
We have shown that if ( 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 | 2 the rationing method r ( N ; ; ) is a priority
method.
Step 3 An equivalent formulation of Consistency
We claim that the rationing method r defined on
for all T N
N0 , all x in X ( N ):
N0
satisfies consistency if and only if
(T ; x T ) [( N , x )]T
(8)
In other words, Consistency amounts to the commutativity of the mapping x ( 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. To see this, observe that the projection satisfies
[[( N , x )]T ]S [( N , x )]S whenever S T N .
Next we introduce two notations. Given a sequence w in N of length at least t, we write
( 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:
O( i; ( t; ( N , x ))) ri ( N ; t; x )
(9)
The following equivalence follows from (9) as well as from the definition of , and the
projection operator:
( N \ j, x N \ j ) [( N , x )]N \ j for all t ,1 t x N
[ ( t; ( N , x ))]N \ j ( t rj ( N ; t; x ); ( N \ j, x N \ j ))
We are now ready to prove the equivalence of Consistency and of property (8). Assume the
latter. Then we compute from (8), (9) and (10)
ri ( N \ j; t rj ( N ; t; x ); x N \ j ) O( i; ( t rj ( N ; t; x ); ( N \ j, x N \ j )))
O( i,[ ( t; ( N , x ))]N \ j O( i, ( t , ( N , x )) ri ( N ; t; x )
Conversely, assume Consistency and use the same computation as above to derive
(10)
30
O(i,[ (t; ( N , x ))]N \ j ) O(i; (t rj ( N ; t; x ); ( N \ j, x N \ j )))
(11)
As the above equality holds for all i in N \ j and all t, 1 t x N , an easy induction argument on
t shows the desired equality namely
[ (t , ( N , x ))]N \ j (t rj ( N ; t; x ); ( N \ j, x N \ j ))
Indeed for t=1, if ( N , x ) starts by j then the two above sequences are empty, whereas if
( N , 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 ( N , x ) is j, both sequences do not change (because
2 rj ( N ;2; x ) 1 rj ( N ;1; x )) , whereas if the second term of ( N , x ) is i, i j , this same i is
added to both sequences. And so on.
Step 4 End of proof
By Step 2 we can define a complete binary relation in
{i j} iff
N0
as follows:
{r ({ij}, , ) gives priority to i over j}
This relation is clearly antisymmetric. We show by contradiction that it is transitive. If
, , ) . By (2) and the definition of
1 2,2 3, and 3 1 consider ( N , x ) for N {123} and x (111
we have:
[( N , x ){12}
({12},(11
, )) {1,2}
[( N , x )]
({23},(11
, )) {2,3}
[( N , x )]{13}
({13},(11
, )) {31
,}
{23}
a contradiction. Therefore is an ordering of
corresponding priority method. Fix N in
N0
N0 .
It remains to check that r is the
and x in X ( N ) and consider the sequence
( N , x ) . By property (8) applied to N and T {ij} we know that all occurrences of i in ( N , x )
must precede those of j. The desired conclusion follows at once.
QED
Proof of Remark 1
Assume X i 1 for all i . Then the sequence ( N , x ) (see Step 1) is an ordering of the support of
x, namely N x {i N / xi 1} . Assume | N x | m and denote: ( N , x ) {m, m 1,,2,1}
By (8) applied to T N \ 1 : ( N \ 1, x N \1 ) {m, m 1,,2} . Hence for all t , t m 1 , we have
for all i N , i 1: ri ( N ; t; x ) ri ( N \ 1; t; x N \1 )
31
Denote by y the following vector in {0,1}N :
yi 0, yi xi for all i N \ 1 . Then, by Consistency
again and the fact that r1 ( N ; t; y ) 0 , we get:
for all i N , i 1: ri ( N ; t; y ) ri ( N \ 1, t; x N \1 )
Combining the above two equalities and the fact that r1 ( N ; t; x ) 0 (because t m 1 ), we have:
r ( N ; t; x ) r ( N ; t; y )
Note that y r ( N ; x N 1; x ) so that the above equation expresses Distributivity for the pair
t , t x N 1 . The full D property now follows from an obvious induction argument.
Finally, recall that C is a self-dual property, namely that r* meets C iff r does. Therefore
the above argument shows that r* satisfies D and so r satisfies D*, as was to be proved.
2.
Proof of Lemma 4
The direct statement is straightforward: the rationing method associated with an ordered
covering
C of S{12} meets D, D* and SI.
Scale Invariance is immediate. As for D, we fix an
arbitrary x in R{+12} \ {0} and show r ( t , x ) r ( t , r ( t , x )) for all t , t ,0 t t x N . If
C( x )
1
x , the claim is obvious. If C( x ) [e1, e2 ] and x 1e1 2e2 , we use (4) and
xN
distinguish two cases:
if t , then r( t , x ) t e1 so that r( t , r( t , x )) te1 as desired
if 1 t then r( t , x ) 1e1 ( t 1 )e2 therefore
r( t; r( t ; x ))
te1 for t 1
1e1 ( t 1 )e2 for 1 t t
and the claim is proved. The similar proof of D* is omitted.
Conversely, we fix a rationing method r satisfying D, D* and SI and we show that r is
associated with an ordered covering of
S{12} as stated in Lemma 4.
For simplicity, here and in the proof of Theorem 2, we assume X i so that
individual demands vary in R . The careful reader will check that all Steps of the proof are
unchanged when some X i are finite. Moreover, we write r ( t ; x ) instead of r ({12}; t; x ) as no
confusion will arise.
32
Step 1
x in R{12}
x ) is made of one or
We fix an arbitrary vector ~
and show that the path t r ( t; ~
+
x1 or ~
x2 is zero, the claim is obvious, so we can
two linear pieces as stated in Lemma 4. If ~
xi 0 for i 1,2 . For all t ,0 t ~
x N , we define
assume ~
( t ) ~
x1 r2 ( t; ~
x) ~
x2 r1 ( t; ~
x)
x , therefore if ( t ) 0 for all
x ) is proportional to ~
Note that ( t ) 0 if and only if r ( t; ~
t ,0 t ~
x N , the path t r ( t; ~
x ) follows the interval [0, ~
x ] . Assume next that ( t ) 0 for
x N , we have ( t ) 0 as well. The proof is by
some t. We show that for all t ,0 t ~
contradiction. Suppose that:
( t ) 0 and ( t ) 0 for some t , t ]0, ~
xN [
x N ] : this is well defined,
Let t be the largest number achieving the maximum of on [0, ~
x N ) 0 , we
because Definition 1 implies that r, hence as well, is continuous in t; as ( 0) ( ~
know that 0 t ~
xN . By continuity of there exists a t 0 , 0 t 0 ~
xN , such that ( t 0 ) 0 .
Denote x r( t ; ~
x ) and x 0 r( t 0 ; ~
x ) . For some ,0 1, we have x 0 ~
x . Assume first
t t 0 , so that D implies x r( t ; x 0 ) . Then SI implies:
x r ( t ; x 0 ) r (
(where
t
t ~
t
, x ) ( t ) . ( )
~
x N because t t 0 ) . The last equality implies ( t / ) ( t ) , a contradiction
of the definition of t .
The second case to consider is t 0 t . In this case we define the function * :
* ( t ) ~
x1 r2* ( t; ~
x) ~
x2 r1* ( t; ~
x)
x N ] at ~
xN t (this easy step is omitted).
and check that * is negative and minimized over [0, ~
Next we invoke ~
xN t ~
xN t 0 and Distributivity of the dual method r * :
r*( ~
xN t ; r * ( ~
xN t 0 ; ~
x )) r * ( ~
xN t ; ~
x)
xN t 0 ; ~
x) ~
x x 0 (1 ) ~
x . Therefore the above equation and Scale Invariance
Compute r * ( ~
of r * imply
33
~
x t
(1 ) r * ( t; ~
x ) r*( ~
xN t ; ~
x ) where t N
1
x N follows from t 0 t ). Therefore:
(Note that t ~
(1 )* ( t ) * ( ~
xN t ) where 0 (1 ) 1
a contradiction of the fact that ~
xN t minimizes * on [0, ~
x N ] (recall that * is negative at
~
xN t ).
x N [ it must be
Up to this point we have shown that if is positive somewhere on ]0, ~
positive everywhere on this interval. A similar argument shows that if is negative somewhere
x is either the
x N [ , it is negative everywhere. Therefore, the path t r ( t; ~
x ) from 0 to ~
on ]0, ~
x ] , or is everywhere above the corresponding line, or is everywhere below.
interval [0, ~
x x ) where
Fix t ,0 t ~
xN and apply the same argument to the path t r ( t; ~
x x ] or be everywhere above this interval or
x r( t ; ~
x ) . This path must be the interval [0, ~
everywhere below it. But D* implies:
r ( t; ~
x ) x r( t t ; ~
x x ) for t t ~
xN
x ) between t and ~
x N follows the chord [ x , ~
x ] or is everywhere above this
therefore the path r ( t; ~
chord, or is everywhere below.
x ), 0 t ~
x N is never
As the choice of t was arbitrary, it follows that the whole path r ( t , ~
below any of its chords, and/or is never above any of its chords (i.e., a parametrization x1 x2
of this path must be a concave function or a convex function). The claim follows from the
x is everywhere above [0, ~
x ] and contains x , then the path
observation that if the path to ~
x cannot be everywhere below [ x , ~
x ] . Suppose it is below and pick x between
between x and ~
x ; then by continuity of the path we can pick x below x and such that x and x are on
x and ~
x ] , as shown on Figure 6. This is a contradiction.
both sides of [ x , ~
x is always above [0, ~
x ] and never below any
From now on, we assume that the path to ~
x ) the highest point on this path maximizing the
of its chords. We denote, as above, x r ( t ; ~
x ] ( t maximizes on [0, ~
x N ] ). We claim that the path follows [ x , ~
distance to [0, ~
x ] between
t and ~
xN . If it is not, it lies strictly above [ x , ~
xN , such that
x ] ; we pick x1 r( t1; x ), t t1 ~
its distance to [ x , ~
x ] is maximal. See Figure 7. Denote x 2 x1 x and observe that x 2 is on
34
the dual path from 0 to x1 , namely x 2 r * ( t 2 ; x1 ) , where t 2 t1 t . As x is the highest point
x with maximal distance to [0, ~
x ] , it follows that x 2 x1 x is below [0, ~
x].
on the path to ~
x ] , and ends at x1 above
See Figure 7. So the dual path to x1 goes through x 2 , a point below [0, ~
this segment: by continuity, there exists x 3 r * ( t 3; x1 ) , with t 2 t 3 t1 , and proportional to
~
x : x 3 ~
x ,0 1 .
x contains ~
Now the dual path to ~
x x1 (because x1 is on the path to ~x ) hence by Scale
Invariance of r * , the dual path to x 3 contains the point ( ~
x x1 ) . As the dual path from 0 to x1
goes through x 3 , D* implies that it contains ( ~
x x1 ) as well. This in turn means that the path
from 0 to x1 contains the point x 4 x1 ( ~
x x1 ) (see Figure 7) and, by D, so does the path to
~
x . If x 4 lies below x on this latter path, we contradict the definition of x ; if it lies above, we
contradict the definition of x1 : see Figure 7.
x equals the interval [ x , ~
We have shown that the path to ~
x ] above x . Define
x ] . We assume
x r(t ; ~
x ) to be the lowest point on this path maximizing the distance to [0, ~
x x and derive a contradiction. The path to ~
x must follow [ x , x ] between t and t ,
because it is never below any of its chords and a point above [ x , x ] has a higher distance to
[0, ~
x ] . Therefore the dual path to x follows [0, x 0 ] , where x 0 x x , up to t 0 t t , see
Figure 8. By D*, the dual path to x 0 equals [0, x 0 ] , hence the path to x 0 equals [0, x 0 ] as well.
x so by Scale Invariance, the path to ~
x must be [0, ~
x ] , that was ruled
But x 0 is proportional to ~
out in the first place. This proves x x .
x : because r * satisfies all three properties C, D, D*,
Finally, we consider the dual path to ~
the properties uncovered above apply to the path r * ( t , ~
x ) as well. In particular, the dual path is
x ] because it contains the point ~
x ] . Moreover the dual path
always below [0, ~
x x , below [0, ~
x ] , namely above ~
x x ~
x x ,
is linear above the highest point with maximal distance to [0, ~
and so the dual path is linear beyond ~
x x ; thus the path to ~x is linear below x .
35
x consists of the two intervals [0, x ] and [ x , ~
We have shown that the path to ~
x ].
Denoting by e1 and e2 respectively the directions of these two segments, normalized to be in
S{12} , we have shown that either the path to
~
x follows [0, ~
x ] , or it is given by formula (5).
Step 2
From Step 1, we can define for all nonzero x
1
x} if the path to x follows [0, x ]
xN
C( x ) {
C( x ) [e1, e2 ] if the path to x is given by (5).
In Step 2 we show that the sets C( x ) , when x varies, constitute an ordered covering of
S{12} and
that r is the associated rationing method.
We already know that these sets cover
S{12} , and must show that their relative interior are
mutually disjoint. To see this, we take any ~
x 1e1 x e2 and C( ~
x ) [e1, e2 ] ,
x , e1, e2 such that ~
and we prove that for all x such that x 1e1 2e2 , i 0 for i 1,2 , we must have
C( x ) [e1, e2 ] as well, namely the path to x is given by formula (5) (where i replaces i ).
Consider a point x on the half-line borne by e2 at x 1e1 , excluding x . See Figure 9.
x x 2e2 1e1 2e2 where 2 0
Is x is on [ x , ~
x ] , D implies that the path to x follows
(12)
[0, x ] at first, then [ x , x ] as required by
x on the halfline (i.e., 2 2 ), there is a number ,0 1 , such that
(5). If x is beyond ~
x x, ~
x ] : see Figure 9. Thus by Distributivity of r * , the dual path to
x is on the interval [ ~
x x ] then [ ~
x x , x ] . Hence the path to x is as required by (5); by Scale
x follows [0, ~
Invariance, the same holds for x. So the path to any point on the halfline L given by (12) takes
the required form. By Scale Invariance the same holds true for any point on a halfline L , for
any 0 . Such points cover the cone {1e1 2e2 / i 0, i 1,2} and the proof that all points
in this cone have C( x ) [e1, e2 ] is now complete.
It remains only to check that if an ordered interval [e1, e2 ] belongs to our covering, so do
{e1} and {e2} . If C( ei ) is not {ei } , then C( ei ) is an interval [e3 , e4 ] containing ei in its interior,
36
hence overlapping with [e1, e2 ] , a contradiction of the above argument. This completes the proof
of Lemma 4.
3. Proof of Theorem 2
We already know that every method in
H ( N0 ) meets C, D, D* and SI.
method meeting these 4 axioms and show that it belongs to
subset N of
N0
H ( N0 ) .
and show the existence of an ordered covering of
SN
Conversely, we fix a
In Step 1 we fix a finite
generalizing the
coverings described by Lemma 4. In Steps 2, 3 and 4, we exploit the full force of Consistency,
successively in the case | N | 3 and | N | 4 , to show that the ordered covering must have the
structure corresponding to a method in
H ( N0 ) .
SN
Step 1 The ordered covering of
We fix N, finite, and we say that a set {e1,, e K } of vectors in
SN
is of full rank if these
vectors are linearly independent. Given an ordered sequence {e1,, e K } in
SN
and of full rank,
we denote by ( e1,, e K ) the relatively open cone
x ( e1,, e K ) iff
k , k 0, k 1,, K such that x 1 k ek
K
Note that the decomposition is unique. In Step 1 we prove the following claim: for all
~
x in R N \ {0} , there exists an ordered sequence, of full rank, in
SN
such that ( e1,, e K )
x and for all x in ( e1,, e K ) , the rationing method is as follows:
contains ~
if x 1 k e k , write k 1 k then
K
k
r(N; t; x) = 1 k ek ( t k ) ek for all t such that k t k 1
k
(13)
x varies, the (relatively open) cones ( e1,, e K ) form a
Clearly, the claim implies that when ~
partition of R N \ {0} , that we call the ordered covering of
SN
induced by r. This terminology
generalizes that of Lemma 4, from the case n 2 to an arbitrary n.
37
Before proving the claim, we state two useful mathematical properties. Given the finite
set N, and a point ~
x in R N \ {0} , a path ( N ; x ) is a subset of R N , connecting 0 to x , that is
equal to the range of a nondecreasing mapping , from [0, x N ] into R N such that
( 0) 0, ( x N ) x, N ( t ) t
for all t ,0 t x N
(the usual terminology is “monotone path” but we will not consider nonmonotone paths). A path
( N , x ) has a canonical projection
[ ( N , x )]M on R M , for every subset M of N: it is an
easyalthough not entirely trivialmatter to check that the projection of a N-path is a M-path.
This fact implies at once the following property:
Uniqueness lemma: Given a finite N, with | N | 3 , a point x in R N \ {0} and for each i N , a
path i ( N \ i; x N \ i ) , there can exist at most one path ( N ; x ) such that
[ ( N ; x )]N \i i ( N \ i; x N \ i ) for all i N
Moreover, if each path i is piecewise linear (with finitely many pieces), so is .
In order to state the second mathematical fact, we introduce some more notations. For N, M
finite and M N we define a projection operator from the simplex
for all e
SN :
SN
into
SM {0} :
p M ( e ) 0 if e M 0
p M (e)
(14)
1 M
e if e M 0
eM
Next for any sequence {e1 ,, e K } in SN , we denote by p M ( e1,, e K ) the sequence in
SM
obtained from { p M ( e1 ),, p M ( e K )} by removing zero vectors and merging consecutive elements
if they are equal. Thus the sequence p M ( e1,, e K ) might have fewer than K elements (e.g., a
single element if e1 e K and ( e1 ) M 0) ; it might even be empty, if every ek , k 1,, K ,
projects to 0 on R M .
Full rank lemma: Given N finite, with | N | 3 , and a sequence {e1 ,, e K } in SN , of which two
consecutive elements are not equal, suppose that for all i N , the sequence pN \ i ( e1,, e K ) is of
full rank in R N \ i . Then the sequence {e1,, e K } is of full rank in R N . In particular, K n .
In the above statement we adopt the convention that the empty set is of full rank. The proof
of this fact is relegated to Step 6 below.
38
Now to the proof of the claim. We denote by ( N , x ) the path associated with our rationing
method r (namely the image of r ( N ; t; x ) when t varies in [0, x N ] ). Observe that Consistency is
equivalent to the following property (analogous to property (8) in the discrete model):
for all finite M , N , with M N and for all x R N \ {0}
( M , x M ) [( N , x )]M
(15)
Indeed, by definition of Consistency, we have for all j
( r( N ; t; x ))N \ j ( N \ j, x N \ j )
[( N , x )]N \ j
( N \ j , x N \ j )
Now the two sets on each side of the inclusion are (monotone) paths from 0 to x N \ j , hence they
must be equal. Repeated applications of this argument yield (15). Conversely, property (15)
applied to M N \ j implies for all t ,0 t x N :
there exists t ,0 t xN \ j : (r( N ; t; x ))N \ j r( N \ j; t ; x N \ j )
The equality t t rj ( N ; t; x ) follows at once.
We prove the claim by induction on | N | . Lemma 4 established the claim when | N | 2 so
we now assume n 3 and that it holds for all M such that | M | n 1 . We fix arbitrarily x in
R N \ {0} . By the induction assumption applied to x N \ i , for any i N , we know that the path
( N \ i, x N \i ) is either trivial (if x N \ i 0 ) or is a piecewise linear path with successive gradients
{e1( i ),, e Ki ( i )}, a full rank sequence of K i vectors in
SN \i , with
K i n 1 . Property (14)
implies that the projection of the path ( N , x ) on every subspace N \ i is either trivial or is
piecewise linear. Therefore ( N , x ) is piecewise linear (Uniqueness Lemma) with successive
gradients {e1, e K }
Moreover, the sequence of gradients {e1( i ),, e Ki ( i )} obtains from the sequence
{( e1 )N \ i ,,( e K )N \ i } by deleting zero vectors and merging consecutive elements if they are equal.
Thus the sequence {e1,, e K } satisfies the assumptions of the Full rank Lemma, and we deduce
that it is a sequence of full rank (and K n ). By definition of this sequence, the vector x is a
strictly positive linear combination
K
1
k ek and the rationing path r ( N ; t; x ) is given by (13)
(recall that ek is the sequence of successive gradients starting from t=0).
39
It remains to show that for any other element x in ( e1,, e K ) , with decomposition
x k ek , the path r ( N ; t; x ) is given by (13). Fix such an x and observe that ( x )N \ i is in
( e1( i ),, e Ki ( i )) for all i N . By the induction assumption, this means that the path
( N \ i;( x )N \i ) is computed by (13):
( x ) N \ i 1 k ( i ) e k ( i ) and r ( N \ i; t;( x ) N \ i )
Ki
follows the direction e k ( i ) on the interval [ k ( i ), k 1 ( i )], for all k 1,, K i
Now consider the path ( N ; x ) constructed, as in (13), by following successively the direction
ek on the interval
[ k , k 1 ], k 1,, K . It is straightforward to check that its projection on
N \ i is precisely the path r( N \ i; t;( x )N \i ) just described above (because the projection is linear,
and by definition of pN \ i ). On the other hand, (15) implies that ( N , x ) has precisely the same
projection on every N \ i . By the uniqueness Lemma, these two paths coincide, and the proof of
the claim is complete.
Step 2 A reformulation of Consistency
In Step 1 we showed that to each point x in R N \ {0} we can associate a relatively open
cone ( e1,, e K ) containing x and such that the rationing method r is given by (13) in this cone.
The ordered sequence {e1,, e K } in
SN
is uniquely defined for a given x, so we denote
C( x ) C0( e1,, eK ) the ordered polytope with ordered vertices e1,, eK . Step 1 implies that
when x varies, the relative interiors of the polytopes C( x ) form a partition of
SN
. We call
dimension of x the dimension of its ordered polytope, namely K.
To complete the proof of Theorem 2, it remains to apply the full force of Consistency on
these ordered coverings of
H ( N0 ) .
SN
, and to show that they are indeed generating a method in
The key observation is that Consistency can be expressed directly in terms of the
ordered polytopes just defined, namely as follows:
for all x R N \ {0}, all M N :
C( x ) C0( e1,, e K ) C( x M ) C0( p M ( e1,, e K ))
(16)
40
Recall that p M is defined by (14) and that p M ( e1,, e K ) stands for the sequence in
SM
obtained from { p M ( e1 ),, p M ( e K )} by removing zero vectors and merging consecutive elements
if they are equal.
To check that (16) is equivalent to property (15), notice that if the path ( N , x ) is
piecewise linear with successive gradients {e1,, e K } , then its projection ( N , x ) M is piecewise
linear with successive gradients p M ( e1,, e K ) .
A useful consequence of (16) is that if x is of dimension 1, namely C( x ) {
1
x} and
xN
( N , x ) [0, x ] , then x M is zero or is of dimension 1 as well.
Step 3 End of the proof when | N | 3
We fix N {1,2,3} and write simply
S
instead of
SN
. First we deduce from (16) that
a triangle C0( e1, e2 , e3 ) in the covering associated with r must have a very special shape. Indeed
choose x in the relative interior of C0( e1, e2 , e3 ) and note that C( x{1, 2} ) is of dimension 1 or 2
(Step 1). By (16) this means that the sequence p{12}(e1, e2 , e3 ) is of rank 1 or 2; it cannot be of
rank 1 because ( e1, e2 , e3 ) is of rank 3. Therefore the sequence ( p{12} (e1 ), p{12} (e2 ), p{12} (e3 ))
contains either one zero element or two equal consecutive (nonzero) elements.
Applying the above property for all three projections on the 2-faces of
S
(i.e., the faces
[ i , j ] ), an easy argument shows that the triangle C0( e1, e2 , e3 ) must have one of the following
two forms:
e1, e2 on a face {i, j} and e2 , e3 aligned with the vertex k (i.e., e2 is the projection of e3
on the face {i, j}); or the symmetrical configuration from exchanging e1 and e3
(17)
Figure 10 describes these configurations; the formal argument is omitted. Next we distinguish
three cases, depending on the dimension of the ordered polytopes covering
Case 1: All points on all 2-faces of
S
Case 2: At least one point on a 2-face of
of dimension 2 or 3.
S
.
are of dimension 1.
S
is of dimension 2 and all interior points of
S
are
41
Case 3: At least one point on a 2-face of
S
S
is of dimension 2 and at least one point interior to
is of dimension 1.
Assume we are in Case 1. Pick any x interior to
S
. If C( x ) is of dimension 2 or more,
the projection of C( x ) on at least one 2-face is of dimension 2; thus all points in
S
are 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 claim by contradiction: the intersection z of [ 3 , x ] and [ 2 , y ] would be interior to
S
and 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 11.
The claim implies that there are at least two 2-faces, say {12} and {13}, 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 r(12) 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 [ 1, x ] (say the midpoint of this interval). The polytope C( y ) must
be within [ 1, x ] (by (16) applied to {23}) and must project on {12} as { 1, 2} (by (16) applied
to {12}). Hence C( y ) [ 1, x ] . But its projection on {13} is not [ 3 , 1 ] as required by (16) and
our assumption that r(13) gives priority to 3 over 1. The claim is proven, and implies that r( 23)
is a priority rule as well. Take now an arbitrary x, interior to
S
, and check that the only
polytope C( x ) projecting on each face as the full face is the whole simplex
S
, namely
C( x ) C0( i , j , k ) . Thus our method is a priority rule (actually, it follows the priority
ordering 3, 1, 2).
Case 2.b
Suppose next that r(12) and r(13) both give priority to 1. It is easy to check, by (16) again, that
the covering of r( 23) can be any covering described in Lemma 4. Pick any x interior to
S
and
42
assume C( x{23} ) C0( e1, e2 ) . As the projections of C( x ) on the other two faces are
[ 1, 2 ] and [ 1, 3 ] , it follows that C( x ) C0( 1, e1, e2 ) . Similarly, if C( x{23} ) {e} , then
C( x ) [ 1, e] . Thus in Case 2.b the method r(123) is the composition of the method r( 23) in
H2 ({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 composition of r( 23) with the priority ordering giving
{23} priority over 1.
Finally we consider Case 3. We can pick a 2-face, say {2,3} and an interval [e1, e2 ] in
[ 2 , 3 ] such that [e1, e2 ] or [e2 , e1 ] is in the covering of r( 23) . We can also pick z, interior to
S
and of dimension 1. Clearly, z{23} cannot be interior to [e1, e2 ] , hence [e1, e2 ] is a strict
subset of [ 2 , 3 ] . Assuming without loss of generality that z{23} is between 2 and [e1, e2 ] , we
construct a point a at the intersection of the line borne by [ 2 , z ] and of [ 1, ei ] , where ei is the
vertex of [e1, e2 ] closest to 2 . As the projection of C( a ) onto {23} is {ei } , and onto {13} is
{z{13}} , it follows that a is of dimension 1 as well: see Figure 12.
Summarizing, we have now a point a interior to
S
and of dimension 1, and a proper
subinterval [e1, e2 ] of [ 2 , 3 ] such that [e1, e2 ] or [e2 , e1 ] is in the covering of r( 23) , and such
that one of its endpoints, say e1 , is the projection of a on {23}.
We claim that the other endpoint e2 must be one of 2 or 3 . We prove the claim by
contradiction, assuming that e2 is strictly between e1 and 3 as shown on Figure 13. We
construct the points b and c as the intersection of [ 1, e2 ] with the lines 2a and 3a respectively:
see Figure 13. The triangle [abc] is of full dimension (a, b, and c are not aligned) and we pick
an interior point x. If x is of dimension 3, the triangle C( x ) must be contained in [abc] because
all points on the faces of [abc] are of dimension 1 at most; because ei 3 , i 1,2 there is no
such triangle with two of its vertices on one face, hence our point x can be of dimension 2 at
most. If x is of dimension 1, its projection x{23} is of dimension 1 as well, which is impossible
because it is a point in ]e1, e2 [ . Thus x is of dimension 2, and we set C( x ) [h1, h2 ] . This
43
interval is contained in [abc] and its projection on {23} is [e1, e2 ] or [e2 , e1 ] . This implies that
[h1, h2 ] or [h2 , h1 ] is the interval [a, x] depicted on Figure 13, with x on [b, c ] . The announced
contradiction follows from considering two points x, y interior to [abc] and aligned with 2 : the
projection of the two intervals [a, x] and [a , y ] on {1,3} do not coincide, a violation of (16).
Thus the claim e2 3 is established. The above proof also shows that any x interior to
the triangle C0( a, b, 3 ) is of dimension 3. Now consider the ordered triangle C( x ) : it is
contained within the triangle {a, b, 3} , its projection on {23} must be [e1, 3 ] or [ 3 , e1 ] and its
shape is as described in property (17). The reader will check that the only two possibilities are
C0( a, b, 3 ) and C0( 3 , b, a ) . It is now an easy matter to develop similar arguments for the six
triangles cut by the three lines a i : see Figure 14. That is to say, the covering of
S
contains
either the ordered triangle C0( i , e j , a ) or C0( a, e j , i ) , for all pairs i, j .
Finally we check that the ordering of the six triangles must coincide on their common
faces. For instance, say C0( a, e2 , 3 ) is in the covering of
S
: (16) implies that C0( a, e2 ) is in
the covering as well; therefore C0( 1, e2 , 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 a.
Step 4 End of proof when N is finite and | N | 4 .
We use an induction argument on the size of N. Fix N, with | N | n 4 , and assume
Theorem 2 holds for all N of cardinality at most n 1 . For each i N consider the restriction
of our method r to N \ i . By induction it is a method in
H ( N \ i ) and we denote by
Ri its
priority preordering of N \ i .
Any two such preorderings Ri , R j coincide on N \ {ij} with the priority preordering of
the method on N \ {ij} . Therefore, there exists a unique preordering
~ on N of which the
restriction to N \ i equals Ri , for all i (note that this implication holds only when n is at least 4).
Assume first that
~ is not the overall indifference, and let N1 N 2 be a partition of N
such that R ranks all agents in N1 above all agents in N 2 . We check that r must give priority to
N1 over N 2 (Definition 3). Suppose not: we can find x R N , two agents 1,2 with 1 N1 ,2 N 2
44
and a point y ( N , x ) such that y1 x1 and x 2 0. By (15), the path ( N \ {3}, x N \{3} ) contains
y N \{3} , a contradiction of the fact that R3 gives priority to 1 over 2. The claim is proven. Now
the induction argument shows that the restrictions of r to N i is in
H (N )
H ( Ni ), i 1,2 hence r is in
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 N . As | N \ i| 3 , this leaves only the proportional,
weighted gains and weighted losses methods. We distinguish two cases. 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
SN
: 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 g wi ( N \ i ) or l wi ( N \ i ) . Assume that
r ( N \ 1) g wi ( N \ 1) and notice that r ( N \ {12}) is the weighted gains method with weight vector
w1N \ 2 . This, in turn, implies that r ( N \ 2) must be a weighted gains method, namely
r( N \ 2) g w2 ( N \ 2) for some w2 . Moreover w2N \1 and w1N \2 are parallel.
Thus we have for all i a weight vector wi , wi R N\ i , such that wiN \ j is parallel to w Nj \ i
for all i, j . In view of n 4 , this implies the existence of a weight vector w, w R N , with a
projection on each N \ i parallel to wi . Now we have identified the method g w ( 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.
Step 5 Proof of Theorem 2 when
N0
is infinite
Once Theorem 2 is established for any finite subset N of
infinite society
N0
is straightforward, hence omitted.
Step 6 Proof of Full Rank Lemma
N0 , its extension to a countably
45
The K-lemma says: if a sequence {e1 ,, e K } in SN 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 ( e1,, e K ) is of full rank in R N \ i .
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 {e1,, e K } meeting P( K ) .
For simplicity, we denote ek i pN \ i ( ek ), for k 1,, K . We suppose that {e1,, e K } is not of
full rank and derive a contradiction. There is a nonzero vector in R K such that
K
k 1
k ek 0
(18)
Note that the sequence {e2 ,, e K } meets the P( K 1) assumption: two consecutive elements are
different and for all i, pN \ i ( e2 ,, e K ) is a subsequence of pN \ i ( e1,, e K ) , so it is full rank. The
induction assumption shows that {e2 ,, e K } is of full rank, whence 1 0 .
We now compare e1 i and e2i . Assume for some i, we have e1 i 0, e2i 0 and e1 i e2i .
Then the sequence pN \ i {e1, e2 ,, e K } has e1 i and e2i as its first two elements. Denoting this
sequence {a1, a 2 , a 3 ,, a Ki } (where K i K ), equation (18) yields, upon projecting on
SN \i :
1 ( e1 ) N \ i 2 k ( e k ) N \ i 0
K
where
1
e
{
1 e1 i
1 a1 2 k a k 0
1
N \i
K
2
}
k ( ek )N \i 0
Ki
2 obtains by summing k over those k such that e2i ,, ek i contains only zero and e2 i ,
etc.
The above equation, given 1 0 , contradicts the full rank of {a1,, a Ki } . We have
shown for all i N :
{e1 i and e2 i } e1 i e2i
To end the proof, we look successively at the following cases:
Case 1: for all i, e1 i and e2 i . Then (19) implies e1 e2 , contradiction.
Case 2: for all i, e1 i and e2 j for some j. Here for all i j , we get e1 i j i so that
e1 j (as one checks easily), again a contradiction.
(19)
46
Case 3: e1 j for some j and for all i, e 2 i : similar to Case 2.
Case 4: e1 i and e2 j for some i and some j. Note that ( 1 ) k ( 2 ) k for all k i, j , hence
a contradiction of property (19).
This completes the proof of Step 6, and of Theorem 2.
Step 7 Proof of Corollary
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
H ( N0 ) meeting ETE must be irreducible.
If |
N0| 3 , the irreducible methods are
the Proportional, Weighted Gains and Weighted Losses. Clearly ETE forces equal weights for
every agent.
