M .M. Kaminski Mathematical Social Sciences 40 2000 131 –155
133
Fig. 1. Hydraulic metaphor. Note: a problem t , t , t , t ; g is solved by assigning to types t , t , t , t
1 2
3 4
1 2
3 4
appropriate vessels and pouring g water into the main reservoir.
properties a rule should satisfy. Among the most fundamental properties of rationing rules, which in fact are satisfied by almost all real-world rules, are continuity, symmetry
and consistency. It turns out that for the class of continuous rules in a separable topology, a rule is symmetric and consistent if and only if it is hydraulic. If continuity is
substituted by monotonicity, then, for any space of types a rule is symmetric, monotone, and consistent if and only if it is ‘semi-hydraulic,’ i.e. if it is generated by a standard
which is not necessarily hydraulic. This result leads to a corollary that provides simple necessary and sufficient conditions in order for any bilateral rationing rule to have a
unique extension to a consistent rule.
The plan of the paper is as follows. Sections 2 and 3 introduce the formal framework and present the main results. Section 4 describes examples of hydraulic rules in claims
settlement and bankruptcy, and develops applications to social choice. Section 5 discusses the relationship between the present results and the work of Dagan and Volij
1997, and Young 1987. The last section concludes with some general comments. Proofs omitted in the text are given in Appendix A.
2. Model
n
I begin with notational conventions. For any two points x, y [ R , x y means x y
i i
for i 5 1, . . . ,n and x . y means x y and x ± y for some i. The non-negative orthant
i i
n n
in R is denoted as R 5 hx:x 0j. The framework introduced below is based on
1
Young’s 1994 definitions. Let T, a non-empty set of types, be any mathematical structure, i.e. a set with
n
]] relations. T may be
R or R , a family of real-valued functions, a topological space, etc. A single type is denoted as
t and a finite vector of types is denoted as t or h. The
i
cardinality of a vector t is denoted by
utu, and the set of all vectors of such cardinality as
ut u n
T . S 5
T is a set of all finite vectors. The number max t denotes the maximal
n [N i
134 M
.M. Kaminski Mathematical Social Sciences 40 2000 131 –155
amount type t can get and the function max: t [ 0, 1 `] is called the restriction.
i i
]]] When max
t 5 1 `, no effective restriction is imposed on the amount type t can get.
i i
Dom t denotes all portions t can get and is equal to the interval [0,maxt ] when
i i
i
max t , 1 ` and to the interval [0,1` when maxt 5 1 `. Maxt denotes
i i
ut u
o max
t , or the maximal total amount that can be divided among types in t. A
i 51 i
rationing problem, or simply a problem, is defined as a pair t;g where t [ S and g is
]]] the amount of a good to be divided; g max
t when maxt , 1 ` and g , maxt when max
t 5 1 `. The class of all problems defined by a specific set T and a specific restriction maxT is denoted by
P. Examples of P include claims and bankruptcy O’Neill, 1982; Aumann and Maschler, 1985, surplus sharing Moulin, 1985, taxation
Young, 1987, or social choice problems Section 4.3. Rationing problems considered in this paper resemble compensation problems studied in Fleurbaey 1994, 1995.
For any P, I adopt the following definitions:
n
A vector g 5 g ,g , . . . ,g is an allocation for t;g if
utu 5 n, o g 5 g and
1 2
n i 51
i
]]]
n
g [ dom t for 1 i n. A rule is a mapping F:P
→ R
that assigns to any
i i
n [N 1
problem t;g [ P an allocation g. Therefore, the value of F for a problem t;g is a
vector F t;g
.
k k 51, . . . , ut u
SYMMETRY SYM: A rule F is symmetric if for all problems t;g [ P, and for all
pairs of types t ,t [ t, if t 5 t then Ft;g 5 Ft;g .
i j
i j
i j
A rule is symmetric if identical types get equal portions regardless of their position as ]]]
a coordinate in a vector. We say that a vector of types h is a restriction of t if h results
]]] from deleting some coordinates in
t. CONSISTENCY CONS: A rule F is consistent if for all problems
t;g [ P and for all vectors of types
h [ o such that h is a restriction of t, if t [ t appears in h as h
j m
then F t;g 5 Fh;
o F
t;g .
j i :
t [h i m
i
Consistency Harsanyi, 1959; Balinski and Young, 1978; Aumann and Maschler, 1985 specifies properties of the rule when the vector of types is shrunk. Assume that
some uhu types from t get a certain amount of a good o
F t;g . Consistency
i : t [h
i
i
requires that the agents from this subset divide this amount in the same way the good was divided in the larger set. A review of literature on consistency is provided by
Thomson, 1996. MONOTONICITY MON: A rule F is monotonic if for all pairs of problems
t;g,t;g [
P, if g . g then F
t;g . Ft;g .
Monotonicity Balinski and Young, 1982; see also Kalai and Smorodinsky, 1975; Kalai, 1977 says that when the total amount increases, nobody loses. Note: the second
‘.’ denotes a vector inequality. The axioms introduced so far are defined for all type spaces. Below, T is a topological
space. CONTINUITY CONT: A rule F is continuous if, for every sequence of problems
k k
k k
k
t ;g such that, for all k, ut u 5 utu and t converges to t , and g converges to g in the
i i
k k
usual topology on R, then Ft ;g converges to Ft;g in the product topology on
ut u
T 3 R.
Whenever continuity appears, it is assumed that T is a separable topological space, i.e. ]]]
that T contains a countable subset whose topological closure is T. Examples of separable
n
spaces include R or certain spaces of real-valued functions. Except for pathological
cases, a subspace of a separable space with the induced topology is also separable. All real-world descriptions of claimants, creditors, taxpayers, applicants, patients, etc.
M .M. Kaminski Mathematical Social Sciences 40 2000 131 –155
135
constitute separable spaces. This makes separability a sound informational condition for all rationing problems.
The results of this paper will use a weaker, bilateral form of the four axioms. These weaker
axioms, bilateral symmetry
B.SYM, bilateral consistency
B.CONS, ]]]]]
]]]]]] bilateral continuity B.CONT, and bilateral monotonicity B.MON, require that the
]]]]] ]]]]]]
respective conditions hold for all pairs of agents. While each of the bilateral axioms alone is essentially weaker than its multi-agent version, the sets of bilateral axioms
assumed in the results together are equivalent to their strong versions. B.SYM, B.CONS and B.MON imply anonymity Lemma 1f in Appendix A:
ANONYMITY AN: A rule F is anonymous if for all problems t;g [ P, and for all
permutations of coordinates in t, s, Ft + s;g 5 Ft;g + s.
Anonymity is stronger than symmetry and says that renaming types does not change their portions. Bilateral anonymity B.AN requires that the condition holds for all
vectors t such that
utu 5 2. Anonymity will be assumed in some subsequent definitions. The axioms introduced so far concerned rules. A second notion will lead to the next
set of axioms. Let Y 5 ht ,g : t [ T, g [ domt j denote all types and the portions
i i
i i
i
they could get. Let R be a weak ordering over Y, i.e. a complete and transitive binary relation.
Let I
be the
indifference part
and P
the strict
part of
R, i.e.
t ,g It ,g ↔
ht ,g Rt ,g and t ,g Rt ,g j; t ,g Pt ,g ↔
ht ,g Rt ,g and
i i
j j
i i
j j
j j
i i
i i
j j
i i
j j
| t ,g Rt ,g
j. R is a standard if it is strictly monotone in the amount of good Young,
j j
i i
]]] 1994:
STANDARD STRICT MONOTONICITY SSM: For all types t [ T, and all
i
allowable amounts of good g , g [ dom t , if g , g then t ,g Pt ,g .
1 2
i 1
2 i
1 i
2
A standard is interpreted as a comparison of priorities of different types to all portions of a good they can get. We read
t ,g Pt ,g as: t has a higher priority to g than t to
i i
j j
i i
j
g . SSM means that a type has a higher priority to get less than to get more. R is
j
numerical if there exists a real-valued representation r:Y →
R such that t ,g Rt ,g iff
i i
j j
]]] r
t ,g rt ,g .
i i
j j
SSM sets priorities for a given type and various amounts of the good. To define a specific standard, priorities between different types and different amounts of the good
must be specified. In many cases, the definition of a standard can be completed in a very simple fashion. For instance, the egalitarian standard can be defined for all types with no
restriction by asserting that ‘Everybody has an equal priority to the same amount of the
E
good.’ Formally, it is sufficient to say: R is a binary relation on Y that satisfies SSM
E
and such that t ,g I t ,g for all t ,t [ T and all g 0. A numerical representation of
i i
j i
i j
i E
E
R is simple as well, and we can define r
t ,g 5 2 g for all t and g . Informal
i i
i i
i
definitions of other familiar standards can implicitly require that a type carries some specific information, e.g. ‘Claimants have priorities proportional to their claims’ or ‘An
agent who derives less utility from his portion of the good has a higher priority than the agent who derives more utility from his portion.’
The final axiom for standards specifies some technical topological requirement. Let us define B
t ,g ,t 5 hx [ domt : t ,g Rt ,xj. Bt ,g ,t is the set of all amounts x such
i i
j j
i i
j i
i j
that t has not higher priority to x as t to g .
j i
i
CLOSURE BELOW CB: For all types with an allowable portion t ,g [ Y, and for
i i
all types t [ T :
j
136 M
.M. Kaminski Mathematical Social Sciences 40 2000 131 –155
a if max t 5 1 ` then Bt ,g ,t 5 [ or inf Bt ,g ,t [ Bt ,g ,t ;
j i
i j
i i
j i
i j
b if max t , 1 ` then Bt ,g ,t is closed and non-empty.
j i
i j
Condition CB specifies topological requirements on R with no reference to any
structure on T. This condition is closely related to one of Debreu’s 1954 classical conditions applied to the space T 3
R and Young’s 1994, p. 191 axiom see Section 3 for a more detailed discussion of the relationship.
Standards and rules can be linked via a concept of equity Young, 1994. Let g , . . . , g be an allocation for
t;g and 0 , ´ g . A transfer of ´ from t to t is
1 n
j j
i
R-justified if t ,g Pt ,g 2 ´.
i i
j j
]]] EQUITY: A rule F is R-equitable if for all
t;g [ P, no transfer is R-justified for ]]]
F t;g.
Equity means that all allocations generated by a rule conform closely to the standard. In the hydraulic language, a transfer from
t to t is R-justified if the water level in the
j i
vessel t is higher than in t . The reader is requested to go back to Fig. 1, fill the vessels
j i
with virtual water, and check how the metaphor works. Consider a simple example of
E
the unconstrained equal division rule E and the egalitarian standard R . Since E t;g 5
i E
1 utug for all i, for 0 , ´ 1 utug we can check that t ,g 2 ´P t ,g for all
i i
j i
E E
t ,t [ T and no transfer is R -justified. The rule E is R -equitable.
i j
Now, the ‘hydraulic’ and ‘semi-hydraulic’ properties will be defined formally for any rule F. Both names denote concisely two classes of rules constructed on the basis of the
concept of equity. SEMI-HYDRAULIC PROPERTY: F is semi-hydraulic if it is R-equitable for some
]]]] standard R.
HYDRAULIC PROPERTY: F is hydraulic if it is R-equitable for some numerical ]]]
standard R. There exist semi-hydraulic rules that are not hydraulic. An example is the lex-
icographic priority rule X, defined for T 5 R and no restriction, which divides the whole
X X
good equally among agents with the lowest type. X is R -equitable for the standard R
X
defined as ; t ,t [ T, ;g [ domt , ;g [ domt , [t , t
→ t ,g P t ,g ; t 5
i j
i i
j j
i j
i i
j j
i X
X
t →
t ,g P t ,g iff g , g ]. R is not numerical and it is clear that X cannot be
j i
i j
j i
j
R-equitable for any numerical standard R.
3. Results
