M .M. Kaminski Mathematical Social Sciences 40 2000 131 –155
143
ship between social choice problems and bargaining seems to be close indeed. The precise extent to which the results from one framework can be applied to the other
framework depends on the assumed space of types. With every social choice problem u; g, where u 5 u , . . . ,u and g . 0, the
1 n
following bargaining problem can be associated: Bu; g 5 hu g , . . . ,u g : all
1 1
n n
]]]]]]
n
g 0 and o
g g j. Young 1994, pp. 210–212 noted that Lensberg’s 1987
i i 51
i
results obtained for bargaining can be applied in the social choice setting, and he offered a preliminary discussion of the properties of anonymous, continuous and bilaterally
WC
consistent rules for T with respect to maximization of separately additive objective
functions. However, the results for bargaining cannot be mechanically restated for the social choice setting, and vice versa, since the function B is not 1–1 and onto Billera
and Bixby, 1973. Moreover, almost all results in bargaining theory are obtained under the assumption of weak convexity of the bargaining set. This assumption rules out from
consideration the utilitarian summation since for a non-strictly convex bargaining set, U may be multi-valued and makes the domain of L unnecessarily narrow since some
I
bargaining problems corresponding to T are non-convex. More work remains to be done on the formal relationship between the two frameworks.
5. Related results
Below, I discuss informally the relationship between the present results and Dagan and Volij’s 1997 criterion for a consistent extension of a bilateral claims rule, and
Young’s concept of parametric representation. The discussion exemplifies further applications of Theorem 2 and Corollary 1.
5.1. Consistent extension of a bilateral claims rule For every vector of types
t 5 t ,t , . . . ,t , a bilateral rule H, and any allocation
1 2
n H
g 5 g ,g , . . . ,g , the following binary relation is defined: A : 5 hi, j [ I 3 I:
1 2
n g
H
H t ,t ;g 1 g g
j. iA j is interpreted as ‘g treats i no better than j according to H.’
i i
j i
j i
g
H is consistent with g and t if for all t ,t [ t, H t ,t ;g 1 g 5 g . Dagan and Volij’s
i j
i i
j i
j i
main Theorem, slightly reformulated, connects the possibility of a consistent extension
H
of a bilateral claims rule and the transitivity of the strict part of A :
g
Theorem [Dagan and Volij, 1997]. Let T 5 R 2 h0j and maxt 5 t . For any
1 i
i
anonymous bilateral rule H that satisfies B.MON and any problem t;g [ P, there
exists a unique allocation g consistent with H and t if and only if for every allocation
H H
H
x 5 x ,x , . . . ,x , A is quasi-transitive, i.e. when a , the strict part of A , is
1 2
ut u x
x x
transitive. For claims, the Dagan–Volij’s theorem provides a criterion alternative to that of
Corollary 1. We will show that both criteria are equivalent for all types. The relation of ‘no better treatment’ is formulated for specific allocations, but we can
144 M
.M. Kaminski Mathematical Social Sciences 40 2000 131 –155
H H
extend it in a natural way to a global relation. Let us define A as the sum of all A ,
g H
H H
i.e. A : 5 A
. Since A depends only on types t and t , and all possible
n
g [R , n[N g
i j
H
portions allotted to these types, we can interpret A as a binary relation on Y 5 ht ,g :
i i
t [ T and g [ domt j.
i i
i H
H H
Since A satisfies SSM, for pairs , t ,g ;t ,g . A
; R . Whenever H is locally
i i
i j
H H
strictly monotone, A ; R as well. Thus, for strictly monotone bilateral rules, both
criteria are identical this fact is reflected in the formulation of Dagan and Volij’s
H H
Theorem 4.9. In general, A substitutes some strict parts of R by indifferences. An
example is provided by the CG rule and t 5 100, t 5 200. All pairs , t ,g ;t ,g .
i j
i i
j i
for g , 50, and all pairs , t ,g ;t ,100 1 g . for g 50, belong to both relations
i i
i j
i i
CG CG
| and I
; while the pairs , t ,50;t ,g . for 50 g , 150 belong to relations
i j
j j
CG CG
| and P
.
H
The Dagan and Volij’s criterion, as well as the relation A , do not assume any type-specific information. This fact suggests that a generalization is possible. In fact, the
H
criterion is valid for any space of types. Proposition 1 states the relationship between A
H
and R formally. The formulation of the criterion remains unchanged but the restrictions
imposed on T and max can be dropped.
H
Proposition 1. For any
P and any anonymous bilateral rule H that satisfies B.MON, R
H
is transitive if and only if a is transitive for all
g.
g H
Proof outline. The proof is based on the properties of binary relations a Lemma A
g H
H H
and R Lemmas C and D, and the relationship between R
and a Lemmas B and
g
E.
H H
H
Lemma A. If for some
t , t , t , and g 5 g , . . . ,g , i a j, j a k, kA i, then there
i j
k 1
n g
g g
H H
H
exists ´ . 0 such that for x 5 g 1 ´,g ,g 2 ´ i a j, j a k, and k a i.
i j
k x
x x
Lemma B. ;
t ,t [ T, g [ domt ,g [ domt , and g such that g allocates g to t and
i j
i i
j j
i i
H H
g to t , then
hi a j →
t ,g P t ,g j.
j j
g i
i j
j H
Lemma C.
For all
t ,g ,t ,g [ Y, ht ,g P t ,g j
→ hg [ domt . g
i 1
j 2
i 1
j 2
3 i
1 H
s.t. t ,g P t ,g
j.
i 3
j 2
H
Lemma D.
For all
t ,g ,t ,g ,t ,g [ Y, ht ,g R t ,g
and g ,
i 1
j 2
j 3
i 1
j 2
2 H
g j
→ ht ,g P t ,g j.
3 i
1 j
3
Lemma E. For all
t ,g ,t ,g ,t ,g ,t ,g [ Y, hg , g
and g , g and
i 1
i 2
j 3
j 4
1 2
3 4
H H
t ,g P t ,g j
→ ht ,g a t ,g j.
i 2
j 3
i 1
j 4
Comment. In the proof of Lemma A, ´ must be sufficiently small to preserve
H H
inequalities that are implied by i a j and j a k; the proof of Lemma B follows from
x x
H
the definition of R . The details and the proofs of Lemmas C–E are left to the reader.
H
Only if : Let us assume now that a is not transitive for some
g, i.e. for some t , t ,
i j
M .M. Kaminski Mathematical Social Sciences 40 2000 131 –155
145
H H
H
t , and g 5 g , . . . ,g , we have i a j, j a k, but kA i. By Lemma A, we can find x
k 1
n g
g g
H H
H
such that
i a j, j a k,
but k a i.
This, by
Lemma B,
implies
x x
x H
H H
t ,g P t ,g P t ,g P t ,g .
i i
j j
k k
i i
H
If : Let us assume now that R is not transitive, i.e. that for some
t ,g , t ,g ,
i 1
j 2
H H
H H
H
t ,g [ Y , t ,g R t ,g R t ,g P t ,g . We will find a a -intransitive se-
k 3
i 1
j 2
k 3
i 1
g
quence for some g. By Lemmas C and D, we construct a following string indices reflect
the order of applying Lemma C:
H H
H H
H H
H H
t ,g P t ,g P t ,g P t ,g P t ,g P t ,g P t ,g P t ,g P t ,g
i 1
i 10
i 11
j 7
j 8
j 9
k 4
k 5
k 6
H
P t ,g
i 1
H H
H
By Lemma E, t ,g a t ,g a t ,g a t ,g , where g 5 g ,g ,g . h
i 10
g k
5 g
j 8
g i
10 10
5 8
5.2. Parametric representation of a rule For any
P, let ft ,l be any function defined over T 3 [a,b], where [a,b] is a closed
i
interval of the extended reals, with values in R h 1 `j. Following Young’s 1987, pp.
400–401 definition, f is called a parametric representation if for each t : a ft ,l is
i i
]]]]]]] weakly monotone increasing; b f
t ,l is continuous in l when ft ,l [ R; and c it
i i
satisfies f t ,a 5 0 and ft ,b 5 maxt . When maxt 5 1 `, we assume that, in
i i
i i
addition to Young’s postulates: d if f t ,l 5 1 ` then lim
f t ,x 5 1 `. By
i x
→ l
i
a–d, f has the Darboux property and for any t , . . . ,t ;g [ P, there exists l such
1 n
g n
n
that o
f t ,l 5 g. Moreover, if for some other l it holds that
o f
t ,l 5 g,
i 51 i
g h
i 51 i
h
then a implies that f t ,l 5 ft ,l for i 5 1, . . . ,n. Thus, f can be used to define a
i g
i h
parametric rule F in the following way: F t , . . . ,t ;g 5 ft ,l , . . . , ft ,l where
1 n
1 g
n g
]]]
n
l satisfies o
f t ;l 5 g. Young 1987, p. 401, Theorem 1 introduced parametric
g i 51
i g
representation for T 5 R 2 h0j and maxt 5 t , and proved that for every continuous
1 i
i
claims rule F, F satisfies SYM and B.CONS iff it is representable by a continuous parametric function. Among parametric rules discussed by Young, who used the taxation
setup, there are flat, head, leveling, and equal sacrifice taxes, as well as many other families of tax schedules. All these rules are hydraulic.
The concepts of a parametric representation and a numerical standard are in fact closely related. In terms of connected vessels mounted between levels a and b, f
t ,l
i
can be imagined as the volume of the vessel assigned to t up to the level l.
i
Alternatively, the diameter of vessel t at level l may be set as the left-hand side
i
derivative of f t ,l.
i
Formally, in many cases, we can simply obtain a parametric representation of F from a numerical standard r that is equitable relative to F with f
t , 2 rt ,g 5 g . In general,
i i
i i
the range of l can be defined as follows:
i a 5 inf h 2 rt ,y: t ,y [ Yj and b 5 sup h 2 rt ,y: t ,y [ Yj.
i i
i i
The parametric representation f can be constructed in the following way: ii For a
l , 2 rt ,0, ft ,l 5 0.
i i
146 M
.M. Kaminski Mathematical Social Sciences 40 2000 131 –155
iii For 2 r t ,0 l , 2 rt ,maxt , ft ,l 5 sup
hx [ domt : 2 rt ,x lj.
i i
i i
i i
iv For 2 r t ,maxt l b, ft ,l 5 maxt .
i i
i i
Let us check what happens when the above construction is applied outside of Young’s
I
domain of claims, e.g. for T and the leximin rule L introduced in Section 4.3. Since ru , y 5 2 u y and maxu 5 1 `, we have:
i i
i I
I
i a 5 inf hu y: u [ T and y [ R j 5 2 ` and b 5 suphu y: u [ T and y [
i i
1 i
i
R j 5 1 `.
1
ii For 2 ` l , u 0, fu ,l 5 0.
i i
21 21
iii For u 0 l , lim
u y, fu , l 5 u
l, where u is the inverse
i y
→ `
i i
i i
function of u. iv For lim
u y l 1 `, fu ,l 5 1 `.
y →
` i
i
It is easy to check that f is a parametric representation of L according to the definition stated above. This fact suggests that Young’s result can be applied to a wider space of
types. Let T be any separable space. If a continuous rule F is representable by a parametric function, then it is clearly symmetric and consistent. On the other hand,
Theorem 2 implies that if F is continuous, symmetric, and a bilaterally consistent rule, then it is equitable relative to a numerical standard r. It is a mechanical exercise to check
that applying i–iv to this particular standard r brings a parametric representation of F. Therefore, the following proposition can be formulated:
Theorem 4. Let T be any separable topological space and F be a continuous rule. F satisfies SYM and B.CONS iff it is representable by a parametric function.
Theorem 4 provides a characterization similar to Young’s 1987 Theorem 1 for a wider space of types but the family of parametric functions considered in this theorem is
not necessarily continuous in the topology on T 3 R. When continuity is assumed, the
‘if’ part makes a weaker claim, while the ‘only if’ part makes a stronger claim than the relevant parts of Theorem 4. Whether the equivalence holds when the term ‘parametric
function’ is substituted by ‘continuous parametric function’ remains an open question.
6. Conclusion
