302 H
. Norde et al. Mathematical Social Sciences 40 2000 297 –311
can use the axiom of approximation consistency in a meaningful way. Basically this is the issue addressed in Section 4.
3. Axioms and examples
For approximation consistent solutions we have the following proposition.
Proposition 3.1. Let
b be an approximation consistent solution on 3 and let A, u, B, v [ 3 be such that uA 5 vB . Then there is a subset T of uA 5 vB such that
21 21
bA, u 5 u T and
bB, v 5 v T .
Proof. Take T [ubA, u.
21
If a [ u T then ua 5 ua9 for some a9 [
bA, u and hence, by approximation
21 21
consistency, a [ bA, u. So, u
T bA, u. The inclusion bA, u u
T is obvious.
21
If b [ v T then vb 5 ua9 for some a9 [
bA, u. Since A, u and B, v are
21
sup-equivalent we get, by approximation consistency, b [ bB, v. So, v
T bB, v.
If b [ bB, v then, since uA 5 vB, there is an a [ A such that ua 5 vb and hence,
by approximation consistency, a [ bA, u. Therefore, vb 5 ua [ T and hence b [
21 21
v T . So,
bB, v v T .
h Proposition 3.1 shows that, if
b is approximation consistent, the set bA, u only depends on the range uA of u
. Although much information about the optimization problem A, u is lost by considering only the values of u, this approach is an extremely
common one. So, if we are interested in approximation consistent solutions only, we may identify an optimization problem A, u with uA, the range of u, which is a subset
of R. In this and the following sections we focus on this approach.
Let 6 be a non-empty collection of non-empty subsets of R. A solution s on 6 is a
map which assigns to every S [ 6 a subset sS of S.
A solution s on 6 satisfies MON monotonicity if for every S , S [ 6 with sup
1 2
S sup S the following statement is true:
1 2
if s [ sS and s [ S is such that s s then s [ sS
2 2
1 1
1 2
1 1
A solution s on 6 satisfies AC approximation consistency if for every S , S [ 6
1 2
with sup S 5sup S the following statement is true:
1 2
if s [ sS and s [ S is such that s s then s [ sS
2 2
1 1
1 2
1 1
The reason for introducing AC is given by Proposition 3.1: it is immediate to see that, if a solution
b on 3 is approximation consistent, as defined in the previous section, then the induced solution
s on the family 6 of ranges uA A, u [ 3 , satisfies AC. Conversely, a solution
s on 6, satisfying AC, induces, for every 3 with ranges in 6, an approximation consistent solution
b.
H . Norde et al. Mathematical Social Sciences 40 2000 297 –311
303
A solution s on 6 satisfies WAC weak approximation consistency if for every
S [ 6 the following statement is true: if s [
sS and s9 [ S is such that s9 s then s9 [ sS The axioms of non-emptiness, translation invariance and multiplication invariance, as
defined for solutions on a class 3 of optimization problems in Section 2, can be extended in a straightforward way to solutions on a collection 6 of non-empty subsets of
R. A solution
s on 6 satisfies NEM non-emptiness if for every S [ 6 we have: sS ± 5
The collection 6 is closed under translation CL1 if for every S [ 6 and t [ R we have t 1 S[ht 1 s: s [ Sj [ 6. A solution s on 6, obeying CL1, satisfies TI
translation invariance if for every S [ 6 and t [ R we have: st 1 S 5 t 1 sS
The collection 6 is closed under multiplication CL if for every S [ 6 and l . 0
we have lS[hls: s [ Sj [ 6. A solution s on 6, obeying CL, satisfies MI
multiplication invariance if for every S [ 6 and l.0 we have:
slS 5 lsS In this paper we will also characterize solutions on 6 making use of the axiom CCA
Chernoff ’s Choice Axiom see Chernoff, 1954. This axiom is defined as follows: a
solution s on 6 satisfies CCA if for every S, T [ 6 with S T one has:
sT S sS So, if
s satisfies CCA, selection by s of an element s [ T, implies selection by s of s in any subset S of T with s [ S. The axiom CCA is weaker than the independence of
irrelevant alternatives axiom used, for example, in Kaneko 1980 and Peters 1992.
Example 3.1. For the following examples suppose that 6 is the collection of all non-empty subsets of R.
a The solution s
defined by:
mix
hs [ S: s sup S 2 1j if sup S 0 hs [ S: s . sup S 2 1j if sup S [ 0, 1 `
s S [
mix
5
hs [ S: s 22j if sup S 5 1 `
satisfies AC and NEM. b The solution
s , defined by:
rat
304 H
. Norde et al. Mathematical Social Sciences 40 2000 297 –311
S if S Q
s S[
H
rat
hs [ S: s 22j if S≠Q satisfies WAC and CCA.
c The solution s , defined by:
tot
s S[S
tot
satisfies MON, NEM, TI, MI and CCA. d The solution
s , defined by:
max
s S [hs [ S: s s9 for every s9 [ Sj
max
satisfies MON, TI, MI and CCA. e The solution
s where ´ . 0, defined by:
´
s S[hs [ S: s sup S 2 ´j
´
satisfies MON, TI and CCA.
k
f The solution s where k [ R, defined by:
k
s S[hs [ S: s kj satisfies MON and CCA.
g The solution s
where ´.0, k [ R, defined by:
´,k
s S
if s
S ± 5
max max
s S if
s S 5 5 and
s S ± 5 s S[
´ max
´ ´,k
5
k
s S otherwise
satisfies WAC and NEM. ˆ
h The solution s
where ´ . 0, k [ R, defined by:
´,k
s S if sup S k 1
´
´
ˆ s S[
H
k ´,k
s S if sup S . k 1
´
k
ˆ satisfies MON, NEM and CCA. Notice that
s S 5 s S s S.
´,k ´
i The solution s
S where a . 1, b , 1, defined by:
pro a, b
a s
s S if supS 5 s [ 2`, 0
S if supS 5 0
s S [
b s pro
a, b
s S if supS 5 s [ 0, 1 `
5
S if supS 5 1 `
satisfies AC, NEM and MI. The following table summarizes the statements above.
H . Norde et al. Mathematical Social Sciences 40 2000 297 –311
305
MON AC
WAC NEM
TI MI
CCA s
2 1
1 1
2 2
2
mix
s 2
2 1
2 2
2 1
rat
s 1
1 1
1 1
1 1
tot
s 1
1 1
2 1
1 1
max
s 1
1 1
2 1
2 1
´ k
s 1
1 1
2 2
2 1
s 2
2 1
1 2
2 2
´,k
ˆ s
1 1
1 1
2 2
1
´,k
s 2
1 1
1 2
1 2
pro a, b
4. Characterizations for translation and multiplication invariant solutions
