This paper gives a formulation for the condition of preservation of preference proximity which, unlike previous formulations, respects the spirit of anonymity pervading social choice theory. Proximity preservation is however shown to be inconsistent with a very weak condition guaranteeing a minimal non-trivial compensation of pivotal changes.

Daniel Eckert

English

Research Papers in Economics

Proximity Preservation in an Anonymous Framework 
Daniel Eckert
Institute of Public Economics, University of Graz
Abstract
This paper gives a formulation for the condition of preservation of preference proximity
which, unlike previous formulations, respects the spirit of anonymity pervading social choice
theory. Proximity preservation is however shown to be inconsistent with a very weak
condition guaranteeing a minimal non−trivial compensation of pivotal changes.
An earlier version of this paper was presented at the joint Research Seminar of the Department of Economics and the Institute of
Public Economics, University of Graz. The author wishes to thank Nick Baigent, Christian Klamler, Johann Mitlöhner, and
Ulrich Pferschy, for valuable comments.
Citation: Eckert, Daniel, (2004) "Proximity Preservation in an Anonymous Framework." Economics Bulletin, Vol. 4, No. 6 pp.
1−6
Submitted: March 1, 2004.  Accepted: April 21, 2004.
URL: http://www.economicsbulletin.com/2004/volume4/EB−04D70002A.pdf1 Introduction 
Since Baigent (1987) has established the impossibility of a social aggregation rule which 
preserves preference proximity and satisfies typical social choice conditions, various 
impossibility results have shown proximity preservation to be a particularly strong inter-
profile consistency condition, as it can be shown to be inconsistent with a very weak no-veto 
condition and the weakest possible form of non-imposition (Baigent and Eckert 2001, see 
also Eckert and Lane 2002, and Grafe and Grafe 2001). This strength might however be 
attributed to the fact that proximity preservation has always been formulated in a way which 
does not respect the equivalence of profiles which are permutations of each other and hence 
is not consistent in an obvious way with the spirit of anonymity pervading social choice 
theory. By contrast, this paper uses a formulation of proximity preservation which not only is 
consistent with, but even implies anonymity. In this framework an impossibility result 
involving a weak compensation property is established. 
 
2 Formal  framework 
Let I denote a countable set of agents and let Θ  denote an arbitrary set of characteristics. A 
profile is an ordered tuple  12 ( , ,..., )
I  determined by an index function   
assigning a characteristic to every agent. Let U
: uI →Θ
= Θ  denote the set of all profiles considered 
as such index functions. A Social Aggregation Rule (SAR) is a function   that 
assigns, as a group outcome, a characteristic in 
: fU →Θ
Θ  to profiles in U. The set of characteristics 
need not be endowed with any particular structure. In particular, the characteristics need not 
be binary relations. Interpretations are not therefore limited to those that are usual in social 
choice theory as especially preference relations. 
I xx x ∈Θ
I
Two conditions central to social choice theory can be formulated with the help of equivalence 
classes of the set of profiles. A SAR f induces a partition U  of U into f-level sets. Thus, 
the elements of 
f
U  are [] , for all  f :{ | ( ) () } uu U f u f u ′′ =∈ = uU f ∈ . Our first property is a 
weakening of a property, which was introduced by Wilson (1972) as a substitute for the 
Arrovian Pareto condition. A SAR f has the non-imposition property if  1 Uf> . Thus, 
SAR's that are not imposed are not constant functions and exhibit some response, possibly 
minimal, to agent's characteristics. 
Another useful partition of U will be denoted by U /Σ, where Σ denotes the set of all 
permutations on I. The elements of U  are [ ] : | ( ) ( ( )), } U u i u i i I { uu : Σ ′ ∈ Σ , 
for all u . 
/Σ σ σ ′ = ∈∃ = ∀ ∈
U ∈
] [ ]
With the help of these two partitions the familiar property of anonymity can easily be 
formulated as [ f uu ⊆ U Σ , for all u ∈ . The invariance of the SAR f  with respect to 
permutations captures the idea of an equal, symmetric treatment of the agents. 
Two other partitions will also be used: First, U  is the family of subsets of U, which consist 
of profiles which have the same characteristic for agent i. Thus the elements of 
i
U  are  i[]:{ : ( ) ( ) } uu U u i u i ′′ =∈ = iI i , for all  ∈  and u . Second, U  is the family of subsets of 
U, which consist of profiles that are i-variants of a given profile uU ∈ . Thus, the elements of 
Ui − :{ | i u u U − ′′ =∈
: fU
22 d ⊆Θ ×Θ
22 DUU ⊆×
 a r e  [] , for all   and  .  ( ) ' () { } } u iu u j Ii ≠ ∀ ∈ − iI ∈ uU ∈ ( ) i ∧ () , ju j =
→Θ
', , ' uu vv
() () ( ') ), ( aa fv d
2 Θ
,
( , ') ( , ') ( ), uuDv v fv ⇒¬
2 ⊆Θ ×
( fu ') fu
, xy ∈Θ x y ≠ ( , ) ( , ) x yd xx
2 Θ
2 Θ
U ∈ i −
 
3 Proximity  preservation 
The condition of proximity preservation was introduced by Baigent (1987) into the 
framework of Arrovian social choice theory in analogy to the condition of continuity in the 
framework of topological social choice theory initiated by Chichilnisky (1979, 1982). (For 
surveys on the latter see Lauwers 2000 and Baigent 2004.) The other conditions mainly used 
in the topological framework are unanimity and anonymity, the former being weaker than the 
corresponding Pareto condition, while the latter is stronger than the corresponding Arrovian 
condition of non-dictatorship. Like the continuity condition, proximity preservation can be 
motivated by the intuitively plausible requirement that smaller mistakes in registering 
individual characteristics should not result in larger changes in the group outcome than larger 
mistakes (Kelly 1988). A recent generalization of previous results has shown proximity 
preservation to be inconsistent with two extremely weak conditions in the spirit of unanimity 
and anonymity respectively, namely the weakest possible form of non-imposition and a no-
veto condition (Baigent and Eckert 2001). All the previous formulations of proximity 
preservation are however prone to criticism because they are not in an obvious way 
consistent with the spirit of anonymity, as they do not respect the equivalence of profiles 
which are permutations of each other. An analogous critique has been raised by Baigent 
(1985) himself against the use of the product topology in topological social choice. 
The condition of proximity preservation has been formulated in various ways in the previous 
literature. The ordinal formulation in Eckert and Lane (2002) is followed here. Its primitives 
are distance rankings, i.e. binary relations comparing pairs of objects with respect to the 
distance between these objects. 
Definition: A SAR   satisfies proximity preservation with respect to a distance 
ranking   for characteristics (with asymmetric part denoted da) and a distance 
ranking   for profiles (with asymmetric part denoted Da) if for all  , U ∈ : 
. 
Essentially, proximity preservation requires that the ranking of distances among profiles shall 
not be inverted by the ranking of the corresponding group outcomes. 
The inconsistency of proximity preservation with various other social choice conditions has 
been established for very weak conditions on these distance rankings. In particular, the 
distance ranking d  for characteristics is only assumed to fulfill the condition of 
other-dissimilarity which requires that for all     implies  a . In this 
case it will be called a regular distance ranking. This condition requires that the distance 
between identical outcomes be strictly less than the distance between distinct outcomes. This 
requirement is very weak and does not exclude binary relations on   that cannot be 
represented by any real valued function on  , let alone by a metric. (See Suppes et al. 1989, 
and Eckert and Lane 2002.) 
 2On the distance ranking   for profiles only a monotonicity condition is applied. 
For all uu , let 
2 DUU ⊆×
, '): { | uu i I =∈
' u
( , '') uu ∆
≠
,' ,' uu u , uu ( ', '') u u
 denote the set of agents that have different 
characteristics in profiles u and  . If the distance ranking D on profiles is such that, for all 
,   and  'U ∈ ( ') ∆⊂ ( , ') uu ∆ ∩∆ =∅ ( , '') ( , ') uu D uu
iI ∈ , ' [ ] uU u u
 implies  , it is 
called a monotonic distance ranking on profiles in Baigent and Eckert 2001. This condition 
is essentially a ceteris paribus condition which requires that the distance from a given profile 
increases with the number of components which are different. 
a
2
, ' U ∈ ( ( ) '( )} ui u i ∆
This formulation of monotonicity, which is implicit in most previous formulations of 
proximity preservation, however does obviously not satisfy the spirit of anonymity, as it does 
not respect the equivalence of profiles, which are permutations of each other. It is easily 
verified that for all  ,  i − ∈∈ '' [ '] [ ] u  and uu i Σ,  , though   is 
a permutation of u . 
a ∈ ∩ ( , '') ( , ') uu D uu '' u
Thus we will define an anonymous distance ranking on profiles in terms of other collections 
of characteristics. Such collections, where the order of the elements does not matter, while 
repetitions are allowed, are known as multisets (see e.g. Stanley 1997). Formally, a multiset 
M on a set S is a pair  , M M SN = , where   is a function which counts for every 
element 
: M NS → `
 the number   of its occurrences in the mulstiset M. While a profile 
 is an ordered tuple 
( ) M Nx
12 ( , ,..., ) I x xx of elements of Θ , the corresponding multiset, 
denoted by u, is the pair  , u =Θ u , where, for every characteristic  ,  N x∈Θ () u Nx  counts 
the number of occurrences of x in the multiset corresponding to profile u. The relation 
between any profile   and the corresponding multiset  uU ∈ u is given, for all  , by  x∈Θ
1 − () () u Nx u x = . In particular, all profiles  ' [ ] uu Σ ∈  determine the same multiset u. 
x S ∈
uU ∈
Hence we define a weakly anonymous distance ranking on profiles by a condition in the 
spirit of other-dissimilarity which states that for all   and the corresponding 
multisets  ,' ,' ' u uu ,  '' ' uu u =≠ , '') ( , ') uu D uu
22 ⊆×
 implies ( . a
1 With respect to a weakly anonymous 
distance ranking for profiles and a regular distance ranking for characteristics, proximity 
preservation is not only consistent with anonymity, but can even be shown to imply 
anonymity of the SAR. 
, ', '' uu u U ∈
Lemma. If a SAR   satisfies proximity preservation with respect to a regular 
distance ranking   for characteristics and a weakly anonymous distance ranking 
 for profiles, then it satisfies anonymity. 
: fU →Θ
22 ⊆Θ ×Θ d
DUU
Proof. Consider two profiles u  and uu U ∈ ' [ ] [ ] f u Σ ∈ − , and assume to the contrary that there 
exists a profile  '' [ ] [ ]f uu . By weak anonymity of the distance ranking D for profiles 
, while for any regular distance ranking d for characteristics 
u
) )
Σ
( () , ('
∈−
( , ') ( , '') uu D uu a
() () , (' ' ) a f u fu d fu fu , which violates proximity preservation. □ 
                                                 
1 This condition is termed weakly anonymous as we only require that the distance between all profiles that are 
permutations of each other be smaller than the distance between profiles that are not permutations of each other 
and not that the distance between all profiles that are permutations of each other be minimal. 
 3The framework of multisets can also be used to formulate a monotonicity condition, which is 
not sensitive to permutations and hence respects the spirit of anonymity. By defining for all 
multisets  M and  ' M  on some set S,  ' ', MM MM S N ∩ ∩=  by 
' () :m i n () , ) MM M Nx N x x ∩ = ' ( M N , for all x∈Θ, we can define the multiset difference 
' ', MM MM S N − −= Nx  by  ' () : () M MM N N −∩ ' x () x MM = − , for all x . Finally, multiset 
inclusion is defined as  ' M M ⊆  if  '() M Nx () M Nx ≤ , for all x∈Θ, and   if 
'' M MM M ⊆∧ ¬⊆. 
{ }
∈Θ
' M M ⊂
Definition: A distance ranking   for profiles is monotonic if for all uu
2 DUU ⊆×
2 , ', '' u U ∈  
and the corresponding multisets  ,' ,' uu u' : 
' '' ( , '') ( , ') uuuu u uD u u −⊂ −⇒
2
a . 
Essentially, this monotonicity condition requires that the distance from a given profile 
increases with the corresponding multiset differences. 
Proximity preservation may now be expressed as follows: 
Definition: A SAR   satisfies proximity preservation relative to a regular distance 
ranking   for characteristics and a weakly anonymous monotonic distance ranking 
 for profiles if, for all uu
: fU →Θ
2 d ⊆Θ ×Θ
22 U ⊆× DU , ', '' u U ∈ : 
() ( )
f u ∈−
( , ') uu D( , '') uu ⇒¬ ( ), ( fu fu '') ') ( ), ( aa d fu fu . 
 
4  Minimal non-trivial compensation: an impossibility result 
Anonymity guarantees that a pivotal change from some profile   to some 
profileuu  can be compensated if there exists 
uU ∈
' [ ] [ ] i − {} such that uj . In 
this case a profile uu
( ) '( )
'' [ '] [ ] i Σ (which is a permutation the original profile u) can be 
obtained from profile u  by changing in turn individual j’s characteristic from 
 to uj . This possibility is however trivial (as it is an immediate 
implication of anonymity) and limited at the same time. We thus formulate a condition of 
minimal non-trivial compensation to guarantee that pivotal changes can be compensated 
otherwise than via a permutation of the original profile. 
'
''( ) ( ) u i = ( ) '( ) u j u i ==
j Ii ∈ − ui =
u ∈ ∩
'( ) uj
[ ] uU u − '] [ ] [ ] uuu
Definition: A SAR   satisfies minimal non-trivial compensation if it is not the 
case that for all  , [
: fU →Θ
, ' [ ] u u − ∈∈ ifif Σ .  ∩ −= ∅
This condition is an extremely weak analogue to the compensation properties familiar from 
multicriteria decision making, which allow differences on some attribute to be compensated 
by “sufficiently large” differences on other attributes (see e.g. Bouyssou et al. 1997), and it is 
hard to imagine a reasonable SAR that would not satisfy it. However, the following theorem 
establishes its inconsistency with proximity preservation. 
Theorem. There is no non-imposed SAR   that satisfies the condition of minimal 
non-trivial compensation and which preserves proximity relative to a regular distance ranking 
: fU →Θ
 422 d ⊆Θ ×Θ
2 DUU ⊆×
 for characteristics and a weakly anonymous monotonic distance ranking 
 for profiles. 
( , '') uu
() ( ), ( ') f u fu
2
Proof. From non-imposition, there exists a pair of profiles  , ' uu U ∈  such that uu  
and, for at least one such pair, minimal non-trivial compensation implies that there exists 
another profile   such that uu
' [ ] [ ] if u − ∈−
'' [ ] uu ∉ '' [ '] [ ]f u Σ i . By the anonymity of the SAR, which is 
implied by our formulation of preference proximity, u '] u Σ ∉ , so that it is easily verified that 
in this case  ' uuu −⊂ −
( , ') a uu
' ' u
) ( ')
 holds for the corresponding multisets. By monotonicity it follows 
that  . On the other hand,  D ( ) ( '' f uf u f u = ≠ , and hence 
( ( ( ), ) ( a '') d fu fu . Thus, proximity preservation, relative to all regular distance 
rankings for characteristics and weakly anonymous monotonic distance rankings for profiles, 
is violated. □ 
∈ ∩
'' [
 
5 Discussion 
 
Previous formulations of the condition of proximity preservation have failed to respect the 
equivalence of profiles, which are permutations of each other. From an interpretational point 
of view, they thus are not consistent in an obvious way with the spirit of anonymity, which 
pervades social choice theory. This inconsistency can, however, not be considered the reason 
for the tension of proximity preservation with other conditions for social aggregation rules. 
Even if the condition of proximity preservation is formulated in a way which implies 
anonymity, it can be shown to be inconsistent with a very weak condition guaranteeing a 
minimal non-trivial compensation of pivotal changes. 
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Proximity Preservation in an Anonymous Framework

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Proximity Preservation in an Anonymous Framework

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