DOI 10.1007/s00355-008-0342-8 O R I G I NA L PA P E R
Is a continuous rational social aggregation impossible
on continuum spaces?
Wu-Hsiung U. Huang
Received: 12 February 2006 / Accepted: 15 September 2008 / Published online: 4 November 2008 © Springer-Verlag 2008
Abstract This paper provides a global topological setting for the social choice theory on continuum spaces of alternatives, in contrast to the local differentiable setting of Chichilnisky. Chichilnisky proved that a rational continuous social choice must be discontinuous in her setting. Our paper revisits her theorem to trace the source of this discontinuity. We find that the discontinuity is irrelevant to social aggregation, per se. The main theorem states that there exist a number of continuous social utility maps which are anonymous and satisfy the Pareto condition. As a corollary, we show that there exist corresponding continuous social welfare functions, if singularity is not separated from regular preferences in social preference topology. This extends the possibility result of Jonnes-Zhang-Simpson on linear preferences, to the general ones. The notion of singularity of preferences, relative to the given mathematical structure of an alternative space, is carefully studied.
0 Introduction
Following Arrow’s nonexistence theorem (Arrow 1951) of a rational social choice for the finite discrete case of alternatives, Chichilnisky set up a local differentiable fra-mework on continuum alternative spaces and analogously proved (Chichilnisky 1980,
1982) that there exists no continuous social welfare function which is anonymous and respects unanimity.Baigent(1987) applied Chichilnisky’s continuity back to the finite discrete case to derive his notion of proximity preservation, and, like Chichilnisky, showed its inconsistency with unanimity and anonymity. InHuang(2004), the author of this paper analyzed the notion of proximity preservation both on the discrete and continuum cases, finite or infinite, and questioned its rationality by proving the rigidity.
W.-H. U. Huang (
B
)Department of Mathematics, National Taiwan University, Taipei 106, Taiwan e-mail: [email protected]; [email protected]
Jones et al.(2003) considered linear preferences by revising Chichilnisky’s topo-logy for social preferences to allow null preference. In this particular case, they obtai-ned a possibility result that there exists continuous, anonymous social aggregations respecting unanimity. But a possibility result about linear preferences does not gene-rally support the possibility of a continuous rational social aggregation. To show the impossibility, it is logically sound to restrict oneself to linear preferences, as in Chi-chilnisky(1980,1982). However, to claim the possibility, we need to consider general preferences, since linear preferences constitute only a very small portion of speci-fic preferences (with measure zero). Besides, the meaning of the revised topology allowing null social preference in the given sense requires further study.
We first examine local differentiable settings of the continuum framework, which was first considered by Antonelli (Debreu 1972) and developed by Chichilnisky to derive her impossibility theorem. We examine certain weakness in the model and remark that singularities such as saturation points or plainly indifference sets are not consistently defined. In contrast to the local differentiable model, we establish a global topological setting that allows us to study singularities in a systematic way. In this setting it is shown that the impossibility of a continuous rational social aggregation is not a problem of social aggregation itself. In fact, we prove an existence theorem of a continuous “social utility map”,
U : PN−→ Co(X),
which is anonymous and satisfies the strong Pareto condition—and thus respects unanimity (see TheoremAin Sect. 1.4and Theorem3 in Sect.3.2). Here, utility space Co(X) of continuous functions defined on X is equipped with the most com-mon and natural topology for Co(X), i.e., the compact-open topology, which cor-responds to the uniform convergence on compact sets. Preference space P is equip-ped with a canonical topology , which is essentially analogous to Chichlnisky’s topology in the local differentiable setting. Further comments on the result are as follows:
(I) Based on the existence of social utility map U, it is evident that the nonexistence of a continuous rational social welfare function is caused by the discontinuity of mapπ from Co(X) to preference space P. Therefore the nonexistence theorem of Chichilnisky is irrelevant to social aggregation itself. In other words, if we decompose her social welfare function intoπ ◦ U, i.e.,
F : PN−→ CU o(X)−→ P,π
the discontinuity does not occur on the first map, i.e., social aggregation U . It occurs on the second mapπ, which is simply the mathematical quotient map obtained by discarding the cardinality of social utility functions while preserving the ordinality for social preferences.
The above analysis raises two questions:
(i) Is a continuous “rational” social utility map U : PN → Co(X) less desirable than a continuous “rational” social welfare function F : PN → P? Which one
is more satisfactory from the point of view of social aggregation itself? As a corollary of the existence of U , we prove that there exists a continuous rational social choice map in a weak sense (see CorollaryCof Sect.1.5).
(ii) In reality, does the social preference space carry a topology same as that of individual preference space? Given that social aggregation apparatus can be much more accurate and sophisticated, should we consider some other topo-logy for social preference space? We will try the zero order topotopo-logyo(see Definition 2.5.1) as a possible topological model for social preference space. By a quotient theorem (Huang 1996),ois equivalent to Co(X)/ ∼ (The latter was also denoted by F/ ∼ in Chichilnisky’s notation (Chichilnisky 1980), yet not well studied). TheoremB(of Sect.1.5) shows the existence of infinitely many social welfare functions
F : PN−→ (P, o)
which are unanimous, anonymous and continuous relative to social prefe-rence topologyo.It is interesting to note that the revised topology, defined by Jones-Zhang-Simpson for linear preferences, is basically the subtopology of the zero order topologyo.Their related possibility result about linear preferences is extended to general preferences in TheoremBof this paper.
(II) Generally speaking, a topology on a set of objects means a consistent way to
define how the objects vary continuously, or equivalently, to define how a sequence of the objects converges. That is why topology is a significant mathematical tool to describe various natural and social phenomena. In this paper, we not only consider the given topology on alternative space X , but also introduce a suitable topology for preference space P, the totality of preferences, in order to study how preferences vary continuously, based on the given topology of X .
A preference considered in this paper is a (continuous) preference order globally defined on X (See Sect. 1.3for precise definition). This definition is conventional and has been given byMas-Collel et al.(1995). It can be regarded as equivalent (by Proposition2.3.2) to a decomposition of alternative set X into (closed) indifference sets Iαwith a linear order. We call such a decomposition a stratification of X . Accordingly, an adequate topology that we introduce to preference space P should respect the stratification structure.(For precise definition, see Sect.2.3.)
In other words, a topology for P is adequate if it is defined by the following criterion:
Criterion A A sequence of preferences pn of P converges in the topology to a preference po∈ P, if and only if the following two conditions are satisfied:
(A1) For any two indifferent sets Iαand Iβ of po(not of pn), it holds that Iα Iβ in poiff Iα Iβ in pn, for all large n. (This is equivalent to say that: given x and y in X, x y in poiff x y in pnfor all large n.)
(A2) The indifference sets of pnapproximate to those of pouniformly on compact sets. (The precise meaning of approximation is given in Theorem1of Sect.3.)
The stratification topology introduced in this paper is defined on the basis of Criterion A. (It can be regarded as the first order topology of P, in contrast to the zero
order topologyo.) But Chichilnisky’s topology c1does not meet this criterion—see Example1to4in Sect.1.2.
We will define the concept of “singularity” (“topological singularity” for precision) in the global topological setting. A preference p in P, on which topology is equipped, is call singular, if p has an indifference set which is not thin, i.e., the indifference set contains an open subset of alternative space X. Otherwise, p is called regular.
Contrast to our setting, Chichilnisky’s setting is local and differentiable, and hence the concept of “singularity” in her setting is relative to the differentiable structure of X . It means that a preference in her setting is regular if all the indifference sets are differentiable hypersurfaces of X. Otherwise, it is singular.
The difference between the two topologies and cis illustrated particularly by the fact thatcmaintains the separation property which states that
(a) a sequence of singular preferences can never tend to a regular preference, and (b) a sequence of regular preferences can never tend to a singular one,
while satisfies only the semi-separation property, which is defined just by (b). Note that the semi-separation property is a direct result of condition (A2).
(III) The prime difficulty in obtaining a continuous rational social welfare function
is that the semi-separation property holds for topologycand, when either of the two topologies is imposed upon the social preference space. For example, inputting a sequence of individual profiles into a social aggregation apparatus, let the outcome be a sequence of social utility functions:
fn(x) = x
n, x ∈ [0, 1] ≡ X,
Evidently, fn(x) converges to function fo(x) = 0. The semi-separation property forces preference pn, which corresponds to utility function fn(i.e., π( fn) = pn), staying away from(and therefore not tending to) preference po, which corresponds to fo(i.e., π( fo) = po), since pn is regular and yet pois singular. In other words, pn does not converge to po, although fnconverges to fo. This is the essential reason for discontinuity of mapπ. And it makes continuity of a rational social welfare function impossible.
However, it is not the chosen topologycor that should be blamed, because at least the semi-separation property(b) is a logical consequence of acceptable condition (A2) of Criterion A. The discrepancy between continuity of the cardinality-discarding functionπ and the semi-separation property of the preference topology is innate.
Now there are two reasonable stances to be held while viewing the discrepancy. First, we observe that the prime difficulty occurs only on mapπ, which projects the utility functions to the corresponding preferences, after social aggregation has been accomplished. So the discontinuity of social welfare functions has nothing to do with social aggregation itself. In this light, TheoremAin Sect.1.4, as well as CorollaryC
1 Chichilnisky’s original topology is not well-defined, when singularities are included. Topology c(see
Sect.1.1for precise definition) is a modified form based on her original topology. For the analysis, see RemarkAin Sect.1.1and RemarkBin Sect.1.3of his paper.
in Sect.1.5, support the stance and clarify the long confusion about the impossibility of continuous rational social choice.
The second stance is to discard the semi-separation property, and define another topology for the social preference space by requiring only (A1) of Criterion A, which we call the zero-order topology and denote byo. It is then not difficult to see the existence of a required continuous social welfare function F: PN → (P, o), which is stated in TheoremBof Sect.1.5. However, isoa topology more acceptable for the social preference space than (and than c)? In Examples8and10, we give certain practical evidence to support the second stance. Nevertheless, the question remains open for further investigation.
(IV) The idea of proving TheoremAis based on the global topological setting of our model of preference space P. Chichilnisky’s framework is a local differentiable setting in the sense that a preference is defined by directions of differentiable local preferences. Instead, preference space P adopted in this paper is the space of continuous “preference orders” (also called weak orders), defined globally on alternative space X .
The global topological setting may fit reality more closely, at least in a conventional way. It happens to have an advantage that the inferior set Qx(p) of a preference p (see Sect.2.1for definition) to a given alternative x in X, is consistently defined and, therefore, can be measured, so that the lifting of an individual preference p to a utility function f is possible (see Theorem2). Thus the average rule can be selected as a candidate for the desired social utility map U .
However, given x in X , inferior set Qx(p) is not continuous in p. Technically, this is the major problem that we have to solve for showing the existence of utility map U. It makes the main arguments of this paper nontrivial even mathematically.
Nevertheless, the measure2of inferior set Qx(p) is lower semi-continuous in p, i.e., lim infµ(Qx(pn)) ≥ µ(Qx(po)),
as pntends to poin, where “lim inf ” denotes the limit infimum and µ(∗) denotes the measure. Consider the singular set(p) of p, which is defined by the union of all the indifference sets containing interior singular points of p (see Definition 2.4.2). We prove that
µ(Qx(p) − (p))
is continuous in p. It means that a modification with the term(p) makes the measure
µ(Q
x(p − (p)) continuous in p. By letting
f(x) = µ(Qx(p) − (p)) + 1 > 0
we may lift each individual preference pi to a utility function fi continuously (see Theorem2). Then applying the average rule, for example, by
U(p1, . . . , pN) = ( f1+ · · · + fN)/N,
2 The notion of measure of a set may be regarded as a generalized concept of the “area” of a set in a plane.
the desired social utility map U is obtained. TheoremsA, the main result, as well as TheoremBand CorollaryCstated at the end of Sect.1are then proved.
(V) To investigate the possibility of a rational social aggregation, it seems that
singu-lar preferences should not be precluded from consideration. Chichilnisky’s impossibi-lity is the extension of Condorcet paradox to the continuum framework. However, the impossibility occurs because of the preclusion of singularities. Singularities deserve to be carefully treated, as they always exist in reality, though complicated technically. In our model, we study systematically the role of singularity, noting the compatibility between the notion of singularity and the concerned structure of the alternative space (Sect. 2.6). Based on the study of singularities, we establish the lifting theorem by adjusting the term of the inferior set Qx(p) to Qx(p) − (p), i.e., subtracting from Qx(p) singular set (p), and thus finding continuous social utility maps with certain principles of rationality. This makes the existence of the required continuous social utility map possible.
Since the setting and the proofs given in this paper involve many technical concepts of topology, we illustrate the ideas and main results of the paper with many examples and figures in Sect.1. In fact, Sect.1 is provided for general readers, who are not particularly familiar with mathematical language of topology. We include Table at the end of Sect.1to summarize the contents.
Section2 is devoted to rigorous formulation of the framework, pointing out the difficulties involved. In Sect.3, we prove the theorems, again presenting examples to explain certain subtle points. Since parts of the arguments are too technical to be included in the text, we leave them for Appendix.
Throughout this paper, one may assume alternative space X a closed n-cube Inof real n-space Rn, a closed or open ball in Rn, or a quadrant Rn+= {(x1,x2, . . . , xn); xi ≥ 0, ∀i}. However, the assumption is made only for easier understanding.3In fact, all the theorems are valid for topological manifolds with or without boundary(partial or total), except that when X is non-compact, certain standard modifications of the state-ments of the theorems are required. For instance, when X is quadrant R+n, the Jordan measureµ is infinite, but we may convert it into a finite one by taking arctan r as the converting density function, so that Theorem2of Sect.3.2is still true. By a topological manifold,4we mean a topological space, every point of which has an Euclidean-like neighborhood. An n-cube (closed or open), a quadrant in Rn, any domain in Rn, an n-torus, and an n-sphere are the examples.
The case of infinite discrete alternatives has been considered in a joint work of Chen with this author (Chen and Huang 2008). The case is noteworthy, because it incorpo-rates both the continuum case and the finite discrete case. The totality P of preferences on an infinite discrete alternative space X has non-discrete topology, although X itself is discrete. Stratification topology of P as well as the zero order topology o intro-duced in this paper, would make the corresponding forms easier to be understood in that case. As X itself is discrete, one may avoid technical topological formulation on X
3 Most of the theorems are valid for general topological spaces with mild conditions such as Tychonoff T3,
local-compactness, locally-connectedness, and connectedness.
4 Precisely, a topological manifold X is a topological space in which each point x∈ X has a neighborhood
and still somehow get the idea intuitively. However, the forms defining the topologies of P for the case are derived from the theory established in this paper.
The author would like to express his gratitude to the referee and H.J.Chen for having brought the works of previous contributors to his attention.
1 Background and results
1.1 A sketch of Chichilnisky’s theorem
Assume that alternative space X has a differentiable structure. Following Chichilnisky’s formulation (Chichilnisky 1982), one consider a local differentiable setting Sc = (Pc, c), which consists of preference space Pcon X and topologyc imposed on Pc. A preference in Pcis defined by an unit vector fieldv differentiable on X, except v may be zero at some points of X.5The points wherev is zero are called singular points of v, and the rest are called regular.6A metric dcis introduced to Pc by
dc(v, w) = sup x∈X
|v(x) − w(x)|, ∀v, w ∈ Pc. (†)
Metric dcinduces a topologycon Pc.7
Chichilnisky proved that there exists no continuous function F : PcN −→ Pc,
which is anonymous (AN)8and respects unanimity (UN),9 where PcN denotes the product space of N copies of Pcand N denotes the number of individuals involved in social aggregation F . Her proof is elegant and simple. For easier understanding of our later content, let us sketch the idea in a few lines. Let X = R+n or In, and let L ⊂ Pc be the totality of linear preferences in Pc,10i.e.,
L ≡ {v ∈ Pc: v is either the zero field or constant in xwith|v(x)| = 1, ∀x ∈ X}.
5 Chichilnisky considered only two cases thatv may be zero on the whole X or on a set of measure zero.
As we will study singularity systematically here, we do not consider this restrain.
6 Note thatv represents the most desirable direction which is perpendicular to the differentiable indifference
hypersurfaces of the preference. Also note thatv is zero at a point of X, if and only if around that point the indifference set is not a differentiable hypersurface.
7 When the singular sets of preference vectors are non-empty, there is a slight difference between
Chichil-nisky’s formulation and the above setting Sc. The latter can be regarded as a completion of the former. For
details, see RemarkAat the end of this subsection.
8 (AN): F(p σ1, pσ2, . . . , pσN) = F(p1, p2, . . . , pN), ∀(p1, p2, . . . , pN) ∈ P N c and∀ permutation σ of the indices 1,2,…,N. 9 (UN): F(p, p, . . . , p) = p, ∀p ∈ P c.
10 For X a general manifold, we may choose a point x
o, with a coordinate neighborhood U, regarded as
a part of Rn. Consider a constant vector field v in U which is of unit length or zero. Denote v(xo) by vo.
Letv be zero outside U. Call such a vector field a linear preference relative to xoand denote its totality by
It is clear that L is topologically equivalent11 to Sn−1∪{0}, where Sn−1 is the
(n −1)-dimensional unit sphere in Rnand 0 means the zero vector of Rn. Suppose that F is continuous. Then F restricted to LN ⊂ PcN, should also be continuous, where LN = L × L × · · · × L (of N copies). We illustrate the proof by considering the case that N = 2 and n = 2, depicting the idea in a sketch to show how the impossibility occurs. For general N and n, it needs more sophisticated mathematical argument.
Select an interior point xo∈ X. Consider the localization map ψ : Pc−→ S1∪{0} defined by
ψ(v) = v(xo) ∈ S1∪ {0}.
Obviously,ψ is continuous. This is seen by the definition of dcwhich provides uniform convergence of preference vector fields on X . Let F = F|L2be the map of F restricted
to its subset L2of Pc2. We have the composition map
F= ψ ◦ F: L2−→ PcF −→ Sψ 1∪ {0},
which is continuous, since both Fandψ are continuous. Now L2is topologically equivalent to (S1∪ {0})2which contains four components. Among them there is a torus S1×S1. Since the image of a connected set under a continuous map must be connected, Feither maps S1×S1into S1or into{0}, i.e.
F: S1×S1−→ S1, or F: S1×S1−→ {0},
But the unanimity of F precludes the latter case.12 It then suffices to consider F :
S1×S1−→ S1. Clearly, Fis still anonymous and respects unanimity in its mathe-matical form. Let Fbe interpreted by the following figure :
Fig. 1
Here segment AB is pasted on CD, and AC on BD. By this, rectangle ABDC is regarded topologically the torus S1×S1. By anonymity, F(M) = F(R), for any two points M ∈ AB and R ∈ AC which are symmetric with respect to axis AD. (Also, F(R) = F(Q), since AC is pasted on BD.) Hence the map of Fon AB duplicates the map Fon BD. Therefore Fon AB∪ BD must cover range S1an even number of times. However, by unanimity, Fon diagonal AD covers range S1exactly once. Thus the two maps Fon AD and Fon AB∪ BD can not be homotopic to each other.
11 Two topological spaces X and Y are topologically equivalent, if there exists one to one correspondence
f : X −→ Y such that both f and f−1are continuous. By f continuous, we mean∀A open in Y , f−1(A) is open in X . A topological equivalence is also called a homeomorphism.
12 Any elementv
oin S1∪ {0} has a v ∈ L2 such thatv(xo) = vo. By unanimity of F, we have
This contradicts the fact that the former can be deformed along triangle ABD, to the latter, as Fis continuous on S1×S1. Therefore (AN), (UN) and the continuity of F are logically inconsistent.
Remark A Chichilnisky considered (Chichilnisky 1980,1982) various subspaces Pc∗ of Pc with topology∗c slightly different fromc. Here we explain why in place of her setting Sc∗= (Pc∗, ∗c), we have to take Sc = (Pc, c) as the model representing various local differentiable settings to be compared with our global topological setting S= (P, ). First we list the various forms of her setting as follows.
Formα : Pc∗ = the space of Cr codimension-1 transversely oriented foliations of choice space X = the space of locally integrable Crvector fields of X which is nowhere zero. The corresponding topology∗cis given by dcwhich is defined in(†).
Formβ : Pc∗= the locally integrable Cr-vector fields of X whose singular set may be the whole space X or have measure zero. She defined topology∗c by
pj → p iff supx∈X−∪∞
1 Sj | p
j(x) − p(x) |→ 0, (**)
(Chichilnisky 1982, pp 349), i.e.,∗c is given by “dc outside the singular set” as she claimed.
Formγ : Pc∗ = F/∼, where F is the space of Cr+1real valued functions on X and∼ is the obvious equivalent relation. Topology ∗cis inherited from the Cr+1sup norm of F .
Formδ : Pc∗= the subspace of all convex preferences of F/∼ in form γ.∗cis the corresponding subspace topology.
Setting Sc= (Pc, c) is introduced to modify S∗c = (Pc∗, ∗c) and provides a unified form of local differentiable settings. When singularities are precluded,∗c of formα is the same asc. However, to include singularities, setting Sc∗seems invalid, and that is why we have to replace it with Sc. Here are the comments about Sc∗:
(1) Pc∗ of formβ allow only special kinds of singularities to be considered. That means singularities can not be studied systematically. Furthermore,∗c of form
β is not a well-defined topology. There exists a sequence pj which converges to p in ∗c , but has subsequences not convergent to p in ∗c. For instance, there exists a sequence {p1, p2, p3, . . . , pn, . . .} convergent to p in ∗c, but {p2, p3, . . . , pn, . . .} is divergent in ∗c. This makes the notion of continuity not consistently defined.
In fact, given a well-defined topology, defined by a family of open sets {Uα; α ∈ an index set I}, sequence pj converges to p, if and only if for any Uαcontaining p, there exists N such that pj ∈ Uα, for all j>N. As a consequence, any subsequence of a sequence pj convergent to p must converge. However, ∗c fails to satisfy this subsequence convergence property(See Appendix (AIVa) for proof).
(2) ∗c fails to satisfy SCP, the last property, partly because the corresponding sup norm,
dc∗(p, q) = sup x∈X−Sq
| p(x) − q(x) |, (*)
using which Chichilnisky wrote down formula(∗∗), does not define a distance. Clearly, dc∗is not symmetric in ( p, q).
(3) If dc∗(p, q) is adjusted by a symmetric form, supx∈X−Sp∪Sq|p(x) − q(x)|, it
still does not define a distance, since the triangular inequality does not hold (For proof, see Appendix (AIVb)). Therefore Chichilnisky’s idea of defining a topology which intends to ignore singularities is not realized. This is why a modified distance dc has to be defined by the above formula(†) to replace dc∗. Correspondingly, topologycdefined by dc is to replace topo logy∗c defined by dc∗.
(4) The sup norm formula(∗∗) implies that a singular point of p must be a singular point of pnfor some n, if pnconverges to p in∗c.This is unacceptable as well, because convergence concerns only eventual behavior of the sequence.
(5) Topology∗c of formγ and δ are invalid as well. If these forms are conside-red, Chichilnisky’s proof of her impossibility theorem will not be valid. (See Appendix (AIVc) for explanation.)
1.2 The preference convergence
The local differentiable setting Sc = (Pc, c) will be examined when comparing it with our global topological setting S = (P, ) . First we illustrate that many good convergent preferences are divergent in Sc.
Example 1 Consider a sequence{pn : n = 1, 2, . . .} in Pc and po ∈ Pcwith their indifference curves as in Fig.2,where arrowsvnandvodenote their preference vector fields respectively. Herevnandvoare indicated by unit vector fields, perpendicular to the indifference curves with the directions illustrated, except they are zero at Onand O respectively. Let On → O, as n → ∞. Practically, sequence pnis regarded convergent to po. As pn and po satisfy Criterion A, pn converges to po in our stratification topology.
Fig. 2
However, pndoes not converge to poin local differentiable topologyc, since dc(pn, po) = sup|vn(x) − vo(x)| = |vn(O) − vo(O)| = 1
for any n. Note that vn(O) = a unit vector pointing to the right, and vo(O) = the zero vector, hence the last equality is 1. (If we use Chichilnisky’s formula (∗), we also have
dc∗(pn, po) = sup x∈X−∪Sn
|vn(x) − vo(x)| = |vn(O) − vo(O)| = 1,
where the singularity set Snof pnis{On}. Note that O ∈ X = X − ∪Snand hence O is counted in the last formula. Thus the sup norm does not tend to zero and pndoes not converge to po.)
Example 2 Let X = [−1, 1] × [−1, 1]. Consider preferences pn and po on X as described in Fig.3, where the indifference sets of pnand poare given by the concentric circles with center (0,0) and by the dark areas
(x, y) ∈ I2; x2+ y2≤ a+1 n 2 and {(x, y) ∈ I2; x2+ y2≤ a2} of pnand po, respectively. Note that a ≥ 0 is a given real number. When pnand po are regarded as preferences in Pc, preference vectorsvnandvoare either all outward along radial directions, or all zero on the dark areas.
Fig. 3
In our topology, pnconverges to po, as Criterion A is satisfied. But it diverges in c. (As for S∗
c = (Pc∗, ∗c), where forms α and β are the possible models, pnand po are not even considered as preferences, since they have singular sets that are neither the whole space X nor of measure zero.)
Example 3 Let X = [0, 1] × [0, 1] ⊂ R2, or let X be the infinite first quadrant in the plane. Consider preferences pnin Fig.4such that: (1) the indifference sets are exactly the hyperbolas, (x, y) ∈ X; (x − α)(y − α) = 1 n, x − α ≥ 0 and y − α ≥ 0 ,
Namely, for each fixed n, the hyperbolas may be obtained by translating x y= 1nalong vectors(α, α), for any real number α.
(2) the corresponding preference vectorsvnpoint away from the origin(0, 0). Define posimilar to pn, except the indifference hyperbolas having their equations replaced with(x − α)(y − α) = 0. Again pnconverges to poin our topology, since they satisfy Criterion A, but not in setting Sc. (Also, pndoes not converge to poin setting Sc∗, since the singular set of pnis empty, and the sup norm in formula (∗∗) is attained, for any n, by diagonal points xof X, where |vn(x) − vo(x)| = 1.)
Definition Given a continuous function f on X → R, which may be regarded as a
utility function on X, we say a preference p on X corresponds to f , when x y in p iff f (x) ≥ f (y), ∀x, y ∈ X.
We also say that p is defined by utility function f .
Example 4 Consider X = [0, 1] ⊂ R for easier illustration. Suppose f is a smooth function on[0, 1]. Let n be a positive integer. Divide [0, 1] into 2n intervals of equal length. Define fnby a continuous function which satisfies
fn(x) = f k n , f or x ∈ k n, k n + 1 2n and k= 0, 1, 2, . . . , n − 1, making it linear on the remaining subintervals as in Fig.5.
Fig. 5
Let pn, p be the preferences defined by utility functions fn, f , respectively. Then fnconverges uniformly to f . As pnand p satisfy Criterion A, we have pnconvergent to p in, but divergent in c. (In setting Sc∗, pnis not included in preference space Pc∗.)
The inadequacy of Scshown by the above examples is due to topologycgiven by dc(v, w) = sup {|v(x) − w(x)|; x ∈ X}, which forces two preferences far apart (to a distance of least 1) from each other, as long as the singular sets ofv and w are not completely identical.
For instance, in Example1, origin O = (0, 0) is a singular point of po, but it is regular for pn. According toc or any of∗c , pndoes not converge to po, although singular point Ontends to O. In Example2, the singular set, i.e., the dark area in the figures of pn, deforms evenly into the singular set of poas n → ∞, but in c, pn does not converge to po. This occurs because in order to have pn convergent to po, c requires that∀n >some no, the singular set of each pn should be the same as that of po. The same happens in Examples 3and4. The strict requirement makes local differential topologyc“too strong” (i.e., having too many open sets) to contain considerable cases of convergence.
1.3 Globalization of vector fields
A continuous preference order (also called a global preference hereafter) on a conti-nuum alternative space X is normally defined by a complete transitive binary relation 13on X such that for each x in X, the weak preferred set {y ∈ X; y x} and the
weak inferior set{y ∈ X; y x} are closed in X. Preference space P considered in this paper is the totality of such continuous preference orders.
Debreu(1972) has analyzed the cases where a local preference may be integrated into a global preference order, i.e., a vector fieldv in Pcmay be integrated to obtain a preference order in P by a trajectory argument. More precisely, whenv satisfies some integrability condition,14we first define x∼ y, iff x and y are in the same indifference set as given byv, or in mathematical terminology, iff x and y are on the same integral manifold with tangent spaces perpendicular tovat each point of the manifold. One may then use the trajectory argument to define the preference order basically by stating that x y if there exists a trajectory curve from y to a point z with z ∼ x.
However, even when the required integrability condition is satisfied, there may be no adequate preference order in P, corresponding to v in Pc.
Example 5 Define a vector fieldv ∈ Pcon X= [−1, 1] × [0, 1] by
v(x1,x2) = ⎧ ⎨ ⎩ (−1, 0), x1< 0 (1, 0), x1> 0 (0, 0), x1= 0
then for any two alternatives x = (x1, x2) and y = (y1, y2) with x1y1 < 0, we can not determine whether x y or x y, although v has been clearly defined. Thus the completeness condition is not satisfied.
Fig. 6
Surely one may assign x y or x y to the pair x, y in a consistent way, e.g., let f1 be defined by f1(x1, x2) = |x1| , ∀(x1, x2) ∈ X. Then preference order p1, “defined by” utility function f1(as in the definition before Example4), is consistent with vector fieldv. However, there are infinitely many preference orders corresponding
13 By a binary relation on X, we mean a subset G ⊂ X × X. This induces defined by x y iff (x, y) ∈ G. We call complete, if ∀x, y ∈ X, x y or y x or both. If x y and y x, we say
x and y are indifferent, and denote it by x ∼ y. We also call transitive, if ∀x, y, z ∈ X, x y and y z ⇒ x z. A binary relation which is complete and transitive is usually called a weak order or a rational preference relation (seeMas-Collel et al. 1995).
14 Such as Frobenius integrability condition (see any introductory text book on differential geometry,
e.g., S.Sternberg: Lectures on Differential Geometry, Prentice-Hall, p132); for further analysis see Debreu
to the same vector fieldv. For instance, let p be the preference order defined by utility function f which is given by
f (x1, x2) =
−x1, x1≤ 0
x1 x1> 0 ,
where is a small positive number. Let Lh denote the vertical straight line segment {(h, x2); x2∈ [0, 1]}. Given each h ∈ (0, 1] , L− h∪ Lhis an indifference set of p (see Fig.6). Thus we have many different preference orders p ∈ P corresponding to the samev ∈ Pc, which leads to certain ambiguity.
Example 6 Let X = I2⊂ R2andv ∈ Pcbe the unit vector fields pointing counter-clockwise along the concentric circles with center(0, 0), except at (0, 0) it is zero. Let x = (12, 0), y = (−12, 0). Then according to the trajectory argument, x y and y x occur simultaneously. Hence there is no preference order corresponding to v. Remark B As we have noticed in RemarkAof Sect.1.1, Chichilnisky attempted to define “globally integrable foliations” by considering F/ ∼. But once singularities are included, the related topology makes Pc∗a non-Hausdorff space and her theorem fails. Thus the attempt to consider global preferences is unsuccessful, and her formulation remains local when singularities are included in preference space.
1.4 The global topological setting
We now compare and c. As the topological structure is the focus of our concern, we define in the following the notion of singularity for preferences in P relative to the topological structure of X , rather than relative to its differentiable structure.
A topological structure of X can not detect differentiability of level sets. For ins-tance, in Example3, the cusps of level sets of poappeared at the diagonal points of [0,1]×[0.1] are not detectable. Thus the cusps are not regarded singular relative to the topological structure of X .
On the other hand, the saddle points in Example1are not abnormal compared to other points, unless a manifold structure has been introduced to X . In this sense, saddle points are not regarded as singular points relative to a given topological structure.
However, whether a point is an interior point of the level set is detectable. It is natural to define singularity relative to topological structure as follows. Given p∈ P, a point x∈ X is called an interior singular point of p if x has an open neighborhood contained in indifference set Ix(p) ≡ {y ∈ X; y ∼ x in p}. If there exists such a point x in X, preference p is called (topologically) singular. Otherwise, p is regular.
Let(p) be the union of the indifference sets each of which contains an interior singular point. We call each point of(p) a singular point of p and the set (p) the singular set of p. For example, preferences pnand po(when a> 0), as described in Examples2and4, are singular ,with their singular sets denoted by the dark circular disks in Fig.3and at least the odd subintervals in Fig.5, while those in Examples1
and3are regular.
Comparing with local differentiable settings which include Chichilnisky’s fra-mework, one can see a difference. If an indifference set has an interior singular point,
the preference direction at that point must be zero. Therefore, an interior singular point in our global topological setting is also singular in a local differentiable setting, but not vice versa. In fact, in a local differentiable setting, On for pn and O for poin Example1are singular points, which make pnand posingular, but they are regular in our global topological framework.
As stated in (II) of Introduction, we consider Criterion A as the criterion of convergent preferences. Based on Criterion A, stratification topology is defined (see Definition 2.5.2 for precise definition).
Now we introduce the notion of the separation property to study and cas follows. For pnto be convergent to poinc, it is required that the singular set of pomust be identical with that of pn for all large n. Thus a sequence of singular preferences never converge to a regular(i.e., non-singular) preference inc. Yet it is not so in, since we allow the (topological)singular sets of pndeforming from their boundaries gradually to become those of po. In Example2 where a= 0, or in Example4, we have seen a sequence of singular preferences converging to a regular preference, if we adopt.
Namely, the following separation property holds inc, but not in.
Separation property: The set of singular preferences and the set of regular
preferences are separated15from each other, i.e., there exist two disjoint open sets
and in the preference space, with ⊂ and ⊂ , or equivalently, a sequence of singular preferences never converges to a regular preference, and vice versa. Example 7 Let u and un : I2= [0, 1] × [0, 1] → R be defined by uo(x, y) = f (x), un(x, y) = fn(x), where f and fnare given in Example4and it is assumed that f is strictly monotone. Denote by poand pn the preferences on I2, defined by utility functions uo and un respectively. Then po is regular, while pn is singular and pn converges to poin, but not in c. The dark stripes in Fig.7 are the singular sets
(pn). We have po∈ and pn ∈ . The convergence shows that there is no open set
containing poin, such that is disjointed from the set of singular preferences.
Fig. 7
This example illustrates that the separation property does not hold in , although it does inc.
However, both and cadopt the following semi-separation property.
Semi-separation Property A sequence of regular preferences never converges to
a singular preference, i.e.,∃, open in the preference space, such that ⊂ and
∩ are empty.
15 Two sets A and B in a topological space are called separated from each other, if there exist two disjoint
As an example, a sequence of linear preferences never converge in bothcand to the null preference o.
Definition The null preference o is the preference with indifference set Ix(o) = X, ∀x ∈ X.
Remark Given two non-empty disjoint subsets A and B of a general topological space
such that A ∪ B = . If A and B are separated from each other, then is (at least)
disconnected. But the semi-separation property defined by the existence of an open set U ⊃ A such that U ∩ B = φ, may keep still connected. This general topological fact is seen by letting = [−1, 1] , A = (0, 1] and B = [−1, 0], and choosing U = A. Note that 1/n ∈ A, and 0 ∈ B, but 1/n converges to 0, where n = 1, 2, 3, . . .. Considering that A represents the set of singular preferences and B represents the set of regular ones, we see that ∪ may still be connected. However if we also require to be closed in the preference space, then ∪ is disconnected, since now is open and closed in ∪ .
The semi-separation property is compatible with reality, when preference space P is the totality of individual preferences. This is because individuals are insensitive to very slight change of their preferences for distinct alternatives from indifference(∼) to strict preference(). We construct an example to explain this observation. Example 8 Consider an individual who in the beginning has his initial preference po indifferent on all alternatives of X = I2, i.e., po = o, the null preference. After a while, his preference begins to change very slightly on the alternatives, for example, taking a linear preference p which is defined by a linear utility function f with extremely small gradient|∇ f | = > 0. If at this moment, he was asked to declare precisely his preference in front of an apparatus of social aggregation, e.g., he was asked to vote, it can hardly be imagined that he would claim p rather than po, since he is not so sensitive, when is arbitrarily small, say, = 0.0000 . . . 01. (Note that no matter how small a positive number is, p is the same as p1such that the preferences on any two alternatives with different x-coordinates are different.) Namely, p should not be regarded by him as a gradual change of po. This is whycand do not accept p as convergent to pofor individual preferences. The indifference sets of p and po are illustrated in Fig.8.
Fig. 8
Using as the preference topology on P, we will prove our main theorem in Sect.3.
Theorem A There exist infinitely many continuous social utility maps,
which are anonymous and satisfy the strong Pareto condition16 (and hence respect unanimity.17)
1.5 The discontinuity
TheoremAshows the existence of continuous rational social utility maps. In fact, it shows the existence of a rational continuous social aggregation which provides social utilities distributed on the given set X of alternatives. Why our result seems contradic-tory to the discontinuity theorem of Chichilnisky?
(1) The discontinuity of the social welfare function proved by Chichilnisky arises from the separation property applied to social preference spaces byc. Recall her work sketched in Sect.1.1. The separation property makes the range space of
F: S1×S1−→ S1∪ {0}
a disconnected set S1∪{0}, where S1represents the totality of linear preferences (which are regular) and{0} represents the null preference (which is singular). The contradic-tion is then derived from the disconnectedness of S1∪ {0}. And the impossibility theorem is therefore established.
In other words, the essential reason why discontinuity occurs in Chichilnisky’s theorem is that the possible singularities of social preference are precluded from the regular ones. The preclusion is based on the semi-separation property.
Generally speaking, social aggregation may naturally lead to mutual cancellation among individual preferences with various preference directions. For example, linear preferences of opposite directions, or of Condorcet triple, cancel out each other. The cancellation would result in singularity on many occasions. Namely, the resultant preference would possibly become degenerate and null, for which no particular direc-tion is preferred. Thus it becomes singular. However, the semi-separadirec-tion property of topologyc forces the singularity to remain separated from regular preferences. If social aggregation cancels the opposite preferences gradually, and tends to a singula-rity with all directions indifferent, then discontinuity is automatically the outcome.
That is, any topology of social preference space admitting the semi-separation pro-perty will automatically induce discontinuity in a social aggregation. This is what Chichilnisky’s theorem really says.
(2) Nevertheless, if we consider the composition of maps F : PN−→ CU o(X)−→ P,π
16 By U satisfying the strong Pareto condition, we meanπ ◦ U satisfies the strong Pareto condition,
or equivalently, if x y in pi, ∀i, then f (x) ≥ f (y) where f ≡ U(p); also if x y in pi, ∀i, then
f(x) > f (y). Here π is the projection map given in (I) of Introduction.
we have a social welfare function F, where F = π ◦ U and π is the projection map from the utility function space to the preference space, i.e.,π ( f ) ∈ P is the preference defined by the given utility function f ∈ Co(X). Obviously, it is map U , not mapπ, which involves social aggregation. In our topological global setting, recall we have the advantage, depicted in (IV) of Introduction, that Qx(p) can be defined and measured. With this advantage, we use the continuity of the measure of Qx(p)−(p) to prove TheoremA, which shows the existence of U with required continuity and other rationality. Consequently, the nonexistence of F is reduced to discontinuity ofπ. However, discontinuity ofπ is caused by the (semi-)separation property imposed on the topology chosen for social preference space P. Or more precisely, discontinuity ofπ happens because of the incompatibility of the two topologies of utility function space Co(X) and social preference space P.
We explain the incompatibility by the following example.
Example 9 Consider X = [0, 1] and two sequences of utility functions fn(x) = 1 n + 1 x, gn(x) = x n.
Clearly, fn(x) tends to f (x) = x and gn(x) tends to g(x) = 0. However, the two sequences of preferences pnand qndefined by fnand gnrespectively are unfortunately the same sequence of preferences, i.e.,
pn= π ( fn) = π (gn) = qn,
while preferences p = π ( f ) and q = π (g) defined respectively by their limits f and g of fnand gnare distinct.
In fact,∀x, y ∈ [0, 1] ,
x≺ y in p, iff x < y.
but x∼ y in q. Thus p = q . Clearly, p is regular but q is singular. Now we face the dilemma, whether we should define sequence pn(= qn) as convergent to p, or define it as convergent to q? Practically, only one case can be chosen. In both topologiesc and, where the semi-separation property holds, it is required that
lim pn(= lim qn) = p, but not = q, which says that
qn = π (gn) does not tends to q = π (g) , although gntends to g.
(3) The above analysis might change the long accepted interpretation of Chichilnisky’s impossibility theorem. Now we see that it is not a continuous ratio-nal social aggregation impossible. The impossibility reduces to discontinuity of map
π, which is irrelevant to social aggregation. And the discontinuity is caused by the
semi-separation property imposed on the topology of P.
But do we have a better choice of a topology for P, such thatπ is continuous and hence a rational social welfare function F is continuous?
Example9says that even this is impossible, unless we accept a very weak topology for the social preference space, which is different from the topology of the individual preference space. More precisely, the dilemma of selecting p or q as the limit of pn(= qn) always exists, unless we accept our topology to permit the existence of multiple limits of a single sequence and discard the semi-separation property.
For instance, when we accept zero ordered topologyofor P defined by (A1) of Criterion A (see detail in Sect.2.5), the projection map
π = Co(X) → (P, o)
is continuous. The proof is based on a quotient theorem ofHuang(1996), which was proved earlier by this author. It states thatoequipped on P is exactly the quotient topology of Co(X) under π. So oand the topology of Co(X) are now compatible18 with each other. Combining the quotient theorem and TheoremA, we immediately obtain:
Theorem B There exist infinitely many continuous social welfare functions
F : PN −→ CU o(X)−→ (P, oπ ) which are anonymous and satisfy the strong Pareto condition.
(4) To accept a sequence of regular preferences convergent to a singular preference, such as qnconvergent to q in Example9, is not purely based on compatibility of the two topologies chosen for utility space and preference space. It might be previously built in the given topology of alternative space X. Here is an example:
Example 10 In Example5, let podenote the preference obtained by substituting = 0 in the definition of f . Indifference sets of p with > 0 are vertical straight lines, while the indifference set of poat(0, 0) is the whole square I+ = {(x1, x2) ∈ X, 0≤ x1≤ 1}. Given x, y in I+, we have for p ,
x = (x1, x2) ∼ x = (− x1, x2) y= (y1, y2) ∼ y = (− y1, y2).
As is arbitrarily small, x and y are only very slightly different in p . Note that the notion of the “slightness” mentioned here is not defined by certain “assumed” utilities, but by the fact that x tends to xo = (0, x2), y tends to yo = (0, y2) and xo ∼ yo in p , for any ≥ 0. The notion of the “slightness” has been defined precisely by the given topology of alternative space X itself. Therefore when a topology introduced for
18 This is because that U is open in(P,
social preferences is sufficiently sophisticated, it is expected to accept convergence of p to po. And hence the semi-separation property (which rejects the convergence of p to po) should be discarded from the social preference topology.
Example10gives evidence to supportoas a possible model of social preference topology, since it simply reflects the topology of alternative space X . Nevertheless, topologyomight not be accepted as a good model, because there exist multiple limits for a single sequence and thereforeois not even a Hausdorff space.
(5) We may now claim that a continuous rational social choice is possible if we accept singularity. To make it precise, we define a social choice map by
C: PN → 2X
where∀ profile ¯p = (p1, . . . , pN) ∈ PN , C( ¯p) is a closed subset of X, and 2X is the power set of X, i.e., the family of all the subsets of X. We say C is continuous if the given ¯pnconverges to ¯p in PN, and xn∈ C( ¯pn), it holds that
lim
n→∞xn∈ C( ¯p). Then we have:
Corollary C There exist infinitely many continuous social choice maps which are
anonymous and respect unanimity.
The proof is evident in the virtue of TheoremA. Given a utility f(x) on X, we define the maximal set by
M( f ) ≡ {x ∈ X; f (x) ≥ f (y) , ∀y ∈ X} .
Since X is assumed compact, we see that M( f ) is a non-empty compact set in X. Then C= M ◦ U is evidently the required social choice map.
We use the following Table to summarize the notions and results considered in this paper and compare the three topologiesc, and oin terms of them.
2 The framework
2.1 Definition of preferences
Let X be a space of alternatives, which mathematically is a pure set, or more generally a topological space, noting that a pure set may also be regarded as a discrete topological space. A preference order p on X is a binary relation defined on some pairs of elements of X such that
(i) (Completeness);∀ x, y in X, either x y or y x, or both,
(ii) (Transitivity);∀ x, y, z in X, x y and y z ⇒ x z,where x, y, z may be distinct or non-distinct.
Table Notions and results Topologyτ = · · · → Modified Chichilnisky’s topologyc Stratification topology Zero order topologyo
Criterion A (see Sect. 0) N Y N
A1 as Nec.Cond. A2 as Nec.Cond. Y† Y * Y Y‡ N A1 as Suff.Cond. A2 as Suff.Cond. N Y Y‡ N Semi-Separation Prop. Y Y N Separation Property Y N N Continuity of U ifτadopted as individual preference topology N Y (TheoremA) N Continuity ofπ and F ifτ adopted as social preference topology N N Y ** (TheoremB) Y =Yes; N = No; A1 and A2 are given in Criterion A of Sect. 0
Y† = Yes, when A1 is restated for cequivalently by “preference vectors pntends to poateach point
of X , in the sense of vector convergence in Euclidean space Rn”
Y *= Yes, and moreover, a condition A2a (which is stronger than A2) holds, where A2a means: “the singular sets of pnand poare all identical for large n”.
Nec. Cond.= Necessary Condition of preference convergence pn→ p under topology τ
Suff. Cond.= Sufficient Condition of preference convergence pn→ p under topology τ
Y‡= Yes, and moreover, A1 is a necessary/sufficient condition of pn→ p under o
Y **=Yes, when is still adopted as individual preference topology
Let p∈ Poand be the binary relation defined by p. If specification is necessary, we write “x y in p”, otherwise “x y” is written for simplicity. By “ x is preferred to y (in p)”, we mean x y but not y x, and such a case is denoted by “x y ”. On the other hand, when x y and y x, x is called indifferent to y (in p), and is denoted by “x ∼ y. Given U , V subsets of X , if∀ x ∈ U, y ∈ V , x y, then we write
U V. Also by x V , we mean x y, ∀ y ∈ V .
It is evident that both of the binary relations and ∼ are also transitive. Furthermore, we see
x y and y ∼ z ⇒ x z and
x ∼ y and y z ⇒ x z.
A preference order p on X can also be regarded as a subset G of X× X, where
(x, y) ∈ G iff x y, such that the corresponding conditions of completeness and
Definition 2.1.1 For a topological space X , a preference order p respects the topology
of X if the corresponding graph G is closed in X× X with product topology. A preference order p respecting the topology of X is also called “a continuous preference on X ” in economics terminology,19 which is defined by the equivalent
condition that for two sequences xn, ynwith xn → x, yn → y in X, xn yn in p implies x y in p.
The totality of such preference orders respecting the topology of X is called the continuous preference space on X , and denoted by P(X). For a given p ∈ P(X) and x∈ X ,we call
Rx(p) := {y ∈ X; y x in p} by the preferred set of p at x, and
Qx(p) := {y ∈ X; y ≺ x in p} by the inferior set of p at x. Clearly, Rx(p) and Q
x(p) are open in X. Similarly,we call
Rx(p) := y∈ X; y x in p by the weak preferred set of p at x, and
Qx(p) := {y ∈ X; y x in p}
by the weak inferior set of p at x, both of which are evidently closed in X. We also define the indifference set by
Ix(p) := {y ∈ X; y ∼ x in p} = Rx(p) ∩ Qx(p),
which is evidently closed in X . When there arises no confusion, we may simplify the notations by Rx, Qx, Rx, Qxand Ix respectively.
By the notation “:= or “≡”, we mean “denoting” or “being defined by”. This convention will be repeatedly used in the paper. In addition, we introduce several topological notations. Given a set A in X , the closure A of A is defined by the minimal closed set containing A, while the interior I nt A of A is the maximal open set contained in A.
We remark that Rx ⊂ Rx and R
x ⊂ I nt Rx, but it may happen that Rx = Rx or Rx = I nt Rx, unless p is “regular” relative to the topological structure (see Sect.1.4 for definition). The boundary∂ A of A is the difference A − I nt A. It is clear that
∂ A consists of the points, of which each neighborhood intersects both A and X − A.
Therefore A= ∂ A ∪ I nt A. Also notice that for a topological singular preference p in P(X), it is false that ∂ Rx = Rx− Rxat singular points x ∈ (p)( see also Sect.1.4). The singular sets play a significant role in our framework.
The term“preference” is used as a general term to mean a preference order in Po, or a continuous preference order in P(X) , if there arises no confusion.
2.2 Graphs and continuity
Consider a preference p in P(X) on X . The corresponding graph G is closed in X× X. Given a set S in X × X , let Sx and Tx denote the subsets
Sx ≡ {(x, y) ∈ S; y ∈ X} Tx ≡ {(y, x) ∈ S; y ∈ X} .
In general, if Sxand Tx are closed in X ,∀ x ∈ X , the set S is not necessarily closed in X× X. However, for the graph G corresponding to a preference order p ∈ Po, we claim that if Rx and Qx are closed in X ,∀ x ∈ X, then G is closed in X × X. The claim would help us to ensure that we have used the full strength of the assumption of p respecting the topology of X, whenever we apply to p the condition that Rx and Qx are closed in X ,∀ x ∈ X. Namely, we do not need to check whether G itself is closed in X× X, which usually is complicate.
Proposition 2.2.1 Let p∈ Po. Then p ∈ P(X) (i.e., p is continuous) if and only if
for each x ∈ X, Rxand Qx are closed in X. In other words, the graph G≡ (x, y) ∈ X × X; x y in p
is closed in X× X, if and only if
Rx ≡ y∈ X; y x in p , Qx ≡ {y ∈ X; y x in p} . are closed in X, ∀x ∈ X.
Proof The proof of the necessity is evident. We show the sufficiency in the following steps.
Step 1 For K a compact set in X , we first claim∃ xo ∈ K such that xo K . Suppose there exists no such xo in K , then given x ∈ K, we have yx ∈ K with yx x. By Ryx closed in X , Q
yx ≡ {z ∈ X; yx z} is open in X. There exists an
open neighborhood Uxof x such that Ux ⊂ Qyx, i.e., yx Ux. Let Ux1, . . . , UxN
be a finite subcover of{Ux; x ∈ K }. Let yo∈ K be an element in yx1, . . . , yxN such that yo yxi, ∀ i = 1, 2, . . . , N. Then yo Uxi ∀ i = 1, 2, . . . , N. But Uxi; i = 1, 2, . . . , N covers K . We have yo K , contradicting to yo∈ K .
Step 2 Let Gc be the complement X × X − G. We claim that Gc is open in X× X, i.e., given (x, y) ∈ Gc,∃ neighborhoods V , W of x, y respectively, such that V × W ⊂ Gc. First, we remark that(x, y) ∈ Gciff x ≺ y in p. Since Ry is closed in X , Qyis open, there exists a neighborhood U of x such that U ⊂ Q
y. Let V be a compact neighborhood V of x with V ⊂ U.20 By Step 1,∃ z ∈ V such that
z V.
On the other hand, Qzis closed in X . Hence Rz is open. But y U, z ∈ V ⊂ U, we have y z, i.e., y ∈ Rz. There exists a neighborhood W of y such that W ⊂ R
z, i.e., W z.
The transitivity of p implies that W V , i.e., V× W ⊂ Gc.
The proof is completed.
Proposition 2.2.2 Given p∈ P(X) and two points x, y in X with x ≺ y in p, there
exist neighborhoods U, V of x, y respectively such that U ≺ V in p.
As X is locally compact, we may require U and V compact in the last statement. Proof Since(x, y) ∈ Gc, there exists a neighborhood of (x, y) in X × X which lies in Gc. By the definition of product topology of X × X, the neighborhood contains U × V where U and V are neighborhoods of x and y in X respectively. Thus U and V
are as required.
Proposition 2.2.3 Given p∈ P(X) and x, y ∈ X. Suppose xn→ x and yn→ y in
X with xn ynin p,∀ n. Then x y in p.
Proof It follows directly from the closeness of the graph G.
Proposition 2.2.4 Given p ∈ P(X). Let K be a compact set in X, then the sets
defined by
RK(p) ≡ {x ∈ X; K ≺ x} QK(p) ≡ {x ∈ X; x ≺ K } are open in X.
Remark The proposition2.2.4provides us a convenient technique that we shall use often: Given K compact in X , x /∈ K, there exists a neighborhood V of x, such that K ≺ V. Also, we may assume V a compact neighborhood of x, as long as X is locally compact.
Proof In Step 1 of Proposition2.2.1, we have shown the existence of a “maximal” element xoof K , i.e., xo K in p. The set Rxo(p) is known open. Now RK(p) = Rxo(p). Hence RK(p) is open, as claimed. Similarly, QK(p) is open. 2.3 Stratification structure and Examples
A complete binary relation p defined on an alternative space X is a relation between any pair (x, y) in X which is independent of the relation between the pairs other than(x, y). But when the complete binary relation p also satisfies the transitivity, p stratifies the space into indifference sets with a linear order. Namely, a preference order on X defines a stratification of X.
The converse is also true, i.e., a stratification of X corresponds uniquely a preference order on X . The equivalence between a stratification and a preference order is valid, even when X is a topological space. Of course, as the topology of X is now concerned, the stratification structure should also respect the topology of X, in the sense that the indifference sets are closed in X. We define an abstract stratification structure as follows.
Definition 2.3.1 Given a topological alternative space X , a stratificationσ is a
decom-position of X into non-empty closed subsets Iα, α ∈ A, associated with an injective assignment of a non-empty subindex set Aαof A to eachα ∈ A, where A is an index set, such that
(S1) X = ∪
α∈AIα, and Iα∩ Iβ = φ, ∀ α, β ∈ A with α = β, (Decomposition)
(S2) Rαand Qαare closed in X , where (Topological coherence) Rα ≡ ∪
β∈AαIβ, Qα ≡ (X − Rα) ∪ Iα,
(S3) Aα∩ Aβ = Aα or Aβ, (Linear ordering of inclusion)
(S4) α ∈ Aα, ∀ α ∈ A ; Aβ ⊂ Aα, whenever β ∈ Aα. (Minimality ofα in Aα) We denote the stratificationσ by (X, Iα; A, Aα).
We give some examples to illustrate certain notions previously defined.
Example 11 Let X be the real plane with coordinate(x1, x2) and let λ ≡ (λ1, λ2) be a given unit vector in X . We define a linear preference p on X by
x ≡ (x1, x2) y ≡ (y1, y2) in p iff
whereλ, x ≡ λ1x1+ λ2x2is the inner product ofλ and x. Then the indifference sets are the straight lines:
λ, x = α; α ∈ A, where A ≡ (−∞,∞), the index set. Given a point x= (x1, x2) in X, we have
Ix = {y ∈ X; λ, y = λ, x} Rx = {y ∈ X; λ, y ≥ λ, x} Qx = {y ∈ X; λ, y ≤ λ, x}
where Rx and Qx are closed (Ix = Rx ∩ Qx is consequently closed). Therefore p is continuous, i.e., it respects the topology of X and hence p ∈ P(X). On the other hand, for eachα ∈ R ≡ A, let
Aα = [α, ∞) ⊂ R,
α = λ, x and
Iα = {y ∈ X; λ, y = α} Rα = {y ∈ X; λ, y ≥ α} Qα = {y ∈ X; λ, y ≤ α} ,
then Ix = Iα, Rx = Rα and Qx = Qα. Clearly,σ ≡ (X, Iα; A, Aα) is the stratifi-cation corresponding to p.
Example 12 Let X = [a, b] × [−1, 1],both a and b are < −1 . We define a binary relation p by x ≡ (x1, x2) y ≡ (y1, y2) in p, iff (i) x1≤ y1, when x2= y2 = 0, and (ii)|x2| ≤ |y2|, otherwise. Then X is stratified into the points on x1-axis and the horizontal line segments in X (except the segment in x1-axis). The index set A = [a, b] ∪ (−1, 1]. For α ∈ [a, b], Aα = [α, b] ∪ (−1, 1]. For α ∈ (−1, 1], Aα = (α, 1]. This is not a stratification respecting the topology of X as defined in Definition2.3.1, because (S2) is not satisfied. In fact, Rα is not closed in X for
α ∈ (a, b], although ∀α ∈ A, Qαis. Furthermore, we see that p is a preference order, but p is not continuous, i.e., p∈ Po− P(X), This example shows that Po P(X) Example 13 Let X be the torus S1× S1where S1 denotes a unit circle. Given the product coordinate(θ, ϕ) for each point x of X such that
X = {(θ, ϕ); 0 ≤ θ < 2π, 0 ≤ ϕ < 2π} . Define a preference order p on X by
