Topological methods in social choice: an overview

Topological methods in social choice: an overviewAuthor(s): Paras MehtaSource: Social Choice and Welfare, Vol. 14, No. 2 (1997), pp. 233-243Published by: SpringerStable URL: http://www.jstor.org/stable/41106206 .

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Soc Choice Welfare (1997) 14: 233-243 ^ TTTiü ¡ Sodai Choice © Springer- Verlag 1 997

Topological methods in social choice: an overview Paras Mehta

Balliol College, Oxford OXl 3BJ, United Kingdom

Received: 30 December 1994/Accepted: 22 April 1996

Abstract. This paper gives an exposition of the topological framework for social choice theory developed by Chichilnisky, and reviews the mathematical concepts needed for understanding this framework. Within that context, this paper also discusses some classic results of Chichilnisky.

I. Introduction

In Arrow's framework for social choice theory, a society of k individuals seeks to decide among a set X of alternatives, where X is simply a listing of the discrete alternatives with no additional mathematical structure. Each individual, then, has a preference over X, consisting of a complete, reflexive, and transitive binary relation over X> and a social choice function gives a rule for aggregating any particular profile of individual preferences into a statement of group preference.

In this paper, however, we adopt Chichilnisky's topological framework for social choice, which is motivated by the concept of economic decision making taking place over a commodity space such as Euclidean Rn, rather than a discrete set of alternatives. In this framework, the basic social choice scenario is formulated as follows:

We have a society of k > 2 individuals who are faced with deciding among a set of alternatives in the choice space (or commodity space) X = Un, n > 2. Associated with each individual is a preference p over X9 where p is a codimen- sion one, oriented foliation of X, so that the leaves of p define the indifference hypersurfaces of the preference, and the orientation indicates direction of increasing preference. Alternatively, we can view p as a normalized (to unit Euclidean length) vector field over X, where for each xeX, p(x) is the unit vector normal to the leaf of the foliation passing through x, and pointing in the direction given by the orientation of the foliation. At each xeX, then, p(x) is the vector normal to the indifference hypersurface of the preference, and



234 P. Mehta

indicating the direction of increasing preference in the choice space. We will further require that the preferences be continuously differentiable and locally integrable, so that the space of all possible preferences P is simply the space of all continuously differentiable, locally integrable unit-normalized vector fields over X.

A social choice function, then, is a map f:Pk -+ P that associates to each /c-tuple of preferences p = (p,, ... ,pk), representing individually the k members of society, an aggregate preference /(p) representative of the entire society.

In this paper, we begin by discussing some of the tools of algebraic topology, and homotopy theory in particular, used in topological social choice theory. Then, we review some classical results of Chichilnisky inves- tigating the possibility of requiring social choice functions to satisfy various conditions of ethics and rationality.

II. Results from algebraic topology

To each topological space Y (which we will assume for simplicity is path connected) we can associate homotopy groups n¡(Y' where the i-dimensional homotopy group n¡(Y) "counts" the number of i-dimensional "spherical holes" in Y. Consider, for instance, the first homotopy group 7^(7). We fix a point y0 in Y and look at paths in y which begin and terminate at y0. Then, we consider the equivalence classes of paths under the relation which says that for two paths y{ and y2, yi ~ y2 if 7i can be continuously deformed into y2. Formally, a path y is a continuous map y: [0, 1] -► Y with y(0) = y(') = y0. Then yi ~ y2 if there exists a continuous map G:[0, 1] x [0, 1] -> Y with G(x, 0) = )>i(x) and G(x, 1) = y2(x). The set of paths in Y modulo this equivalence relation defines n^Y), with the group operation being composition of paths; i.e., for paths yu y2, yx °y2 will be that path obtained by traversing yt first and then y2.

For example, in R2, any path can be continuously deformed into the constant path which maps [0, 1] into the single point y0. Hence, all paths are equivalent, and nx (R2) is trivial. Now puncture the plane at the origin, so that y = R2'{0}. If we consider a path y which traverses a circle around the origin once in a clockwise direction, we can see that this path cannot possibly be deformed into the constant path because we cannot collapse the circle into a single point without passing through the origin. Similarly, a circular path that passes around the origin twice counterclockwise cannot be continuously deformed into a path passing around the origin only once, nor can it be continuously deformed into a path passing around the origin twice clockwise. In fact, a path in Y can be identified, up to the homotopy equivalence relation, by the number of times it passes around the origin, and the direction in which it does so. Hence, n^Y ) is simply the infinite cyclic group of integers Z under addition. Note that nl here has one generator, and y has precisely one 1 -dimensional hole.

For our purposes, it will be useful to use a different definition of paths: a path y can be seen as a continuous map y.S1 ->Y, where S1 is the one- dimensional unit sphere (i.e., the unit circle). Using this definition and the appropriate homotopy equivalence relation (yi ~ y2 if there exists a continuous deformation G: S1 x [0, 1] -> y with G(x, 0) = y^x) and G(x, 1) = y2(x)),



Topological methods in social choice 235

we obtain the same homotopy group Ui(Y), Then, we define the higher homotopy groups in an analogous fashion: n¡(Y) consists of equivalence classes of paths y : S1

f -► Y where two paths are equivalent if they can be continuously deformed into one another. And, in analogous fashion, these will measure the number of higher dimensional holes in Y.

In this paper we need only concern ourselves with the homotopy groups of i-dimensional spheres S1 and of Cartesian products of; ¿-dimensional spheres (Siy. We will use the fact that n^S') = Z. This makes some intuitive sense: S* has one i-dimensional hole, so we would expect 7i,(Sl) to be the group generated on one generator. Furthermore, from a more general result which states that the homotopy group of a Cartesian product of path connected spaces is the direct sum of the homotopy groups of the individual factor spaces, we have that n,-((Sl')/') = ©/UiTC,(S") = ©í=iZ. Essentially, this says that elements of the homotopy group of a product of spheres can be identified uniquely by the action of the coordinate projection maps.

Now if we have a continuous map m:X -► y, this induces a map m* :Ui(X) -► 71,(7) on the homotopy groups which maps 0L€n¡(X) to ßen^Y) as follows: If /: S' -> X is a path contained in the class of paths a, then ß is the class of paths containing m °f. In fact, the unique map m* is a homomorphism; i.e., it commutes with the composition of group elements.

As we noted above, 7i,(S') = Z (under addition), so any map m : Si f -> S1 between spheres defines a homomorphism of the integers m* : Z -> Z. Under these circumstances, we say that m*(l) is the degree of the map m, which we shall denote d(m). Intuitively, the degree tells us how many times the sphere in the domain is "wrapped around" the sphere in the range. For example, consider a map m : S 1 -> S l. If m is the identity map, its degree is 1 since the domain sphere is wrapped around the range sphere precisely once. Now we think of S1 as the unit circle in U2 and let m be the map that takes the vector forming angle 6 with the positive horizontal axis to the vector forming angle 2n - 8. In this case, the domain sphere is wrapped once around the range sphere, but the orientation is reversed; i.e., as we traverse the domain sphere counterclockwise, we traverse the range sphere clockwise. Hence, the degree of this map is - 1, negative because it is orientation reversing. Another way of thinking about the degree, given the above notion of wrapping the domain sphere around the range sphere, is that the degree counts the number of points in the preimage of a "suitably nice" point in the range, taking into account whether the map preserves or reverses orientation.

There are a few facts we will want to invoke about the degree of maps. First, given maps m1,m2: S' -* S' if mj is homotopic to m2 (i.e., if ml can be continuously deformed into m2 via a homotopy G:Si x [0, 1] -* S* with G(x, 0) = mi(x), G(x, 1) = m2(x)), then the degree of mx and m2 are equal. That is to say, the degree is invariant among maps in the same homotopy class, and in fact, the degree defines a homomorphism of 7i,(S') into Z. Second, we note that any nonsurjective continuous map on spheres is of degree zero.

III. An induced map on spheres

We now return to our social choice function /: Pk -* P, for aggregating the preferences of k > 2 individuals over the choice space X = R", n > 2. In this



236 P. Mehta

section, we construct a map (/>:(Sn~1f ̂ S"'1 induced by the social choice function f:Pk^>P and examine the degree of <f> when restricted to subspaces homeomorphic to a sphere. The following construction and argument is due to Chichilnisky (e.g., in [3], where it is used to prove the non- existence of continuous, majority-type rules).

For any veSn~' we denote by kv the linear preference given by the direction v; i.e., kv is the constant vector field with kv(x) = v for all xeX (alternatively, the linear preference kv is the foliation whose leaves are the hyperplanes with normal v and whose orientation is given by the direction of v). Then, we define the map k : Stt~ 1 -> P which continuously embeds S"~ l

in P by mapping veS"'1 to the linear preference kv. Finally, we fix a point yeX and define <t>:(Sn'lf -> Sn~l by

4>:(vl9 .-.,vk)y-+[f(kVi, ...,>U)]()>). One can interpret <f> as acting, in accordance with /, on the linear preferences given by vÌ9 ... 9vk9 and then approximating the preference f(kVi, ... ,kVk) by a linear preference at the point y. Alternatively, one can simply view <t> itself as a social choice function on the subspace of linear preferences. In any case, <j> is continuous by the continuity of A, of/ and of preferences f(kVl, ... , kVh) over X.

Next, fix a point voeStt~l and consider the following subspaces of {Sn xf. Denote by D the "diagonal" of (Sn

" * )* given by D = { (v9 v, . . . , v9 v) : v e Sn ' ' }, and by £¿ the "¿th edge" of (Stt~l)k given by E{ ■ = {(vu ... 9vh ...9vk):Vj = v0 for j ^ i, and têS"""1} (in the case n = %k = 2, (Sn~1)k = S1 xS1 is a torus and can be depicted as a square with edges identified, and then D and EUE2 are the diagonal and edges of the square, respectively). Finally, we define the following canonical inclusion and projection maps:

TD'S»-l^{Sn-l)' xD:v^{v9 ...,i7)eD,

xi:Sn~l-^(SH~í)' Tí:v'->{v0, ... 9vo,vhvO9 ... 9v0)eEhVi = v9

gr.iS'-^^S"-1, Qt:(vu-,vk)»vh These maps induce the homomorphisms t£, xf9 and gf, respectively, on {n - l)th homotopy groups. The identity map on any domain will be denoted i.

Our next step is to investigate the degree of </> restricted to the subspaces D, Ef by examining the homotopy classes of </> restricted to those subspaces and invoking the homotopy invariance of the degree discussed in the last section. We make some preliminary observations prior to examining the degree of <f>'D and of </>|£i, denoted d{<f)'D) and d{<t>'E). First, D and each £, are homeomorphic to S""1, so it is reasonable to speak of the degree of these maps. Strictly speaking, however, references to d(<j)'D) and d{(j>'E) actually mean d(<t>°xD) and d(<t>°T¡' respectively, but we will feel free to use the two notations interchangeably. Finally, the remaining discussion in this section is based solely on the topology of (Sn~ l )k, with no particular properties of / or 0 (except, of course, continuity) invoked.

Maps enclosed in brackets ([ ]) denote members of (n - l)th homotopy groups; since the maps we will consider have domain Sn ' by our comments in Sect. II, we can consider these maps as representative of homotopy group elements. It will be clear from context whether the homotopy classes are elements of nn _ ! (Sn

" 1 ) or nn _ , ((Sn ' * )k).




Recalling that the maps x, are the canonical inclusions of Sn~l into EiCz(Stt~l)k, we next choose a map m.S"'1 -»(S"1"1)* such that l>] = l5«i M e ̂ ((S""1)*). Then, for all i, fo-m] = #([*]), by the definition of the induced homomorphism, and

Qf(im-]) = Q?( I [t,])= E ^([Tj) = [^°Tl]e7cn,1(S't-1). (1)

The penultimate equality follows from the fact that gf is a homomorphism. The last equality follows from the fact that £>/([t,]) = [£,oT/], and g¡°Xj is a constant mapping onto v0 for i ̂ j, so that [t,] is in the kernel of gf for i ̂ j. Moreover, since g> ° t^ = &• ° t,- = i : S"1

~ * -► S" ~ * (the identity map) for all i, we conclude that '_gi°TD'] = [g,0!,] = [i] for all i, and hence by Eq. (1) that for all i

[e,°m] = [ftoTi] = [ôTd] = [,] en^^S"-1).

But since [pf ° m] = '_g¡ ° t^] means precisely that £*([m]) = pfdjß]) (for all i), we conclude that [m] = [ix>], i.e., that m is homotopic to td. This follows as a consequence of the fact cited in Sect. II that elements of nn- '((Stt~ l)k) are uniquely determined by the action of the projection homomorphisms gf. The composition of homotopic maps with homotopic maps produces composites which are themselves homotopic, and hence

[tf>om] = [4>oTD]. (2)

Recalling that [m] = £Î= j [t;], we have that

[>°m] = tf>*([m]) = <t>*( ¿ [t;]) = ¿ 4>*{lij-]) = I [0°tj. (3)

From the homotopy invariance of degree, and the fact that the degree defines a homomorphism between nn- i(Sn~ l) and Z, Eqs. (2) and (3) give us the desired conclusion:

k

E d((j) ° xj) = d((¡> ° Tp), or equivalently, (4)

¿ d(0|£j) = d(0lx,). (5)

This line of argument, culminating in Eqs. (4) and (5), is not altogether suprising intuitively if we note that the diagonal D can be continuously deformed into the union of the edges 'Jkj=lEj. For example, consider the simplest case, when n = 2, k = 2. Then we have D, Eu E2^S1xS1. But S1 x S1 is just the torus, which we can consider as a square in the plane with sides identified; in this case, D is the diagonal of the square, and Ei, E2 perpendicular edges. Then we clearly see that by "stretching" the diagonal to one pair of perpendicular edges, we can continuously deform D into £iu£2- This gives some insight into why the degree of <j> restricted to the diagonal should be equal to the sum of the degrees of <j> restricted to each edge.



238 P. Mehta

IV. Selected topologica! results in social choice theory

This section is dedicated to presenting and proving three classic results, due to Chichilnisky, concerning the possibility of requiring a social choice function to satisfy certain selected criteria of ethics and rationality. In all three results, we will require that a social choice function f:Pk -+P be continuous. We can view continuity as a requirement of stability on the social choice functions, so that a suitably small change in individual preferences does not produce a wild change in the aggregate preference of the society.

A social choice function / is said to satisfy unanimity if and only if / satisfies an "identity" condition along the "diagonal" of Pk; formally, unanimity requires that for all preferences p e P, /(p, p, ... ,p, p) = p. Una- nimity merely ensures that the aggregation rule defined by / has some minimal relationship to the preferences on which it acts. An anonymous social choice function is invariant under permutation of the k entries in its argument; i.e.,

f(Pu •••>Pk)=f(P«l), "-,Pn(k)) for all permutations n and all profiles of preferences (pl9 ... ,pk). Anonymity offers a strong condition of egalitarianism in which the rule of preference aggregation takes note only of the preferences that are held by individuals in the society, and not of the identity of the particular person who holds a specific preference. The following impossibility result is due to Chichilnisky [1, 2, 4]: Theorem 1. No continuous social choice function f:Pk-+P(k> 2) satisfies the conditions of unanimity and anonymity.

Proof Suppose that a continuous social choice function / respecting unanimity and anonymity exists. Then consider the continuous mapping </> : (Sn~ l )k -► Sn~ 1 induced by f' <j> inherits respect of unanimity and anonymity from /:

<Kv9...,v)=f(XO9...,Xv)(y) = Xv(y) = v

and

=/(^, ~',K)(y) = <l>(vu •••>*>*) for all permutations n.

By Eq. (5), we have that

¿ d(<l>'Ej) = d(<t>'D). Now d(<¡)'D) = 1, since by unanimity <t> ° td is the identity, and the degree of the identity map is 1. And by anonymity, the degree on each edge is the same: d(<t>'Ei) = d(<j>'E) = X for some X e Z and for all ¿,;' But now we have that kX = 1 for some X s Z, which has no solutions in A over Z for integral k>2. □




One way of overcoming the impossibility result of Theorem 1 is to relax the conditions placed on / For example, we can certainly construct a continuous social choice function f:Pk^P that respects unanimity; simply pick a dictatorial rule. Say, for instance, that /:(pi, ... ,p*) h-> p'. Then this rule, which makes the first individual a dictator, satisfies both continuity and unanimity. We make one further observation about this social choice function. If we consider the induced map 0 : (Sn

" í )k -> Sn ~ 1 (which is also continuous and respects unanimity) at any point, then <j> restricted to Ex and D both "cover" Sn~~ l once, suggesting that both have degree one, while </> restricted to Ej, j '^ 1, is simply a constant map (and hence of degree zero). This is precisely the result we would expect given our study of the degree of (f> (in particular, Eq.(5)).

Producing a continuous, anonymous social choice function is a bit trickier. For ease of description, we restrict our attention to the case k - 2 and n = 2. Furthermore, we consider only the space of linear preferences PL = S1. Then a social choice function is a mapping on the torus /: Sl x Sl -* S1. If we consider S1 to be the unit circle centered about the origin in U2, then an element of S1 is given by an angle, and we can define / by angular addition. Then / is clearly anonymous and continuous, and as we would expect, / restricted to each edge covers the image Sl once, and / restricted to the diagonal covers the image twice.

Rather than weakening either anonymity or unanimity, another way of overcoming the impossibility result of Theorem 1 is to restrict the domain P of possible preferences. For example, Chichilnisky and Heal [5] show that a necessary and sufficient condition for the existence of continuous, anonymous, and unanimous social choice functions /: Pk -> P, for all k > 2, is that the space of preferences P be topologically contractible. In [6], Heal considers several examples of contractible preference spaces and interprets contractibility in terms of some sort of "limited agreement." For example, if everyone always agrees on some direction in the commodity space which is undesirable (if there exists some direction veSn~i such that for all preferences p e P, p(x) # i; for all x in the commodity space), then P is contractible.

In the next result, we consider continuous social choice functions that satisfy a stronger requirement than that of unanimity called the Pareto criterion. Of course, in light of Theorem 1, such a social choice function is definitely not anonymous. The Pareto criterion on f:Pk -> P simply requires that, for any /c-tuple of preferences (pl9 ... ,pk) e Pk, if x e X is preferred to y g X according to each pi9 1 < i < k, then x is preferred to y according to /(Pi, ••• ,PkY This definition naturally demands some explanation of what it means for one alternative x to be preferred to y according to a preference p. Since we have assumed our preferences to be locally integrable, we can view p as the gradient, locally, of a utility function, and define comparability, locally, in terms ofthat function. Specifically, suppose at each x e X there exist a neighborhood N of x and a utility function u:N -> IR such that p'N = Vu. Then for any y e N, we say x > p y if and only if m(x) > u(y).

We now note the following consequence of the Pareto criterion (as noted in [3, 4]): At any point x e X, if p = (pu ... ,pk) e Pk satisfies the condition that for some v, w e S""1, p,(x) = v or p,(x) = w for all i, then /(p)(x) is contained in the "cone" spanned by v and w. Specifically, suppose that we have a p such that, at x g AT, p,(x) = v or w for some v,w e S"'1 and for all /. We



240 P. Mehta

think of Sn~ 1 as an object centered at the origin in Euclidean R". Then, p¡(x) for each i is the normal to the surface formed by those points y e X such that preference p¿ is indifferent between choices x and y. Such an indifference surface separates (in some local neighborhood) X into two disjoint subsets of those choices y g X that are strictly preferred to x by ph and of those choices y e X to which x is strictly preferred by /?,. Hence, for all e > 0 sufficiently small, this means that

X + £-V >pi X

for all v g Sn" l satisfying <v, p,(x)> > 0, where < • , • > is the standard Euclid- ean inner product. Now we return to the case under consideration, in which each px f = v or w at x e X. The preceding discussion implies that for all e > 0 sufficiently small, x + £-v is preferred to x by all /?, for all v eS"1"1 having nonnegative inner product with both i; and w. By the Pareto criterion, then, /(p)(x) must have nonnegative inner product with all v g Sn~l having nonnegative inner product with v and w. In particular, this means precisely that /(p)(x) is contained in the closed cone generated by v and w. Moreover, this consequence implies respect of unanimity (just take v = w in the preceding argument).

We further note that this consequence, that /(p)(x) lies in the cone generated by v, w if p,(x) = v or w for all i, is strictly weaker than the Pareto criterion itself. In fact, it is the only consequence of the Pareto criterion that we will use in this section and the next. Therefore, in Theorems 2 and 3 below, we can replace the requirement that / satisfy the Pareto criterion with the requirement that / satisfy the general consequence of the Pareto criterion discussed above.

Finally, we define the nonnegatiue association axiom. Suppose that there exist a point x e X, a fc-tuple of preferences p = (pi, ... >p*)> and an index i, such that /(p)(x) = - p¡(x). Then / satisfies the nonnegative association axiom if and only if for any fc-tuple of preferences q = (qÍ9 ... ,qk) such that ^.(x) = - q¡(x) for all j ^ i, /(q)(x) ^ q.(x). Intuitively, the nonnegative association axiom states that whenever everyone in the society has the same preference locally at some alternative x e X, with the exception of a single individual who has the exact antipodal preference, the aggregate preference should not locally at x coincide with the single individual in the minority if there is even one fe-tuple of preferences which gives rise to an aggregate that, at x, is antipodal to that same single individual in the minority of one. The axiom can be seen as a very mild condition of rationality or consistency. Then we have the following characterization of Chichilnisky [4].

Theorem 2. Let f:Pk -+P be a continuous social choice function satisfying the Pareto criterion and the nonnegative association axiom. Then f is homotopic to a dictatorial social choice function.

As its name suggests, a dictatorial social choice function is simply one that projects p = (pi, ... ,Pk) onto its ith coordinate for a fixed i.

Proof The main strategy behind this proof is to prove that under the hy- potheses of the theorem, there is some coordinate i such that /(p) is never equal to - ph and then to conclude that / is homotopic to the map /which projects the profile (pl5 ... ,pk) onto p¡.




We fix a point y e X, a direction v0 e S"1*"1, and consider the induced map <j>' (Sn~x)k^Sn^1 and the subspaces D and £,. By Eq.(5), Z*=i¿(0Ui) = d((f>'D). By our previous discussion, respect of the Pareto criterion implies that /, and hence <£, respects unanimity. So, <j> ° xD is the identity onS"-1, andd(0|D)= 1.

Now we consider d(<t>'E). The point <fr{v0, ... ,Uo, ty» ̂o, ••• ,v0) must be contained in the cone spanned by v0 and v¡ by the Pareto criterion. But this, together with continuity, implies that <t>(vOi . . . , v0, - v0, v0, . . . , v0) e W, - v0}. A priori, according to Pareto, <j)(vOi ... ,v0, - v0, ... ,Uo) can be any dgS""1, since the cone generated by vOi - v0 consists of all of S""1. However, suppose that <t>(vOi ... ,t?o, - v09v0, •• - ,v0) = w${vo, - v0}. Then consider a sequence {vj}JL i converging to - v0 and such that the closed cone generataed by v0 and v j does not contain w for any j (this is possible because the cone generated by two points on the sphere is contained within a hemi- sphere unless those two points are antipodal). Then lim;_»oot;J = - u0, but limôoô, ... ,vOivj,vO9 ... ,v0) ¥" w, in violation of continuity. So we conclude that <l>(vOf ... ,u0, -t>o> v09 ...,vo)e{vo,-vo}.

Now if Vi^-vo, then <l>{v09 ... 9vO9vi9vO9 .- ,v0) *£ -v0, since for vi "£ - v0, the cone generated by t;0, v¡ is contained in Sn~ x'{ - i;0}. Hence, if </>(uo, .-.,«0,-^0^0. ••.,ô) = ô, then (f>'Ê(l(- v0) = 0, ¿Ie, is not surjective, and hence its degree is zero. On the other hand, if 0(i;o, •-., Vo,-vo,vOi ...9v0)= -v0, then 0|¿1(-i;o) = {(t;0, ... ,vo,-vo,vo, ...,t>0)} andd(</>|£j)=l.

By Eq. (5), and since d(<t>D) = 1, there exists an index i* such tht

,an fl if and only if i = i*,

[ 0 if and only if i ̂ i*,

or, equivalently,

ai-i/ ^_ J^0' •••»»o,üí»,üo, ..ô), u,*= ~ô if and only if i = i*, 0k ai-i/

(-»o)-|0 ^_

if and only if i #i*.

(7)

Moreover, this implies that

^.,£,(^1» •••)^)^ -Vi*. (8)

To understand why (8) must be the case, consider first the edge £, when iî*. Then for all (vl9 ... , vk) e Eh - v(* = - i?0; but by (7), for iî*, <f)'E.(E¡)n{- v0} = 0, or in other words, - ü/.£0|e.(ííí) when i ^ i*. Next we consider the edge £, when i = i*. In this case vr is allowed to vary over Sn~' so we have two possibilities. First, when tv ^ - vOi then Îe.'ô» •••,w«*, ...ô)^ -Vi* since - tv is not in the cone spanned by v0 and Vi* as long as t;f» # - Uo- Second, when v{* = - v0, then by Eq. (7), 4>'eì(vq9 ... , i7o,Vi», iô, .. >tfo) = Vi*. So we conclude (8).

But now we have, following the above reasoning, that for any v'o e Sn~ ' if we define E' = {(vu ... 9vk) e {S^îvj = v'o for j ï i, and v¡ ¡g S"'1}, there is an index j* satisfying

4>^mxE[(vu ...9vk)* -Vj*, (9)



242 P. Mehta

analogous to Eq. (8) above. Moreover, by continuity considerations, we must have that j* - z*, for if j* # i*9 we would have

= 17,*

4>(vo* ~-,v0, -vOyvo, ...,fo)= -Vq,

4>(v'o, ».,vro, -Vo9v'o, ...9v'o) = v'o,

in violation of continuity (the first equality follows from Eq. (7); the second follows from continuity, Pareto, and Eq. (9)).

Furthermore, given any v = (vu ... ,vk) e (S""1)*, if </>(v) = - i;,*, then by the nonnegative association axiom, (f>(- v¡*, ... , - v¡*, v¡*, - i>, ... , - v¡0) = - v¡., which contradicts the property of i* established in (8) and (9). Hence, we must have that <£(v) # - iv for all v g (S""1)*.

So far we have been considering a map </> induced at a point y g X. The above discussion implies that for any / e X, there exists a map </>' analogous to (/>, and an index ï analogous to i*. Specifically, we have that <t>'(vu ... , vk) # - Vi'. Again, however, continuity considerations require that Ï = i*, for if not, for some we?"1:

= v*

4>(w, ...,w, - w, w, ...,w)= - w,

0'(w, ...,w, - w, w, ... ,w) = w,

or alternatively, recalling the construction of 0, <t>' and the use of X to denote linear preferences,

/(¿w, ... ,¿M,, - Aw, XW9 ... ,iiv)(y) = - w,

violating the continuity of the preference that is the image of/ All of this leads to the conclusion that there exists an index i* such that

f(Pi> -•• >Pk)(y) 7e - Pi*(y)for all fc-tuples of preferences pe Pk and all y e X. In other words, if we denote by /the dictatorial rule f(p' , . . . , p*) = p,*, then we have /(p) / - /(p) for all p g P' But now we have that / is homotopic to /by the homotopy F:Pkx [0, 1] -> P:

F(M)O0- ^-^^ + t{^, □ B(i-t)/(p)(y) + i/(p)(j')ll

In Theorem 3, we present the final characterization of social choice functions in this paper, again using the construction of Sect. III. The social choice functions under consideration in Theorem 3 are continuous and satisfy the Pareto criterion. We also introduce a new condition called the decisive majority axiom. Essentially, the decisive majority axiom requires that, whenever a fc-tuple of preferences p = (pu ... ,pk) can be partitioned locally at a point xG X into two classes that hold completely opposite preferences from one another, then the aggregate /(p) locally at x will agree with the majority. Formally, for all xgX and for all p = (Pi, ...,pk)GP' if there exist a




number r and a direction veSn~l such that # {p,:p,(x) = v} = r and # { Pi * P«M = - v} = k - r, and if r > fc - r, then /(p)(x) = t>.

We make some observations about the decisive majority axiom. First, it is strictly weaker than a majority rule requirement. Second, we will not, in fact, use the full power of the decisive majority axiom. We will only be interested in the axiom for the case r = k - 1; in fact, we can replace the decisive majority axiom in the statement of Theorem 3 with the requirement that the decisive majority axiom need hold only in the case r = k - 1. We then have the following [3]. Theorem 3. No continuous social choice function satisfies the Pareto criterion and the decisive majority axiom.

In particular, since the decisive majority axiom is implied by a majority rule requirement, Theorem 3 immediately implies that there is no continuous, Pareto, majority rule social choice function. This consequence should not be surprising: in a majority rule decision-making process, a single change in preferences can lead to a large change in outcome, at odds with the demands of continuity.

Proof. Suppose f:Pk -> P is continuous and satisfies the Pareto criterion and decisive majority axiom. Then fix some t;0 e Sn~' y e X, and consider the induced map <t> and subspaces D, E¡. By Eq. (5), d(<¡>'D) = £*=1 d(<t)'Ej). Since the Pareto criterion implies unanimity, d(<j>'D) = 1.

Now consider d{<j>'E). By Pareto, 4>'El(vo, ••• ,vOy vh v0, ... ,v0) = w must be in the cone generated by v0, v¡. In particular, w ̂ - v0 except possibly when Vi = - Vq. But (¡>'Ei(vOi ... , v0, - v0, v0, ...,v0) = - v0 would violate the decisive majority axiom. So we conclude that image(<£|£<) Ç: Sn~l'{- v0}. Then (f>'E. is not surjective, and hence d(<i>'E) = 0 for all i. The last statement gives the desired contradiction:

1=<WId)= Z d(^k) = 0. D 7=1

Acknowledgements. The author wishes to thank Daniel Goroff, Gracida Chichilnisky, Geoffrey Heal, and Amartya Sen for their assistance in this study of social choice theory.

References

1. Chichilnisky G (1979) On fixed point theorems and social choice paradoxes. Econ Lett 3: 347-351

2. Chichilnisky G (1980) Social choice and the topology of the spacse of preferences. AdvMath37: 165-176

3. Chichilnisky G (1982) Structural instability of decisive majority rules. J Math Econ 9: 207-221

4. Chichilnisky G (1983) Social choice and game theory: recent results with a topological approach. In: Pattanaik PK, Salles M (eds) Social Choice and Welfare. North- Holland, Amsterdam

5. Chichilnisky G, Heal G (1983) Necessary and sufficient conditions for a resolution of the social choice paradox. J Econ Theory 31: 68-87

6. Heal G (1983) Contractibility and public decision-making. In: Pattanaik PK, Salles M (eds) Social Choice and Welfare. North-Holland, Amsterdam



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Topological methods in social choice: an overview