1

Notes on Continuous Social Choice

1. Introduction to topological social choice

2 Harmless homotopic dictators

2

An Introduction to Homotopy Theory and Topological Social Choice.

by

Nick Baigent* (February 25, 2006).

1. Consider 2 linear preferences and a given commodity bundle with 2 commodities, as in the diagram. At this bundle, place two vectors (arrows), each of length 1, and each perpendicular to one of the 2 indifference curves of the linear preferences. Each vector determines a point on a circle of radius 1. Now, re-centre this circle at the Euclidean origin. From this construction, the set of all linear preferences may either be regarded as the set of all unit vectors or as the set

{ }1 2 2 2( , ) : 1S x y x y= ∈ + =ℝ of points on the unit circle. A point v in 1S is

given by its anti clockwise distance from the point (0,1) and this point will also be written 0v . If notation is abused by treating v as a vector with Cartesian

coordinates, this will be clear from the context.

2. { : 2 , }v x x k v kπ= ∈ = + ∈ℝ ℝ ℤ for all 1v S∈ are the equivalence classes of real

numbers induced by points in 1S where ℤ is the set of integers.

3. Given 2 agents, a social welfare function is a function 1 1 1:f S S S× → , assumed to be continuous.

3

4. In view of 1, 1S may be represented by [0,2 ]π with the end points, 0 and 2π ,

identified in the sense that the end points, 0 and 2π correspond to the same 0v in

point in 1S . Similarly, 1 1S S× may be represented by [0,2 ] [0,2 ]π π× with all 4 corners identified and points on opposite sides identified.

5. A loop in a set X is a continuous function : [0,2 ] Xα π → such that (0) (2 )α α π=

or, equivalently, 1: S Xα → .

6. For two loops α and β in X, the product of α and β is the loop α β⋅ in X such that for all [0, ]v π∈ , ( ) (2 )v vα β α⋅ = , and for all [ ,2 ]v π π∈ ,

( ) (2( ))v vα β β π⋅ = − .

7. The following loops in 1 1S S× play a key role in continuous social choice: 1 1 1: : ( , )T S S S v v v∆ → × → ; 1 1 1

1 0: : ( , )T S S S v v vλ → × → ; 1 1 1

2 0: : ( , )T S S S v v vλ → × → ; and

1 2T Tλ λ⋅ . (T is used because 1 1S S× is known a the torus.)

The image of these loops in [0,2 ] [0,2 ]π π× may be given respectively by the diagonal, a horizontal edge, a vertical edge, and by a combination of horizontal vertical edges. In the diagram of point 4 above, these loops are respectively, D, 1E ,

2E and 1 2&E E .

8. Continuous functions , :g g X Y′ → are homotopic if and only if there exists a continuous function : [0,1]h X Y× → such that, for all x X∈ , ( ,0) ( )h x g x= and

( ,1) ( )h x g x′= . This is written, g g′≃ .

2π

2π

E2

E1

D

4

9. 1 2T T Tλ λ∆ ⋅≃ . The images of two such homotopies, one polynomial and the other

piece-wise linear are given below.

5

10. The following loops in 1S play a key role in continuous social choice: Tf∆ = ∆� ,

1 1Tfλ λ= � and 2 2

Tfλ λ= � .

11. 1 2λ λ∆ ⋅≃ , since homotopies are preserved by composition.

12. Since, from 2, vℝ for all 1v S∈ are the equivalence classes of real numbers

induced by points in 1S from 2, a loop 1: [0,2 ] Sα π → in 1S may be represented

by the unique function : [0,2 ]α π →ɶ ℝ such that min vα α+= ɶℝ � . Thus, α may be

illustrated in a diagram of αɶ . Note, (0) (2 )α α π=ɶ ɶ since the points 0 and 2π in [0,2 ]π are identified. See the diagram.

0 2π

2π

D

E1

E2

0 2π

2π

D

E1

E2

6

13. For any loop α in 1S , the degree of α is the integer deg( ) (2 ) (2 )α α π π= ɶ . It

gives the net number of times α wraps around 1S . In the diagram 1α begins by

going 4 times anti-clockwise around 1S , then twice clockwise around 1S , then twice anti-clockwise around 1S and finally once clockwise around 1S . The degree of 1α , equal to the net number of times it winds around 1S is therefore equal to 4 –

2 + 2 – 1 = 3.

14. For all loops ,α β in 1S : deg( ) deg( )α β α β⇔ =≃ . Thus in the diagram in 12,

1 2α α≃ and 1 2β β≃ are the only pairs of loops that, having the same degree, are

homotopic.

15. 1 2 1 2deg( ) deg( ) deg( )λ λ λ λ⋅ = + . This is intuitively plausible and is clear if the

product of two loops are drawn on a diagram like the one in 13.

16. 1 2deg( ) deg( )λ λ∆ = ⋅ from 11 and 14.

2π

8π

6π

4π

2π

0

−2π

−4π

−6π

−8π

˜ α 1

˜ α 2

˜ β 1

˜ β 2

˜ γ

7

17. 1 2deg( ) deg( ) deg( )λ λ∆ = + from 15 and 16.

This is the fundamental equation of topological social choice theory. It follows from continuity alone. All of the major results are obtained by deriving restrictions on it from properties imposed on 1 1 1:f S S S× → as in points 18, 19 and 20.

18. Unanimity: ( ( , )f v v v= for all 1v S∈ ) implies deg( ) 1∆ = .

19. Anonymity: ( ( , ) ( , )f v v f v v′ ′= for all 1,v v S′∈ ) implies 1 2deg( ) deg( ) kλ λ= = .

20. The equation in 17 then implies 1 2k= which is a contradiction since it does not hold for any integer k. Thus, there is no Continuous, Unanimous, Anonymous function 1 1 1:f S S S× → . This is Chichilnisky’s impossibility theorem. This result is easily extended to take account of some non linear preferences. If a Continuous function with appropriately defined Unanimity and Anonymity properties exists since such non linear preferences may be approximated by linear preferences.

21. Domain restriction question: What points on the circle (linear preferences) may be deleted so that continuous, Unanimous and Anonymous functions may be constructed? Answer: any single point.

22. Consider the punctured circle 1 { *}S v− , 1*v S∈ and reposition it so that the deleted

point is at the Euclidean origin. For all pairs of points 1, { *}v v S v′∈ − , the mean of

v and v′ is given by: 1

( , ) ( )2

v v v vµ ′ ′= + , where v and v′ are viewed as vectors.

While 1 { *}S v− is not convex, all mean values lie in a convex set, the disk

{ }2 2 2 21 2 1:D x x x ≤= ∈ +ℝ that has 1S as its boundary. Therefore such mean

values may be continuously retracted on to 1 { *}S v− by a function r as shown in

the diagram. 1 2 1: ( { *}) ( { *})f S v S v− → − such that f r µ= � is the composition of r and µ is Continuous, Unanimous and Anonymous. The “retraction function” r projects the mean from *v on to the circle.

8

23. Remark: This construction does not work for 1S because there does not exist a continuous retraction of the unit disk onto its boundary.

24. Question: What property does the punctured circle have that the circle does not have, that permits the construction of Continuous, Unanimous and Anonymous aggregations? Answer: The property of contractibility . Intuitively, a contractible set can be continuously contracted to one of its points. The following points give the general theory precisely.

25. A subset X of Y is a retract of Y if there exists a continuous function :r Y X→ such that, for all x X∈ , ( )r x x= .

26. A subset X is contractible if and only if there is a continuous function : [0,1]F X X× → and a point 0x X∈ such that ( ,0)F x x= and 0( ,1)F x x= .

Equivalently, a subset is contractible if and only if the identity function on it is homotopic to a constant function. Note that the circle is not contractible but that the punctured circle is contractible. Obviously, a convex set if contractible.

27. Proposition: If X is contractible then it is a retract of some convex set Y containing X.

v′

v

( , )v vµ ′

( , )f v v′

r

9

28. For any convex set X, and a finite number of elements 1, , nx x X∈… , their mean is

given by 11

1( , , )

k

k ii

x x xk

µ=

= ∑… . Note that : kX Xµ → is Continuous and has

analogues of the Unanimity and Anonymity properties appropriately redefined.

29. Proposition: Let X be a contractible retract of a convex set Y. Then f r µ= � , the composition of r and µ , is Continuous and has analogues of the Unanimity and Anonymity properties appropriately redefined.

30. Chichilnisky and Heal (1983) proved that, for a specific class of preferences, contractibility is necessary and sufficient for the existence of Continuous, Unanimous and Anonymous aggregations. This is known as the resolution theorem. Furthermore, any such aggregation is homotopic to a generalised mean.

31. Concluding remark 1: All of the other main results in topological social choice theory are obtained in the same way as Chichilnisky’s impossibility theorem. That is, Continuity is assumed, giving the fundamental equation of topological social choice theory, as in 17. Then, other social choice properties are imposed that place further restrictions on this equation, as in 18 and 19. Furthermore, all of these results drop Anonymity, and weaken Unanimity to the usual Weak Pareto property.

32. Concluding remark 2: Outstanding issues include the justification of Continuity and sufficient conditions for dictatorship.

10

Harmless homotopic dictators*

by

Nicholas Baigent**

Abstract Continuous and Paretian social welfare functions are constructed for which one agent is a homotopic dictator and yet another is, in a precise sense, almost all powerful. Thus, the well known homotopic dictatorship theorem of Chichilnisky cannot be regarded as a genuine Arrow-type impossibility in the sense of showing that some desirable properties entail an undesirable concentration of power.

11

1 Introduction

This paper constructs continuous Paretian social welfare functions for which one agent is

a homotopic dictator but another is, in a precise sense, almost all powerful. The

significance of this arises from the widely differing viewsi that have been expressed about

a theorem in Chichilnisky (1982) showing that, for all continuous Paretian social welfare

functions there must be a homotopic dictator. What the analysis in this paper therefore

shows is that Chichilnisky’s theorem is not a genuine Arrow type impossibility theorem

in the sense that desirable properties are not shown to entail some undesirable

concentration of power.

While this does not necessarily mean that Chichilnisky’s theorem is not significant, at

least it calls for a reappraisal. One possible argument for the significance of this theorem

starts from the fact that a homotopic dictator is also a strategic manipulator. However, as

argued below, this argument does not establish the independent significance of

Chichilnisky’s theorem. At best, its significance seems to be derivative.

Section 2 introduces the main concepts and definitions. Section 3 provides an informal

overview drawing heavily on diagrams. Section 4 presents results and a final section 5

concludes with a summary and a few remarks towards a reappraisal of Chichilnisky’s

theorem.

2 Concepts and definitions

Consider parallel linear indifference curves on a two dimensional space of alternatives,

and call the underlying preferences linear preferences. See figure 2.1 which shows two

indifference curves for each of two linear preferences. For a given linear preference,

draw a vector of length 1 perpendicular to an indifference curve at an arbitrary

alternative. Such vectors are called unit normals. Since they are independent of the

arbitrary alternative, a linear preference may be represented by such a unit normal. Also,

since each unit normal takes a point in the Euclidean plane to another point on a circle of

radius one, the set of all preferences may be taken as the set of points on a unit circle.

12

For convenience, re-centre this circle at the origin as in figure 2.2. Thus, the set of all

linear preferences will be taken as: { }1 2 2 21 2 1 2( , ) : 1S x x x x= ∈ + =ℝ .

For all vectors, 11 2( , )x x x S= ∈ , its polar coordinates are (1, )xρ where xρ is the distance

around the circle 1S from the vector (1,0) to x in the positive (anticlockwise) direction as

shown by a bold arc in figure 2.2.

Let [0,2 ]π denote the closed interval of real numbers from 0 to 2π , and let (0,2 )π

denote the open interval from 0 to 2π .ii For all 1x S∈ and all [0,2 ]δ π∈ , let 1( , )s x Sδ ∈

denote the point in 1S that is a distance of δ around 1S from x in an anticlockwise

direction. Thus, for all 1,x y S∈ , ( , )s x yδ = if and only if y xρ ρ δ− = . See figure 2.2.

That is, ( , )s x δ determines an anticlockwise rotation from 1x S∈ . Since the

circumference of the unit circle is equal to 2π , it follows immediately that:

(2.1) ( ,0) ( ,2 )s x s x xπ= =

figure 2.1

x

x1

x2

xρ1

figure 2.2

( , )y s x δ= y xδ ρ ρ= −

13

For simplicity, consider the case of only two agents. A social welfare function is then a

function 1 1 1:f S S S× → that assigns a group preference 1( , )f x y S∈ to all pairs of

individual preferences 1 1( , )x y S S∈ × . Since the domain and range of a social welfare

function are subsets of Euclidean spaces, when continuity is required it is taken in the

usual sense for functions between subsets of Euclidean spaces.iii

3 Overview

Continuous Paretian social welfare functions on a two dimensional space of alternatives

may be illustrated in a simple diagram. This diagram is used in this section in order to

offer an informal presentation of the main point of the paper that is presented more

precisely in section 4.

The Weak Pareto property of social welfare functions requires that the group preference

rank one alternative strictly above another whenever every individual does. In diagram

2.3, an indifference curve for each agent is given in bold for which a is ranked above b.

This is also the case for the indifference curve of the group preference, shown by the

dotted line. Indeed, for the social preferences illustrated, any alternative ranked by both

individual agents above another is also ranked above it by the group preference. In fact,

this must be the case for all group preferences whose unit normal is contained in the cone

spanned by the agents’ unit normals. This is shown by the arrows in figure 2.3. Now,

consider the case shown in diagram 2.4. Both agents rank a* above b*, but the group

ranks these alternatives in the opposite way. In this case, the unit normal for the group is

not in the cone spanned by the agents’ unit normals.

a

a*

b*

b

figure 2.3 figure 2.4

14

Alternatively but equivalently, for agents’ preferences 1,x y S∈ , the group preference

must be on the shortest arc in 1S from x to y. For example, in the case shown in figure

2.2, the group preference must be on the bold arc going anticlockwise from x to y, and its

distance δ ′ from x along this arc must satisfy 0δ δ′≤ ≤ .

To illustrate a continuous Weakly Paretian social welfare function, consider an arbitrary

1x S∈ , and ( , ( , ))f x s x δ as δ varies from 0 to 2π . This is shown in figure 2.5 in which

values of δ are shown on the horizontal axis and the anticlockwise distance,

( , ( , )f x s x xδρ ρ− , of the social preference from x is shown on the vertical axis.

The relevant details are all shown in the square with sides of length 2π , which is sub-

divided into 4 sub-squares each with sides of length π . As δ goes from 0 to 2π on the

2π

2π π

π

δ

1t =

2t =

( , ( , ))f x s x xδρ ρ−

figure 2.5

15

horizontal axis, the height of the S-shaped curve, shown by a continuous line from the

point (0,0) to the point (2 ,2 )π π , shows the anticlockwise distance of the social

preference around 1S from x. At the point (0,0) agents 1 and 2 both have preferences

given by 1x S∈ , and this is also the case at the point (2 ,2 )π π . At δ π= , agent 2’s

preference is exactly opposite 1’s preference in 1S and exactly the same as the social

preference since the S-curve goes through the point ( , )π π .

Now consider values of δ between 0 and π . In this case, the height of the S-curve is

less than the height of the diagonal. This implies that in 1S , the anticlockwise distance

from x to the social preference is less that that to 2’s preference, thus satisfying the

requirement of the cone restriction. This is also true for values of δ between π and 2π .

In this case, the height of the S-curve is greater than the height of the diagonal. This

implies that in 1S , the anticlockwise distance from x to the social preference is greater

than that to 2’s preference, and again the requirement of the cone restriction is satisfied.

Another crucial feature of the social welfare function illustrated by the S-curve is that the

social preference is never the exact opposite of 2’s preference. That is, the point in 1S

that gives the social preference is never exactly opposite the point that gives agent 2’s

preference. If it were, it would intersect the diagonals of the northwest or southeast sub-

squares in figure 2.5, shown by dotted lines.

Now consider the case of a social welfare function that is illustrated by the diagonal of

the square. In this case, as the preference of agent 2 rotates anticlockwise from x, the

social preference also goes through exactly the same rotation. That is, the preferences of

society and agent 2 are always identical. If this is the case for all possible preferences

1x S∈ that agent 1 could have, then this social welfare function is dictatorial and agent 2

is the dictator.

Finally, a crucial role is played by two continuous deformations of the S-curve. In one of

these, the continuous S-curve is continuously deformed into the diagonal. Just

continuously raise the S-curve for all (0, )δ π∈ and lower it for all ( ,2 )δ π π∈ . Such

pairs of functions that can be continuously deformed into each other are called homotopic

16

functions. Thus, the social welfare function illustrated by the S-curve in figure 2.5 and

the social welfare function illustrated by the diagonal are homotopic. Furthermore, agent

2 is then called a homotopic dictator for the social welfare function illustrated by the S-

curve. Indeed, there must be a homotopic dictator by Chichilnisky’s theorem.

The other important observation is that the social welfare function illustrated by the S-

curve may also be continuously deformed as shown by the broken lines in figure 2.5. For

this continuous deformation, for all (0, )δ π∈ , the heights of the curves shown by broken

lines decrease towards 0 and for all ( ,2 )δ π π∈ , the heights of the curves shown by

broken lines increase towards 2π . For this class of deformations of the S-curve, apart

from its end points, only the point ( , )π π remains constant. In other words, it may be

concluded that, if agents do not have opposite preferences, the group preference may be

made arbitrarily close to the preference of agent 1, even though agent 2 remains a

homotopic dictator.

4 Results

This section makes precise concepts that are used informally in the previous section and

the results given in this section justify the conclusion of the previous section.

Projection functions on 1 1S S× are continuous social welfare functions that have a special

role. They are functions 1 1 1:ip S S S× → , 1,2i = such that, for all 1,x y S∈ , 1( , )p x y x=

and 2( , )p x y y= . For the social welfare function ip , 1,2i = , i is called the dictator.

Note that if agent 2 is a dictator then, for all 1x S∈ and all [0,2 ]δ π∈ ,

( , ( , )) ( , )f x s x s xδ δ= iv.

The concept of homotopic dictatorship first requires the concept of homotopic functions.

For arbitrary continuous functions , :F G A B→ from A to B, F and G are homotopic if

and only if there is a continuous function : [0,1]h A B× → such that, for all a A∈ ,

( ,0) ( )h a F a= and ( ,1) ( )h a G a= . Thus, homotopic functions F and G may be

continuously deformed into each other. For a social welfare function 1 1 1:f S S S× → ,

17

agent i, 1,2i = , is a homotopic dictator if and only if f and ip are homotopic. A dictator

is a homotopic dictator but not necessarily vice versa.

Next, the cone restriction is made precise. For 2 agents the satisfaction of the cone

restriction is equivalent to the Weak Pareto property, though for more than two agents it

is strictly weaker though still sufficient for Chichilnisky’s theorem.

For all 1x S∈ and [0,2 ]δ π∈ , the closed circular cone ( , ( , ))C x s x δ spanned by x and

( , )s x δ is defined as follows:

(3.1) 1

1

{ : ( , ) 0 } if 0( , ( , ))

{ : ( , ) 2 } if 2

y S y s xC x s x

y S y s x

δ δ δ δ πδ

δ δ δ π π δ π′ ′ ∈ = ∧ ≤ ≤ ≤ >= ′ ′∈ = ∧ ≤ ≤ < ≤

A social welfare function 1 1 1:f S S S× → satisfies the cone restriction if and only if, for

all 1x S∈ and [0,2 ] \ { }δ π π∈ , ( , ( , )) ( , ( , ))f x s x C x s xδ δ∈ . That is, as long as agents do

not have opposite preferences, the social preference is on the shortest arc between them.

Note that if agents have opposite preferences so that δ π= , the cone restriction does not

restrict the social preference. Finally, as noted already, a social welfare function has the

Weak Pareto property if and only if it satisfies the cone restriction.

The class of social welfare functions that are illustrated in figure 2.5 may now be defined

as follows. For all real numbers t, 1t ≥ :

(3.2) 1

1

( , ) if [0, ]( , ( , ))

( ,2 (2 ) ) if [ , 2 ]

t t

t t t

s xf x s x

s x

π δ δ πδ

π π π δ δ π π

−

−

∈= − − ∈

1 1 1:tf S S S× → are easily shown to be continuous, and their properties are established

by the following results.

Proposition 3.1. For all 1t ≥ and all 1x S∈ : (i) ( , ( ,0))tf x s x x= ; (ii)

( , ( ,2 ))tf x s x xπ = and (iii) ( , ( , )) ( , )tf x s x s xπ π= .

18

Proof: (i) Substituting 0δ = into the first part of (3.2) and then using (2.1) gives

( , ( ,0)) ( ,0)tf x s x s x x= = . A similar argument substituting 2δ π= into (3.2) and again

using (2.1) proves (ii). For (iii), substitute δ π= into both parts of (3.2) gives what is

required. For example, substituting into the first part gives 1( , ( , )) ( , )t ttf x s x s xπ π π−= =

( , )s x π .

Proposition 3.2. For all 1t ≥ , 1 1 1:tf S S S× → satisfies the cone restriction.

Proof: There are four cases to consider.

(i) 0δ = : Substituting 0δ = into (3.1) gives ( , ( ,0)) { }C x s x x= . Using (2.1) and (3.2)

now gives ( , ( ,0))tf x s x x= , so that ( , ( ,0)) ( , ( ,0))tf x s x C x s x∈ .

(ii) 2δ π= : A similar argument as used in (i) but beginning by substituting 2δ π= into

(3.1) leads to ( , ( ,2 )) ( , ( ,2 ))tf x s x C x s xπ π∈ .

(iii) (0, )δ π∈ : (3.2) implies that 1( , ( , )) ( , )t ttf x s x s xδ π δ−= . Therefore satisfying the

cone restriction in this case requires that 10 t tπ δ δ−≤ ≤ from (3.1). Since π and δ are

both positive, 10 t tπ δ−< . Since δ π< , it follows that 1 1t t t tπ δ δ δ δ− −< = .

(iv) ( ,2 )δ π π∈ : (3.2) implies that 1( , ( , )) ( , 2 (2 ) )t ttf x s x s xδ π π π δ−= − − . Therefore

satisfying the cone restriction in this case requires that 12 (2 ) 2t tδ π π δ π−≤ − − ≤ from

(3.1). Since ( ,2 )δ π π∈ , it follows that 0 2π δ π< − < . Using the argument in (iii) with

2δ π δ′ = − instead of δ , it follows that 1 (2 ) 2t tπ π δ π δ− − < − or, rearranging,

12 (2 )t tδ π π δ−≤ − − which is part of what is required. For the other part, note that

1 (2 ) 0t tπ π δ− − > since both π and 2π δ− are positive. Therefore,

12 (2 ) 2t tπ π π δ π−− − < and this completes the proof.

Corollary of propositions, 3.1 and 3.2: For all [0,2 ]δ π∈ , ( , ( , )) ( , )tf x s x s xδ δ π≠ − + .

That is, The social preference is never the exact opposite of 2’s preference.

Proposition 3.3 For all (0, ) ( , 2 )δ π π π∈ ∪ , ( , ( , ))ttLim f x s x xδ

→∞= .

19

Proof: There are two cases to consider.

(i) (0, )δ π∈ . In this case (3.2) implies 1( , ( , )) ( , )t ttf x s x s xδ π δ−= , so that

1( , ( , )) ( , )t tt

t tLim f x s x Lim s xδ π δ−

→∞ →∞= . From continuity, 1 1( , ) ( , )t t t t

t tLim s x s x Limπ δ π δ− −

→∞ →∞= .

Since 1 ( )t t tπ δ π δ π− = , 1 0t t

tLimπ δ−

→∞= since ( ) ( )t t

t tLim Limπ δ π π δ π

→∞ →∞= and ( ) 1δ π < .

Therefore, using (2.1), 1 1( , ( , )) ( , ) ( , ) ( ,0)t t t tt

t t tLim f x s x Lim s x s x Lim s x xδ π δ π δ− −

→∞ →∞ →∞= = = = .

(ii) ( ,2 )δ π π∈ . In this case, (3.2) implies 1( , ( , )) ( , 2 (2 ) )t ttf x s x s xδ π π π δ−= − − .

1 1( , 2 (2 ) ) ( , (2 (2 ) ))t t t t

t tLim s x s x Limπ π π δ π π π δ− −

→∞ →∞− − = − − from continuity, and

furthermore, 1 1(2 (2 ) ) 2 ( (2 ) )t t t t

t tLim Limπ π π δ π π π δ− −

→∞ →∞− − = − − . Also

1 2(2 )

tt t π δπ π δ π

π− − − =

and

2 2t t

t tLim Lim

π δ π δπ ππ π→∞ →∞

− − =

. Therefore, since

21

π δπ− < , it follows that

20

t

tLim

π δπ→∞

− =

, and this implies that

1( , (2 (2 ) )) ( , 2 )t t

ts x Lim s xπ π π δ π−

→∞− − = . Therefore, ( , ( , )) ( ,2 )t

tLim f x s x s x xδ π

→∞= =

which completes the proof.

Since, for all t, 1t ≥ , 1 1 1:tf S S S× → is continuous and satisfies the cone restriction, it

follows from Chichilnisky’s theorem that either agent 1 or agent 2 must be a homotopic

dictator. The final result shows that the homotopic dictator is agent 2.

Proposition 3.4 For all t, 1t ≥ , tf and tp are homotopic.

Proof: First, it will be shown that ( , ( , )) ( , )tf x s x s xδ δ≠ . If δ π= then this follows

from part (iii) of proposition 3.1. If δ π≠ then ( , ( , )) ( , )tf x s x s xδ δ= − and the cone

restriction would not be satisfied, contrary to proposition 3.3. Therefore, for all 1x S∈

and all [0,2 ]δ π∈ , ( , ( , )) ( , )tf x s x s xδ δ≠ . Since 2( , ) ( , ( , ))s x p x s xδ δ= , it follows that

20

2( , ( , )) ( , ( , ))tf x s x p x s xδ δ≠ . Given this, the following homotopy between tf and 2p is

well defined. For all 1x S∈ , all [0,2 ]δ π∈ and all [0,1]λ ∈ :

1

1

( , ( , )) (1 ) ( , ( , ))( , ( , ))

( , ( , )) (1 ) ( , ( , ))t

tt

f x s x p x s xh x s x

f x s x p x s x

λ δ λ δδλ δ λ δ

+ −=+ −

.

It is straightforward to check that, for all 1x S∈ and [0,2 ]δ π∈ ,

( , ( , ),1) ( , ( , ))t th x s x f x s xδ δ= and 2( , ( , ),0) ( , ( , ))th x s x p x s xδ δ= , and also that th is

continuous as required.

Propositions 3.3 and 3.4 justify and make precise the claim that concludes section 2.

Namely, if agents do not have opposite preferences, the group preference may be made

arbitrarily close to the preference of agent 1, even though agent 2 is a homotopic

dictator.

5 Conclusion

One possible reservation about the analysis in this paper is that it is limited to two agents.

However, given the nature of the issue, it is only necessary to establish the conclusion for

a simple case, and this has been accomplished. Indeed, Chichilnisky’s theorem is not an

Arrow type impossibility result in the sense that it shows that desirable properties entail

an undesirable concentration of power.

It may be argued that a homotopic dictator is also a strategic manipulator in the sense of

being able to get any particular social preference, for all preferences of other agents. This

is indeed the case. It can be seen from figure 2.5 and easily checked from (3.2), that, for

all t, 1t ≥ , and all 1x S∈ , 1( , ( ,[0,2 ])tf x s x Sπ = . Thus, for any possible preference agent

2 can choose a preference so that the former is the social preference. This does

concentrate a certain sort of power in agent 2. However, if strategic manipulation is of

concern, then conditions for its existence can be given directly, and there seems to be no

purpose served by tying it to an analysis of homotopic dictatorship.

21

References

Baigent, Nicholas: “Topological Theories of Social Choice”, 2003, in Handbook of

Social Choice and Welfare: vol 2, edited by K.J. Arrow, A.K. Sen and K. Suzumura,

Elsever, forthcoming.

Chichilnisky, Graciela; "The Topological Equivalence of the Pareto Condition and the

Existence of a Dictator"; Journal of Mathematical Economics; Vol. 9, No. 3; March,

1982; 223-234.

Heal, Geoffrey; "Social Choice and Resource Allocation: A Topological Perspective";

Social Choice and Welfare; Vol. 14, No. 2; April, 1997; 147-160.

Lauwers, Luc;"Topological Social Choice"; Mathematical Social Sciences; Vol. 40, No.

1; July, 2000; 1-39.

MacIntyre, Ian D. A.; "Two-person and Majority Continuous Aggregation in 2-good

Space in Social Choice: A Note"; Theory and Decision; Vol. 44, No. 2; April, 1998; 199-

209.

Saari, Donald G.; "Informational Geometry of Social Choice"; Social Choice and

Welfare; Vol. 14, No. 2; April, 1997; 211-232

Sen, Amartya K.; "Social Choice Theory"; Handbook of Mathematical Economics, Vol.

III; edited by Kenneth J. Arrow and Michael D. Intriligator; Amsterdam; North-Holland;

1986; 1073-1181.

i See Chichilnisky (1982), Heal (1997), Sen (1986) on the one hand and Saari (1997), Lauwers (2000), McIntyre (1998), Baigent (2003) on the other hand. ii Though the same sort of parenthesis is used for both intervals of real numbers and vectors in 2

ℝ , confusion is avoided by explicitly designating vectors, for example by writing, “the vector (0,1)”. iii That is, with respect to the relative topologies given the Euclidean topologies on 4ℝ which contains the

domain and 2ℝ which contains the range.

iv Dictatorship of agent 1 would require ( , ( , ))f x s x xδ = .