Linear preferences and spheres

18 L . Lauwers Mathematical Social Sciences 40 2000 1 –39 Remark 3.4.8. Candeal et al. 1992, Efimov and Koshevoy 1994, and Horvath 1995 study Chichilnisky rules on other structures such as topological vector-spaces, semi-lattices, metric spaces, . . . We list three simple examples of spaces admitting a Chichilnisky rule although they are not contractible or not a CW-complex: • 3 5 the set of rational numbers and F 5 convex mean, • 3 5 the set of irrationals and F 5 minimum, • 3 5 the comb space CS and F x , y , . . . , x , y 5 min hx j, min hy j. s d 1 1 n n k k , ,

4. Linear preferences and spheres

4.1. A fundamental relationship , 11 The normalized gradient field of a linear preference over A , R we assume that , 1 is a constant vector field and can be represented by a single point on the sphere , , 11 n S , R of radius one. Restricting an aggregation rule G: 3 → 3 which is defined , on the whole space 3 of non-satiated and differentiable preferences to the space S of , n linear preferences, results in a rule G: S → 3. Then, fix some alternative a [ A , , 11 R and define the map , n , F : S → S : p∞Gpa where Gpa is the value of the social preference i.e. the aggregated gradient field at , the point a, and since gradient fields are normalized, the image Gpa belongs to S . Hence, the behaviour of the rule F on the sphere is clearly connected with the behaviour of the rule G on 3. As a consequence, the study of aggregation rules on spheres has been indicative for the general case. This section explains the notion of the degree of a continuous map between spheres and its use in different set-ups of the social choice problem. We start with a lemma that gives a sufficient condition for two maps into a sphere being homotopic. The lemma already reveals the specific behaviour of spheres. , Lemma 4.1.1. Let X be a topological space and let f, g: X → S be two continuous maps such that for all x [ X we have fx ± 2 gx. Then f and g are homotopic. Proof. The following map is well defined and is a homotopy between f and g: 1 2 tfx 1 tgx , ]]]]]] qx, t: X 3 I → S : x, t ∞ . i1 2 tfx 1 tgx i , , Next, we define the notion of degree of a continuous map f : S → S . Intuitively, the degree of f indicates how many times the sphere the domain is wrapped around itself the sphere in the range. For example, the degree of the identity map is equal to one. , Before we introduce the formal definition let us repeat that the group p S Z is , L . Lauwers Mathematical Social Sciences 40 2000 1 –39 19 , abelian. The operator in p S is therefore denoted by ‘ 1 ’, the neutral element a , constant map by ‘0’, and a generator see Example 3.2.3 ii by ‘1’. , , Definition 4.1.2. The degree of a continuous map f : S → S is the unique integer deg f that satisfies the equation f 1 5 deg f 3 1, where f is the induced homomorphism on the , th homotopy group. Properties 4.1.3. , , • If the map f : S → S is not onto then the degree of f is equal to zero. • The degree of a map is a homotopy invariant e.g. Dugundji, 1989, Thm 7 4: two , , maps f, g: S → S are homotopic if and only if deg f 5 deg g. , , , Let us now return to the aggregation problem. Let F : S 3 ? ? ? 3 S → S be a continuous map. A vector p 5 p , p , . . . , p of points on the sphere is said to be a 1 2 n profile. The vector p 5 p , . . . , p , p , . . . , p collects the preferences of 2k 1 k 21 k 11 n individual k’s opponents. The notation p , p is an alternative for p. Let p be some 2k k , , n fixed profile. One can define a continuous embedding i : S → S : p∞p , p for k 2k each individual k [ N. These embeddings induce continuous maps between spheres: , , F + i : S → S : p∞Fp , p k 2k 9 for k [ N. For different profiles p and p9 the maps p∞Fp , p and p∞Fp , p are 2k 2k homotopic and have therefore the same degree. Hence, we can define the degree of individual k as the degree of the map F + i and this without explicit reference to the k vector p. Note that in case the degree of individual k is different from zero, the map p∞Fp , 2k , p is onto, i.e. ImF + i 5 S . For any given vector of preferences of her opponents, k such an individual is able to generate any outcome by choosing a suitable preference for herself. By means of the diagonal embedding D: p∞ p, p, . . . , p another map between spheres can be defined: , , F + D: S → S : p∞F p, p, . . . , p. Now we are able to prove a relationship between the individual degrees and the degree of this diagonal map. , n , Lemma 4.1.4. Let F : S → S be a continuous map and let D, i , i , . . . , i be as 1 2 n above. Then degF + D 5 deg1 1 deg2 1 ? ? ? 1 degn, 2 where degk is the degree of individual k [ N. 20 L . Lauwers Mathematical Social Sciences 40 2000 1 –39 Proof. Consider the induced maps at the homotopy level and observe that D 1 5 1, 1, . . . , 1 H i 1 5 0, . . . , 0, 1, 0, . . . , 0 with the 1 at the kth place. k It follows that D 1 5 i 1 1 i 1 1 ? ? ? 1 i 1. Apply the group homo- 1 2 n morphism F and conclude the proof. For an alternative proof, based upon integration along the sphere, we refer to Aumann 1943 and Lauwers 1999a. The following subsections are devoted to the implications of the above fundamental relationship. 4.2. The topological social choice paradox Chichilnisky 1979, 1980, 1982a questions the existence of a topological aggregator , in case the space 3 of preferences coincides with a sphere S . From the resolution theorem we already know that the answer is negative: spheres are not contractible. An alternative proof goes as follows. , n , Theorem 4.2.1. There does not exist a map F : S → S that combines continuity, anonymity, and unanimity. Proof. Suppose there exists a Chichilnisky rule F on the sphere. Anonymity implies that the individuals all have the same degree: degi 5 deg1 for i 5 1, 2, . . . , n. Unanimity , implies that F + D p 5 p for all p [ S . Hence, F + D is the identity map and has degree equal to one. Apply the fundamental relationship 2 and obtain a contradiction: 1 5 n deg1. Remark 4.2.2. The special case with two individuals and a two dimensional choice space is equivalent to i Brouwer’s two dimensional fixed point theorem Chichil- ¨ ´ nisky, 1979 and ii a particular property of the Mobius strip Candeal and Indurain, 1994a. Remark 4.2.3. The present set-up excludes vanishing gradient fields. Concerning this phenomenon, we insert some results of Chichilnisky 1982a and Le Breton and Uriarte 1990a. Chichilnisky proves that in the framework of Section 2 her results extend to the set 3 of gradient fields that might vanish on a subset of A of measure zero. Hereby, the set 3 is equipped with a topology similar to the original set-up. Then, a Chichilnisky rule on 3 is restricted to the set of linear preferences and applied upon one alternative a. This results in a continuous with respect to the Euclidean topology map , n , w : S → S h 0j: p∞Fpa the image set is the union of the unit sphere and the origin that satisfies unanimity and L . Lauwers Mathematical Social Sciences 40 2000 1 –39 21 , n anonymity. Since w is continuous and S is connected, the image of w is connected , 17 and coincides use unanimity with S . Now Theorem 4.2.1 applies. Le Breton and Uriarte observe that the existence of the above rule w crucially , depends upon the topology the set S h0j is equipped with. For a particular non- 18 Hausdorff topology the map , n , c: S → S h0j: p 1 ? ? ? 1 p i p 1 ? ? ? 1 p i if p 1 ? ? ? 1 p ± 0 1 n 1 n 1 n p ∞ H otherwise is continuous, anonymous, and unanimous. Furthermore, they show that the topology on 3 considered by Chichilnisky also violates the Hausdorff separation axiom. 4.3. Pareto rules and homotopic dictators This section confronts the Pareto axiom as it appears in Arrow’s theorem with the topological approach to social choice. Recall that an aggregation rule satisfies the Pareto condition when the following holds for an arbitrary profile of individual preferences and for an arbitrary pair a, b [ A of alternatives: if every individual ranks a higher than b, then so does the social preference. The Pareto condition implies respect of unanimity. An aggregation rule is dictatorial and individual k is said to be a dictator if for an arbitrary profile the social preference coincides with the preference of individual k. Such an aggregation rule is a projection: , n , P : S → S : p 5 p , p , . . . , p ∞ p . k 1 2 n k Notice that a dictatorial rule satisfies the Pareto condition. The next lemma is a first step in obtaining a proof of the reverse statement. , n , Lemma 4.3.1. Let F : S → S be a continuous map that satisfies the Pareto ] condition, let p be the constant profile p, p, . . . , p, and let k [ N. Then the ] aggregated preference q 5 Fp , 2 p belongs to the pair h p, 2 pj. 2k ] Proof. The Pareto condition implies that the aggregated preference q9 5 Fp , p9 with 2k p9 ± 2 p is contained in the circular convex hull of p and p9. This is the smallest arch on the great circle through p, 2 p and p9 that contains p and p9 indicated in bold in the figure. Now let the point p9 approach 2 p then continuity implies that q belongs to one particular half ‘a’ convex hull of p and 2 p of the great circle through p and 2 p. The 17 Chichilnisky 1985 uses a similar connectedness argument to show that the space of von Neumann– Morgenstern utility functions over a finite set of choices is not contractible and does not admit topological aggregators. 18 With respect to this particular topology full indifference cannot be distinguished from the rest of the , preferences. Or, the only open neighbourhood of 0 is the whole space S h0j. 22 L . Lauwers Mathematical Social Sciences 40 2000 1 –39 same argument applies to the other half of the great circle. Hence the aggregate belongs to the intersection of both halves. This is exactly the pair h 2 p, pj. The combination of Lemmas 4.1.1 and 4.3.1 results in the topological equivalence of the Pareto condition and the existence of a dictator in the special case of two individuals. , , , Corollary 4.3.2. Let F : S 3 S → S be a continuous map that satisfies the Pareto condition. Then F is homotopic to a dictatorial rule. Proof. Continuity and the Pareto condition imply that , , ; p [ S we have F p, 2 p 5 p or else ; p [ S we have F p, 2 p 5 2 p. In the first case it follows that ; p , p : F p , p ± 2 p and in the second case that 1 2 1 2 1 ; p , p : F p , p ± 2 p . Apply Lemma 4.1.1 and conclude the proof. 1 2 1 2 2 The following corollary will help us in moving towards the general result. , n , Corollary 4.3.3. Let F : S → S be a continuous map that satisfies the Pareto condition. Then i the degree of an individual is either one or zero, and ii there is exactly one individual with degree equal to one. Proof. i Recall that the degree of individual k does not depend upon the preferences of ] her opponents. Hence let us determine the degree of the map F + i : p9∞Fp , p9 k 2k where p is equal to the constant profile p, p, . . . , p. According to the previous lemma ] we have that Fp , 2 p is either 2 p or p. In the former case it follows that use 2k Pareto ] ; p9 we have Fp , p9 ± 2 p9 2k which implies that F + i is homotopic to the identity map Lemma 4.1.1. Hence, the k degree of k is equal to one. In the latter case it follows that ] ; p9 we have Fp , p9 ± 2 p. 2k In this case the map F + i is homotopic to the constant map p9∞ p Lemma 4.1.1 and k the degree of k equal to zero. ii Since the Pareto condition implies unanimity, the degree of the diagonal F + D is equal to one. The fundamental relationship 2 indicates that exactly one individual has a degree equal to one. L . Lauwers Mathematical Social Sciences 40 2000 1 –39 23 Now we are almost ready to prove the topological equivalence between the Pareto condition and dictatorship in the case of an arbitrary number of individuals. First, we state an extra axiom that will be used in the argumentation. Definition 4.3.4. An aggregation rule F satisfies the weak positive association condition WPAC if Fp 5 2 p for some k and some profile p, implies that k ] ] Fp , p ± p with p 5 2p , 2 p , . . . , 2 p . 2k k k k k k The interpretation is clear. If for a particular profile the social outcome is the exact opposite 2 p of the preferences of individual k, then a move of the preferences of the k opponents of individual k towards 2 p cannot change the outcome into p . Obviously, a k k dictatorial rule does satisfy WPAC. , n , Theorem 4.3.5 Chichilnisky, 1982c. Let F : S → S be a continuous map that satisfies the Pareto condition and WPAC. Then F is homotopic to a projection P : p∞ p j j for a unique j [ h1, 2, . . . , nj. Proof. From Corollary 4.3.3 we know that exactly one individual has a degree equal to one, and that the others have a degree equal to zero. Consider for each individual k [ N the following condition: , n ;p [ S we have Fp ± 2 p 5 2 P p. 3 k k Observe that • if for individual k condition 3 holds, then the maps F and P are homotopic use k Lemma 4.1.1, • if for individual k condition 3 does not hold, then the degree of k is equal to zero. We clarify the second assertion. Let Fp 5 2 p for some profile p. WPAC implies that k ] for the particular constant profile p 5 2p , 2 p , . . . , 2 p the map k k k , n , ] F + i : S → S : p∞Fp , p k 2k is not onto p does not belong to the image of F + i . Hence the degree of individual k k k is equal to zero. Apparently condition 3 holds for exactly one individual. This completes the proof. Remark 4.3.6. Koshevoy 1997 shows that in the two dimensional case the WPAC can 1 n 1 be dropped: a continuous map F : S → S that satisfies the Pareto condition is homotopic to a dictatorial rule. Remark 4.3.7. Chichilnisky’s theorem states that an aggregation rule that satisfies the Pareto condition and WPAC can be continuously deformed into a dictatorial rule. Again, as such homotopies might be ‘wild’, the interpretation of this statement is not obvious. Saari 1997 uses the notion of ‘dominant’ voters to interpret this theorem. 24 L . Lauwers Mathematical Social Sciences 40 2000 1 –39 Such an individual plays a dominant but not total role in determining the outcome. Let 1 1 1 us illustrate his approach in the two person two dimensional case S 3 S → S . The 1 circle S can be seen as the interval I the two endpoints of which are identified. 1 1 Similarly, the torus S 3 S results from a square I 3 I after identifying the lines [a, b] with [a9, b9] and [a, a9] with [b, b9]. Now we consider two aggregation rules, the dictatorial rule P : p , p ∞ p and a 2 1 2 2 Pareto rule F which is homotopic to P . The next picture shows some of the level sets 2 1 1 for both maps. The domain S 3 S is represented by the square, the edges of which have to be identified as described above. Observe that, because both rules satisfy the Pareto condition, they also satisfy the weaker unanimity condition. Therefore, each level set intersects the diagonal exactly once. 21 Consider first the dictatorial map. The level sets P p 5 h p , p uP p , p 5 pj 2 1 2 2 1 2 are horizontal lines. Indeed, as the preferences of individual 1 do not matter, the 9 profiles p , p and p , p belong to the same level set. 1 1 1 1 1 The second part of the picture deals with a non-dictatorial rule F : S 3 S → S that satisfies the Pareto condition. As the rule F is homotopic to P , the level sets are 2 deformations of the horizontal lines. Note that locally in particular in the neigh- bourhood of the diagonal this rule F gives more ‘power’ to the first player the level sets are almost vertical . The following lemma shows that the concept of ‘topological’ dictatorship becomes empty in case the space of preferences is contractible. Lemma 4.3.8. Let Y be a contractible topological space and let X be any topological space. Let f, g: X → Y be two continuous maps. Then f and g are homotopic. L . Lauwers Mathematical Social Sciences 40 2000 1 –39 25 Proof. We show that any continuous map f : X → Y is homotopic to a constant map. Since the relation ‘homotopic to’ is an equivalence relation the lemma follows. The contractibility of Y implies the existence of a homotopy q : Y 3 I → Y with q y, 0 5 y and q y, 1 5 y where y [ Y. Use this homotopy q to define a new map: q 9: X 3 I → Y: x, t∞ q 9x, t 5 q fx, t. The map q 9 establishes a homotopy between f. 5 q 9., 0 and C 5 q 9., 1. y If the topological space 3 of preferences is contractible, an ‘anonymous’ aggregation rule exists use the resolution theorem but such a rule is homotopic to a ‘dictatorial’ rule. In addition, all dictatorial rules are homotopic to each other In other words, the concept of topological dictator is only significant in case the space of preferences is not contractible. This subsection ends with the statement of two results by Baryshnikov 1994. He narrows the concept of homotopy and refines the analysis of Chichilnisky: let 3 be a n space of preferences and let f, g: 3 → 3 be two continuous maps that satisfy the Pareto condition. Then f and g are said to be Pareto-isotopic if there exists a homotopy q between f and g such that the intermediate maps q., t satisfy the Pareto condition 19 for all t [ I. Equipped with this new concept, he establishes the following results. , n , Theorem 4.3.9. Let F : S → S be a continuous map that satisfies the Pareto condition and WPAC. Then F is Pareto-isotopic to a projection P : p∞ p for a unique j j j [ h1, 2, . . . , nj. 2 4 2 Theorem 4.3.10. There exists a continuous map F : S → S that satisfies the Pareto condition and that is not Pareto-isotopic to a dictatorial rule. Baryshnikov interprets this second result as a possibility theorem. It shows the existence of a Pareto rule which cannot be deformed into a dictatorial rule unless one moves away from the Pareto condition during the continuous deformation. 4.4. Variations on the theme: no-veto and manipulation This subsection shows that without the introduction of ‘homotopic dictators’ some interesting results were obtained in the framework of continuous aggregation maps on the sphere. The results we have in mind involve the notion of ‘no-veto’ and of ‘manipulation’. In view of the following list of properties, the proofs are straightforward. • degF + D 5 deg1 1 deg2 1 ? ? ? 1 degn, 2 • if F is unanimous, then degF + D 5 1, • if F is Pareto, then degi [ h0, 1j and degF + D 5 1. Let us now repeat the definition of the ‘no-veto condition’. 19 Replacing the Pareto axiom with a different set of axioms results in a different concept of isotopy. 26 L . Lauwers Mathematical Social Sciences 40 2000 1 –39 Definition 4.4.1. A rule F satisfies the no-veto condition if for all constant profiles ] ] p 5 p, p, . . . , p we have Fp , 2 p ± 2 p. 2k This axiom states that if one individual holds completely opposite preferences from the others, then the aggregate will not agree with this single individual. Theorem 4.4.2 Chichilnisky, 1982b; Mehta, 1997. There does not exist a continuous , n , map F : S → S with n 3 that satisfies the Pareto condition and the no-veto condition. Proof. The no-veto condition implies that the degree of each individual is equal to zero. ] Indeed, consider the embedding i : p9∞p , p9. The image of the composite map F + i k 2k k does not contain the point 2 p use the Pareto condition. Hence degk 5 0 for all k. This is in contradiction to 2. A reformulation of this theorem reads as follows: an aggregation procedure that combines the Pareto and the no-veto conditions must be unstable in the sense of 20 non-continuous. Secondly, we present the notion of ‘manipulator’. , Definition 4.4.3. An individual k is said to be a manipulator if for all p [ S and for all , n profiles p [ S there exists a p9 such that Fp , p9 5 p. 2k Obviously dictators are manipulators. Also, rules homotopic to a dictatorial one induce a unique manipulator. The converse of the previous statement is not true: the existence of a manipulator does not imply that a rule can be continuously deformed into a dictatorial rule. Furthermore, an individual with degree different from zero is a manipulator. Theorem 4.4.4 Chichilnisky, 1983, 1993a. Let the aggregation map F be continuous and unanimous. Then there exists at least one manipulator. In Chichilnisky and Heal 1984 the unanimity condition in the previous theorem is weakened to ‘citizen’s sovereignty’. This condition states that F + D is of nonzero degree, and implies that by coordinating their strategies the individuals can achieve all outcomes. Replace in the previous theorem unanimity with the stronger Pareto condition, and obtain: Theorem 4.4.5 Chichilnisky, 1983; Baryshnikov, 1994; Koshevoy, 1997. Let the aggregation map F be continuous and satisfy the Pareto condition. Then there exists a unique manipulator. 20 Chichilnisky 1982a and Mehta 1997 use a ‘decisive majority axiom’ which is stronger than the no-veto axiom. MacIntyre 1998 obtains a result similar to Theorem 4.4.2 in case , 5 1. L . Lauwers Mathematical Social Sciences 40 2000 1 –39 27

5. The topological approach to Arrow’s theorem