A model of elitist qualification

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1 Group Decision and Negotiation manuscript No. (will be inserted by the editor) A model of elitist qualification Miguel A. Ballester 1, José Luis García-Lapresta 2 1 Dep. d Economia i d Història Econòmica, Universitat Autònoma de Barcelona, Bellaterra (Barcelona), Spain. miguelangel.ballester@uab.es 2 Dep. de Economía Aplicada (Matemáticas), Facultad de Ciencias Económicas y Empresariales, Universidad de Valladolid, Avda. Valle de Esgueva 6, Valladolid, Spain. lapresta@eco.uva.es Received: date / Revised version: date Abstract This paper deals with the collective qualification of members of society as belonging to a certain category or group based on a fixed attribute. Our model contains three main features: the existence of individual gradual opinions, the notion of elitism (only the opinions of certain individuals are taken into account to delineate the reference group), and the idea of sequentiality (elites are successively created by using the previous elites opinions on a social decision scheme). The main results of the paper characterize when this sequential procedure converges for some intuitive ways of aggregating individual opinions. Finally, we analyze the role of convergence for two extra basic properties (symmetry and contractiveness) that elitist rules should possess. Key words social choice; voting systems; liberalism; consensus; social identity; sequential elitism; gradual assessments 1 Introduction In this paper we deal with how individuals within a society are collectively perceived as belonging to a certain category or group. In particular, we discuss a set of rules that rest on the idea that qualified members might have special significance when evaluating other individuals. That is, certain members opinions deserve favored treatment by virtue of their perceived superiority (elitism). However, elitism only arises from social evaluations Send offprint requests to: José Luis García-Lapresta.

2 2 Miguel A. Ballester, José Luis García-Lapresta in such a way that in each step the elite is built from the opinions of the previous elite (sequential elitism). The delineation of identity seems to be crucial in several economic and social problems. Some axiomatic studies have been built on this issue, starting with Kasher and Rubinstein [10]. 1 As discussed in Çengelci and Sanver [4], most of the rules introduced thereafter, to which we refer as simple rules, (see Samet and Schmeidler [12], Sung and Dimitrov [13], Houy [7, 8] or Miller [11]) are shown to share a common group of properties. However, there exists a literature dealing with a more complicated set of rules (multi-stage rules), that cannot be expressed as simple rules. They initially determine a group of individuals by means of a simple rule. This set of qualified individuals may be modified, taking into account the opinions of the individuals qualified in the previous stage. This recursive process finally generates the qualified set of individuals within the society. Dimitrov, Sung and Xu [5] and Çengelci and Sanver [4] provide axiomatic characterizations of some of these recursive procedures based on the original ideas of Kasher [9] and Kasher and Rubinstein [10]. The main features of our model are the following. First, each member of society gradually assesses all individuals, including herself. In the previous literature, for any individual i, the judgement regarding if an individual j possesses a certain attribute is a dichotomous assessment, i.e., individual i either does or does not qualify individual j. Our model is more general, since opinions can include cardinal information. Secondly, this information is aggregated and those individuals who reach a given threshold are qualified. As in the multi-stage rules, in the next stage only the opinions of this emergent elite are taken into consideration and a new subset of qualified individuals is selected; and so the procedure continues. As pointed out, we first focus on the convergence of the sequence of elites. In Kasher [9], Dimitrov, Sung and Xu [5] or Çengelci and Sanver [4], the issue of convergence was not substantially covered. In their specific rules, looking at convergence is not complicated, since all their procedural rules are intuitively convergent. 2 Our main results provide some necessary and sufficient conditions for the convergence of sequential elitism when using other classes of aggregators that reflect natural ways of combining individual opinions. We complement these results by analyzing two further properties of sequential elitism, symmetry and contractiveness. The former refers to classical notions of anonymity in Social Choice. The latter is related to the idea that stricter committees should select less individuals as qualified. 1 In a non-axiomatic approach, Kasher [9] first considered (Jewish) identity and proposed some rules in order to decide the subset of individuals who belong to that group. 2 The corresponding dichotomous procedures are based on invitation rules: in each stage qualified members can qualify as many members as they desire. Thus, the set of qualified members does not decrease in any stage and, consequently, the procedures are convergent. For a more technical reasoning, see Section 3.

3 A model of elitist qualification 3 The paper is organized as follows. Section 2 introduces the main features of our model of sequential elitism, providing some archetypal social evaluations as the basis of the model. In Section 3, we provide necessary and sufficient conditions for the convergence of our sequential elitism model. Finally, Section 4 analyzes two extra properties of the rules that our model defines and concludes the paper. 2 Collective assessment and qualification functions 2.1 Gradual assessments A society is conformed as a finite set of individuals N = {1, 2,..., n} with n 2. We use 2 N to denote the power set of N, i.e., the set of all the subsets of N, and for any S N, S is the cardinal of S. A profile is an n n matrix P = (p ij ) with values in the unit interval, where p ij is interpreted as the assessment with which individual i evaluates individual j as being qualified to belong to the group in question. The set of profiles is denoted by P. 2.2 Sequential elitism A social qualification consists merely of a rule or mechanism that, given a profile of opinions, specifies the set of qualified members. Definition 1 A rule is a mapping f : P 2 N which assigns, to any profile P P, the set of qualified members of the society, f(p ) N. We are interested in sequential elitist rules, where the final set of qualified individuals (the final elite) is the result of a sequential process in which an initial elite determines which members should be included in the next elite or reference group. To formalize these ideas we first need to specify the method that qualifies individuals on the basis of the assessments of any subset of N. A natural way to do so combines a collective assessment function, which provides an evaluation of any individual according to the (reference) set of individuals S N, and the profile P P, and a threshold family (to be explained below). Definition 2 A collective assessment function is a mapping ( ) v : 2 N \ { } P N [0, 1]. A collective assessment function determines individual qualification only in gradual terms. A very natural way to convert a gradual opinion into a dichotomic assessment is by means of thresholds.

4 4 Miguel A. Ballester, José Luis García-Lapresta Definition 3 A family of values {α S }, with = S N and α S [0, 1] for every S, is called a threshold family. Given a collective assessment function v and a threshold family {α S }, the function g {α S} v : 2 N P 2 N defined by: { {j N v(s, P, j) αs }, if S, g {α S} v (S, P ) =, otherwise, for all S N and P P, is called the collective qualification function associated with {α S } and v. Remark 1 One may convert each individual opinion into a dichotomous one before the aggregating process. However, given that thresholds can be different across different elites, a unique matrix of dichotomous opinions would not necessarily arise. Non-constant thresholds are precisely a key of the following results. Moreover, even with constant thresholds, gradual assessments are far richer than dichotomous assessments. The former can be aggregated in plenty of different ways before the application of the threshold. Thus, the case of gradual opinions becomes substantially richer than the one of dichotomous opinions, for which parallel results to ours can be derived easily. Our model should also specify the idea of sequentiality. First of all, given a profile, our society determines a first set of qualified individuals. This can be done through the collective qualification function g {α S} v, merely calculating the set S 1 = g {α S} v (N, P ). Once the group S 1 has been qualified as a significant elite, the recursive process moves from this set to another one, considering again the collective qualification function g {α S} v, S 2 = g {α S} v (S 1, P ), and this process is iterated. 3 In this way, a sequence of subsets of N is obtained, and the limit set of such sequences can be considered the solution of a sequential elitist model to the problem of qualifying the members of a society. At this point, a natural question arises. Do collective qualification functions always specify a final set of qualified members? In other words, is it possible to assign an (elitist) rule f to any collective qualification function? To answer this question, the following definition makes sense. Definition 4 Given a collective qualification function g {α S} v and a profile P P, the sequence {S t }, where S 0 = N and S t+1 = g {α S} v (S t, P ), is called a chain. A chain is said to be convergent if {S t } has a limit lim S t, i.e., there exists a positive integer q such that S t = S q for every t q (and it is also said that the chain converges to lim S t ). The collective qualification 3 Notice how the profile of opinions remains the same in each of the stages of the process. The modification of opinions would also arise interesting economic questions, as other models of committees show. See for instance Barberà, Maschler and Shalev [1].

5 A model of elitist qualification 5 function g {α S} v is said to be convergent if every chain generated by g {α S} v is convergent. Remark 2 A variant of Definition 3 is that only the members of the elite can be qualified by that elite: g {α S} v (S, P ) = { {j S v(s, P, j) αs }, if S,, otherwise, for all S N and P P. Now, if an individual is expelled from the elite, this individual cannot enter in a subsequent stage. Obviously, this collective qualification function converges to a final set of qualified individuals. Due to it being a trivial case, we do not consider it. Remark 3 A variant of Definition 4 is that the initial elite is any subset S 0 of N. However, the convergence of the elitist model is seriously flawed. The following example shows how, for any natural collective qualification function, convergence cannot be ensured. Consider = S 0 N and the profile P defined by: ) 1, if (i S 0 and j N \ S 0 p ij = 0, otherwise. or (i N \ S 0 and j S 0 ), Most natural qualification functions should determine S 1 = N \ S 0. 4 Similarly, once N \ S 0 is the elite, the new group of selected individuals should be S 0, thus provoking a cycle. In Kasher [9] and Dimitrov, Sung and Xu [5], the initial sets S 0 are determined using (in our terminology) a different collective qualification function than the one used in subsequent stages. To study the cases considered in the above mentioned papers, very minor modifications should be included in our model. 2.3 Some remarkable collective assessment functions To test the validity of the (sequential) elitist methods, we now present some specific cases that constitute natural ways to aggregate evaluations. In Section 3, we look at the convergence of the elitist procedure when using such methods. 4 Notice that the cases presented in 2.3, excluded the extreme liberal case, verify this fact.

6 6 Miguel A. Ballester, José Luis García-Lapresta The extreme liberal case Similarly to Kasher and Rubinstein [10], we consider the extreme liberal case, where the only opinion taken into account to reach a decission is that of the individual being judged: v 1 (S, P, j) = p jj. Given a threshold family {α S }, an individual j is qualified by the elite S if the opinion of j about herself reaches the corresponding threshold: j g {α S} v 1 (S, P ) p jj α S. Remark 4 The extreme liberal case is, in some sense, a degenerated elitist function, as elites play almost no role in judging individuals. Nevertheless, there exists a partial influence of elitism, as different elites may impose different thresholds. The results within the paper give the hint that very mild conditions over the thresholds allow the convergence of any of these degenerated sequential procedures. We have included this case for the sake of illustration, although for elitism to be fully explored, it seems natural to think that a set of members should use only their own opinions to provide an assessment. In this spirit, the remaining list of collective assessment functions tries to cover a wide range of intuitive and philosophical aggregation methods The benevolent case Following the view of Kasher [9], the benevolent case corresponds to making social opinion be the opinion of the most enthusiastic member of the elite: v 2 (S, P, j) = max{p ij i S}. So, given a threshold family {α S }, an individual j is qualified by the elite S if the opinion of some member of the elite reaches the corresponding threshold: j g {α S} v 2 (S, P ) p ij α S for some i S. This function generalizes the dichotomous case corresponding to an invitation rule, where one member of the elite is sufficient to qualify an individual The consensus We could consider a more restrictive collective assessment, reached through consensus: v 3 (S, P, j) = min{p ij i S}. Given a threshold family {α S }, an individual j is qualified by the elite S if the opinion of each member of the elite reaches the corresponding threshold: j g {α S} v 3 (S, P ) p ij α S for every i S. This function generalizes the dichotomous case corresponding to the (elitist) consensus idea, where individuals are qualified only if they are qualified by each member of the elite.

7 A model of elitist qualification The average Balancing the two previous extreme cases, additive methods try to combine all the opinions into a compensative collective view. As a special case, we consider that the social opinion is neither the most optimistic nor the most pessimistic, but the average of all the elite opinions: v 4 (S, P, j) = 1 S p ij. Given a threshold family {α S }, an individual j is qualified by the elite S if the average of the elite opinions reaches the corresponding threshold: j g {α S} v 4 (S, P ) 1 S i S p ij α S. We note that the arithmetic mean, as an aggregation operator, satisfies interesting properties, which have been characterized in the Social Choice framework (see Candeal et al. [2], Candeal and Induráin [3] and García- Lapresta and Llamazares [6]). i S Liberalism and elitism Liberalism and elitism may be combined in a very natural way. Whenever an individual is part of the elite (but only in this case), it is possible to assign a special weight for this person s evaluation of herself. Consider the collective assessment function: γ S p jj + 1 γ S p ij, if j S and S {j}, S 1 (S, P, j) = 1 S p ij, i S i S\{j} otherwise, where γ S [0, 1], for every S N such that S 2. In the case of S = 1, the second part of the definition guarantees that only the opinion of the individual in S is considered, i.e., ({i}, P, j) = p ij. Whenever γ S = 1 S for any non-empty subset S N with at least two individuals, v 4 is recovered. The γ S = 1 case essentially considers for some cases only the opinion of the individual herself as the collective assessment of the group S. Notice that, while v 1 (S, P, j) = p jj for every S, it is the case that (S, P, j) = 1 p ij whenever j S. S i S Although not from a liberal point of view, the γ S = 0 case may have some interest. Suppose that a committee is evaluating the merits of its members. It is common practice to exclude a member s opinion of herself, and then assign zero weight to her opinion. Finally, we also note that v k ({j}, P, j) = p jj for all k {1,..., 5}, j N and P P.

8 8 Miguel A. Ballester, José Luis García-Lapresta 3 Analysis of convergence Probably the most natural threshold families are those where all the elites use the same threshold α, i.e., α S = α for every S N. In this case, we denote by gv α the collective qualification function associated with α and v. Individuals are qualified whenever the elite opinion about them reaches the corresponding threshold. However, as Corollaries 1, 2, 3 and Theorem 2 highlight, such assumption would extremely restrict the convergence of sequential elitism. These facts may be interpreted as a partial impossibility result. Whenever constant thresholds are at stake, convergence of collective qualification functions leads to a very relaxed notion of group identification. Indeed, only the degenerated case (the extreme liberal) and the benevolent cases ensure convergence under any constant threshold. This result also shows the reason why the rules introduced in Kasher [9], Dimitrov, Sung and Xu [5] and Çengelci and Sanver [4] are well-defined. 5 In the following results, we establish necessary and sufficient conditions on the (general) family of thresholds to guarantee the convergence of collective qualification functions g {α S} v k, for k = 1,..., 5. Theorem 1 Given a family of strictly positive thresholds {α S } S N, the following conditions are equivalent: 1. g {α S} v 1 is convergent. 2. g {α S} v 2 is convergent. 3. For all non-empty sets S, T N, α S α T whenever S T. Proof: To prove that 1 (and 2) implies 3, suppose, by contradiction, that 3 is not true. In this case, there exist two non-empty sets S T such that α S < α T. Consider the profile P defined by: 0, if j N \ T, p ij = α S, if j T \ S, 1, if j S. It is easy to see that for v 1 (and v 2 ), either S 2 = T (whenever α S α N ) or S 2 = S (whenever α S < α N ), and after that, the qualification chain oscillates cyclically between these two sets. Therefore, g {α S} v 1 (and g {α S} v 2 ) is not convergent, which is absurd. To prove that 3 implies 1 (and 2), it is sufficient to note that any qualification chain is decreasing for v 1 (and v 2 ), and therefore, convergent. Convergence in the case of constant thresholds can be directly derived from Theorem 1. Corollary 1 The functions g α v 1 g α v 2 converge for every α [0, 1]. 5 Although the rules introduced in these papers only apply the benevolent case after S 1 is defined, minor modifications have to be included to derive the previous intuition.

9 A model of elitist qualification 9 As we show in the following theorem, convergence is not achievable in the consensus case. Theorem 2 For any family of strictly positive thresholds {α S } S N, g {α S} v 3 is not convergent. Proof: Take a profile P P in which all individual assessments are 1 except p ii = 0 for a given individual i N. It is not hard to see that S 2t 1 = N \ {i} and S 2t = N, for every positive integer t. Therefore, the qualification chain associated with P does not converge. Theorem 3 Given a family of strictly positive thresholds {α S } S N, the following two conditions are equivalent: 1. g {α S} is convergent. 2. For all non-empty sets S, T N, (1 γ T ) S α S ( T 1) α T whenever S T. Proof: See Appendix A. Convergence in the case of constant thresholds can be directly derived from Theorem 3. Corollary 2 The function g α converges if and only if α = 0 or (n = 2 and γ N = 0). Since v 4 is included in the family analyzed in Theorem 3, we can derive the following result. Corollary 3 Given a family of strictly positive thresholds {α S } S N, the following two conditions are equivalent: 1. g {α S} v 4 is convergent. 2. For all non-empty sets S, T N, S α S T α T whenever S T. In particular, g α v 4 is not convergent for any α (0, 1]. Proof: The function g {α S} v 4 is a specific case of g {α S} with γ S = 1 S for every S N such that S 2. Substituting this value in the convergence in Theorem 3, we obtain the condition S α S T α T for every pair of subsets S T N. In order to interpret the above results, we present some brief remarks condition of g {α S} that follow from the necessary and sufficient conditions for the convergence of g {α S} v k. Remark 5 Given a family of strictly positive thresholds {α S } S N, for g {α S} to be convergent, elitism cannot be freely combined with all kinds of liberalism. First, it is necessary that γ T 1 for every T N such that T 2: Taking into account Theorem 3, if γ T = 1, then 0 = (1 γ T ) S α S ( T 1) α T,

10 10 Miguel A. Ballester, José Luis García-Lapresta with S T, i.e., α T = 0, which is absurd. Other upper bounds can be found. Consider, for instance, the family of subsets of T with T 1 elements, i.e., those sets obtained by removing one individual. In this case, for each of these sets S, it must be that 1 γ T α T α S, or equivalently, γ T 1 α T α S. Therefore, the value α T 1 min{α T \{k} k T } is an upper bound for γ T. Naturally, other values can be obtained by considering subgroups of different cardinality. Remark 6 Higher thresholds allow a lower degree of liberalism (when the individual is in the elite group), and viceversa. In extreme cases, if γ T = 0, for every T, it is possible to have α T > 1 T, although it will be α T 1 T 1. If, on the other hand, making thresholds to vary considerably allows us to establish γ T as close to 1 as desired. The function g {α S} v 4 can be seen as a compromise result, in which thresholds vary restrictively, and the opinion about herself yet plays a significant role. Remark 7 Convergence might require the use of very relaxed thresholds. Especially remarkable is the case of g {α S} ˆ, where α T 1 T 1 for every T N, as mentioned in Remark 6. Thus, the described qualification process could end up with all the individuals in the limit set, even in those cases in which there are some individuals who are not highly evaluated by anyone. For instance, a society in which all assessments are α N leads to the qualification of every individual. To correct the problem to some extent, we can introduce a refinement of which enables us to obtain the same results on convergence, while at the same time partially avoiding the above-mentioned issue. To do this, consider an increasing and bijective function ϕ : [0, 1] [0, 1] and the collective assessment function ˆ defined by ˆ (S, P, j) = (S, ϕ(p ), j), where ϕ(p ) ij = ϕ(p ij ). Theorem 3 can be reproduced for g {α S} ˆ. In this case, it is possible to consider an increasing bijective function in the process, such that individual assessments are lowered before the aggregation process. Take, for instance, values p r ik with r > 1. In this case, in the introductory example, no individual passes the first qualification step. In the case of common assessments, an individual would require an evaluation of ϕ 1 (α N ) to be qualified, where this value could be as close to 1 as desired. Notice, however, that an individual j such that p jj = 1 would still be qualified at any step. 4 Symmetry and contractiveness The previous section contains necessary and sufficient conditions for the convergence of collective qualification functions. Under these conditions, for any profile we may interpret the limit set of the associated chain as the final set of qualified individuals in society. Thus, it is possible to conform an (elitist) rule.

11 A model of elitist qualification 11 Definition 5 Let g {α S} v : 2 N P 2 N be a convergent collective qualification function. The rule associated with g {α S} v, f : P 2 N, is defined by means of f(p ) = lim S t for every P P, where S t is the chain associated with g {α S} v and P. We now study two properties of rules associated with convergent collective qualification functions: symmetry and contractiveness. First, we consider that qualification should be symmetric, i.e., not depend on the individuals names. The following axiom reproduces the symmetry axiom in Samet and Schmeidler [12]. We consider permutations π of the set of individuals, N, and interpret π(i) as the former name of the individual whose new name is i. Given S N, we have πs = {π(j) j S}. Moreover, for any profile P P, we denote by πp the profile permutation obtained as (πp ) ij = p π(i)π(j). Definition 6 A rule f : P 2 N is symmetric if f(πp ) = πf(p ) for every P P and every permutation π. In the following proposition we establish a necessary and sufficient condition for symmetry in the case of rules associated with convergent qualification functions. Proposition 1 Let f : P 2 N be the rule associated with a convergent collective qualification function g {α S} v i, i = 1,..., 5. The following two conditions are equivalent: 1. f is symmetric. 2. α S = α T for all S, T N such that S = T. Proof: To prove that 1 implies 2, suppose by way of contradiction that f is symmetric but there exist S, T N such that S = T and α S α T. Without loss of generality, suppose that α S < α T. Consider a permutation π over the set of individuals such that πs = T, and the following profile: { αs, if j S p ij = 0, otherwise. Since f is convergent, Theorems 1, 2 and 3 guarantee that α N α S. Therefore, for any convergent f it must be clearly f(p ) = S and therefore πf(p ) = T. However, given that α S < α T, it is straightforward to see that f(πp ) =. This is an absurd and therefore, 1 implies 2. The proof that 2 implies 1 is trivial, concluding the proof of the proposition. To motivate the following property, consider a society endowed with a certain collective assessment function. Suppose that an individual has not been qualified when the society uses a threshold family which guarantees the convergence of sequential elitism. If this society becomes more demanding by rising all the thresholds it uses for selecting qualified individuals, preserving convergence, such individual should not be qualified either. In other words, the higher the thresholds a society uses, the smaller the set of qualified individuals is.

12 12 Miguel A. Ballester, José Luis García-Lapresta Definition 7 A collective assessment function v is said to be contractive if, for any pair of strictly positive threshold families {α S } S N, {α S } S N, such that g {α S} v and g {α S v } are convergent, then the following holds: α S α S for every S N f (P ) f(p ) for every P P, where f and f are the rules associated with g {α S} v and g {α S v }, respectively. To conclude our analysis, we provide necessary and sufficient conditions for contractiveness. Theorem 4 The collective assessment functions v 1 and v 2 are contractive. Proof: Let {α S } S N and {α S } S N be two strictly positive threshold families such that g {α S} v 1, g {α S} v 2, g {α S v } 1 and g {α S v } 2 are convergent, and α S α S for every S N. By Theorem 1, α S α T and α S α T whenever S T. Since g {α S v } 1 (N, P ) g {α S} v 1 (N, P ), we have f (P ) f(p ) and, consequently, v 1 is contractive. In order to justify that v 2 is contractive, we proceed by induction over each step of the qualification chains associated to g {α S} v 2 and g {α S v } 2. Obviously, S 1 = g {α S v } 2 (N, P ) g {α S} v 2 (N, P ) = S 1, for every P P. Suppose S t S t for a step t 1. If j S t+1 = g {α S v } 2 (S t, P ), then p ij α S for some i S t. Taking into account α t S α t St, we have p ij α St for some i S t, i.e., j S t+1 = g {α S} v 2 (S t, P ). Consequently, S t+1 S t+1 and v 2 is contractive. In the following result we establish a necessary condition for contractiveness for. Proposition { 2 Suppose γ} S = γ s for all S N such that 2 S = s < n and δ j = min 1 γ j, j 1 j for j = 2,..., n 2. If the collective assessment function is contractive, then γ S 1 γ S δ 2 δ s 1, for every S N such that 2 S < n. Proof: See Appendix A. The previous result can be generalized to the case in which thresholds associated to sets with the same cardinality do not necessarily coincide. However, the case presented here is the most natural and, according to Proposition 1, the one that corresponds to symmetric rules. Theorem 5 The collective assessment function v 4 is contractive. Proof: Let {α S } S N and {α S } S N be two strictly positive threshold families such that g {α S} v 4 and g {α S v } 4 are convergent, and α S α S for every S N. By a direct application of Theorem 3, the following conditions S α S T α T and S α S T α T whenever S T are necessary and

13 A model of elitist qualification 13 sufficient for the convergence of the specific v 4 case. For any profile P P, consider the limit sets S = f(p ), S = f (P ) associated with g {α S} v 4 and g {α S v } 4 and P, respectively. We prove that S is contained in S by induction over the sets of the chain that generates S. It is obvious for S 0 = N. Suppose it is true for certain S t. Then, for every j S, it holds that: p ij p ij S α S S α S S t α St i S t i S which implies that j S t+1. Consequently, S S t+1 and f (P ) f(p ). Thus, the induction reasoning proves that v 4 is contractive. Acknowledgements The authors are grateful to Dinko Dimitrov (who introduced us to the problem of Who is a J? and related literature), Carlos Hervés, Jordi Massó, Jorge Nieto, Pedro Rey-Biel, José Ramón Uriarte and two anonymous referees for their suggestions and comments. This work is partially financed by the Spanish Ministerio de Educación y Ciencia (Projects SEJ /ECON and SEJ /ECON), ERDF, Generalitat de Catalunya (grant 2005SJR00454) and Barcelona Economics Economics-XREA. Appendix A: Proofs Proof of Theorem 3 We first show that 1 implies 2. Suppose, by contradiction, that there exist S, T N such that S T and (1 γ T ) S α S < ( T 1) α T. Consider the profile P defined by: ( ) 0, if j N \ T or i N \ S and j T \ S, p ij = α S, if i S and j T \ S, 1, if j S. It is straightforward to see that 1, if j S, (N, P, j) = (1 γ N ) S α S, if j T \ S, n 1 0, if j N \ T. Consequently, S 2 = g {α S} (N, P ) = S, if α N > (1 γ N) S α S, n 1 T, if α N (1 γ N) S α S. n 1

14 14 Miguel A. Ballester, José Luis García-Lapresta Since 1, if j S, (S, P, j) = α S, if j T \ S, 0, if j N \ T, we have g {α S} (S, P ) = T. On the other hand, 1, if j S, (T, P, j) = (1 γ T ) S α S, if j T \ S, T 1 0, if j N \ T. Since (1 γ T ) S α S < ( T 1) α T, we have g {α T } (T, P ) = S. Hence, the collective qualification function g {α S} does not converge. To prove that 2 implies 1, take a family of thresholds satisfying (1 γ T ) S α S ( T 1) α T, whenever S T N. Given a profile P, we prove by induction that the qualification chain is decreasing, hence, convergent. Since S 0 = N, obviously S 1 = g {α S} (S 0, P ) S 0. Suppose S k+1 = g {α S} (S k, P ) S k is true for k = 1,..., t 1 (in particular S t S t 1 ). In order to prove S t+1 S t by contradiction, suppose there exists j S t+1 = g {α S} (S t, P ) such that j S t. In this case, it must be that i.e., (S t, P, j) = 1 p ij α St, S t i S t i S t p ij S t α St. Consider the greatest integer m such that j S m, with m t 1. Notice that this is well-defined, since S 0 = N. By the induction hypothesis, it must be that S t = S t \ {j} S m \ {j} and p ij p ij S t α St. i S t i S m\{j} Therefore, we can use this inequality to ensure that (S m, P, j) = γ Sm p jj + 1 γ S m S m 1 1 γ S m S m 1 i S m \{j} p ij i S m\{j} p ij 1 γ S m S m 1 S t α St.

15 A model of elitist qualification 15 Noticing that = S t S m, we can apply the hypothesis to the family of thresholds, by which we get 1 γ Sm S m 1 S t α St α Sm. Consequently, (S m, P, j) α Sm, i.e., j g {α S} (S m, P ) = S m+1, which either contradicts the definition of m or the fact that j S t. Therefore, the hypothesis that there exists j S t+1 \ S t is false, concluding the induction argument and the proof. Proof of Proposition 2 Suppose, by contradiction, that there exists T N, with 2 T = t < n, such that γ T 1 γ T < δ 2 δ t 1. Let R be a minimal subset satisfying the above property, i.e., γ R 1 γ R < δ 2 δ r 1 and γ S 1 γ S δ 2 δ s 1, for every S R such that S = s 2, and R = r. We now show that there exists a family of thresholds {α S } S N that α s = δ 2 δ s 1 (1 γ s ), such for every S R and S = s, allowing the convergence of. Taking into account Theorem 3, we need to justify that α j s j 1 α s (1 γ j ) for S = s = 2,..., j 1 and j = 3,..., r. First we note that α 2 = 1 γ 2. j = 3 and s = 2: α 3 = δ 2 (1 γ 3 ) (1 γ 2 ) (1 γ 3 ) = α 2 (1 γ 3 ). j = 4 and s = 2: α 4 = δ 2 δ 3 (1 γ 4 ) (1 γ 2 ) 2 3 (1 γ 4) = 2 3 α 2 (1 γ 4 ). j = 4 and s = 3: α 4 = δ 2 δ 3 (1 γ 4 ) δ 2 (1 γ 3 ) (1 γ 4 ) = α 3 (1 γ 4 ). It is straightforward to complete the proof for the rest of values of s and j. We choose α 1 = 1 and the remaining thresholds α S in order to satisfy the convergence conditions of Theorem 3. Now let α s = α s for all s < n and α N = α N ε for a sufficiently small value of ε > 0. Since α S α S for every S N, in order to obtain a contradiction, we now justify why is not contractive. We need to prove that f(p ) is not included in f (P ) for some profile P. Select the profile P defined by: p kk = 1, for some k R, p jj = α N, for every j R \ {k},

16 16 Miguel A. Ballester, José Luis García-Lapresta p ij = 0, for every j N \ R, p ij = 0, for all i, j R \ {k} such that i j, p ij = λ, for all i N \ R and j R, p ik = µ, for every i R \ {k}. The value λ [0, 1] is selected in such a way that: γ N α N + 1 γ N n 1 (n r) λ [α N, α N ). To see that this is possible, notice that, for λ = 0 it holds that: γ N α N + 1 γ N n 1 (n r) λ = γ N α N < α N < α N. On the other hand, for λ = 1 it holds: γ N α N + 1 γ N n 1 (n r) λ 1 γ N n 1 = (1 γ α 1 N) n 1 α N α N. The value µ is selected as follows: If γ N > 0, take µ = 0. If γ N = 0, let µ simultaneously satisfy the following inequalities: n r n 1 λ + r 1 n 1 µ α N = α N + ε µ < α r γ R 1 γ R, i.e., γ R + (1 γ R ) µ < α r = α r. Notice that the first inequality is possible because ε can be as small as we want. On the other hand, α r γ R > 0 is guaranteed by the selection of R and the family of thresholds. In order to see that g {α S v } 5 (N, P ) = R, it is necessary to justify that: (N, P, j) = γ N p jj + 1 γ N n 1 i N\{j} Some simple calculations prove this statement: (N, P, k) = γ N + 1 γ N ((r 1) µ + (n r) λ) n 1 γ N α N + 1 γ N n 1 (n r) λ α N. p ij α N j R. j R \ {k}: (N, P, j) = γ N α N + 1 γ N n 1 (n r) λ α N. j N \ R: (N, P, j) = 0 < α N. We now justify that g {α S } (R, P ) =, i.e., (R, P, j) < α r for every j N:

17 A model of elitist qualification 17 (R, P, k) = γ R + 1 γ R r 1 (r 1) µ = γ R + (1 γ R ) µ < α r (when either γ N = 0 or γ N > 0). j R \ {k}: (R, P, j) = γ R α N γ R < δ 2 δ r 1 (1 γ R ) = α r. j N \ R: (R, P, j) = 0 < α r. Then, f (P ) =. We now justify that g {α S} (N, P ) = {k}: (N, P, k) = γ N + 1 γ N ((r 1) µ + (n r) λ). n 1 1. When γ N > 0, taking into account that µ = 0 and α N < 1, an adequate selection of ε (as small as desired, allowing all requisites) guarantees that: (N, P, k) > γ N α N + ε + 1 γ N n 1 (n r) λ α N + ε = α N. 2. When γ N = 0, it holds that: (N, P, k) = n r n 1 λ + r 1 n 1 µ α N. j R \ {k}: (N, P, j) = γ N α N + 1 γ N n 1 (n r) λ < α N. j N \ R: (N, P, j) = 0 < α N. On the other hand, it is clear that g {α S} ({k}, P ) = {k}. Then, f(p ) = {k} is not included in f (P ) =, contrary to the hypothesis. References 1. Barberà, S., Maschler, M. and Shalev, J.: (2001), Voting for voters: a model of electoral evolution, Games and Economic Behavior 37: Candeal, J.C., Induráin, E. and Uriarte, J.R.: (1992), Some issues related to the topological aggregation of preferences, Social Choice and Welfare 9: Candeal, J.C. and Induráin, E.: (1995), Aggregation of preferences from algebraic models on groups, Social Choice and Welfare 12: Çengelci, M.A. and Sanver, M.R.: (forthcoming), Simple collective identity functions, Theory and Decision. 5. Dimitrov, D., Sung, S.C. and Xu, Y.: (2007), Procedural group identification, Mathematical Social Sciences 54: García-Lapresta, J.L. and Llamazares, B.: (2000), Aggregation of fuzzy preferences: Some rules of the mean, Social Choice and Welfare 17: Houy, N.: (2006), I want to be a J!: Liberalism in group identification problems, Hitotsubashi University. 8. Houy, N.: (2006), He said that he said that I am a J, Economics Bulletin 4(4): 1 6.

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