Working Papers Institute of Mathematical Economics September 2006 383 Procedural Group Identification Dinko Dimitrov, Shao Chin Sung and Yongsheng Xu IMW · Bielefeld University Postfach 100131 33501 Bielefeld · Germany email: imw@wiwi.uni-bielefeld.de http://www.wiwi.uni-bielefeld.de/˜imw/Papers/showpaper.php?383 ISSN: 0931-6558 Procedural group identi…cation Dinko Dimitrov Institute of Mathematical Economics Bielefeld University, Germany Email: d.dimitrov@wiwi.uni-bielefeld.de Shao Chin Sung Department of Industrial and Systems Engineering Aoyama Gakuin University, Japan Email: son@ise.aoyama.ac.jp Yongsheng Xu Department of Economics, Andrew Young School of Policy Studies Georgia State University, U.S.A. Email: yxu3@gsu.edu and Graduate School of Public Finance and Policy Central University of Finance and Economics Beijing, China September 14, 2006 Abstract In this paper we axiomatically characterize two recursive procedures for de…ning a social group. The …rst procedure starts with the This is a substantial revision of our CentER Discussion Paper 2003-10, Tilburg University, with the same title. D. Dimitrov gratefully acknowledges …nancial support from the German Research Foundation (DFG). 1 set of all individuals who are de…ned by everyone in the society as group members, while the starting point of the second procedure is the set of all individuals who de…ne themselves as members of the social group. Both procedures expand these initial sets by adding individuals who are considered to be appropriate group members by someone in the corresponding initial set, and continue inductively until there is no possibility of expansion any more. JEL Classi…cation: D63, D71. Keywords: consensus, liberalism, procedure, social identity 1 Introduction The problem of group identi…cation serves as a background in many social and economic contexts. For example, when one examines the political principle of self-determination of a newly formed country, one would like to de…ne the extension of a given nationality. Or when a newly arrived person in Atlanta chooses where to live, the person is interested in …nding out a residential neighborhood that would suit her: “Are they my kind of people? Do I belong to this neighborhood?” In all those contexts, it is typically assumed that there is a well-de…ned group of people who share some common values, beliefs, expectations, customs, jargon, or rituals. Consequently, questions like “how to de…ne a social group” or “who belongs to the social group” arise. In very recent papers (see Billot (2003), Çengelci and Sanver (2006), Houy (2006a,b), Kasher and Rubinstein (1997), Miller (2006), Samet and Schmeidler (2003)) this problem has been related to formal models from social choice and voting theory. Kasher’s (1993) paper on collective identity can be considered as a …rst, non-formal attempt to look at the group identi…cation problem as an aggregation task. In that paper the author views that each individual in a society has an opinion about every individual, including oneself, whether the latter 2 is a member of a group to be formed1 . The collective identity of the group to be formed is then determined by aggregating opinions of all individuals in the society. The formal link between Kasher’s approach and the theory of aggregators mainly developed in economic theory was made by Kasher and Rubinstein (1997). For this purpose, they provide, among others, an axiomatic characterization of a “liberal” aggregator whereby the group consists of those and only those individuals who each of them views oneself a member of the group (see also Çengelci and Sanver (2006), Houy (2006a), Miller (2006), Samet and Schmeidler (2003), Sung and Dimitrov (2005)). The purpose of this paper is to extend the study of the group identi…cation problem by adding a procedural view in the analysis. This procedural view allows us to see a collective as “a family of groups, subcollectives, each with its own view of who is a member of the collective, its own sense of tradition and its own underlying conceptual realm, but each bearing some resemblance to the other ones” (Kasher (1993, p. 70)). More speci…cally, we axiomatically characterize two recursive procedures for determining “who is a member of a social group”: a consensus-start-respecting procedure which is the one introduced by Kasher (1993) and a liberal-start-respecting procedure which adds a procedural view to the “liberal” aggregation of Kasher and Rubinstein (1997). The structure of both procedures consists of two components: an initial set of individuals and a rule according to which new individuals are added to this initial set. As the names of the procedures suggest, the initial set of the …rst procedure consists of all individuals who are de…ned as group members 1 An analysis of the question how the identity of an individual has been formed or what is the impact of identity on economic behavior is out of the scope of this paper. With respect to those questions the interested reader is referred to Sen (1999) and Akerlof and Kranton (2000), respectively. 3 by everyone in the society, while the initial set of the second procedure collects all individuals who de…ne themselves as members of the social group. The extension rule for both procedures is the same: only those individuals who are considered to be appropriate group members by someone in the corresponding initial set are added. The application of this rule continues inductively until there is no possibility of expansion any more.2 An initial set can be interpreted for example as a set of society founders who choose new society members from a …nite set of candidates (see Berga, Bergantiños, Massó, and Neme (2004)), and the extension rule (the voting rule) is “voting by quota one”, i.e., it is enough for a candidate to receive one vote in order to be admitted (see Barberà, Maschler, and Shalev (2001)). In contrast to the cited papers, we study here the problem of group formation in a social choice setting and do not consider a predesignated set of society founders. We allow rather for the possibility that the views of all individuals in the society determine endogenously who is a society founder. The rest of the paper is organized as follows. In Section 2, we present the basic notation and de…nitions. Sections 3 discusses the axioms that are necessary and su¢cient to reach logically the consensus-start-respecting procedure and presents our characterization result. Section 4 is devoted to the corresponding axioms and characterization of the liberal-start-respecting procedure. We conclude in Section 5 with some …nal remarks. 2 For an axiomatic characterization of the aggregator selecting the agents that are indirectly designated by all individuals in the society the reader is referred to Houy (2006b). 4 2 Basic notation and de…nitions Let N = f1; : : : ; ng denote the set of all individuals in the society and assume that n 2. The set of all subsets of N is denoted by P (N ). Each individual i 2 N forms a set Gi N consisting of all society members that in the view of i have the social identity G. It may be noted that it is possible to have Gi = ; for some i 2 N . For all i 2 N , when i 2 Gi , we also say that i considers himself as a G. A pro…le of views is an n-tuple of vectors G = (G1 ; : : : ; Gn ) where Gi N for all i 2 N . Let G be the set of all pro…les of views, i.e., G = (P (N ))n . A collective identity function (CIF) F : G ! P (N ) assigns to each pro…le G 2 G a set F (G) N of socially accepted group members. In what follows, we denote by F the set of all collective identity functions. Kasher (1993) o¤ers a neutral method for de…ning the collective identity, i.e., a method which is “... free of any commitment to some partisan view of the nature of the collective”. This method is introduced as follows. For any G 2 G, let K0 (G) = fi 2 N : i 2 Gk for all k 2 N g. We de…ne a CIF being consensus-start-respecting, to be denoted by K(G), as follows: for each positive integer t, let Kt (G) = Kt k 2 Kt 1 (G)g; and if for some t 1 (G) [ fi 2 N : i 2 Gk for some 0, Kt (G) = Kt+1 (G), then K(G) = Kt (G). For each G 2 G the procedure K starts with K0 (G) which consists of all individuals who are viewed by everyone in the society as group members. Kasher (1993) calls the set K0 (G) the “incontrovertible core” of a collective to be de…ned and he considers it as an initial approximation to an appropriate de…nition of the group identity. Notice that K0 (G) does not re‡ect the di¤erences in views of “who is a G” held by those who are unquestionably Gs. Because one is interested in a neutral aggregation rule, an “improved approximation” is needed. For each G 2 G, the CIF K now expands the set 5 K0 (G) as follows. If, according to some individual i 2 K0 (G) an individual k 2 N is viewed as a G, then k should be a G collectively. By adding all such ks to K0 (G), we obtain the set K1 (G). We then repeat the above process with K1 (G) by adding those individuals who are considered as Gs by some individual in K1 (G) to K1 (G) to obtain K2 (G). Since n is …nite, at a certain step t, we must have Kt (G) = Kt+1 (G), i.e., the set Kt (G) can no longer be expanded. The intuition behind each step of the expansion is in line with Kasher’s (1993) argument: every socially accepted G as being newly added brings a possibly unique new view of being a G collectively with him; since a collective identity function is supposed to aggregate those views, it must pay attention to this new individual’s G-concept in order to cover the whole diversity of views in the society about the question “what does it mean to be a G?”. We turn now to the liberal-start-respecting procedure mentioned by Kasher and Rubinstein (1997). For any G 2 G, let L0 (G) = fi 2 N : i 2 Gi g. With the help of L0 (G), we de…ne a CIF being liberal-start-respecting, to be denoted by L(G), as follows: for each positive integer t, let Lt (G) = Lt N : i 2 Gk for some k 2 Lt 1 (G)g; and if for some t 1 (G) [ fi 2 0, Lt (G) = Lt+1 (G), then L(G) = Lt (G). Notice that the extension rule for L and K is the same (and so the intuition behind it), but the initial set is di¤erent: the liberal-start-respecting procedure starts with L0 (G) which consists of all members of the society who view themselves as Gs. Thus, the set L0 (G) re‡ects a weak notion of selfdetermination: if one considers oneself a member of G, then one should be a member of G collectively. Therefore, the procedure L re‡ects a strong liberal view of collective identity3 . 3 See, for example, Çengelci and Sanver (2006), Houy (2006a), Kasher and Rubinstein 6 To illustrate the above procedures for de…ning collectively accepted group members, consider the following example. Let N = f1; 2; 3g and consider the pro…le G = (G1 ; G2 ; G3 ) with G1 = f1; 2g; G2 = f2; 3g and G3 = f2g. Then, for this pro…le, K0 = f2g, K1 = K0 [ f3g = f2; 3g, K2 = K1 . Therefore, the collectively accepted group members according to the consensus-startrespecting procedure are collected in the set K = f2; 3g. For the same pro…le G of individual views we have L0 = f1; 2g, L1 = L0 [ f3g = f1; 2; 3g, L2 = L1 . Therefore, for the given pro…le of views, and as a result of the application of the liberal-start-respecting procedure, we have L = f1; 2; 3g. It should be noted that, from their respective de…nitions, for all pro…les G 2 G, K(G) 3 L(G). The consensus-start-respecting procedure In this section we o¤er an axiomatic characterization of the Kasher’s method for de…ning a social group. For that purpose we start with the following two axioms a CIF may satisfy. A CIF F 2 F satis…es consensus (C) if for all G 2 G, – [i 2 Gk for every k 2 N ] implies [i 2 F (G)], and – [i 2 = Gk for every k 2 N ] implies [i 2 = F (G)]. A CIF F 2 F satis…es irrelevance of an outsider’s view 1 (IOV1) if for all G; G0 2 G and for all i; j 2 N , (1997), Miller (2006), Samet and Schmeidler (2003), Sung and Dimitrov (2005). If the determination of the membership of a social group is a personal matter, there is indeed some reason to call individuals in L0 as liberals (see Sen (1970)). 7 – G0j = Gj [ fig, and – G0l = Gl for all l 2 N n fjg, imply – [j 62 F (G) and i 62 G0k for some k 2 N ] ) [i 2 F (G) i¤ i 2 F (G0 )]. Consensus is used by Kasher and Rubinstein (1997) to reach logically their “liberal” CIF and, in fact, sounds very plausible when imposed as a requirement on a collective identity function. This axiom says that, if an individual is de…ned as a group member by everyone in the society, then this individual should be considered as a socially accepted group member; and, correspondingly, if no one de…nes this individual as a group member, then he or she should not deserve the social acceptance as a group member. Irrelevance of an outsider’s view 1 is in the spirit of the exclusive selfdetermination axiom introduced by Samet and Schmeidler (2003) and it basically stipulates that if someone is collectively de…ned as a non-G, then this person’s view about any society member is not relevant in deciding his or her collective identity. Note however that there is one case, in which the view of an outsider cannot be deemed as irrelevant; this case corresponds to the situation in which everyone in the society except the outsider j considers i as a G, so that the change of j’s view in favour of i is (via (C)) relevant for the social identi…cation of i. As the reader can see, we exclude this case in (IOV1) by requiring that there is a k 2 N such that i 62 G0k . It should also be noted that (IOV1) is weaker than the exclusive self-determination axiom used by Samet and Schmeidler (2003). Consider now a pro…le G 2 G and the group of socially accepted society members F (G) generated by a CIF F 2 F that satis…es both axioms. Our …rst result relates F (G) with the result of the consensus-start-respecting 8 procedure K at pro…le G. As it turns out, K(G) acts as an upper bound for F (G) at any pro…le G 2 G. Proposition 1 If a CIF F 2 F satis…es (C) and (IOV1), then F (G) K(G) for all G 2 G. Proof. Let F 2 F satisfy (C) and (IOV1). We start by observing that the claim “F (G) K(G) for all G 2 G” is equivalent to “for all G 2 G and for all i 2 N : i 62 K(G) implies i 62 F (G)”. Hence, we prove this equivalent claim by induction of jGj, where jGj = jG1 j + jG2 j + : : : + jGn j. Basis Step: When jGj = 0, we have Gl = ; for all l 2 N . Thus, i 2 = Gj for all i; j 2 N . From (C), F (G) = K(G) = ;. Induction Step: Let g be a non-negative integer such that g < n2 . Assume that the claim holds for all G 2 G with jGj = g, and we show that the claim holds for all G 2 G with jGj = g + 1. Let G 2 G be such that jGj = g + 1, and let i 2 N be such that i 62 K(G). If i 62 Gj for all j 2 N , then from (C) we have i 62 F (G). Suppose there exists j 2 N such that i 2 Gj . By de…nition of K, from i 62 K(G) and i 2 Gj , it then follows that j 62 K(G). Moreover, from i 62 K(G), there also exists k 2 N such that i 62 Gk . Observe that k 6= j. Let G0 2 G be such that G0 = (G1 ; : : : Gj 1 ; Gj n fig ; Gj+1 ; : : : ; Gn ). By de…nition of K, we have K(G0 ) ously, jG0 j = jGj K(G), which implies i; j 62 K(G0 ). Obvi- 1 = g. By induction hypothesis, we have F (G0 ) K(G0 ). Thus, we have i; j 62 F (G0 ) from i; j 62 K(G0 ). Notice that for the pro…le G0 we have i 62 G0j and i; j 62 F (G0 ), and for the pro…le G we have Gj = G0j [ fig, i 62 Gk for some k 6= j and Gl = G0l for all l 2 N n fjg. Hence, applying (IOV1) with G0 and G in the roles of G and G0 , respectively, we conclude that i 62 F (G). 9 In order to complete the characterization of K we have to show also the reverse inclusion to the one in Proposition 1. For that purpose, we introduce our third axiom. A CIF F 2 F satis…es equal treatment of insiders’ views (ETIV) if for all G; G0 2 G and for all i; j; k 2 N , – i 2 Gj , – G0j = Gj n fig, and G0k = Gk [ fig – G0l = Gl for all l 2 N n fj; kg, imply – [j 2 F (G) and k 2 F (G0 )] ) [i 2 F (G) i¤ i 2 F (G0 )]. Equal treatment of insiders’ views requires that if an individual i is considered to be an appropriate group member by an individual j, i 2 Gj in a given pro…le, and if in a new pro…le j does not consider i as an appropriate group member anymore but a third individual k does, and nothing else has changed, then, when j is a collectively accepted group member in the original pro…le and k is a collectively accepted member in the new pro…le, it must be true that i is a G collectively in the original pro…le if and only if i is a G collectively in the new pro…le. This axiom essentially requires that a CIF should treat the views of all the members who are considered to be Gs collectively equally. Proposition 2 If a CIF F 2 F satis…es (C), (ETIV) and (IOV1), then K(G) F (G) for all G 2 G. Proof. Let F 2 F satisfy (C), (ETIV) and (IOV1). Note …rst that the claim “K(G) F (G) for all G 2 G” is equivalent to “Kt (G) 10 F (G) for all G 2 G and for all non-negative integers t induction of g = n2 n”. We prove this equivalent claim by jGj and t. Basis Step (g = 0): When jGj = n2 , we have Gl = N for all l 2 N . Thus, i 2 Gj for all i; j 2 N , i.e., F (G) = N follows from (C). Therefore, we have Kt (G) F (G) for all non-negative integers t n. 0): Let g be a non-negative integer such that g < n2 . Induction Step (g We assume that F (G) for all G 2 G with jGj = n2 K(G) F (G) for all G 2 G with jGj = n2 and show Kt (G) (IH1) g 1 and for all 0 and t = 0): From (C), we have K0 (G) F (G) for all non-negative integers t Basis Step (g g; n. G 2 G. Induction Step (g 0 and t 0): Let t be a non-negative integer such that t < n. We further assume that Kt (G) F (G) for all G 2 G with jGj = n2 and we show Kt+1 (G) F (G) for all G 2 G with jGj = n2 Let G 2 G be such that jGj = n2 we have K(G) = Kt (G) g g 1; g (IH2) 1. 1. If Kt+1 (G) n Kt (G) = ;, then F (G). Suppose there exists i 2 Kt+1 (G) n Kt (G). Then, by de…nition of K, there exists j 2 Kt (G) such that i 2 Gj . Hence, we have j 6= i, and by induction hypothesis (IH2), we conclude that j 2 F (G). Moreover, since i 62 K0 (G), there exists k 2 N such that i 62 Gk . Observe that k 6= j. Let H 2 G be such that H = (G1 ; : : : ; Gk 1 ; Gk [ fig ; Gk+1 ; : : : ; Gn ). We consider now the following two cases: 11 (1) Suppose k 2 F (H). Notice from i 62 Gk that jHj = jGj + 1 = n2 and thus, by induction hypothesis (IH1), K(H) tion 1, we have F (H) K, K(G) g, F (H). From Proposi- K(H), and thus, F (H) = K(H). By de…nition of K(H), we have i 2 Kt+1 (G) K(G) K(H) = F (H). Then, notice that for the pro…le H we have i 2 Hk and i; k 2 F (H), and for the pro…le G we have i 2 Gj , Gk = Hk n fig, Gl = Hl for all l 2 N n fkg, and j 2 F (G). Hence, applying (ETIV) with H, G, i, k, j in the roles of G, G0 , i, j, k, respectively, we conclude that i 2 F (G). (2) Suppose k 62 F (H). From K(H) = F (H), we have K(G) F (H), and thus, k 62 K(G). From Proposition 1, we have F (G) thus, k 62 F (G). Moreover, from i; j 2 Kt+1 (G) K(H) = K(G), and K(G) and k 62 K(G), we have k 62 Gi and k 62 Gj . Let H 0 2 G be such that H 0 = (G1 ; : : : ; Gi 1 ; Gi [ fkg; Gk+1 ; : : : ; Gn ). From k 62 Gi , we have jH 0 j = jGj + 1 = n2 g. By induction hypothesis (IH1) and Proposition 1, we have F (H 0 ) = K(H 0 ). From i 2 K(G) K(H 0 ) and k 2 Hi0 , we have k 2 K(H 0 ), and from F (H 0 ) = K(H 0 ), we have k 2 F (H 0 ). Notice that for the pro…le G we have k 62 Gi and k 62 F (G), and for the pro…le H 0 we have Hi0 = Gi [ fkg, Hl0 = Gl for all l 2 N n fig, k 2 F (H 0 ), and from j 6= i, we have k 62 Gj = Hj0 . Then, applying (IOV1) with G, H 0 , k, i, j in the roles of G, G0 , i, j, k, respectively, we conclude that i 2 F (G). Theorem 1 A CIF F 2 F satis…es (C), (ETIV) and (IOV1) if and only if F = K. Moreover, all three axioms are independent. Proof. It is easy to check that the consensus-start-respecting procedure satis…es (C), (ETIV) and (IOV1). The combination of Proposition 1 and Proposition 2 proves that, if a CIF F 2 F satis…es (C), (ETIV) and (IOV1) then it is K. Hence, we need only to show that the axioms are tight. The 12 proof consists of three examples, each of which satis…es exactly two of the three axioms. (:(C)) Let F 2 F be such that F (G) = ; for all G 2 G. Clearly, this CIF satis…es all axioms but (C). (:(ETIV)) Consider the CIF F 2 F with F (G) = K0 (G) [ fi 2 N : i 2 Gj for all j 2 K0 (G)g for all G 2 G. The following example shows that this aggregator does not satisfy (ETIV). Let N = f1; 2; 3g, G = (f1; 2; 3g ; f1g ; f1g), and G0 = (f1; 3g ; f1g ; f1; 2g). For these pro…les of views we have F (G) = f1; 2; 3g and F (G0 ) = f1; 3g. In this case (ETIV) is violated because 1 2 F (G), 3 2 F (G0 ), 2 2 G1 , G01 = G1 n f2g, G03 = G3 [ f2g, but 2 2 F (G) and 22 = F (G0 ). (:(IOV1)) Take the liberal-start-respecting procedure that de…nes the CIF L 2 F. If we set j = i in the formulation of (IOV1) we immediately see that this axiom is violated. 4 The liberal-start-respecting procedure For the axiomatic characterization of the liberal-start-respecting procedure de…ned in Section 2 we have …rst to modify the irrelevance of an outside’s view 1 axiom. A CIF F 2 F satis…es irrelevance of an outsider’s view 2 (IOV2) if for all G; G0 2 G and for all i; j 2 N with i 6= j, – G0j = Gj [ fig, and 13 – G0l = Gl for all l 2 N n fjg, imply – [j 62 F (G)] ) [i 2 F (G) i¤ i 2 F (G0 )]. This axiom basically says, like (IOV1), that if someone is collectively considered as a non-G, then this person’s view about any society member is not relevant in deciding his or her collective identity. Recall that in the formulation of (IOV1) it was crucial to avoid the case, in which everyone in the society except the outsider j considers i as a G, so that the change of j’s view in favour of i is relevant for the social identi…cation of i (i.e., in this way a possible tension between (IOV1) and (C) was excluded). Notice that the liberal-start-respecting procedure does not satisfy (IOV1) because one’s self-determination de…nes immediately one’s social status: there is no consensus needed in this case. Therefore, in order to avoid the situation in which an individual becomes crucial for his own social determination (from being outsider to being insider) we require i 6= j in the formulation of (IOV2). As it turns out, the combination of (C) and (IOV2) plays a similar role for a CIF F 2 F as the role of the combination of (C) and (IOV1): it produces an upper bound for F at any pro…le G 2 G. This upper bound is exactly the set of socially accepted group members at G according to the liberal-startrespecting procedure. Proposition 3 If a CIF F 2 F satis…es (C) and (IOV2), then F (G) L(G) for all G 2 G. Proof. Let F 2 F satisfy (C) and (IOV2). Observe again that the claim “F (G) L(G) for all G 2 G” is equivalent to “for all G 2 G and for all i 2 N : i 62 L(G) implies i 62 F (G)”. Hence, we prove this equivalent claim by induction of jGj, where jGj = jG1 j + jG2 j + : : : + jGn j. 14 Basis Step: When jGj = 0, we have Gl = ; for all l 2 N . Thus, i 2 = Gj for all i; j 2 N . From (C), F (G) = L(G) = ;. Induction Step: Let g be a non-negative integer such that g < n2 . Assume that the claim holds for all G 2 G with jGj = g, and we show that the claim holds for all G 2 G with jGj = g + 1. Let G 2 G be such that jGj = g + 1, and let i 2 N be such that i 62 L(G). If i 62 Gj for all j 2 N , then from (C) we have i 62 F (G). Suppose there exists j 2 N such that i 2 Gj . Note that j 6= i. By de…nition of L, i 62 L(G) and i 2 Gj imply j 62 L(G). Let G0 2 G be such that G0 = (G1 ; : : : Gj 1 ; Gj n fig ; Gj+1 ; : : : ; Gn ). By de…nition of L, we have L(G0 ) jG0 j = jGj L(G), which implies i; j 62 L(G0 ). Obviously, 1 = g. By induction hypothesis, we have F (G0 ) L(G0 ). Thus, we have i; j 62 F (G0 ) from i; j 62 L(G0 ). Notice that for the pro…le G0 we have i; j 62 F (G0 ), and for the pro…le G we have Gj = G0j [ fig and Gl = G0l for all l 2 N n fjg. By noticing that i 6= j and applying (IOV2) with G0 and G in the roles of G and G0 , respectively, we conclude that i 62 F (G). Finally, we introduce the following monotonicity requirement. A CIF F 2 F satis…es monotonicity (MON) if for all G; G0 2 G, – [Gk G0k for every k 2 N ] implies [F (G) F (G0 )]. In this axiom, used also by Samet and Schmeidler (2003), pro…les G and G0 are considered such that every individual who deserves to be a group member according to someone in the pro…le G is de…ned as a group member by the same person also in G0 . Then, (MON) requires that in this case every individual who is socially accepted in G is accepted in G0 as well. 15 It turns out that combining (C), (MON), (ETIV) and (IOV2) results in the existence of a lower bound for a CIF F 2 F at a given pro…le G 2 G; this lower bound is exactly the result of the liberal-start-respecting procedure at the same pro…le. Proposition 4 If a CIF F 2 F satis…es (C), (MON), (ETIV) and (IOV2), then L(G) F (G) for all G 2 G. Proof. Let F 2 F satisfy (C), (MON), (ETIV) and (IOV2). In the following, in order to prove L(G) F (G) for all pro…les G 2 G F (G), we prove Lt (G) by induction on t. F (G) for all G 2 G. Suppose to Basis Step: We …rst show L0 (G) the contrary that there exists a pro…le G 2 G such that i 2 L0 (G) but i 62 F (G) for some i 2 N . Notice then that, by i 62 F (G) and (C), the set Q := fj 2 N : i 2 = Gj g is nonempty; clearly, i 2 = Q. Let G0 2 G be a pro…le such that G0k = Gk n Q for all k 2 N . Then, by i 62 F (G) and (MON), i 2 = F (G0 ). By (C), Q \ F (G0 ) = ;. Let j 2 Q and consider the pro…le = F (G0 ) G00 2 G where G00k = G0k for all k 2 N n fjg, and G00j = G0j [ fig. By i 2 and (IOV2), i 2 = F (G00 ). Notice again that, by (C), Q \ F (G00 ) = ;. By repeating the same argument (jQj 1)-times, we arrive at pro…le G 2 G with Gk = G0k for all k 2 N n Q, and Gk = G0k [ fig for all k 2 Q. Thus, by (IOV2), i 2 = F (G ). Observe however that i 2 Gk for all k 2 Q and i 2 G0k = Gk for all k 2 N n Q. By (C), i 2 F (G ), a contradiction. We conclude that L0 (G) F (G) for all G 2 G must hold. Induction Step: Let t be a nonnegative integer. We assume that Lt (G) F (G) for all G 2 G and show that Lt+1 (G) F (G) for all G 2 G. Let i 2 Lt+1 (G). If i 2 Lt (G), then i 2 F (G) from Lt (G) F (G). Assume therefore i 62 Lt (G). From the de…nition of L, there exists j 2 Lt (G) such that i 2 Gj . 16 Note that j 2 F (G), which follows from j 2 Lt (G) F (G), and that i 6= j, which is due to i 62 Lt (G) and j 2 Lt (G). Let G0 2 G be a pro…le such that G0 = (G1 ; : : : ; Gi 1 ; Gi [ fig ; Gi+1 ; : : : ; Gj 1 ; Gj n fig ; Gj+1 ; : : : ; Gn ). From the de…nition of L0 , i 2 L0 (G0 ). From L0 (G0 ) F (G0 ), it follows that i 2 F (G0 ). Noting that j 2 F (G) and i 2 F (G0 ), by (ETIV), we obtain i 2 F (G). Therefore, L(G) F (G) for all G 2 G. Theorem 2 A CIF F 2 F satis…es (C), (MON), (ETIV) and (IOV2) if and only if F = L. Moreover, all four axioms are independent. Proof. It is easy to check that the liberal-start-respecting procedure satis…es (C), (MON), (ETIV) and (IOV2). The combination of Proposition 3 and Proposition 4 proves that, if a CIF F 2 F satis…es (C), (MON), (ETIV) and (IOV2) then it is L. Hence, we need only to show that the axioms are tight. The proof consists of four examples, each of which satis…es exactly three of the four axioms. (:(C)) Let F 2 F be such that F (G) = N for all G 2 G. Clearly, this CIF satis…es all axioms but (C). (:(MON)) Let N = f1; 2g and consider the CIF F de…ned as follows. 8 > ; if G 2 f(;; ;) ; (;; f1g) ; (f2g ; ;) ; (f2g ; f1g)g ; > > > > < f1g if G 2 f(f1g ; f1; 2g) ; (f1g ; f1g) ; (f1g ; ;)g ; F (G) = > f2g if G 2 f(f2g ; f2g) ; (;; f2g)g ; > > > > : f1; 2g otherwise. This CIF satis…es all axioms except (MON). To see that (MON) is violated, take G = (f1g ; f1; 2g), G0 = (f1g ; f2g). According to the proposed aggregator we have F (G) = f1g and F (G0 ) = f1; 2g. Notice that G0k = Gk for k = 1; 3, and G02 G2 . Nevertheless, F (G) 17 F (G0 ). (:(ETIV)) Take Kasher and Rubinstein’s “liberal” aggregator, i.e., consider L0 2 F with L0 (G) = fi 2 N : i 2 Gi g for all G 2 G. In order to see that L0 violates (ETIV), take j 2 L0 (G), i 2 = L0 (G), k 2 L0 (G0 ), and set k = i. (:(IOV2)) Let F 2 F be de…ned as follows: F (G) = fi 2 N : i 2 Gj for some j 2 N g for all G 2 G. Clearly, this CIF does not satisfy (IOV2) because the change of an outsider’s opinion in favour of i changes i’s social status. 5 Conclusion In this paper, we have axiomatically characterized the procedures that de…ne the collective identity functions K and L in the framework proposed by Kasher and Rubinstein (1997). The consensus-start-respecting procedure is characterized by consensus, irrelevance of an outsider’s view 1, and equal treatment of insiders’ views. The axioms (C) and (IOV1) guarantee that any CIF satisfying them selects only socially accepted group members that K would also select. The characterization of K is based on the following simple observation: given a pro…le G 2 G, the axiom (C) guarantees that, for any CIF F 2 F that satis…es it, F (G) contains K0 (G). The application of (IOV1) gives the result that there is no individual in K1 (G) whose social status is determined by someone outside of K0 (G), and (ETIV) implies that only individuals in K0 (G) are crucial for determining an individual’s social status. The induction argument in the proofs completes the characterization. A similar observation can be made with respect to the liberal-start18 respecting procedure that is characterized by consensus, monotonicity, equal treatment of insiders’ views, and irrelevance of an outsider’s view 2. Here (MON) is crucial for guaranteeing, together with (C) and (IOV2), that, for any pro…le G 2 G and any CIF F 2 F satisfying these axioms, F (G) contains L0 (G). It may be noted that the roles an outsider plays in determining someone’s social status in these two procedures are quite di¤erent: for K, an outsider’s change of his opinion in favor of himself is inconsequential, while for L, an outsider who changes his opinion in favour of himself becomes crucial in determining his own social status. References [1] Akerlof, G. and R. Kranton (2000): Economics and identity, Quarterly Journal of Economics 115, 715-753. [2] Barberà, S., M. Maschler, and J. Shalev (2001): Voting for voters: A model of electoral evolution, Games and Economic Behavior 37, 40-78. [3] Berga, D., G. Bergantiños, J. Massó, and A. Neme (2004): Stability and voting by committees with exit, Social Choice and Welfare 23(2), 229-247. [4] Billot, A. (2003): How liberalism kills democracy or Sen’s theorem revised, Public Choice 116, 247-270. 19 [5] Çengelci, M. A. and M. R. Sanver (2006): Embracing liberalism for collective identity determination, Mimeo, Bilgi University. 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