Elsevier

Mathematical Social Sciences

Volume 54, Issue 2, September 2007, Pages 137-146
Mathematical Social Sciences

Procedural group identification

https://doi.org/10.1016/j.mathsocsci.2007.06.001Get rights and content

Abstract

In this paper we axiomatically characterize two recursive procedures for defining a social group. The first procedure starts with the set of all individuals who are defined by everyone in the society as group members, while the starting point of the second procedure is the set of all individuals who define 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.

Introduction

The problem of group identification 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 define the extension of a given nationality. Or when a newly arrived person in Atlanta chooses where to live, the person is interested in finding 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-defined group of people who share some common values, beliefs, expectations, customs, jargon, or rituals. Consequently, questions like “how to define a social group” or “who belongs to the social group” arise. In some recent papers (see Billot (2003), Çengelci and Sanver (2006), Houy, 2006, Houy, 2007, 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 first, non-formal attempt to look at the group identification 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 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 (2007), Miller (2006), Samet and Schmeidler (2003), Sung and Dimitrov (2005)).

The purpose of this paper is to extend the study of the group identification 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 specifically, 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 first procedure consists of all the individuals who are defined as group members by everyone in the society, while the initial set of the second procedure collects all the individuals who define 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 finite set of candidates (see Berga et al., 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 et al., 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 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 definitions. Section 3 discusses the axioms that are necessary and sufficient 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 final remarks.

Section snippets

Basic notation and definitions

Let N = {1,..., n} 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  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  N. When i  Gi, we also say that i considers himself as a G. A profile of views is an n-tuple of sets G = (G1,..., Gn) where Gi  N for all i  N. Let G be the set of all profiles of views,

The consensus-start-respecting procedure

In this section we offer an axiomatic characterization of the Kasher's method for defining a social group. For that purpose we start with the following two axioms a CIF may satisfy.

  • A CIF F  F satisfies consensus (C) if for all G  G,

    • [i  Gk for every k  N] implies [i  F (G)], and

    • [i  Gk for every k  N] implies [i  F (G)].

  • A CIF F  F satisfies irrelevance of an outsider's view 1 (IOV1) if for all G, G′  G and for all i, j  N,

    • Gj = Gj  {i}, and

    • Gl = Gl for all l  N \ {j}, imply

    • [j  F (G) and i  Gk for some k  N]  [i  F (G)

The liberal-start-respecting procedure

For the axiomatic characterization of the liberal-start-respecting procedure defined in Section 2 we need first to modify the irrelevance of an outsider's view 1 axiom.

  • A CIF F  F satisfies irrelevance of an outsider's view 2 (IOV2) if for all G, G′  G and for all i, j  N with i = j,

    • Gj = Gj  {i}, and

    • Gl = Gl for all l  N \ {j}, imply

    • [j  F (G)]  [i  F (G) iff i  F (G′)].

This axiom basically says, like (IOV1), that if someone is collectively considered as not having the corresponding social identity, then this

Conclusion

In this paper, we have axiomatically characterized the procedures that define 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 minimal robustness. 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

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This is a substantial revision of our CentER Discussion Paper 2003-10, Tilburg University, with the same title. We thank two anonymous referees of this journal for their very helpful comments and suggestions. D. Dimitrov gratefully acknowledges financial support from the German Research Foundation (DFG).

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