Endogenous Homophily and the Persistence of Disagreement∗ JOB MARKET PAPER Isabel Melguizo† February 15, 2016

Abstract

We study a dynamic model of attitude formation in which individuals average others’ attitudes to develop their own. In our approach individuals are homophilous, that is, the attention that they pay to each other is based on whether they possess similar attributes. Also, attention co-evolves with attitudes. This co-evolution is based on the salience of attributes. The salience of an attribute is given by the difference in attitudes between individuals possessing and lacking it. Since we assume that more salient attributes deserve more attention, we prove that when there is, initially, a unique most salient attribute, it becomes increasingly salient, deserving growing attention. As a result, we show that individuals eventually interact only with others similar to them in this attribute. This process allows for disagreement to persist. Disagreement materializes in the emergence of two groups of thinking, defined according to whether individuals possess or lack the initially most salient attribute. Finally, we show that the higher the initial salience of this attribute the quicker the convergence to this disagreement scenario. Keywords: disagreement, homophily, salience, average-based updating JEL classification: D83, D85, Z13 ∗ I am grateful to my advisor, Miguel Angel ´ Ballester. I thank Antonio Cabrales, Francesco Cerigioni, Giacomo De Giorgi, Ben Golub, Matthew O. Jackson, Konrad Mierendorff, Juan D. Moreno-Ternero, Pedro Rey-Biel, Tom´ as Rodr´ıguez Barraquer, Javier Alejandro Rodr´ıguez Camacho, Benjam´ın Tello, Jan Z´ apal and the participants at IDEA Micro-Theory Lab, UCL Student Seminar and Barcelona IDGP Workshop for their helpful comments and suggestions. Financial support from the Universitat Aut` onoma de Barcelona through PIF Scholarship 912-0109/2010, from the Spanish Ministry of Economy and Competitiveness through FPI Scholarship BES-2013-06492 and from the Spanish Ministry of Science and Innovation through grants “Consolidated Group-C” ECO2008-04756 and FEDER is gratefully acknowledged. † Universitat Aut` onoma de Barcelona and Barcelona GSE. Department of Economics and Economic History, Universitat Aut` onoma de Barcelona, Building B, 08193, Bellaterra (Barcelona), Spain. Email: [email protected].

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1

Introduction

Disagreement is an everyday life phenomenon. In 2012 the American public was confronted with the reasons as to why people are poor. Circumstances were then blamed, on average, 46.55 % of the times while lack of effort appeared to be the reason, on average, in 37.75 % of the cases. Not only there was an opinion gap in society as a whole, discrepancies were also present when considering sociodemographic divides of society. For instance, women were more prone than men to consider circumstances to be the reason. The following table documents these opinion gaps:1 Table: gaps in opinions about why people are poor More often to blame if a person is poor (%) Men Women Neither Black nor Hispanic Black or Hispanic College grad Non-College grad More than $30k Less than $30k Republican Non-Republican Source: Pew Research Center Value Survey 2012

Circumstances

Lack of Effort

40 52 41 60.5 47 45 40.5 58 28 53.5

46 30 41 27.5 33 40 41.5 31 57 30.5

Disagreements tend to persist, most of the times, over issues that are non-factual. In fact, average differences in opinions regarding a wide range of topics of ethical and ideological content have persisted among the American public from 1987 to 2012.2 Individuals usually form and update their opinions through own experiences, by observing others’ actions and by communicating with others about their opinions and behavior. That is, learning is social and takes place within the individuals’ social network. Despite the aforementioned evidence, existing models of communication and learning, regardless of whether individuals behave as Bayesian or use rules of thumb, 1 We

have dichotomized some of the attributes. Party Affiliation is originally presented as Republican, Democrat and Independent. Since opinions of Independents are more in line with those of Democrats, we collapse both. We also collapse information for income, ethnicity and education. Regarding the latter we use as criterion the idea that (unobservable) characteristics of individuals having a college degree might be considerably different from those of individuals not having it. Finally, we omit the percentage of do not know answers and answers that consider both reasons. Thus, data do not sum up to 100 %. The original information is available at http://www.peoplepress.org/2012/06/04/section-3-values-about-economic-inequality-and-individual-opportunity. 2 Detailed information is available at http://www.people-press.org/2012/06/04/section-2-demographics-andamerican-values.

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typically lead to consensus.3 They are, thus, not suitable for explaining the persistence of disagreement. It is then useful to formulate processes that allow disagreement to persist. In this paper we study the dynamics of opinion formation following DeGroot (1974), a parsimonious and widely used framework in which individuals use the rule of thumb of averaging others’ opinions to develop their own. The computational requirements imposed on agents that behave as Bayesian, updating their (common) priors regarding the true state of the nature according to all relevant information, have placed rules of thumb as a useful and powerful alternative for the understanding of learning and communication processes.4 This reason seems to be borne out by recent evidence supporting, specifically, averaging models as a consistent description of individuals’ updating behavior. For instance, experimental results in Chandrasekhar et al. (2012) and Grimm and Mengel (2014) favor a DeGroot procedure over a Bayesian one.5 In DeGroot (1974), which considers a time independent averaging rule, consensus is always the eventual outcome under fairly mild conditions. Disagreement basically persists when there are groups of individuals that are disconnected to each other.6 In fact, communication models based on the DeGroot procedure, as Golub and Jackson (2010) and Golub and Jackson (2012), work with strongly connected time independent network structures, deriving always consensus results. For instance, Golub and Jackson (2012) discuss the effects of homophily, the robust tendency of individuals to associate disproportionately with similar others, on the speed of convergence to consensus, an eventual outcome that is never precluded no matter the level of homophily.7 In contrast with these papers, we construct a version of DeGroot’s model in which we incorporate the natural idea that the intensity of individual interactions might vary over time. Individuals might intensify interactions with others that think like themselves or that share with them certain sociodemographic attributes, incorporating their opinions and values over time.8 We explore here a particular mechanism for the co-evolution of homophily in sociodemographic attributes and opinions. Our approach is as follows. Time is discrete and infinite. At every point in time individuals consider others’ opinions to form and update their own. An individual’s type is defined as a subset of attributes. We introduce two key assumptions. 3 See

DeMarzo et al. (2003), Acemoglu et al. (2010), Golub and Jackson (2010), Golub and Jackson (2012), Smith and Sørensen (2000), Gale and Kariv (2003) and Banerjee and Fudenberg (2004). We discuss notable exceptions at the end of this section. 4 See Ellison and Fudenberg (1993), Acemoglu and Ozdaglar (2011) and Golub and Jackson (2012). 5 See also Corazzini et al. (2012) and Brandts et al. (2014). 6 See Jackson (2008), chapter 8, for two characterizations of consensus. 7 Homophily is technically defined in Golub and Jackson (2012) as the second largest eigenvalue of the matrix of linking densities among types. See McPherson et al. (2001) for a survey on homophily. 8 See Kossinets and Watts (2006).

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The first one is that individuals are homophilous. Briefly, when considering others’ opinions, common attributes matter. The second one is that homophily varies over time, that is, others’ opinions are incorporated at a different extent as time goes by. This variation is governed by the salience of attributes. The salience of an attribute is given by the difference in opinions between individuals possessing and lacking it. The more salient an attribute, the more the attention that the group of individuals sharing it pay among themselves, that is, the more homophilous towards this attribute they become. Also, the attention that these individuals pay to others not sharing this attribute with them, decreases with the salience of this attribute. There is a large literature in the context of consumer choice supporting the idea that individuals focus in aspects in which their alternatives differ more, that is, in aspects that are salient. For instance, in Bordalo et al. (2013) consumers’ purchasing decisions are driven by either the price or the quality of products, depending on which aspect is furthest from prices and qualities of an average bundle.9 There is also evidence suggesting a negative relationship between differences in opinions and interactions among individuals. For instance, Suanet and Van de Vijver (2009) study the relationship between perceived cultural distance, that is, individual reports of discrepancies in attitudes and values between the home and the host culture, and the acculturation of foreign students in Russia. They find a positive (negative) relationship between perceived cultural distance and interactions among co-nationals (host nationals). Also, Sole et al. (1975) consider experiments in which subjects decide whether to grant help to a stranger. They find a positive relationship between rates of helping and attitude similarity. They also find how one dissimilar attitude was sufficient to cause significantly lower rates of helping. Finally, Rosenbaum (1986) and Singh and Ho (2000) offer support to the idea that repulsion to others with dissimilar attitudes is the main mechanism shaping homophily. With this model at hand we answer the following questions: Q1: Under which conditions does attributes’ salience preclude consensus, and therefore, promote the persistence of disagreement? Q2: How does long-run disagreement look like? Q3: How does salience relate with the speed of reaching this situation? Our results are as follows. We find that disagreement persists if and only if there is, initially, a unique most salient attribute. This attribute becomes increasingly 9 See

also K˝ oszegi and Szeidl (2013) and the references therein.

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salient, in detriment of the remaining attributes. Thus, ties among individuals sharing it, will progressively gain strength in detriment of ties based on the remaining shared attributes. As a result, the society appears eventually divided in two groups of thinking, according to whether individuals possess or lack the initially most salient attribute. The difference in average eventual opinions between the groups of individuals possessing and lacking the initially most salient attribute persists while the differences in average eventual opinions associated to the possession and lack of the remaining attributes vanish. The intuition behind this process is that individuals become progressively homophilous according to one, and only one, dimension. We interpret this process as one by which individuals construct their identity. Initially confronted with several attributes upon which they may build their personality, they progressively focus in only one of them. As we will see, disagreement persists because this dynamics is fast enough. Although the functional form linking homophily to salience determines the result, in the first extension of the model we derive more general conditions guarantying the persistence of discrepant opinions. With respect to the properties of disagreement, we find how the difference in average eventual opinions between the groups of individuals possessing and lacking the initially most salient attribute, is a proportion of the difference in average initial opinions between these two groups. Finally, we find that the higher the initial salience of the attribute across which disagreement persists, the quicker the convergence to this disagreement scenario. Our work is related to previous papers discussing disagreement. Specifically, Krause (2000) and Hegselmann and Krause (2002) study disagreement in a model of bounded confidence in which individuals only consider opinions sufficiently close to their own. There are, at least, two differences with their approach. The first one is that while our primary source of attention are individual types as well as their opinions, they directly focus on similarity in opinions and do not explicitly model homophily in attributes. The second one is that they assume that considered opinions are incorporated with equal weights. This is not generally true in our case. In Acemoglu et al. (2013) disagreement persists because of the presence of stubborn agents, interpreted as leaders or media sources, that never change their opinions. We do not model the presence of such agents. The rest of the paper is organized as follows. Section 2 describes the model. Section 3 derives conditions for disagreement to persist, providing its properties. Section 4 deals with the speed of convergence. Section 5 concludes. Section 6 discusses extensions. Section 7 contains technical proofs.

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2

Preliminaries

Let I = {1, 2, . . . , n} be a finite set of attributes. The type A of an individual is defined by the attributes possessed by this individual, that is, A ⊆ I. Given two types A and B, we say that they are i-similar whenever attribute i is either present or absent in these two types. Otherwise, we say that they are i-dissimilar.10 We denote by I(AB) all the attributes for which A and B are similar, i.e., I(AB) = (A ∩ B) ∪ (Ac ∩ B c ). n The (column) vector of attitudes at time t ∈ Z+ is denoted by at ∈ [−1, 1]2 , where the component relative to type A is aA t . The average attitude across all types is denoted at and the average attitude across all types possessing (respectively lacking) attribute i is denoted at [i] (respectively at [−i]).11 Without loss of generality, let us normalize a0 = 0.12 The evolution of attitudes follows an average-based updating process similar to DeGroot (1974). Namely, each of the components of at+1 is a weighted average of attitudes in at . Let Wt be the 2n × 2n -matrix of weights describing the updating of attitudes from time t to time t + 1. Thus, every entry of Wt is the weight that type A assigns to type B. Let wtA,B denote this weight. In a nutshell, attitudes at t + 1 are given by the following expression: at+1 = Wt at

(1)

As in Golub and Jackson (2012), individuals are homophilous, a characteristic that can be captured as follows: every attribute i has a non-negative value αti and the weight that type A assigns to type B, is the sum of values of the attributes they P P i i . For normalization purposes we set share, that is, wtA,B = α t i∈I(AB) i αt = 1 n−1 notice that a type A is i-similar to exactly 2 types. Then, 2n−1 . To see this, P P P A,B A,B n−1 i = 1, i.e., at every t, Wt is = 2 B wt i αt = 1. Notice that A wt doubly stochastic. In this paper, we study the case of endogenous homophily, i.e., a situation in which the magnitude of αti co-evolves with attitudes. In particular, it depends, at every t, on the difference in average attitudes between individuals possessing attribute i and individuals lacking it, that is, ∆t [i] = at [i] − at [−i] (the salience of attribute i at time t). For the main part of the paper we link homophily and salience by the well-known Luce form: 10 Notice that attributes are dichotomous, that is, either a type possesses an attribute or lacks it. See Schelling (1969) for a discussion of this assumption. Also, as McPherson et al. (2001) point out, the distinction in terms of social distance appears to be of the type same versus different, and not on any more elaborated forms of stratification. P A P P 1 1 11 Formally, a = 1 A A t A at , at [i] = 2n−1 A:i∈A at and at [−i] = 2n−1 A:i∈A / at . 2n 12 This implies that at every t, a = 0. See subsection 7.1. t

6

αti =

∆t [i] . ∆t [j]

1 2n−1

(2)

P j

In the first extension of the model we will work with a more general representation of the homophily values, discussing results in this setting. Finally, we assume w.l.o.g. that differences in average initial attitudes are nonnegative and such that, ∆0 [1] ≥ ∆0 [2] ≥ · · · ≥ ∆0 [n] ≥ 0.13 To illustrate the notation above we consider the following example: Example 1. Consider the 2-attribute case. There are four individual types, namely, {1, 2} {1}, {2} and {∅}. The 4 × 4-matrix of weights at an arbitrary t is: {1, 2} αt1 + αt2 1  Wt =  α t  α2 t 0



{1} αt1 αt1 + αt2 0 αt2

{2} αt2 0 1 αt + αt2 αt1

{∅}  0 {1, 2} αt2   {1} 1 αt  {2} αt1 + αt2 {∅}

To make clear how homophily enters into play consider, for instance, type {2}. It is 1-similar to type {∅} and 2-similar to type {1, 2}. Thus, it pays more attention to types {∅} and {1, 2}, with whom it shares the lack of attribute 1 and the possession of attribute 2, respectively, than to type {1}, with whom it does not share any attribute. Consider also type {1, 2}. It is 1-similar to type {1} and 2-similar to type {2}. Thus, it pays more attention to types {1} and {2}, with whom it shares the possession of attribute 1 and 2, respectively, than to type {∅}, with whom it does not share any attribute. The same happens with the weights that types {1} and {2} impose on others. {2} {1} {1,2} Suppose that initial attitudes are a0 = 0.8, a0 = 0.2, a0 = −0.05 and {∅} a0 = −0.95. Thus, the differences in average initial attitudes are ∆0 [1] = 1 and ∆0 [2] = 0.75, for attributes 1 and 2, respectively. The initial homophily, driven by salience through expression (2), becomes α01 = 0.285 and α02 = 0.215, for attributes 1 and 2, respectively. The 4 × 4-matrix of weights at t = 0 becomes: {1, 2} {1} 0.5 0.285 W0 =   0.285 0.5  0.215 0 0 0.215

{2} {∅}  0.215 0 {1, 2} 0 0.215  {1} 0.5 0.285 {2} 0.285 0.5 {∅}



13 They

will preserve this order and will remain non-negative over time. See step 1 in the proof of Theorem 1.

7

The following graph illustrates this interaction structure. Let types be composed by colors as follows: types possessing attribute 1 are blue and those lacking it are green. Types possessing attribute 2 are white while types lacking it are red. Thus, {1, 2} is a mixture of blue and white, {1} is a mixture of blue and red, {2} is a mixture of green and white and {∅} is a mixture of green and red: Figure 1. Depicting initial interactions 0.5

1,2 0.285

0.215 0.285

0.5

0.215

1

2 0.215

0.5

0.285

0.215

0.285

∅ 0.5

We use this structure as a running example in subsequent sections. We proceed to examine the persistence of disagreement.

3

The persistence and properties of disagreement

To analyze under which conditions disagreement persists, notice that equation (1) QT can be solved recursively to get at+1 = W T a0 where W T = t=0 WT −t . That is, attitudes at an arbitrary t are determined by initial attitudes. Notice that if the matrix describing point-wise interactions, as the one in example 1, was constant over time, consensus will eventually emerge. The reason is that individuals would then be able to incorporate, directly or indirectly, everyone else’s attitudes.14 In our framework, this is equivalent to establish that all eventual attitudes will be equal to zero. Formally, a∞ = limt−→∞ at+1 = 0. However, as the strength of the attention individuals pay to each other varies over time, there is room for disagreement to persist. Clearly, the existence and properties of eventual attitudes can be understood by investigating the existence and properties of the limit of the product of time dependent matrices. We denote this limit by W ∞ . Formally, W ∞ = limT −→∞ W T . 14 Formally,

the matrix of weights is strongly connected and aperiodic. See Jackson (2008), chapter 8.

8

In order to state our main result, we discuss the concept of Dobrushin coefficient of ergodicity. Ergodicity coefficients provide information about the extent to which a matrix is a contraction, that is, the extent to which all its rows become eventually equal.15 The Dobrushin coefficient of ergodicity of a matrix M is defined as: τ (M ) =

X 1 max |mik − mjk |. 2 ij

(3)

k

It lies between zero and one and is different from zero if and only if the rows of M are not the same. Now, we present our main result, that describes the existence, form and extent of disagreement in eventual attitudes. Theorem 1 is as follows: Theorem 1. For every vector of initial attitudes eventual attitudes always exist. They exhibit disagreement if and only if attribute 1 is, initially, the unique most salient one (that is, if and only if ∆0 [1] > ∆0 [2]). Eventual attitudes are such that, for every type A: 1 ∞ |aA ∞ | = τ (W )∆0 [1] 2 where τ (W ∞ ) > 0. Furthermore, aA ∞ > 0 if and only if 1 ∈ A. Several aspects merit further attention. First, disagreement is almost the unique outcome of this process.16 When there is initially a unique most salient attribute, it gains increasing attention in detriment of the attention paid to the remaining attributes. Thus, eventual homophily is based upon one, and only one, dimension. Consensus would emerge if and only if there were, at least, two initially most salient attributes. In the extreme case in which all differences in average initial attitudes were equal, all attributes will have the same initial homophily value, which will be also constant over time. In this case, consensus will eventually emerge.17 Second, disagreement is always partial, across the initially most salient attribute. As Theorem 1 illustrates, types possessing attribute 1 have the same eventual attitudes. The same happens for types lacking attribute 1. Attitudes between the two groups are different. There is no possibility of observing a situation in which eventual attitudes differ by individual. Third, the difference in average eventual attitudes between the groups of types possessing and lacking attribute 1, is a proportion of the difference in average initial attitudes between the groups of types possessing and lacking attribute 1. This 15 See Stachurski (2009) for a reference on the suitability of the Dobrushin coefficient in the study of economic models with a Markovian structure. See also Ipsen and Selee (2011) and Chatterjee and Seneta (1977) for the study of convergence properties of inhomogeneous Markov chains by means of ergodicity coefficients. 16 Since differences in average initial attitudes are real numbers the probability that they are equal is negligible. 17 Specifically, αi = (2n−1 n)−1 for every attribute i and at every t. Also, when all differences in average initial t attitudes are equal to zero, expression (2) is not defined. We also set αit = (2n−1 n)−1 in this case. In both cases the matrix of weights is time independent and consensus eventually emerges.

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proportion is exactly given by the degree of contractiveness of the infinite product of the point-wise matrices of weights.18 The ergodicity coefficient then characterizes the distance from consensus in the long-run.19 The 2-attribute case is pretty informative. In it, the deviation from consensus in the long-run is given by the ratio of differences in average initial attitudes between attributes 2 and 1. Specifically, τ (W ∞ ) = 1−∆0 [2]/∆0 [1]. Also, the difference in average eventual attitudes between the groups of types possessing and lacking any attribute different from 1 is zero.20 Fourth, Theorem 1 still holds for the the general Luce representation λit = δ P∆t [i] δ when δ ∈ (0, ∞) for every attribute i. The literature, see Chen et al. ∆ [j] t j (1997), interprets δ as a rationality parameter. In our case δ reflects the extent to which differences in attitudes across attribute 1 are exacerbated. The following example illustrates Theorem 1: Example 2. Consider example 1’s information. The process evolves as follows: 











0.5 0.285 0.215 0 0.5 0.32 0.18 0 0.5 0.5 0 0      0.285 0.5 0 0.18 0 0.215  0.32 0.5 0.5 0.5 0 0  W0 =   , W1 =   , ..., lim Wt =   t→∞ 0.215 0.18 0  0 0 0.5 0.5 0 0.5 0.285 0.5 0.32 0 0.215 0.285 0.5 0 0.18 0.32 0.5 0 0 0.5 0.5



Also,

W∞

0.3125 0.3125  = 0.1875 0.1875

0.3125 0.3125 0.1875 0.1875

0.1875 0.1875 0.3125 0.3125



0.1875  0.1875 . 0.3125 0.3125





0.8  0.2   W ∞ by a0 =    −0.05 −0.95



yields a∞



0.125  0.125    = . −0.125 −0.125

Types {1, 2} and {1} and types {2} and {∅} hold the same eventual attitudes. Attitudes between the two groups are different. In this case τ (W ∞ ) = 0.25. This process goes hand in hand with segregation in interactions of 1-similar types. 3.1

Segregation in interactions and disagreement

Types similar in attribute 1 eventually interact exclusively among themselves. They reach this situation by weakening their interactions with 1-dissimilar types. To summarize this interaction information, we derive here the Spectral Segregation Index proposed by Echenique and Fryer (2007), for attribute i at time t, henceforth SSIti . As the Spectral Segregation Index has a static nature, we just repeat its computation at every point in time. Being based on the nature of individual interactions, it 18 Let a [i] = lim n−1 types possessing (respectively ∞ t−→∞ at [i] and a∞ [−i] = limt−→∞ at [−i]. Since there are 2 lacking) attribute 1, a∞ [1] − a∞ [−1] = 2−1 τ (W ∞ )∆0 [1] − (−2−1 τ (W ∞ )∆0 [1])=τ (W ∞ )∆0 [1].
t QT P 19 Formally, τ (W ∞ ) = lim −1 2 ]−1 , being fully determined by the initial T →∞ t=0 [1 + i>1 ∆0 [i](∆0 [1]) information. 20 Notice that within the 2n−1 types possessing (respectively lacking) attribute 1, there are 2n−2 types possessing and lacking any other attribute i > 1, respectively. Given the form of eventual attitudes in Theorem 1, the difference in average eventual attitudes between the groups of types possessing and lacking any attribute i > 1, is zero.

10

is particularly suitable in our framework. Other indexes of segregation, as the Dissimilarity Index or the Isolation Index, are based on partitions (census) of a physical unit (a city). In our case individuals are not partitioned into physical units, thus, we do not interpret our interaction process in their terms.21 The higher the value of this index the higher the amount of attention that i-similar types pay among themselves. P Before the result, let us denote, for the ease of exposition, λit = ∆t [i]( j ∆t [j])−1 for every i and t. That is, expression (2) without normalization. Let us also stress the fact that interaction patterns within the groups of types possessing and lacking any attribute i, are the same at every t. Thus, we use the SSIti to describe interactions of both groups. This can be visualized the matrix of example 1. Interactions of individuals possessing attribute 1 take the same form as those of individuals lacking it. The same is true for attribute 2. The following result describes the evolution of interactions of 1-similar types by means of this index. Proposition 1 is as follows:22 Proposition 1. At every t, the Spectral Index of Segregation of 1-similar types is 1 + λ1t n+1 SSIt1 = . It increases over time, with SSI01 > and lim SSIt1 = 1. t−→∞ 2 2n The SSIt1 captures the intensity of interactions of 1-similar types at time t. Due to our assumptions, the groups of 1-similar types have the most intense relations. This is due to the fact that attribute 1 is always the most salient one.These relations are gradually intensified. Thus, the eventual segregation of 1-similar types is maximal, being the limiting value of the index equal to one. Let us examine segregation patterns for attributes other than 1. It turns out that SSIti = (1+λit )(2)−1 for every attribute i. Attributes other than 1 become gradually irrelevant in shaping interactions. The criterion that individuals will gradually use in deciding whom to pay attention, is whether others are 1-similar to them. As a consequence, segregation according to attributes other than 1 decreases over time. If one considers eventual interactions within the groups of 1-similar types and focuses on any attribute other than 1, one would observe that types equally distribute their unit of attention between the two groups, defined now according to whether individuals possess or lack this other attribute. Thus, the limiting value of the index for attributes other than 1 is exactly one half.23 There is also a relationship between the segregation of a group i-similar types and the segregation of its members. By definition, the Spectral Segregation Index is the average of individual segregation indexes. They are computed by distributing the 21 See

Echenique and Fryer (2007) for a discussion. SSIti is computed by looking only at interactions among i-similar types at t. The SSIti is the largest eigenvalue of the matrix describing these interactions. Also, here we consider the case in which λit > 0 for every attribute i. The case in which λit = 0 for some attributes i is addressed in the proof of this proposition. 23 Formally, lim i i −1 . t−→∞ SSIt = 0.5. Also, SSI0 ≤ (n + 1)(2n) 22 The

11

former among the members of the group it considers.24 In our framework, at every t and for every attribute i, the segregation level of every type A is the same and equal to the SSIti . Intuitively, every type A pays the same total amount of attention to i-similar types. As a consequence, it also pays the same total amount of attention to i-dissimilar types. In a nutshell, every type A segregates its interactions at the same extent, equally contributing to the segregation of its group. The following figure illustrates the evolution of interactions among types: Figure 2. Segregation in interactions 0.5

0.5

1,2

1,2

0.285

0.215 0.285

0.5

0.32

0.215

1

2 0.215

0.5

0.5

2 0.18

0.285

0.32

∅

0.5 (a)

= 0.785,

0.5

0.32

0.18

∅

SSI01

0.18

1

0.285

0.215

0.18 0.32

0.5 SSI02

= 0.715

(b)

SSI11

= 0.82, SSI12 = 0.68

0.5

1,2 0.5 0.5 0.5

1

2

0.5

0.5 0.5

∅ 0.5 (c) lim SSIt1 = 1, lim SSIt2 = 0.5 t→∞

t→∞

24 This

is done according to the entries of the eigenvector associated to the largest eigenvalue of the matrix describing interactions of i-similar types. In our case this eigenvector is composed by ones.

12

Observe how 1-similar types eventually interact exclusively among themselves. Observe also how types possessing (respectively lacking) attribute 2, equally split their unit of attention between themselves and others lacking (respectively possessing) attribute 2. The next question is how long it takes to reach this scenario.

4

Speed of convergence to the long-run disagreement

We discuss here the role of salience in determining the speed of convergence to the scenario in which disagreement persists. As previously discussed, having followed the prescriptions of those similar to them in attribute 1, individuals end up stuck at discrepant attitudes. Facing disagreements might have implications in different settings. As Alesina and Tabellini (1990) point out disagreement between policymakers in ideological views about social welfare, specifically regarding the desired composition of government spending between two public goods, might cause the accumulation of inefficient levels of public debt. Also, Voss et al. (2006) show how organizational success of nonprofit professional theatres was affected by the divergent views of their leaders regarding the values that should drive the organizations’ behavior. In a broad sense Friedkin and Johnsen (1999) state that there might be difficulties in arriving at collective decisions when individuals have fixed discrepant preferences. It is worth noticing that studying the time to reach a situation in which individuals exhibit fixed discrepant attitudes is different in nature from studying the time to reach a situation in which individuals are in consensus. In the first situation attitudes evolve and get eventually stuck at different values. In the second, attitudes might be point-wise different but eventually converge to a common value. One motivation for studying the former situation is that, in the presence of a policy intervention aiming to recover consensus, it might be important to know the timing for its implementation, in order, for instance, to avoid conflict. In our framework, consensus could be recovered by changing the attitudes of specific types, in order to make attribute 2 as salient as attribute 1. The speed of convergence of the system to the equilibrium in which disagreement persists, is determined by the relation between the difference in average initial attitudes associated to attribute 1 and the differences in average initial attitudes associated to the remaining attributes. In other words, the salience of attribute 1 determines how long it takes for individuals to become sufficiently homophilous with respect to it. The expression that links homophily towards attribute 1 and salience is given by λ1t , as defined in section 3.1. Recall that eventually 1-similar individuals interact exclusively among themselves, formally lim λ1t = 1. When we are suffit−→∞

13

ciently close to this interaction pattern, we can also say that we are sufficiently close to the equilibrium in which disagreement persists. It is worth mentioning that at every t, λ1t is the second largest eigenvalue of the point-wise matrix of interactions Wt . As widely discussed Golub and Jackson (2010) and Golub and Jackson (2012), the second largest eigenvalue of a stochastic matrix plays an important role in the analysis of the speed of convergence. Our aim in this section is precisely to characterize the time it takes for individuals to become homophilous exclusively with respect to attribute 1, that is, the minimum time it takes for λ1t to be above an  > 0 distance of its limit. For this purpose we formally define this minimum time as: T = min{t : λ1t ≥ 1 − }. The following result will describe its properties. Let us focus on the case in which all differences in average initial attitudes are positive. Before stating the result let ∆ [i] r0i = ∆00[1] for every attribute i > 1. These ratios capture initial relative salience of attribute 1 with respect to any other attribute. The smaller these ratios the higher ∆ [n] the initial relative salience of attribute 1. Let us exceptionally denote r0 = ∆00 [1] ∆ [2]

and r0 = ∆00 [1] . Notice that these two ratios represent extreme cases. Specifically, r0 considers the difference in average initial attitudes associated to attribute n, which is the smallest one. In contrast, r0 considers the difference in average initial attitudes associated to attribute 2, which is the second highest one. Let us set t 1 1 λt = [1+(n−1)(r0 )2 ]−1 and define Tmin = min{t : λt ≥ 1−}. Similarly, let us set t λ1t = [1 + (n − 1)(r0 )2 ]−1 and define Tmax = min{t : λ1t ≥ 1 − }. Notice that both, P t 1 λt and λ1t , are constructed from the recursive expression of λ1t = [1 + i>1 (r0i )2 ]−1 , by substituting all differences in average initial attitudes, by the smallest difference and by the second highest difference, respectively. Proposition 2 is as follows: Proposition 2. For every vector of initial attitudes such that disagreement persists, T is non-increasing in the initial relative salience of attribute 1. Furthermore, T ∈ [Tmin , Tmax ]. It follows from the expression of λ1t that an increase in the difference in average initial attitudes associated to attribute 1, increases the attention that 1-similar individuals pay among themselves. The same happens when differences in average initial attitudes associated to attributes other than 1 decrease. In both cases, attribute 1 becomes relatively more salient than the others.25 Intuitively, some attributes are now weaker competitors for attention. 25 We

consider that the variation in differences in average initial attitudes, takes place for one attribute i at a time. It is such that ∆0 [1] > ∆0 [2] ≥ · · · ≥ ∆0 [n] ≥ 0 is preserved in order, and in magnitude for differences associated to attributes j 6= i. In fact, when we can decrease or increase any ∆0 [i] by decreasing or increasing, in the same magnitude, initial attitudes of both, the type possessing all attributes and the type that only possesses

14

When attribute 1 is fairly salient, individuals exhibit high homophily with respect to 1-similar others to start with. They also form completely inward-looking groups relatively fast. In a nutshell, the higher the initial relative salience of attribute 1, the quicker the process of convergence to the equilibrium in which disagreement persists might become. Not only the speed of convergence is affected by the initial relative salience of attribute 1. The magnitude of disagreement is also sensitive to variations in differences in average initial attitudes. Consider the expression of eventual attitudes in Theorem 1.26 An increase in the initial relative salience of attribute 1, contributes to generate a larger disagreement. It is also worth mentioning that Proposition 2 characterizes the minimum time of convergence for the system as a whole. The reason is that the homophily value associated to attribute 1 is always further away from one, its limiting value, than the homophily values associated to the remaining attributes are from zero, their limiting value.27 We finally discuss how the configuration of initial attitudes matters for determining the speed of convergence. Consider the extreme case in which, although the difference in average initial attitudes associated to attribute 1 is the highest one, it is fairly similar to the differences associated to the remaining attributes. The initial relative salience of attribute 1 is fairly small in this case and it would take a while for individuals to gradually redirect their homophilous behavior towards attribute 1. The time to reach the equilibrium would be considerably high in this case. The other extreme situation is such that the difference in average initial attitudes associated to attribute 1 is, by far, the highest one.28 Being the relative salience of attribute 1 fairly high, individuals would quickly conclude that the possession or lack of this attribute clearly defines two groups in society. Thus, it would not take much time for them to become homophilous exclusively with respect to it. The equilibrium will be reached much more faster than before. In the extreme situation in which differences in average initial attitudes associated to all attributes different from 1 are zero, the equilibrium is reached at t = 1.29 the particular attribute i we consider, differences associated to attributes j 6= i, keep unaltered. The decrease or increase could be marginal such that the order above is preserved. QT 26 We can rewrite τ (W ∞ ) as lim 2t 2t −1 . When r or r decrease, the elements in this 0 T →∞ 0 t=0 [1 + r 0 + ... + r 0 ] product are point-wise higher than before, so it is its limiting product. 27 Technically the time it takes for every λi , for i > 1, to be sufficiently close to 0 is at most as larger as the time t P i it takes for λ1t to be sufficiently close to 1. The following argument makes clear why. Since at every t, i λt = 1, P i 1 2 1 1 then λt ≥ 1 −  implies that i>1 λt ≤ . When only λt and λt are different from zero, then λt ≥ 1 −  implies that λ2t ≤ . When λit is also different from zero for some attributes i > 2, then λ1t ≥ 1− implies that ( n−1)−1 ≤ λ2t < . 28 The first extreme case might correspond to the ranking ∆ [1] ' ∆ [2] = · · · = ∆ [n]. The second extreme case 0 0 0 might correspond to the ranking ∆0 [1] > ∆0 [2] = · · · = ∆0 [n] ' 0. 29 Notice that when ∆ [n] = 0 then λ1 = 1. Notice also that when ∆ [i] = 0 for some attribute i < n then, 0 0 t by assumption, ∆0 [n] = 0 as well. In this case Tmin = 0. Similarly, when ∆0 [2] = 0 then λ1t = 1. In this case Tmax = 0. When both situations hold, in fact, λ1t = 1 and the equilibrium is reached at t = 1.

15

5

Conclusions

On the basis of the observation that disagreement in opinions is a common phenomenon, we propose a model of opinion evolution able to capture its persistence. In our approach individuals exhibit homophily and the attention they pay to similar others varies over time. Specifically, homophily co-evolves with opinions governed by the salience of attributes. We find that disagreement is the long-run outcome if and only if there is a unique attribute that becomes increasingly salient as time goes by. This attribute is precisely the initially most salient one. Thus, eventual homophily is such that individuals only pay attention to others if they are similar to them in that particular attribute. As a product of this behavior, two groups of thinking emerge in the longrun. The time to convergence to this scenario is non-increasing in the initial relative salience of this attribute. We consider our findings to be related to the phenomenon of unidimensionality in opinions, a widely discussed topic in political economy. As DeMarzo et al. (2003) point out, there is a strong debate on whether voting records of Congress and Senate members can be explained by a unidimensional liberal-conservative model. There is, in fact, evidence strongly supporting this model. For instance, Poole and Daniels (1985) find that the voting behavior in the U.S. Congress can be mainly explained by a single liberal-conservative dimension. We also consider that our model has a direct application related to the persistence of the gender pay gap. It is sometimes argued that the reason as to why females consistently self-report to be happier at work than males, relies on the fact that they have traditionally held lower labor reward aspirations than males. This phenomenon is known as The Paradox of Female Happiness. Two references discussing this paradox and related aspects are Bertrand (2011) and Clark (1997). Divergent aspirations between males and females might be able to explain that part of the gender gap that remains unexplained even after controlling for relevant aspects such as skill levels. Our intuition is that a model of wage setting in which individuals are of both sexes and are endowed with gender biased aspirations, will deliver as a result a gender pay gap, provided that the updating of aspirations takes place with the mechanism we describe. Females might end up self-selected into low payment jobs, even without discriminatory behavior from the part of employers. Finally we would like to also mention that our model follows a representative agent approach in which there is one individual by type. We do not deal with the case in which individuals appear in society in different quantities. We leave it for further research.

16

6

Appendix 1. Extensions

In this section we present three extensions of the model. The first one discusses a generalization of Theorem 1. In it, the Luce representation considered up to now is not a requirement for results regarding the persistence of disagreement. The last two set some insights for the 2-attribute case when initial attitudes are random variables and when relations are not symmetric, respectively.

6.1

Extension 1. On a general representation of homophily values

Theorem 1 is based on a more general statement that does not require the Luce representation of the homophily values we consider. To illustrate this point let γti be the non-negative homophily value associated to attribute i at time t. Let this value depend on the differences in average attitudes associated to all attributes. We assume that, at every t: (i)

i i γt

= 1. We also consider that limt−→∞ γti exists for every attribute i and P P that limt−→∞ i γti = i limt−→∞ γti = 1.

P

(ii) ∆t [i] ≥ 0 implies that γti ≥ 0. In particular, γti = 1 if and only if ∆t [i] > 0 and ∆t [j] = 0 for attributes j 6= i. When ∆t [i] = 0 for every attribute i, we set γti = 1/n. Also, that γti = 0 whenever ∆t [i] = 0 and that γti = 1 whenever ∆t [i] > 0 and ∆t [j] = 0 for every attribute j 6= i, are natural normalizations. The former establishes that when the difference in average attitudes is zero for an attribute i, it does not deserve any attention. The latter establishes that when differences in average attitudes are different from zero only for an attribute i, it captures all the attention. (iii) ∆t [1] ≥ ∆t [2] ≥ · · · ≥ ∆t [n] ≥ 0 implies that γt1 ≥ γt2 ≥ · · · ≥ γtn ≥ 0. 1 Finally, to be consistent with expression (2), let βti = 2n−1 γti . In the same spirit of Theorem 1, the following result derives conditions for the persistence of disagreement. It also provides its form. Theorem 2 is as follows:

Theorem 2. For every vector of initial attitudes eventual attitudes always exist. They exhibit disagreement if and only if homophily based on attribute 1 approaches 1 sufficiently fast (that is, if and only if

∞ P

log(γt1 ) exists). Eventual attitudes are

t=0

such that, for every type A: 1 ∞ |aA ∞ | = τ (W )∆0 [1] 2 ∞ A where τ (W ) > 0. Furthermore, a∞ > 0 if and only if 1 ∈ A. 17

In general, disagreement persists whenever the process by which individuals progressively intensify their relations with 1-similar others is fast enough. Intuitively there are two forces playing a role, on the one hand individuals pay increasingly attention to others on the basis of attribute 1 and on the other hand they still pay a positive amount of direct or indirect attention to everyone else. It is the case that the first force dominates the second. As in Theorem 1, the long-run disagreement materializes in two groups of thinking, defined according to whether individuals possess or lack attribute 1. In contrast with Theorem 1, we cannot specify the closed form expression for the ergodicity coefficient, since it depends on the particular functional form of the homophily values. QT We just set τ (W ∞ ) = limT →∞ t=0 γt1 in this case. The following examples illustrate the requirements in this theorem. Using information in example 1, they consider updating rules that are mainly based on modifications of the Luce representation in expression (2), with the exception of example 6. Example 3 deals with a scenario in which consensus is achieved. In examples 4 to 6 disagreement persists. Example 3. Eventual consensus. Consider that the Luce updating rule in the main model keeps constant from some point in time on. That is, once a certain level of homophily based on attribute 1, for instance αt1 = 0.32, as in examples 1 and 2, has been achieved, it remains constant. Notice that from a point in time on the matrix of interactions becomes time independent. As a consequence, individuals do not become homophilous exclusively with respect to attribute 1. The requirements in the theorem above are therefore not satisfied and consensus eventually emerges. Example 4. The persistence of disagreement. Consider γt1 = and γt2 =

∆t [1]δ ∆t [1]δ + ∆t [2]δ

∆t [2]δ with δ = 1.2. Thus, γ01 = 0.586 and γ02 = 0.414, and: δ δ ∆t [1] + ∆t [2] 











0.5 0.5 0 0 0.5 0.293 0.207 0 0.5 0.34 0.16 0    0.293 0.5   0 0.207 0 0.16 0.34 0.5 0.5 0.5 0 0   W0 =    , ..., lim Wt =   , W1 =  t→∞  0 0 0.5 0.5 0.16 0 0.207 0 0.5 0.293 0.5 0.34 0 0 0.5 0.5 0 0.16 0.34 0.5 0 0.207 0.293 0.5



Also,

W∞

0.332  0.332  =  0.168 0.1688

0.332 0.332 0.168 0.168

0.168 0.168 0.332 0.332



0.168  0.168  0.332 0.332



. W ∞ by



0.8  0.2    a0 =   −0.05 −0.95



yields a∞



0.16  0.16    = . −0.16 −0.16

In this case τ (W ∞ ) = 0.323. Notice that the initial relative salience of attribute 1, that is, (∆0 [2]/∆0 [1])1.2 , exacerbates with respect to example 1 in which δ = 1. 18

Example 5. The persistence of disagreement. Consider the case in which β∆t [1] δ∆t [2] γt1 = and γt2 = , with β > δ > 0. Notice that β∆t [1] + δ∆t [2] β∆t [1] + δ∆t [2] the initial relative salience of attribute 1, that is, δ∆0 [2]/β∆0 [1], exacerbates with respect to example 1. This process leads to disagreement for any configuration of differences in average initial attitudes associated to attributes 1 and 2. Examples 3 to 5 basically consider modifications of the Luce form in the main model. Assumptions in extension 1 are therefore satisfied. There is one minor exception in example 5. Notice that when ∆t [1] = ∆t [2] it is still the case that γt1 > γt2 . In this case results hold as well, this exception only precludes consensus even in the case in which ∆0 [1] = ∆0 [2]. Example 6. The persistence of disagreement. Consider the following updating rule:

γt1 =

                    

0.5 1 0 α  ∆t [1] ∆t [2]  β ∆t [2] 1− ∆t [1]

if ∆t [1] = ∆t [2] ≥ 0 if ∆t [1] > ∆t [2] = 0 if ∆t [2] > ∆t [1] = 0 if ∆t [2] > ∆t [1] > 0

.

if ∆t [1] > ∆t [2] > 0

We set α and β such that (∆t [1]/∆t [2])α and (∆t [2]/∆t [1])β are smaller than one half. With respect to the homophily value of attribute 2, let γt2 = 1 − γt1 , at every t. Let us set β = 2.5. 30 The process evolves as follows: 











0.5 0.257 0.243 0 0.5 0.286 0.214 0 0.5 0.5 0 0 0.257 0.5      0 0.243 0 0.214  0.286 0.5 0.5 0.5 0 0  W0 =   , W1 =   , ..., lim Wt =   t→∞ 0.243 0.214  0 0 0.5 0.5 0 0.5 0.257 0 0.5 0.286 0 0.243 0.257 0.5 0 0.214 0.286 0.5 0 0 0.5 0.5



Also,

W∞

0.31 0.31  = 0.19 0.19

0.31 0.31 0.19 0.19

0.19 0.19 0.31 0.31



0.19  0.19 . 0.31 0.31



W ∞ by



0.8  0.2    a0 =   −0.05 −0.95



yields

a∞



0.12  0.12    = . −0.12 −0.12

In this case τ (W ∞ ) = 0.23. 30 Assumptions in extension 1 are satisfied for the information in example 1. Also, at every t, ∆ [1] > ∆ [2] > 0, t t we then do not specify any value for α. Finally, since ∆t [1]/∆t [2] (respectively ∆t [2]/∆t [1]) is decreasing over time whenever ∆t [2] > ∆t [1] (respectively ∆t [2] < ∆t [1]) we set α and β to be constant. See steps 1 and 7 in the proof of Theorem 1.

19

6.2

Extension 2. Initial attitudes as random variables

Up to now we have dealt with the case in which attitudes are real numbers. We analyze in this section the case in which initial attitudes are random variables. A 2 Specifically, the initial attitude of every type A is a ˜A 0 ∼ N (a0 , σ ). We assume that attitudes are independent to each other. Differences in average initial attitudes ˜ 0 [i] ∼ N (∆0 [i], σ 2 ) and ∆ ˜ 0 [j] ∼ N (∆0 [j], σ 2 ). We also assume w.l.o.g that are ∆ the distributions of differences in average initial attitudes have non-negative means, that is, ∆0 [i], ∆0 [j] ≥ 0. With respect to the link between homophily and salience, we have that, at every t: α ˜ ti =

˜ t [i]| |∆ 1 ˜ t [i]|+|∆ ˜ t [j]| 2 |∆

and α ˜ tj =

˜ t [j]| |∆ 1 ˜ t [i]|+|∆ ˜ t [j]| . 2 |∆

Differences in average attitudes come now in absolute value. That is so because homophily values should be defined only in terms of the magnitude of these differences, and not on their sign. As a preview of the results, we find that the persistence of disagreement is robust to the introduction of randomness. However, in contrast with the deterministic case, disagreement might manifest across either attribute with positive probability. The following lemma states the result: Lemma 1. Disagreement persists across either attribute i or j with positive probability. Furthermore, disagreement across attribute i is more likely than disagreement across attribute j if and only if attribute i is, initially, the unique most salient in mean (that is, if and only if ∆0 [i] > ∆0 [j]). Notice that the initially most salient attribute in mean is not always the one determining disagreement, although this event is the most likely. The following results describe eventual attitudes, conditional on disagreement taking place across attribute i or j. For simplicity, let a ˜A ∞ denote this conditional eventual attitude. It will be clearer which case we are dealing with. Results are as follows: Lemma 2. Suppose that disagreement persists across attribute i. Then:


˜ 1 ˜ ˜ 0 [j]| ∆0 [i] if i ∈ A |∆0 [i]| − |∆ ˜ 0 [i]| 2 |∆
˜ 1 ˜ ˜ 0 [j]| ∆0 [i] if i ∈ = a0 − |∆0 [i]| − |∆ / A. ˜ 0 [i]| 2 |∆

a ˜A ∞ = a0 + a ˜A ∞

Furthermore, E(˜ aA aA / A) is different from zero. It is also ∞ : i ∈ A) − E(˜ ∞ : i ∈ positive with probability higher or equal than one half. 20

Lemma 3. Suppose that disagreement persists across attribute j. Then:


˜ 1 ˜ ˜ 0 [i]| ∆0 [j] if j ∈ A |∆0 [j]| − |∆ ˜ 0 [j]| 2 |∆
˜ 1 ˜ ˜ 0 [i]| ∆0 [j] if j ∈ = a0 − |∆0 [j]| − |∆ / A. ˜ 0 [j]| 2 |∆

a ˜A ∞ = a0 + a ˜A ∞

/ A) is different from zero. It is also aA Furthermore, E(˜ aA ∞ : j ∈ ∞ : j ∈ A) − E(˜ positive with probability higher or equal than one half.31 Disagreement manifests in a similar way than in the deterministic case, with two groups holding different eventual attitudes. The differences in expected eventual attitudes between types that possess and lack attribute i (Lemma 2) and between types that possess and lack attribute j (Lemma 3) are positive with the probability that the differences in average initial attitudes across attribute i or j are positive, respectively. This probability is higher for attribute i than for attribute j if and only if the mean of differences in average initial attitudes is the highest for attribute i (that is, if and only if disagreement across attribute i is the most likely). With respect to Lemma 2 (respectively Lemma 3), as in the deterministic case, the difference in eventual average attitudes associated to attribute i (respectively j) persists whereas the one associated to attribute j (respectively i) is zero.32 6.3

Extension 3. Some types perceive differences differently

It is straightforward to think that different groups of individuals perceive build their identity on different basis, that is give more or less attention to certain attributes. Suppose for instance that types lacking attribute 2 pay less attention to differences in attitudes associated to this attribute than types possessing attribute 2. The matrix describing interactions is no longer symmetric. In particular it becomes: 31 Expectations in Lemmas 2 and 3 are conditional on disagreement taking place across either attribute i or j, respectively. Notice also that a0 is a random variable. P P 32 Formally, when disagreement persists across attribute i, (2n−1 )−1 n−1 )−1 ˜A ˜A ∞ − (2 ∞ = A:i∈A a A:i∈A / a n−1 )−1 2n−1 (˜ A : i ∈ ˜ 0 [i]| − |∆ ˜ 0 [j]| (respectively |∆ ˜ 0 [j]| − |∆ ˜ 0 [i]|) (2n−1 )−1 2n−1 (˜ aA : i ∈ A) − (2 a / A) = | ∆ ∞ ∞ ˜ 0 [i] ≥ 0 (respectively ∆ ˜ 0 [i] < 0). Since disagreement across attribute i persists when |∆ ˜ 0 [i]| > |∆ ˜ 0 [j]|, when ∆ the distribution of the difference in eventual average attitudes associated to attribute i has either positive or negative support. For attribute j, notice that within the 2n−1 types possessing attribute i there are 2n−2 types possessing and lacking attribute j, respectively. The same happens within the 2n−1 types lacking attribute i. P P n−1 )−1 −1 (2n−2 )−1 2n−2 (˜ Thus, (2n−1 )−1 A:j∈A a ˜A ˜A aA ˜A / A, j ∈ ∞ − (2 ∞ = 2 ∞ : i ∈ A, j ∈ A + a ∞ : i ∈ A:j ∈A / a −1 n−2 −1 n−2 A −1 (2a − 2a ) = 0. That is, the distribution A) − 2 (2 ) 2 (˜ a∞ : i ∈ A, j ∈ / A+a ˜A : i ∈ / A, j ∈ / A) = 2 0 0 ∞ of the difference in eventual average attitudes associated to attribute j is degenerated at zero. A similar analysis holds when disagreement persists across attribute j. Thus, we omit it here. See the proof of Lemma 1.

21

{1, 2} λ1t + λ2t 1 βt1 Wt =   2 λ2 t 0



{1} λ1t βt1 + βt2 0 βt2

{2} λ2t 0 1 λt + λ2t βt1

{∅}  0 {1, 2} βt2   {1} 1 λt  {2} βt1 + βt2 {∅}

Where at every t, βt1 + βt2 = 1 with βt1 , βt2 ∈ (0, 1] and βt1 > λ1t and thus βt2 < λ2t . Recall also that w.l.o.g we set ∆0 [1] ≥ ∆0 [2]. The following lemma summarizes the result. Lemma 4. For any configuration of initial attitudes eventual attitudes always exists and exhibit disagreement across attribute 1. In particular, types hold the same eventual attitudes if and only if they are 1−similar. Furthermore, eventual disagreement is larger than in the symmetric case. Now differences in attitudes across attribute 1 are exacerbate with respect to the case analyzed in the main part of the paper. Notice that the extreme version is such that at every t, β1t = 1. There is a more interesting case such that βt1 < λ1t at every t and the process starts with ∆0 [1] > ∆0 [2]. Although attribute 1 is initially the most salient, individuals pay now less attention than before to 1-similar others. It might be that initial differences in remain being higher in subsequent periods. It might also be that differences in opinions associated to attribute 2 become higher at some point it time t and disagreement persist across this attribute. A deeper analysis is left for further research.

7

Appendix 2. Proofs

For the ease of exposition let λit = expression all along the proofs.

P∆t [i] j ∆t [j]

for every i and at every t. We use this

Proof of Theorem 1. It is composed by several steps. In step 1 we show how, at every t, λi0 > 0 implies that λit > 0 and λi0 = 0 implies that λit = 0. Steps 2-8 analyze disagreement when λi0 > 0 for every attribute i. In particular, Steps 2-5 identify the eigenvalues and eigenvectors of Wt and diagonalize it. Step 6 deals with the existence of the limiting product of point-wise stochastic matrices, that is, W ∞ . Step 7 provides its form. Step 8 establishes the necessary and sufficient condition for disagreement to persist. It also qualifies disagreement. Finally, step 9 describes the case in which λi0 = 0 for some/all attributes i > 1.

22

Step 1. We prove that at every t, λi0 > 0 implies that λit > 0 and λi0 = 0 implies  P 1 P aA aA that λit = 0. We proceed by decomposing ∆t [i] = n−1 t . t − A:i ∈A / A:i∈A 2 P A,B B A,B A wt−1 Consider a type A such that i ∈ A. By (1), at = at−1 . Since wt−1 = B P 1 i i∈I(AB) λt−1 , then: 2n−1 aA t =

X

1

A,B B wt−1 at−1 =

X

2n−1

B

λit−1 aB t−1 +

B:i∈B

1

X

2n−1

X

λjt−1

j6=i

aB t−1 .

B:j∈I(AB)

i B Since there are 2n−1 types A possessing attribute i, A:i∈A aA t = B:i∈B λt−1 at−1 + P P j B / A, j6=i λt−1 B:j∈I(AB) at−1 . By a similar reasoning, for types A such that i ∈

P

P

A A:i∈A / at

P

1

=

P

B:i∈B /

X

2n−1

aA t −

A:i∈A

λit−1 aB t−1 + 1

2n−1

X

P j B j6=i λt−1 B:j∈I(AB) at−1 .

P

aA t =

A:i∈A /

1

X

2n−1

λit−1 aB t−1 −

B:i∈B

Therefore:

1

X

2n−1

λit−1 aB t−1

B:i∈B /

or equivalently, ∆t [i] = λit−1 ∆t−1 [i].33 From the definition of λit , it follows that at every t, ∆t [i] ≥ 0 implies that λit ≥ 0. Also, ∆t [i] ≥ 0 if and only if λit−1 ≥ 0 and ∆t−1 [i] ≥ 0. With these two observations we conclude that ∆0 [i] > 0 implies that at every t, ∆t [i] > 0 and λi0 > 0 implies that at every t, λit > 0. Also ∆0 [i] = 0 implies that at every t, ∆t [i] = 0 and λi0 = 0 implies that at every t, λit = 0.34 Step 2. At every t, 1 is an eigenvalue of Wt , with associated right-eigenvector u of size 2n × 1, where u has all components equal to 1. This result directly follows from the stochasticity of Wt . Notice that u is time independent. We therefore omit the time subscript. Step 3. At every t and for every attribute i, λit is an eigenvalue of Wt , with associated right-eigenvector ui of size 2n × 1, where ui has the following form: the component of ui associated to type A is equal to 1 if i ∈ A and equal to −1 otherwise. We prove that by showing that the pair (λit , ui ) satisfies the eigenvalue equation, Wt ui = λit ui . Consider an attribute i and an arbitrary type A. Suppose first that i ∈ A. Notice that there are exactly 2n−1 types B possessing attribute i. Also, notice that for every j 6= i, there are exactly 2n−2 types B possessing attribute i that are j-similar to A and 2n−2 types B lacking attribute i that are j-similar to A. Therefore, the row in Wt corresponding to type A, multiplied by ui , is equal to: 2n−1 λit + 2n−2

X B:i∈B

wtA,B −

X

n P

λjt − 2n−2

j6=i 2n−1

wtA,B =

B:i∈B /

n P j6=i

λjt = λit

Q i this expression holds at every t, we recursively write ∆t [i] = t−1 s=0 λs ∆0 [i]. i from the definition of λt it follows that at every t, ∆t [1] ≥ ∆t [2] ≥ · · · ≥ ∆t [n] ≥ 0 implies that 1 2 λ1t ≥ λ2t ≥ · · · ≥ λn t ≥ 0. Additionally, ∆t [1] ≥ ∆t [2] ≥ · · · ≥ ∆t [n] ≥ 0 if and only if λt−1 ∆t−1 [1] ≥ λt−1 ∆t−1 [2] ≥ n · · · ≥ λt−1 ∆t−1 [n] ≥ 0. Since by assumption ∆0 [1] ≥ ∆0 [2] ≥ · · · ≥ ∆0 [n] ≥ 0 then, at every t, ∆t [1] ≥ ∆t [2] ≥ QT QT QT 1 2 n · · · ≥ ∆t [n] ≥ 0 and λ1t ≥ λ2t ≥ · · · ≥ λn t ≥ 0 hold. This also implies that t=0 λt > t=0 λt ≥ · · · ≥ t=0 λt ≥ 0. 33 As

34 Also

23

Since every type A is such that i ∈ A, the RHS of the eigenvalue equation also equals λit . Thus, we conclude that (λit , ui ) is a pair of eigenvalue and right-eigenvector of Wt . The proof for the case in which A is such that i ∈ / A is analogous and hence i omitted. As in step 2, the eigenvectors u corresponding to every λit are also time independent. Step 4. At every t, the rest of eigenvalues of Wt are zero. Consider any type B P A,{i} − such that |B| ≥ 2. We start by proving that for every type A, wtA,B = i∈B wt A,∅ [|B| − 1] wt . By doing so, we are proving that the column vector of weights associated to type B is a linear combination of the column vectors of weights associated to types containing at most 1 attribute and hence, there are at most n + 1 independent 1 columns in Wt . First, notice that, getting rid of the normalization n−1 : 2

X

A,{i}

wt

i∈B

X

=

(λit +

i∈B∩A

X

X

λjt ) +

j∈Ac

X

λjt

i∈B∩Ac j∈Ac ,j6=i

and that this is equivalent to:

X

(λit +

X j∈Ac

i∈B∩A

λjt ) +

X

(

X

X

λjt − λit ) =

i∈B∩Ac j∈Ac

i∈B∩A

λit +

XX i∈B j∈Ac

λjt −

X

λit .

i∈B∩Ac

Second, consider (|B|−1)wtA,∅ = (|B|−1) j∈Ac λjt . Thus, the difference of these P P P two resulting terms is equal to i∈B∩A λit + j∈Ac λjt − i∈B∩Ac λit . This expression P P P P j i i i can be rewritten as j∈I∩Ac λt − i∈B∩A λt + i∈(I\B c )∩Ac λt = i∈B∩A λt +

P

λit . This is equivalent to wtA,B = i∈I(AB) λit where I(AB) = (B ∩ A) ∪ as defined in section 2. Therefore rank(Wt ) ≤ n + 1. Recall that the rank of a matrix is equal to the number of non-zero eigenvalues. Since steps 2 and 3 already identified n + 1 of them, indeed rank(Wt ) = n + 1. Thus, the rest of the 2n − (n + 1) eigenvalues are zero. Step 5. We prove here that Wt is always diagonalizable and provide its form. From symmetry of Wt there is an orthogonal diagonalization Wt = U Λt U 0 , where U is an orthonormal basis. Orthonormal eigenvectors have unitary euclidean norm and are orthogonal to each other. Therefore, for the zero eigenvalues, there exist u and eigenvectors u0 with ku0 k = 1, orthogonal to each other and to both, kuk

P

P

i∈B c ∩Ac (B c ∩ Ac )

i

every kuui k , where kuk = 2n/2 and kui k = 2n/2 , for every i. Since by steps 2 and 3 u and every ui are time independent, every u0 is also time independent. Now, fix the following order of eigenvalues: first eigenvalue 1, afterwards eigenvalues λit , by type, and finally the zero eigenvalues in a fixed order. Then U is such that n u ui U = [ kuk ... u u0 . . . u0 ], and the diagonal matrix at time t is: ku1 k kun k

24



1 0 0  1  0 λt · · ·  0 0 · · · Λt =  0 0 0  ..  .. .. . . . 0 0 0

0 ··· 0 ··· ··· 0 ··· ··· 0 0 · · · λnt .. .. .. . . . 0 0 0

0 0 0 0 .. . 0

··· ··· ··· ··· 0 0



0  0  0  0 



0 0

Since at every t Wt is diagonalizable over the same eigenspace, hence W T = QT QT U ΛT U 0 where ΛT = t=0 Λt with diagonal entries: 1, t=0 λit for every i and zeros. Step 6. Here we deal with the existence of W ∞ and a∞ . By step 5, W ∞ = U limT −→∞ ΛT U 0 , provided that the RHS of this expression exists. We confirm here that this is, in fact, the case. In computing limT −→∞ ΛT we focus on the nonzero diagonal entries of ΛT . Eigenvalue 1 is constant across time, thus its limiting Q∞ product is 1. Since at every t λit ∈ (0, 1) for every i, t=0 λit exists in [0, 1). Thus, U limT −→∞ ΛT U 0 exists and defines both, W ∞ and a∞ = W ∞ a0 , for every a0 . Step 7. We provide here the specific form of W ∞ . Suppose that ∆0 [1] > ∆0 [2]. Consider attribute 1 first. Let rti = ∆t [i]/∆t [1] for every attribute i. We then P P rewrite λ1t = (∆t [1])(∆t [1] + i>1 ∆t [i])−1 = [1 + i>1 rti ]−1 . By step 1, rti = λit−1 ∆t−1 [i]/λ1t−1 ∆t−1 [1]. From the expression of λit it follows that λit−1 /λ1t−1 = i . Thus, r i = (r i )2 and recursively we get that r i = (r i )2t . ∆t−1 [i]/∆t−1 [1] = rt−1 t t 0 t−1 P t Thus, λ1t = [1 + i>1 (r0i )2 ]−1 . It is important to notice that, 0 < r0i < 1 for attributes i > 1. It then follows that limt−→∞ λ1t = 1. This opens the possibility Q∞ for t=0 λ1t 6= 0. We prove that this is indeed the case by equivalently stating that P∞ 1 t=0 log(λt ) exists. In order to do it, we consider r02 , the highest ratio smaller than one, and construct a new homophily value as follows: we replace r0i , for attributes i > 1, with t r02 in λ1t . Specifically, we have that λ1t = [1 + (n − 1)(r02 )2 ]−1 . Since r02 ≥ r0i P∞ 1 for every i > 1, then λ1t ≥ λ1t at every t. We prove that t=0 log(λt ) exists, P∞ 1 so does proceed by testing the absolute cont=0 log(λt ), by comparison. We P∞ 1 vergence (and hence the convergence) of t=0 log(λt ), using the ratio test. It is well known that an adaptation of the L’Hopital rule can be used to find limits of sequences. We thus define f (x) and g(x) as functions of a real variable x and {st } such that at every t, st = f (t)/g(t). Then, we evaluate limx→∞ f (x)/g(x) = x+1 log(1 + (n − 1)(r02 )2 ) limx→∞ . Since 0 < r02 < 1, this limit is indeterminate. By x log(1 + (n − 1)(r02 )2 ) x x 2(r02 )2 (1 + (n − 1)(r02 )2 ) 0 0 L’Hopital limx→∞ f (x)/g(x) = limx→∞ f (x)/g (x) = = x+1 (1 + (n − 1)(r02 )2 ) P∞ 1 0. Thus, limt−→∞ st = limx→∞ f (x)/g(x) = 0. This implies that t=0 |log(λt )| exists. Since at every t, λ1t ≥ λ1t , then |log(λ1t )| ≤ |log(λ1t )|. Thus, by comparison

25

P∞

P∞

exists, so does t=0 log(λ1t ).35 Consider now attributes i > 1. For a given i > 1, let j denote attributes other P t than it and let rtj = ∆t [j]/∆t [i]. Then λit = [1 + j6=i (r0j )2 ]−1 . Notice that r01 > 1. Q∞ Then limt−→∞ λit = 0 and t=0 λit = 0 for attributes i > 1. Summing up we have Q∞ Q∞ that t=0 λ1t = µ1 with µ1 ∈ (0, 1) and t=0 λit = 0 for i > 1. Under this scenario: 1 t=0 |log(λt )|



1

0

  0 µ1  0 0 lim ΛT =  0 0 T −→∞  . .  .. .. 0

0



0 .. . .. . 0 .. .

···

0 0 ···

0

···

0 0 ···

0

0 0

0

0 .. . .. . 0 .. .



  · · · 0 0 · · · 0  and U lim ΛT = T −→∞ · · · 0 0 · · · 0   .. .. · · · . . · · · 0 0 0

0

1  .. .

µ1 .. .

  µ1 1 1 2n/2 1 −µ1   .. .. . .

0 0 ··· .. .. . . ··· 0 0 ··· 0 0 ··· .. .. . . ···

1 −µ1 0 0

0

0

0 0 ··· .. .. . . ··· 0 0 ··· 0 0 ··· .. .. . . ··· 0 0

0

Thus: 

W∞

1 + µ1 · · ·  1 + µ1 · · ·   ..  . ···   1 + µ1 · · ·   1  1 + µ1 · · · = n 2  1 − µ1 · · ·   1 − µ1 · · ·   .. ···  .

1 + µ1 1 − µ1 1 + µ1 1 − µ1 .. .. . . 1 + µ1 1 − µ1 1 + µ1 1 − µ1 1 − µ1 1 + µ1 1 − µ1 1 + µ1 .. .. . .

··· ··· ··· ··· ··· ··· ··· ···



1 − µ1 1 − µ1   ..  . 



1 − µ1    1 − µ1   1 1+µ   1 + µ1   ..  . 

   1 − µ1 · · · 1 − µ1 1 + µ1 · · · 1 + µ1  1 − µ1 · · ·

1 − µ1 1 + µ1 · · ·

1 + µ1

. Notice that since eigenvectors u0 , associated to the zero eigenvalues, occupy the last columns ( respectively rows) of U ( respectively U 0 ), they are not involved in these products. Suppose now that ∆0 [1] = ∆0 [i] for some attributes i > 1. Let e be the number of attributes i > 1 such that ∆0 [1] and ∆0 [i] are equal. Then limt−→∞ λ1t = [e + 1]−1 6= Q∞ 1, This implies that t=0 λ1t = 0. By the same reasoning this is also the case for attributes i > 1 such that ∆0 [1] = ∆0 [i]. For attributes i > 1 such that ∆0 [1] > ∆0 [i], then limt−→∞ λit = 0 by a similar arguments as above and thus Q∞ i ∞ is (2n )−1 . t=0 λt = 0 for them. Under this scenario, every entry in W Step 8. We establish here the necessary and sufficient condition for disagreement to persist. We also qualify disagreement. Recall that a0 = 0. The eventual attitude of a type A is the result of multiplying its corresponding row in W ∞ by the column vector of initial attitudes. Consider first ∆0 [1] > ∆0 [2]. Then W ∞ is 35 Consider

that λit = ∆t [i]δ /

P

j

∆t [j]δ with δ ∈ [0, ∞). Similar algebra and reasoning as above and let us to

conclude that limx→∞ f 0 (x)/g 0 (x) = then

λ1t

2(r02 )δ

2

(δ+1)x (1

+ (n − 1)(r02 )δ

(1 + (n − 1)(r02 )δ

= 1/n at every t.

26

2

(δ+1)x )

2 (δ+1)x+1

)

=0 for δ 6= 0. Notice that when δ = 0



0



0  0  0 



0 0

.

the one depicted in step 7. For the first 2n−1 rows of W ∞ , corresponding to types  1 P  P 1 1 A− 1 A = 1 µ1 ∆ [1]. A such that 1 ∈ A, aA a a n−1 n−1 0 ∞ = 2µ 0 A:i∈A 0 A:i∈A / 2 2 2 n−1 B For the subsequent 2 rows corresponding to types B such that 1 ∈ / B, a∞ = 1 1 1 1 A − 2 µ ∆0 [1]. Thus, in general, for every type A, |a∞ | = 2 µ ∆0 [1] and eventual attitudes are positive if and only if A is such that 1 ∈ A. That is, disagreement persists. We are left to prove that τ (W ∞ ) = µ1 . Consider expression (3). Fixµ1 ing any column in W ∞ , the maximum distance between any two rows is 2n−1 , n 1 which summing across the 2 columns and dividing by 2 yields µ . Finally since µ1 = |aA ∞| =

Q∞

i t=0 λt = limT →∞

1 2

Q∞ h t=0

1+

P

i>1

QT h t=0

1+

P

i>1

∆0 [i](∆0 [1])−1

∆0 [i](∆0 [1])−1


2t i−1


2t i−1

, we have that

∆0 [1].

Consider now that ∆0 [1] = ∆0 [i] for some attributes i > 1. By step 7, every entry of W ∞ is (2n )−1 . In this case aA ∞ = 0 for every type A. That is, consensus eventually emerges. We then conclude that disagreement persists if and only if attribute 1 is, initially, the unique most salient attribute. Step 9. We consider the case in which λi0 = 0 for some/all attributes i > 1. Step 1 relies on the linearity of the updating process. Thus, it still holds. Since at every t, Wt remains stochastic, step 2 holds. For the attributes i such that λit > 0, the statement in step 3 hold as well. Step 4 holds with the difference that now there are 2n − (n + 1 − N ) zero eigenvalues, where N is the number of attributes i such that λit = 0. In the extreme case in which λit = 0 for every attribute i > 1, the column corresponding to the empty type and the n − 1 columns corresponding to the singleton types with attributes different from 1, are the same. In such a case N = n − 1 and there are 2 independent columns. The eigenvalues different from zero at every t are 1 because of stochasticity and λ1t = 1. Since at every t, Wt remains symmetric, step 5 holds. Step 6 deals with the existence of W ∞ , which is based on the existence of the limiting product of non-zero eigenvalues. It also goes through. Since the form of W ∞ depends only on whether ∆0 [1] > ∆0 [2], despite of λit being 0 for some/all attributes i > 1, step 7 holds. Finally step 8, that establishes the necessary and sufficient condition for disagreement to persist, qualifying it, also holds. Notice that when λi0 = 0 for all attributes i > 1 then W ∞ = W0 . Also, µ1 = 1 and the equilibrium is reached at t = 1.

 Proof of Proposition 1. In computing the SSIti , we follow Echenique and Fryer (2007). Before proceeding recall that by step 1 in the proof of Theorem 1, positive (respectively zero) homophily values remain positive (respectively zero) all along the P process. Recall also that i λit = 1 at every t. Consider first the case in which for 27

every attribute i, λit > 0. Consider only types possessing attribute 1. Denote the matrix of their interactions by 1t . Since all types have attribute 1 in common, they pay a positive amount of attention to each other, thus 1t has only one connected component composed by all individuals in 1t . We now compute the largest eigenvalue of 1t . Our claim is Pn that λt = λ1t + 2−1 j6=i λjt , with associated time independent right-eigenvector u of size 2n−1 × 1, where u is composed by ones, is the largest eigenvalue of 1t . We first prove that in fact (λt , u) is a pair of eigenvalue and right-eigenvector of 1t . Second, we argue that λt has to be the largest eigenvalue of 1t . First, notice that every type A shares attribute 1 with 2n−1 types. It also shares the rest of attributes with 2n−2 types. Thus, any row of 1t by u reads (2n−1 λ1t + Pn 2n−2 j6=i λjt )(2n−1 )−1 . This is equivalent to λt × 1. Therefore, the eigenvalue equation is satisfied and (λt , u) is a pair of eigenvalue and (column) eigenvector of 1t . Second, by Perron-Froebenius Theorem, being 1t a positive matrix, it has a unique largest eigenvalue, which is strictly positive (that is, the spectral radius of 1t ). It is bounded above by the maximum sum of the entries of a row in 1t , (see Meyer (2000), chapter 8). Notice that every row of 1t sums up to the same value, which is precisely λt . Suppose that there is other positive real eigenvalue, different than λt , which is the largest. Then it has to be also larger than the maximum sum of the entries of a row in 1t , contradicting the Perron-Froebenius Theorem. Then, λt has to be the largest eigenvalue. We rewrite it as λt = λ1t + 2−1 (1 − λ1t ) = 2−1 (1 + λ1t ). Let us denote SSIt1 = λt . Finally, it directly follows that λt increases with λ1t . Since limt−→∞ λ1t = 1 then limt−→∞ SSIt1 = 1 as well. Also, if every attribute i was initially equally salient, then λi0 = 1/n for each of them. Since attribute 1 is the initially most salient, it has to be that λi0 > 1/n. Thus, SSI01 > (n+1)(2n)−1 . Notice that the analysis is exactly the same when we consider interactions of types lacking attribute 1. In fact, the matrix of interactions is exactly 1t . Also, in computing the SSIti for attributes i > 1, we follow similar arguments. Thus, we omit the proofs. Consider now the case in which for attribute 1, λ1t > 0 and for some/all attributes i > 1, λit = 0.36 We prove here that when for an attribute i, λit = 0 then the SSIti is, at every t, equal to one half.37 Given the evolution of the homophily values, as described in the proof of Theorem 1, this is also its limiting value. Recall that, by step 1 in the proof of Theorem 1, when for an attribute 2 ≤ i ≤ n, λit = 0, then λjt = 0 for all attributes j > i. Let us focus on types possessing attribute i. The analysis is exactly the same when we consider interactions of types lacking attribute 36 When all differences in average initial attitudes are equal, either positive or zero, then λi = 1/n for every i and t at every t. Then, SSIti = (n + 1)(2n)−1 for every attribute i and at every t. See the proof of Theorem 1. 37 Notice that when computing the SSI i for an attribute i such that λi > 0 in the presence of attributes j 6= i t t such that λjt = 0, the matrix of interactions of i-similar types has all its entries positive. Thus, the analysis is the same as before.

28

i. Two cases arise: (i) Suppose that for every attribute 1 < j < i, λjt = 0, then interactions among types possessing attribute i are defined by two connected components, based on the lack or possession of attribute 1. The matrices defining these two connected components are the same and have all their entries positive. One of the matrices has 2n−2 types possessing attribute 1 and the other has 2n−2 types lacking it. The analysis within each matrix is exactly the same as before. In each of them, the sum of every row is 2n−2 (2n−1 )−1 λ1t = 0.5. Thus, within each component the SSIti equals to one half at every t, so it is the average of the two. (ii) Suppose that for some/all attributes 1 < j < i, λjt > 0. In this case there is only one connected component containing all individuals possessing attribute i. The reason is as follows: types possessing (respectively lacking) attribute 1 are always connected among themselves. Furthermore, these two groups are connected between them since both contain types that are similar in attributes j < i with λjt > 0. The sum of the entries of every row of the matrix of P interactions is 2n−2 (λ1t + j6=i λjt )(2n−1 )−1 = 0.5. Thus, the index is equal to one half at every t.38

 Proof of Proposition 2. Consider the case in which all differences in average initial attitudes are positive.39 To start with, we set the bounds for T . First, let ∆ [n] r0 = ∆00 [1] . Now, replace every r0i , for attributes i > 1, with r0 in λ1t . We 1

1

t

1

get λt = [1 + (n − 1)(r0 )2 ]−1 . Notice that λt ≥ λ1t for every t. Solving λt ≥ 1 −  for t, we get the for Tmin , as defined in section 4. That is,  expression  t ≥ log log

 (1 − )(n − 1)

log(r0 )−1

1 . log(2)

When the strict inequality in the previ-

1

ous expression is not satisfied, then λt ≤ 1 − , implying that λ1t ≤ 1 − . Therefore, ∆ [2] Tmin is a lower bound for T . Second, let r0 = ∆00 [1] . Replace every r0i , for attributes t

i > 1, with r0 in λ1t . We get λ1t = [1 + (n − 1)(r0 )2 ]−1 . Notice that λ1t ≤ λ1t for every t. Solving λ1t ≥1 − for t, we get for Tmax , as defined in section 4.  the expression 

That is,

t ≥ log log

 (1 − )(n − 1)

log(r0 )−1

1 . log(2)

When this expression is satisfied,

then λ1t ≥ λ1t ≥ 1 − . Thus, Tmax is an upper bound for T . Finally notice that making Tmin and Tmax positive is always possible, for small enough  > 0. With respect to the comparative statics on the initial relative salience of attribute 1, we prove that λ1t is decreasing in r0i . Recall that we consider that the variation 38 In

this case the matrix of interactions is just non-negative. Since it is irreducible, the Perron-Froebenius eigenvalue is equal to the sum of entries of any row in the interaction matrix, which is here always the same. The associated time independent eigenvector is u of size 2n−1 × 1 with unitary entries. 39 In section 4 we describe the situation in which some/all differences associated to attributes i > 1 are zero.

29

in r0i comes from varying ∆0 [i], one at a time. This is done in such a way that ∆0 [1] > ∆0 [2] ≥ ... ≥ ∆0 [n] ≥ 0 is preserved in order, as well as in magnitude for differences associated to attributes j 6= i. Thus, r0 and r0 are always constructed with differences associated to the same attributes. Consider the expression of λ1t t in the step 7 in the proof of Theorem 1. We have that ∂λ1t /∂r0i = −2t (r0i )2 −1 [1 + P i 2t −2 < 0. Thus, when r i decreases, at every t, λ1 will be higher. The time t 0 i>1 (r0 ) ] to reach its limit, has to be therefore smaller. Being T an integer, we say that the time it takes for λ1t to reach its limit cannot be higher than before.

 Proof of Theorem 2. This proof is based on the proof of Theorem 1. We proceed to explain, one in a row, which of its steps hold here. Step 1 describes a property that mainly relies on the linearity of the updating process and partially on the Luce form, specifically, at every t, ∆t [i] ≥ 0 implies that λit ≥ 0. Given property (ii) of γti , this step holds here. Steps 2-5 do not depend on the Luce form and they also apply here. Step 6 relies on the fact that γti ∈ (0, 1) for every i and at every t and not on its specific form, thus it also holds. Step 7 absolutely relies on the Luce form. The new step is as follows: Step 7. We first consider the case in which the limiting product of the homophily values is different from zero for one attribute i. Notice that this limiting product cannot be different from zero for more than one attribute. The reason is that in P lim γti > 1, this case limt−→∞ γti = 1 for more than one attribute and Qi ∞ t−→∞ contradicting assumption (i) in extension 1. We prove that t=0 γti = µi with Q∞ µi ∈ (0, 1) implies that t=0 γtj = 0 for attributes j 6= i. We also show that if there is such an attribute, it has to be attribute 1. Second, we consider the case in which the limiting product of homophily values is zero for all attributes. Before proceeding recall that, by assumption (i) in extension 1, limt−→∞ γti exists for every attribute i P P and limt−→∞ i γti = i limt−→∞ γti = 1. Q∞ P∞ First, suppose that t=0 γti = µi with µi ∈ (0, 1), or equivalently, that t=0 log(γt1 ) exists. This implies that limt−→∞ γti = 1. Thus, limt−→∞ γtj = 0 for every attribute Q∞ j. Thus, there is no attribute j for which t=0 γtj = µj with µj ∈ (0, 1). In this Q∞ case, t=0 γtj = 0 for every of them. If there is an attribute for which the limiting product is different from zero, it has to be attribute 1. To see this recall that, by property (iii) of γti and by step 1 in the proof of Theorem 1, at every t, ∆t [1] ≥ ∆t [2] ≥ · · · ≥ ∆t [n] ≥ 0 and γt1 ≥ γt2 ≥ · · · ≥ γtn ≥ 0 hold. Suppose, by contradiction, that there is Q∞ an attribute j for which t=0 γtj = µj with µj ∈ (0, 1) holds. This implies that limt−→∞ γtj = 1 and that limt−→∞ γt1 = 0. Thus, for high enough t, γtj would be arbitrarily close to 1 while γt1 would be arbitrarily close to 0. Given that, at every t, 30

γt1 ≥ γt2 ≥ · · · ≥ γtn ≥ 0 holds, the former statement cannot be true. We therefore Q∞ Q∞ conclude that t=0 γt1 = µ1 with µ1 ∈ (0, 1) and t=0 γtj = 0 for attributes other than 1 hold. Under this scenario W ∞ is the one depicted in step 7 in the proof of Q∞ Theorem 1. In this case we just set τ (W ∞ ) = t=0 γt1 . Q∞ 1 1 Second, suppose that t=0 γt = 0 and limt−→∞ γt = 1. This implies that limt−→∞ γtj = 0 for every attribute j 6= i. Thus, no attribute j is such that Q∞ 1 Q∞ j 1 j with µj ∈ (0, 1). Suppose now that γ = µ t t=0 γt = 0 and limt−→∞ γt = α t=0 P j with α ∈ (0, 1). Then, j6=Q i limt−→∞ γt = 1 − α with 1 − α ∈ (0, 1). Thus, ∞ j j j no attribute j is such that t=0 γt = µ with µ ∈ (0, 1). Finally, suppose Q∞ 1 that t=0 γt = 0 and limt−→∞ γt1 = 0. Then, either for exactly one attribute j, P j limt−→∞ γtj = 1, or for all attributes j, j6=i limt−→∞ γt = 1. None of these cases can hold. The reason is that for high enough t, some γtj would be arbitrarily close to a positive number (which is 1 when for exactly one attribute j, limt−→∞ γtj = 1) while γt1 would be arbitrarily close to 0. Since, at every t, γt1 ≥ γt2 ≥ · · · ≥ γtn ≥ 0 Q∞ holds, this cannot be true. We therefore conclude that in all these cases t=0 γti = 0 for all attributes. Under this scenario all entries of W ∞ are (2n )−1 . Step 8. By step 7, the necessary and sufficient condition for disagreement to P∞ persist is that t=0 log(γt1 ) exists. When this is the case, W ∞ is the one depicted in step 7 in the proof of Theorem 1. Eventual attitudes take the same form as in step 8 in the proof of Theorem 1. Here there is no closed form expression for QT τ (W ∞ ) = limT →∞ t=0 γt1 . Step 9. It does not depend on the Luce form, applying also here.

 Proof of Lemma 1. Consider first the case in which ∆0 [i] > ∆0 [j] ≥ 0 or equivalently {i,j} {∅} {i} {j} {i,j} {∅} {j} {i} a0 − a0 + a0 − a0 > a0 − a0 + a0 − a0 . Notice that ∆0 [i] − ∆0 [j] = {j} {i} {j} {i} {j} {i} a0 − a0 . Thus, a0 − a0 > 0 has to hold. Since ∆0 [j] ≥ 0 and a0 − a0 < 0, {i,j} {∅} then a0 − a0 > 0 has to hold as well. Thus, for ∆0 [j] ≥ 0 it has to be that {i,j} {∅} a0 − a0 > 0. ˜ 0 [i]| = |∆ ˜ 0 [j]| is zero. That is It is important to notice that the probability that |∆ {i} {j} {i,j} {∅} so because this expression holds when exactly a0 −a0 = 0 and/or a0 −a0 = 0. Since these differences follow continuous (normal) distributions, these events have zero probability. ˜ 0 [i]| > |∆ ˜ 0 [j]|. Let us examine Disagreement across attribute i persists when |∆ the conditions under which this expression holds. Notice that it is equivalent to {i,j} {∅} {i} {j} {i,j} {∅} {j} {i} |˜ a0 −a ˜0 + a ˜0 − a ˜0 | > |˜ a0 −a ˜0 + a ˜0 − a ˜0 |. This expression is satis{i,j} {∅} {i} {j} {i,j} {∅} fied whenever a ˜0 −a ˜0 ≥ 0 and a ˜0 − a ˜0 ≥ 0 hold or a ˜0 −a ˜0 ≤ 0 and {i} {j} {i,j} {∅} {i} {j} ˜ t [i]| > |∆ ˜ t [j]|) = P (˜ a ˜0 − a ˜0 ≤ 0 hold. Thus P (|∆ a0 − a ˜0 ≥ 0 ∩ a ˜0 − a ˜0 ≥ {i,j} {∅} {i} {j} 0) + P (˜ a0 − a ˜0 ≤ 0 ∩ a ˜0 − a ˜0 ≤ 0). Since a ˜A 0 are independent to each other, 31

this is equivalent to: {i,j}

P (˜ a0

{∅}

−a ˜0

{i}

{j}

≥ 0)P (˜ a0 − a ˜0

{i,j}

≥ 0) + P (˜ a0

{∅}

−a ˜0

{i}

{j}

≤ 0)P (˜ a0 − a ˜0

≤ 0).

˜ 0 [i]| < |∆ ˜ 0 [j]|, On the contrary, disagreement across attribute j persists when |∆ {i,j} {∅} {i} {j} {i,j} {∅} {j} {i} or equivalently |˜ a0 −a ˜0 + a ˜0 − a ˜0 | < |˜ a0 −a ˜0 + a ˜0 − a ˜0 |. This ex{i,j} {∅} {i} {j} {i,j} {∅} pression holds when a ˜0 −a ˜0 ≤ 0 and a ˜0 − a ˜0 ≥ 0 hold or a ˜0 −a ˜0 ≥ 0 {i} {j} ˜ 0 [i]| < |∆ ˜ 0 [j]|) can be expressed as: and a ˜0 − a ˜0 ≤ 0 hold. Then P (|∆ {i,j}

P (˜ a0

{∅}

−a ˜0

{i}

{j}

≥ 0)P (˜ a0 − a ˜0 {i,j}

{∅}

{i,j}

≤ 0) + P (˜ a0 {i}

{∅}

−a ˜0

{i}

{j}

≤ 0)P (˜ a0 − a ˜0

≥ 0).

{j}

Notice that since a ˜0 −a ˜0 and a ˜0 − a ˜0 are normally distributed, there is a positive probability that they take values higher or lower than zero. This implies ˜ 0 [i]| > |∆ ˜ 0 [j]|), P (|∆ ˜ 0 [i]| < |∆ ˜ 0 [j]|) ∈ (0, 1). Furthermore: that P (|∆ {j} {i} {∅} {i,j} ˜ 0 [i]| > |∆ ˜ 0 [j]|) − P (|∆ ˜ 0 [i]| < |∆ ˜ 0 [j]|) = P (˜ ˜0 ≥ a0 − a ˜0 ≥ 0)(P (˜ P (|∆ a0 − a {i} {j} {i,j} {∅} {i} {j} {i} {j} 0)−P (˜ a0 −˜ a0 ≤ 0))+P (˜ a0 −˜ a0 ≤ 0)(P (˜ a0 −˜ a0 ≤ 0)−P (˜ a0 −˜ a0 ≥ 0)). {i,j}

{∅}

{i,j}

{∅}

˜0 is normal with positive mean, implying ˜0 − a Since a0 − a0 > 0, then a {∅} {i,j} −a ˜0 ≥ 0) > 0.5. The (by symmetry of the normal distribution) that P (˜ a0 {i} {j} same argument holds for a ˜0 − a ˜0 . Thus, the expression above is positive, being disagreement across attribute i is the most likely. The case in which ∆0 [j] > ∆0 [i] ≥ 0 follows a similar reasoning and is therefore omitted. In this case disagreement across attribute j is the most likely. Consider now the case in which ∆0 [i] = ∆0 [j] ≥ 0 we have that ∆0 [i] − ∆0 [j] = {i} {j} {i,j} {∅} a0 − a0 = 0 and a0 − a0 ≥ 0 has to hold. This implies (by symmetry of the ˜ 0 [i]| > |∆ ˜ 0 [j]|) − P (|∆ ˜ 0 [i]| < |∆ ˜ 0 [j]|) = 0. In this normal distribution) that P (|∆ case, disagreement across either attribute is equally likely.

 Proof of Lemma 2. Suppose that disagreement persists across attribute i. By the ˜ 0 [i]| > proof of Lemma 1, this happens with positive probability, that is, P (|∆ ˜ 0 [j]|) ∈ (0, 1). Notice that the expression of eventual attitudes is based on the |∆

32

˜ 0 [i]| > |∆ ˜ 0 [j]|. We have that: expression of the deterministic ones.40 Let x = |∆

E(˜aA ∞

1 : i ∈ A|x) = E(a0 |x) + E 2

E(˜aA ∞

1 :i∈ / A|x) = E(a0 |x) − E 2

and





˜ 0 [i]
∆

˜ 0 [i]| − |∆ ˜ 0 [j]| |∆

˜ 0 [i]| |∆

˜ 0 [i]
∆

˜ 0 [i]| − |∆ ˜ 0 [j]| |∆



˜ 0 [i]| |∆

|x

 |x .

Also,

  1

˜ 0 [i] ∆ = ˜ 0 [i]|  |∆ 

˜ 0 [i] > 0 if ∆ .

˜ 0 [i] < 0 −1 if ∆

˜ 0 [i] > 0), we have that E(˜ Then, with probability P (∆ aA ∞ : i ∈ A|x) = E(a0 |x) +

1 1 ˜ 0 [i]| − |∆ ˜ 0 [j]||x and E(˜ ˜ 0 [i]| − |∆ ˜ 0 [j]||x . / A|x) = E(a0 |x) − E |∆ E |∆ aA ∞ : i ∈ 2 2 ˜ 0 [i] > 0) the signs in front of 1/2 change in these expresWith probability 1 − P (∆
˜ 0 [i]| − |∆ ˜ 0 [j]| |x) > 0 since it is the expectation sions. Finally, notice that E( |∆ of a distribution, conditional on itself being positive. Also, since ∆0 [i] ≥ 0 then ˜ 0 [i] > 0) ≥ 0.5. In this case, E(˜ / A|x) > 0. aA P (∆ aA ∞ :i∈ ∞ : i ∈ A|x) − E(˜

 Proof of Lemma 3. It follows a similar reasoning as the one in the proof of Lemma ˜ 0 [i] and ∆ ˜ 0 [j], are normally 2 is therefore omitted. Just notice that since both, ∆ ˜ 0 [i] > 0) ≥ P (∆ ˜ 0 [j] > 0) ≥ 0.5 if and distributed with non-negative means, P (∆ only if the mean of the former is higher or equal than the one of the latter.  Proof of Lemma 4. Let focus first on ∆0 [1] > ∆0 [2], thus λ10 > 12 > λ20 . Let us deλ1 + βt1 note λ∗t = t ∈ (0, 1]. Simple algebra yields ∆t [1] = λ∗t−1 ∆t−1 [1] and ∆t [2] = 2 Qt−1 Qt−1 (1 − λ∗t−1 )∆t−1 [2]. Thus, ∆t [1] = s=0 λ∗s ∆0 [1] and ∆t [2] = s=0 (1 − λ∗s )∆0 [2]. ˜ 1 the homophily value of attribute 1 under For the ease of exposition let us denote λ t the process described in the main part of the paper, where at every t, Wt is symmetλ∗0 ∆0 [1] 1 1 1 ric. Since by assumption βt > λt at every t, then λ1 = ∗ > λ0 ∆0 [1] + (1 − λ∗0 )∆0 [1] ˜ 1 . Also β t > λ1 then ∆2 [1] = λ∗ ∆1 [1] is higher than in the symmetric case λ 1 1 1 1 ˜1 ˜ 2 . In general at every t, β 1 > λ1 > λ (and ∆2 [2] smaller). Consequently λ21 > λ t t t 1 ˜ 2 with λ1 = λ ˜ 1 . Consider now a sequence of ones. Since at and βt2 < λ2t < λ t 0 0 1 ˜ ˜ 1 = 1, every t, 1 > λt ≥ λt and by step 7 in the proof of Theorem 1, limt→∞ λ t 40 Notice that the expression for λ ˜ i = 2α ˜ it and its properties remain the same when we consider differences in t average attitudes in absolute value. See mainly steps 1 and 7 in the proof of Theorem 1. The ergodicity coefficient   ˜ −1 |∆ ˜ 0 [i]| − |∆ ˜ 0 [j]| ∆0 [i] ˜ 0 [i]| − |∆ ˜ 0 [j]|)(|∆ ˜ 0 [i]|)−1 in this case. See section 3. We have that a is (|∆ ˜A ∞ = a0 + 2 ˜ 0 [i]| |∆   ∆ ˜ 0 [i] −1 |∆ ˜ 0 [i]| − |∆ ˜ 0 [j]| if i ∈ A and a ˜A if i ∈ / A. Finally recall that a0 is now a random variable. ∞ = a0 − 2 ˜ 0 [i]| |∆

33

then limt→∞ λt = 1. Also since 1 > βt1 > λ1t , thus limt→∞ βt1 = 1. By the 1 1 P∞ ˜ 1 )| converges. Since at every t, λ∗ = λt + βt > λ ˜ 1 then |log( λ same step, t t t t=0 2 P∞ ∗ ∗ 1 ˜ hence |log(λt )| ≤ |log(λt )|. Thus, by comparison t=0 |log(λt )| converges and Q Q∞ ∗ ∞ ∗ 1 ˜1 ˜ , then δ > µ = at every t, λt > λ t t=0 λt . t=0 λt = δ ∈ (0, 1]. Notice that sinceQ ∞ Finally, since limt→∞ 1 − λ∗t = 0 then t=0 (1 − λ∗t ) = 0. Consider Wt in extension 3. We rewrite it as: {1, 2} 1 + λ1t + λ2t 1 1 + β1 − β2 t t Wt =   4 1 − λ1 + λ2 t t 1 1 − βt − βt2



{1} 1 + λ1t − λ2t 1 + βt1 + βt2 1 − λ1t − λ2t 1 − βt1 + βt2

{2} 1 − λ1t + λ2t 1 − βt1 − βt2 1 + λ1t + λ2t 1 + βt1 − βt2

{∅}  1 − λ1t − λ2t {1, 2} 1 − βt1 + βt2   {1} 2 1 1 + λt − λt  {2} 1 + βt1 + βt2 {∅}

1 1+1i λ1 ∆ [1] + Let us focus on Wt at = at+1 . It follows that aA t t t+1 = at + ((−1) 2 1 1+1i β 1 ∆ [1] − β 2 ∆ [2]) if 2 ∈ λ2t ∆t [2]) if 2 ∈ A and aA / A where t t t t t+1 = at + ((−1) 2 1i is the indicator of type A possessing attribute 1. Notice that ones in every entry in a row of matrix Wt multiplied by at compute average attitudes at t + 1. Similarly, weights associated to attribute 1 (resp. attribute 2), i.e., λ1t or βt1 (resp. λ2t or βt2 ) compute differences associated to attribute 1 (resp. attribute 2). Differences enter with positive sign if and only if the type possesses attribute 1 (resp. attribute 2). Using the expressions for ∆t [1] and ∆t [2] above we rewrite aA t+1 = Q Q 1 t−1 t−1 at + ((−1)1+1i λ1t s=0 λ∗s ∆0 [1]+ λ2t s=0 (1 − λ∗s )∆0 [2]) if 2 ∈ A and aA t+1 = at + 2 Qt−1 Qt−1 1 / A. Thus, limt→∞ aA ((−1)1+1i βt1 s=0 λ∗s ∆0 [1]- βt2 s=0 (1 − λ∗s )∆0 [2]) if 2 ∈ t+1 = 2 1 1 limt→∞ at + δ∆0 [1] if 1 ∈ A and limt→∞ aA / A, t+1 = limt→∞ at − δ∆0 [1] if 1 ∈ 2 2 provided that limt→∞ at exists. We prove that this is the case. Simple algebra yields Pt−1 Qs−1 1 Pt−1 2 1 at = a0 + (λs − βs2 )∆s [2] = a0 + ∆0 [2] s=0 (λ2s − βs2 ) m=0 (1 − λ∗m ). We s=0 4 P∞ 2 Qs−1 4 2) ∗ prove here that (λ − β prove that s s=0 s m=0 (1 − λs ) exists. In doing so we Q P∞ 2 s−1 2 2 2 2 2 every s, λs − βs > (λs − βs ) m=0 (1 − s=0 (λs − βs ) exists. Given that atP Q ∞ s−1 λ∗m ) we conclude by comparison that (λ2s − βs2 ) m=0 (1 − λ∗m ) exists. We s=0 P∞ 2 2 proceed by contradiction. Suppose that or equivalently, s=0 (λs − βs ) diverges, Q∞ Q∞ 2 2 41 that s=0 (1−(λs −βs )) = 0. We rewrite this expression as s=0 ((1−λ2s )+βs2 ) = Q∞ 2 Q∞ ˜ 1 1 ). From step 7 in the proof of Theorem 1, (β + λ λ ∈ (0, 1]. Recall s s s=0 Q∞ 2 s=01 s 1 1 ˜ that at every s > 1, λs > λs , it then has to be that s=0 (βs + λs ) ∈ (0, 1], therefore P∞ 2 P∞ 2 Qs−1 2 2 ∗ s=0 (λs − βs ) exists, so does s=0 (λs − βs ) m=0 (1 − λm ). Let α ∈ [0, ∞) be 1 1 the value of this infinite sum. Thus, limt→∞ aA t = a0 + α∆0 [2] + δ∆0 [1] if 1 ∈ A 4 2 41 See

Apostol (1977), chapter 8.

34

1 1 and limt→∞ aA / A. As stated above δ > µ, that is, t = a0 + α∆0 [2] − δ∆0 [1] if 1 ∈ 4 2 eventual disagreement is higher than in the symmetric case. ˜ 0 = 0.5 = λ ˜ 0 at every t by Focus now the case in which ∆0 [1] = ∆0 [2]. Then λ 1 2 1 ˜ ˜t the proof of Theorem 1. By the same arguments as above λ0 = λ1 and that λt0 > λ and ∆t [1] is higher than the one in the symmetric case at every t > 1. Consider ˜ t . Same arguments as above follow the process starting at t = 1 where λt0 > λ and disagreement persist across attribute 1. It is also higher than in the symmetric case.  7.1

Appendix 3.1

We show here that a0 = 0 implies that at = 0, at every t. By step 5 in the proof of Theorem 1, at every t, Wt is diagonalizable over the same eigenspace. Let G be the projection onto the eigenspace of Wt corresponding to eigenvalue 1. Let Gi be the projection onto the eigenspace of Wt corresponding to eigenvalue λit . By the Spectral Pn QT Theorem, W T a0 = Ga0 + i=1 ( t=0 λit )Gi a0 (see Meyer (2000), pages 517-520). We proceed by describing how row j of Gi looks like. Denote by Gijk the jk entry of Gi . It is constructed using the eigenvectors in U , in step 5 in the proof of Theorem 0 . In constructing a row j of Gi , we fix column i+1 1, as follows: Gijk = Uj(i+1) U(i+1)k in U , i.e., the eigenvector corresponding to λit and consider its j entry. By step 5 in 1 1 the proof of Theorem 1, the j entry takes value 2n/2 if i ∈ A and − 2n/2 otherwise. 0 Entry j is multiplied, by the k entries corresponding to row i+1 in U , one in a turn. Notice that row i+1 of U 0 is the eigenvector associated to λit , now transposed. Thus, row j of Gi is just the eigenvector associated to λit whenever i ∈ A and its negative otherwise. Notice that matrix G is constructed in the same way and it is a matrix Qs 1 Pn of ones. Thus, aA (−1)1+1i ∆0 [i] t=0 λit , where 1i is the indicator s = a0 + i=1 2 of type A possessing attribute i. Since there are 2n−1 types possessing and lacking every attribute i, respectively, when summing aA s for all types, the second term in P P n−1 a . Since the previous expression cancels out. Specifically, A aA s s = A a0 = 2 a0 = 0, for every s, as = 0.

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