Disagreement in a Naive Model of Attitude Formation: Comparative Statics Results/El desacuerdo en modelos de formacion de opiniones: resultados de estatica comparativa/O desacordo em modelos de formacao de opinioes: resultados de estatica comparativa.

• Author: Melguizo, Isabel
• Pages: NA

Introduction

Melguizo (2019) proposes a model of attitude formation where individuals update their attitudes by averaging those of their friends. This model captures the persistence of disagreement, that is, the persistence of a situation in which individuals hold different attitudes about an issue. In that approach, a collection of dichotomous attributes defines the individuals' types. Also, individuals are more prone to interact with others similar to them in those attributes; that is, individuals exhibit homophily. (1) Interactions co-evolve with attitudes, and this feature is central to the persistence of disagreement. The main finding is that disagreement persists if, and only if, individuals develop sufficiently intense relations over time with others, similar in one specific attribute. Thus, society polarizes according to this dimension.

In that framework, individuals update their attitudes as in DeGroot (1974), further: (i) individuals' attitudes are known with certainty, and (ii) homophily relations were symmetric; that is, all individuals were homophilous with the same intensity. This paper explores how previous findings react to natural modifications in these two assumptions. The context is one in which individual types come from the combination of two dichotomous attributes. Some insights for the case of n [greater than or equal to] 2 attributes are provided. The following is a preview of the results.

(i) Random attitudes. It might be that it is better to describe attitudes as random variables. In contexts in which the aim is to learn the true state of the world, randomness might be interpreted as lack of information (noise) regarding the issue at hand, as in Golub and Jackson (2010); as the degree of attitudes' precision, as in DeMarzo, et al. (2003), or as experts having probability distributions about the true state of the world, as in DeGroot (1974). In situations in which individuals deal with ideological issues, randomness might be interpreted as flexibility or lack of stubbornness. In line with these observations, this paper goes further in proposing that initial attitudes draw from symmetric continuous distributions. This is in line with DeMarzo, et al. (2003), which allow for randomness only in the first period. The persistence of disagreement is robust to this modification. In particular, disagreement may now persist across one of the two attributes. It is, however, more likely to persist across the one for which the mean of the distribution of the initial differences in attitudes is the highest.

(ii) Non-symmetric homophily. It might be also natural to think that the (intensity of) relations that individuals establish with others depend(s) on the specific nature of the shared attributes. In fact, McPherson, et al. (2001) document how gender homophily is lower when people are younger than older. Gender homophily is also lower for high educated than for low educated people and for Anglos than for African Americans. This might imply, in particular, that pairs of individuals no longer devote the same amount of attention to each other. As an example, suppose there are four types of individuals, that is, an individual can be either young or old and either a female or a male. Consider that young people establish less intense relations with same-gender others than seniors. This behavior could emerge in the model when individuals have different sensitivity to differences in attitudes between groups. Specifically, when confronted with information about differences in attitudes between males and females, seniors exacerbate the differences in attitudes by gender with respect to young people. Notice then that the intensity of gender relations depends on another attribute defining the individuals involved; that is, on youth. The finding is that, when disagreement persists across the attribute for which initial differences in attitudes are the highest, its magnitude is higher than in the case in which homophilous relations are symmetric. The process also converges faster to the eventual attitudes.

The remaining of the paper is as follows. First, it presents the model in Melguizo (2019) to clarify the baseline setup. Then, it discusses random attitudes and then the non-symmetric homophily. Next, it includes some comments on the co-existence of random attitudes and non-symmetric homophily. After that, it presents the conclusions, and finally, it shows the proofs.

1. A Model on Homophily and Disagreement

   Before the extensions, let us first introduce the baseline model in Melguizo (2019).

   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 [subset or equal to] I. Two types, A and B, are i-similar whenever attribute i is either present or absent in these two types. Otherwise, they are i-dissimilar. Let [A.sup.c] be the complementary set of A. Then I(AB) [equivalent to] (A [intersection] B) [union] ([A.sup.c] n [B.sup.c]) is the set of shared attributes between A and B.

   The (column) vector of attitudes at time t [member of] [Z.sub.+] is denoted [a.sub.t] [member of] [-1, 1][2.sup.n]. Let [a.sup.A.sub.t] be a typical component of [a.sub.t], denoting the attitude of type A. Notice that there are [2.sup.n-1] types possessing (resp. lacking) any attribute. Thus, the average attitude across types possessing (resp. lacking) attribute i is [[bar.a].sub.t] [i] [equivalent to] [([2.sup.n-1]).sup.-1] [[summation].sub.A:i[member of]A] [a.sup.A.sub.t] (resp. [[bar.a].sub.t] [i] [equivalent to] [([2.sup.n-1]).sup.-1] [[summation].sub.A:i[member of]A] [a.sup.A.sub.t]). The difference between average attitudes across attribute i is denoted [A.sub.t] [i] [equivalent to] [[bar.a].sub.t][i] - [[bar.a].sub.t][-i].

   The following example illustrates the notation for the two-attribute case.

   Example 1. Let I = {1, 2}. Types are {1, 2} {1}, {2} and {[empty set]}. Observe, as an illustration, that types {1, 2} and {1} are 1-similar and 2-disimilar. Let [mathematical expression not reproducible] such that [a'.sub.0] = [0.8 0.2 - 0.05 - 0.95]. Notice that [[bar.a].sub.0] [1] = 0.5, [[bar.a].sub.0] [-1] = -0.5, and [[DELTA].sub.0][1] = 0.5 - (-0.5) = 1. Analogously, [[bar.a].sub.0] [2] = 0.375, [[bar.a].sub.0] [-2] = -0.375, and [[DELTA].sub.0][2] = 0.375 - (-0.375) = 0.75.

   Attitudes evolve according to an average-based process similar to DeGroot (1974). That is, current attitudes are weighted averages of previous ones. Let [W.sub.t] be the [2.sup.n] * [2.sup.n] weighting matrix describing the updating of attitudes from t to t + 1. Thus:

   [a.sub.t+1] = [W.sub.t] [a.sub.t]. (1)

   The interpretation of every row in [W.sub.t] is that every type A has one unit of attention to devote to others (and to itself). Then every entry of [W.sub.t] is the weight, i.e., the share of attention, that type A assigns to type B at time t. Let [w.sup.A,B.sub.t] denote this weight. Individuals are homophilous, a behavior that can be captured as follows; every attribute i has a non-negative value [[alpha].sup.i.sub.t]. The weight that type A assigns to type B is the sum of the values of the shared attributes between A and B, that is, [w.sup.A,B.sub.t] [equivalent to] [[summation].sub.i[member of]I(AB)] [[alpha].sup.i.sub.t]. For normalization purposes let [[summation].sub.i] [[alpha].sup.i.sub.t] = [([2.sup.n-1]).sup.-1]. That is the right normalization because any type A is i-similar to exactly [2.sup.n-1] types. Then, [mathematical expression not reproducible]. Let [[lambda].sup.i.sub.t] [equivalent to] [2.sup.n-1] a] [[alpha].sup.i.sub.t], so that [[lambda].sup.i.sub.t] [member of] [0,1] and [[summation].sub.i] [[lambda].sup.i.sub.t] [equivalent to] 1. Notice that sharing at least one attribute is necessary for any pair of types A and B to hold a relation.

   The following example describes the attention structure.

   Example 2. In the two-attribute case, the interaction matrix at time t is:

   [mathematical expression not reproducible].

   To clarify this structure, notice that relations are symmetric and focus on type {2}. It is 1-similar and 2-similar to types {[empty set]} and {1, 2}, respectively. Thus, it pays attention to them, on the basis of attributes 1 and 2, respectively. Since type {2} and {1} does not share any attribute, they pay no (direct) attention to each other.

   Let [[lambda].sup.i.sub.t] depend on the difference in average attitudes between the individuals possessing and lacking attribute i, that is, on [[DELTA].sub.t][i], and on (possibly) all the differences associated with the remaining attributes, that is, on [[DELTA].sub.t][j] for every attribute j [not equal to] i. Let, w.l.o.g,

   [[DELTA].sub.0][1] [greater than or equal to] [[DELTA].sub.0][2] [greater than or equal to] ... [[DELTA].sub.0][n] [greater than or equal to] 0. (2)

   Let further [[lambda].sup.i.sub.t] satisfy three properties:

   Within differences monotonicity (WDM): for every attribute i, [[DELTA].sub.t][i] = 0 implies that [[lambda].sup.i.sub.t] = 0, and [[DELTA].sub.t][i] > 0 implies that [[lambda].sup.i.sub.t] > 0.

   Across differences monotonicity (ADM): [[DELTA].sub.t][1] [greater than or equal to] [[DELTA].sub.t][2] [greater than or equal to] ... [greater than or equal to] [[DELTA].sub.t][n] [greater than or equal to] 0 implies that [[lambda].sup.1.sub.t] [greater than or equal to] [[lambda].sup.2.sub.t] [greater than or equal to] ... [greater than or equal to] [[lambda].sup.n.sub.t] [greater than or equal to] 0. When [[DELTA].sub.t][i] = 0 for every attribute i at time t, set [[lambda].sup.i.sub.t] = 1/n. Well defined limit: for every attribute i, [mathematical expression not reproducible]. A functional form for [[lambda].sup.i.sub.t] that satisfies the above properties is:

   [[lambda].sup.i.sub.t] = [[DELTA].sub.t][[i].sup.[delta]]/[[summation].sub.j][[DELTA].sub.t] [[i].sup.[delta]], (3)

   with [delta] [member of] [0,[infinity]).

   The co-evolution of interactions and...