Erratum: Convergence of cultural traits with time-varying self-confidence in the Panebianco (2014) model – A corrigendum (Journal of Economic Theory (2014) 150 (583–) (S0022053113001488) (10.1016/j.jet.2013.09.003))
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Journal of Economic Theory
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© 2018 Elsevier Inc. Introduction: Panebianco (2014) (from here on “P14”) studies a model of continuous trait transmission with inter-ethnic attitudes through parental (vertical) and non-parental (oblique) socialization. P14 establishes convergence of cultural traits and studies the structure of steady state outcomes. This note demonstrates that the proof of the convergence results in P14 is incorrect in two places. In the first instance, the proof in P14 incorrectly transposes a convergence result for Markov Chains which does not apply to the model in P14. In the second instance, an algebraic argument in P14 contains a mistake. The note provides a new proof that corrects both issues and recovers all affected results of P14. We first highlight the difference between Markov chains and repeated averaging models, which are commonly used in models of cultural transmission and opinion formation, in a simple example in Section 2. In Section 3 we identify the two errors in the convergence proof of P14 and present a new proof that restores all results and offers some novel insights into the steady state properties of the P14 model. We conclude with a brief discussion of convergence in repeated averaging models with time-varying transition matrices of which P14 is an example. Example: Before we present the correction to P14 in detail, we briefly illustrate the repeated averaging setting and its relationship with Markov chains in a simple example. Fix a sequence of row stochastic matrices [Formula presented] that is time-varying by alternating between the two matrices [Formula presented] and [Formula presented] depending on whether the period t is odd or even. [Formula presented] Define by [Formula presented] and [Formula presented] the products resulting from multiplying on the right and left, respectively, a total number of t matrices according to the sequence of [Formula presented] and [Formula presented], starting with [Formula presented]. Multiplication on the right as in [Formula presented] presents a Markov chain. In a Markov chain the dimensions of [Formula presented] correspond to states and [Formula presented] is a transition matrix in which element [Formula presented] describes the probability of transitioning from state i to j. The right product converges towards the matrix [Formula presented] Note that the long-run outcome depends on the first matrix on the left of the sequence. If the sequence started with [Formula presented] the positions of 0.4 and 0.6 would be switched around in the long-run outcome. Multiplication on the left represents a repeated averaging setting that is used in the cultural traits model of P14. This type of model is also used in naive learning and opinion formation literature, including for example, Cavalli-Sforza and Feldman (1973), DeGroot (1974) and, more recently, DeMarzo et al. (2003), Golub and Jackson (2010) and Büchel et al. (2014). Here the transition matrix [Formula presented] acts as an influence matrix that describes how next period attitudes are derived as the weighted average of current-period attitudes with element [Formula presented] giving the weight that individual i assigns to the trait of individual j. In contrast to the Markov chain approach above, [Formula presented] does not converge but instead leads to a limit cycle which alternates between two matrices depending on whether the final matrix on the left is [Formula presented] or [Formula presented]. [Formula presented] Fig. 1 illustrates these dynamics and plots the entry in the first row and the third column of [Formula presented] and [Formula presented]. In the cultural traits setting, this entry corresponds to the trait held by the first agent if we set the initial trait vector to [Formula presented]. The example illustrates that the convergence behaviour of a given sequence of row stochastic matrices depends on whether multiplication is from the right as in Markov chain models or the left as in the cultural traits transmission literature discussed here. Furthermore, for left multiplication, the example is a simple sequence of matrices that does not converge in a cultural transmission context. Both points play an important role in the inaccuracies in the convergence proof of P14 that we describe below. Convergence in the Panebianco (2014) model: The model in P14 presents a model of cultural transmission that can be summarized as follows: [Formula presented] where [Formula presented] is a column vector of inter-ethnic attitudes and [Formula presented] is a time-varying row stochastic square transition matrix. P14 endows [Formula presented] with the following specific structure: [Formula presented] where [Formula presented] is a diagonal square matrix capturing the vertical aspect of socialization within a group and Φ is a row stochastic square matrix with entries [Formula presented] that captures the oblique socialization between groups. Assumption 1 in Panebianco (2014) ensures that the entries of [Formula presented] and thus the diagonal entries of [Formula presented] are non-zero for all time periods t. For off-diagonal entries this structure implies that the pattern of zeros in [Formula presented] is equal to that of Φ. Furthermore, the ratio of any pair of off-diagonal entries of [Formula presented] in the same row, that is, the relative socialization weight that any group puts on a given pair of other groups, is constant for all t. P14 presents one main convergence result for this system and a corollary that presents a generalization of the result to time-varying [Formula presented] under the condition that [Formula presented] has at most one communication class per component and a pattern of zero entries that is constant across time. A component in a repeated averaging setting refers to a group of individuals such that there is a non-zero weight between every pair in the group in at least one direction after a certain minimum number of periods. This positive weight corresponds to the notion of a directed path through a network if one treats the individuals as a set of vertices and [Formula presented] as an adjacency matrix in which [Formula presented] implies that individual i is influenced by individual j. A communication class then refers to a group of individuals that influence each other but put zero weight on individuals outside the group. Such a class is also referred to as an essential class. Furthermore, individuals that themselves influence every other individual that they are influenced by are called essential. Those that are not essential are called inessential. The convergence results in P14 are as follows. (P14) Proposition 2 The system described by Equations (1) and (2) converges for any time-invariant row stochastic matrix Φ.(P14) Corollary 1 The convergence result can be extended to time-varying [Formula presented] if each [Formula presented] has at most one communication class per component and the zero entries of [Formula presented] are fixed for all [Formula presented] for some period T.The proof of Proposition 2 in P14 distinguishes between transition matrices that are irreducible and those that are reducible. Irreducible matrices correspond to influence networks that are strongly connected such that every pair of individuals, either directly or indirectly, influences each other. All individuals thus form a single essential class. By contrast, in a reducible transition matrix, there exist some inessential individuals that are influenced by some other individuals that they themselves do not influence. If a matrix is reducible, it can be written in lower triangular block form as illustrated in Equation (3) where those groups of individuals that are essential are collected in the block matrix [Formula presented] and the second row collects the remaining inessential individuals. [Formula presented] Reducible matrices are then further subdivided according to whether the individuals in block [Formula presented] form one diagonal block and thus a single essential class (Case 1) or more than one diagonal block and thus more than one essential class (Case 2). The convergence proof in P14 is incorrect in its argument for convergence of reducible matrices for both cases. Case 1 – One essential class: In the proof of Case 1, P14 restates Theorem 3.2 from D'Amico et al. (2009) which establishes convergence of single-unireducible non-homogeneous Markov chains, and then builds on this result. However, in restating it as Theorem 2 on p. 602, P14 switches the direction of multiplication from the right as in the original to the left as needed for the model in P14. It thus incorrectly applies a result from Markov chains to a repeated average setting. The two classes of models show different convergence behaviours for time-varying matrices as we show in the example in Section 2. Convergence for the case of reducible matrices with a single diagonal block can be readily recovered by using an appropriate convergence result for left multiplication. Theorem 1.10 in Hartfiel (2006) provides this result. The theorem relies on the notion of regular matrices. A stochastic matrix A is regular if it has exactly one essential class and the upper left block in lower triangular form is primitive, that is, there exists a constant k such that [Formula presented] has all strictly positive entries. (Hartfiel (2006), Theorem 1.10) If [Formula presented] is regular for each [Formula presented] and [Formula presented] uniformly for all [Formula presented] (where [Formula presented] is the minimum over all positive entries), then [Formula presented], a rank one matrix that depends on p. Further there are constants K and β [Formula presented], such that [Formula presented].[Formula presented] denotes the backward product of a sequence of matrices [Formula presented] with elements [Formula presented] and is defined as [Formula presented]. Proof of convergence for Case 1 Case 1 of the P14 model satisfies the conditions of Theorem 1.10 Hartfiel (2006). First, the left multiplication in Equation (1) corresponds to backward products. Second, in Case 1 there is exactly one essential or communication class. Third, the block matrix corresponding to this class is primitive. By the definition of a communication class there exists a path from every individual within the class to every other individual within the class. Furthermore, as [Formula presented] has non-zero diagonal entries, the block matrix is aperiodic and thus there exists an integer [Formula presented] such that all entries of the k-step backward product of [Formula presented] are positive. Finally, Assumption 1 in P14 together with fixed oblique socialization matrix Φ ensures that the non-zero entries of the transition matrix [Formula presented] are bounded away from zero. The left product is thus regular for all t. It follows from Theorem 1.10 Hartfiel (2006) that [Formula presented] converges and the long-run outcome [Formula presented] is a matrix of rank one, implying consensus in cultural traits. Furthermore, there are constants K and [Formula presented] such that [Formula presented]. □ The proof extends to the case of time-varying [Formula presented] covered by Corollary 1. The additional condition of the corollary that the pattern of zeros remains constant ensures that the regularity of the left product is preserved. Thus as long as the non-zero elements in [Formula presented] are bounded away from zero for all t, Hartfiel (2006) Theorem 1.10 continues to apply and the cultural traits converge to consensus. Case 2 – More than one essential class: To show convergence for Case 2, that is, a reducible transition matrix with more than one isolated block matrix in [Formula presented], P14 presents a proof by construction. The proof decomposes each updating step of an individual trait into a weighted average of the previous value of the trait and the long-run outcomes of the essential classes.2The argument in P14 is incorrect because the terms that describe the weight assigned to the long-run outcomes of the essential classes do not converge with arbitrary time-varying entries in the transition matrix. Specifically, the assertion that “[Formula presented] is a monotone increasing [in t] series” in P14 (top of p. 605) is not true without further restrictions. To see why, rewrite this sum term by defining [Formula presented] and then rearrange as follows: [Formula presented] [Formula presented] is strictly monotone increasing in t if and only if it is strictly larger than [Formula presented]. Given that [Formula presented], [Formula presented] can be smaller than [Formula presented] if [Formula presented] is small. In a model with a general time-varying transition matrix [Formula presented] can increase as well as decrease for large t if the individual [Formula presented] switch between large and small values. Note that the proof in P14 does not make use of the specific restrictions on [Formula presented] in that paper and summarized in Equation (2) above. If the proof were valid it would thus apply to a very general class of time-varying transition matrices, including those with time-varying ratios of off-diagonal elements and including the example presented above. As we have shown, this general claim is not true. However, as we argue next, convergence in the model of P14 is preserved, and can be proven by using the restrictions on time variation in the P14 setting. Proof of Case 2 As [Formula presented] is reducible, the left product [Formula presented] is also reducible and can be written in lower triangular form [Formula presented] We discuss convergence of each block in sequence.
AuthorsPanebianco, F; Prummer, A; Siedlarek, JP
- Economics and Finance