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Attitude change in arbitrarily large organizations

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Abstract

The alignment of collective goals and individual behavior has been extensively studied by economists under a principal-agent framework. Two main solutions have been presented: explicit incentive contracts and monitoring. These solutions correspond to changes in the objective situation faced by individuals. However, an extensive literature in social psychology provides evidence that behavior is influenced, not only by situational constraints, but also by attitudes. Therefore, an important aspect of organization is to choose the structures and procedures that best contribute to the dissemination of the desired attitudes throughout the organization. This paper studies how the initial configuration of attitudes and the size of the organization affect the optimal organizational structure and the timing of information flows when the objective is to align the members’ attitudes. We identify and characterize three factors that affect the optimal organizational structures and procedures and the degree of alignment of attitudes: (1) clustering effects; (2) member cross-influence effects; and (3) leader cross-influence effects.

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Notes

  1. There is an extensive literature in the field of social psychology on the relationship between individual attitudes and behavior (e.g., Fishbein and Ajzen 1974; Fazio 1986; Ajzen and Sexton 1999). Although some early studies, in particular the one conducted by LaPiere (1934), indicated that attitudes were largely irrelevant to the prediction of behavior, recent empirical research confirms that, in general, attitudes influence behavior (see, for an overview, Kraus 1995). However, the consistency of attitudes and behaviors has been found to depend on a number of factors, such as the level of effort required to perform a behavior (e.g., Bagozzi et al. 1990), the accessibility of the attitude from memory (e.g., Fazio et al. 1989), the extent to which individual behavior is susceptible to situational or interpersonal cues, as opposed to inner states or dispositions (e.g., Ajzen et al. 1982), and the consistency between the affective and cognitive components of an attitude (e.g., Norman 1975).

  2. We assume that all the influences are positive, meaning that all communications produce results consistent with the source attitudes. This means that when two individuals with the same attitude interact, their attitudes are reinforced. However, people have attitudes not only toward objects or ideas, but also relative to the people with whom they are communicating. According to Balance Theory (Heider 1946), at the extreme one may dislike something because a person he/she dislikes is advocating for it. Thus, influences may be negative. This issue is briefly discussed in the conclusion.

  3. Most likely, no real organization is correctly described by either of these two extreme specifications. In general, we would expect a combination of both dynamics, with some subgroups changing their attitudes simultaneously and others sequentially. However, since all the other possible dynamics are combinations of the two extreme cases, we believe that the discussion of these two cases captures the main features of the dynamics of attitude change in organizations.

  4. Note that the isolated leader case does not correspond to a situation where the top manager is not influenced by its subordinates, but rather to the case where he/she has initially a different attitude from the rest of the organization. Independently of the initial configuration of attitudes, we consider throughout the paper both top-down and bottom-up influences.

  5. As mentioned by Kotter and Heskett (1992), effort toward major change is often initiated by leaders who “either came into their positions from outside their firms, came to their firms after an early career somewhere else, ‘grew up’ outside the core of their companies or were unconventional in some other way” (1992, page 89). As a result, these leaders tend to bring with them perspectives, personal values and attitudes that are different from the ones that are dominant within their organizations. Kotter and Heskett (1992) offer an interesting description and analysis of major change processes that occurred in several large organizations. Other important references on the topic of organizational change are Kanter et al. (1992) and Jick (1993).

  6. In Friedkin and Johnsen’s model, the assumption that individuals revise their positions by taking weighted averages of the influential positions of other members allows for the convergence of the process of opinion change.

  7. Burton and Obel (1988) also contrast different organizational forms. There are, however, two main differences to our paper. First, they focus on a different issue—the effect of opportunistic behavior on the appropriate choice of economic organization. Second, they use a laboratory experiment and, consequently, the interpretation of their results has to recognize the particular laboratory setup.

  8. A similar model has been used by Almeida Costa and Amaro de Matos (2002), focusing on very small organizations with a limited number of hierarchical levels and individuals. In the present paper, by focusing on arbitrarily large organizations we are able to provide a more complete characterization of the forces underlying the dynamic process of attitude change.

  9. Alternatively, attitudes could be modeled as continuous variables, rather than binary ones. Although such a representation of attitudes may seem more natural, it would significantly complicate the analysis. With continuous attitudes, there would be an infinite number of configurations of attitudes to be compared. As mentioned in the Introduction, our binary approach is justified by our focus on the alignment of attitudes.

  10. Note that this assumption does not imply that the influence of i over j has the same intensity as the influence of j over i.

  11. There is an extensive literature in social psychology that analyzes influence in dyadic relationships between an influencing agent and a target (see, for an overview, Eagly and Chaiken 1993, Chap. 13, pp. 634–642). This research largely focuses on the identification of the factors that determine the power the influencing agent has to influence the target (e.g., French 1956; Raven 1965; Kelman 1958, 1974; Cialdini 1988). In other words, this literature discusses the factors that determine the value of a given J ij . Our focus in this paper is different. We take each J ij as given, and discuss the conditions under which the dynamic system of social influence in organizations evolves to a consensus.

  12. When comparing the different organizational structures and the different dynamics, we just look at the final configuration, ignoring the length of the adjustment period. The sequential dynamics typically requires a larger number of interactions than the simultaneous dynamics. However, this does not imply that the length of the adjustment period in the sequential dynamics is larger. The reason is that one step of the simultaneous dynamics may take longer than one step of the sequential dynamics. In fact, in real life situations discussions involving many people at the same time may take longer than a number of discussions in small groups.

  13. The assumption that u>d can also be justified by the fact that managers control several factors that may affect values, beliefs and attitudes of their subordinates (Harrison and Carroll 1991). In the same vein, the influence exercised by superiors over subordinates encompasses not only an element of conformity, whereby an agent simply follows the behavior of another agent, but also an element of obedience, which results from enforcement by an authority (Elsenbroich and Xenitidou 2012).

  14. These specifications can be generalized in several different ways. For example, some parameters could be negative, and different u’s, d’s and e’s could have different values. The number of alternative scenarios is unbounded. For simplicity, we limit our analysis to the above mentioned cases.

  15. Note that the sufficient conditions in Lemma 4.1 and Proposition 4.4 coincide with the sufficient condition in this proposition. This explains why those results do not depend on the dynamics.

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Correspondence to João Amaro de Matos.

Appendix

Appendix

In this appendix, we present the proofs of our results.

Proof of Proposition 4.1

In the network, the i-th individual of level l is under a field

$$\begin{aligned} h_{il} =&u\sum_{k<l}\bigl(n_{k}^{+}-n_{k}^{-} \bigr)+e\bigl(n_{l}^{+}-n_{l}^{-}-s_{il} \bigr)+d\sum_{k>l}\bigl(n_{k}^{+}-n_{k}^{-} \bigr) \\ =&(u-es_{il})+u\sum_{1<k<l} \bigl(n_{k}^{+}-n_{k}^{-}\bigr)+e \bigl(n_{l}^{+}-n_{l}^{-} \bigr)+d\sum_{k>l}\bigl(n_{k}^{+}-n_{k}^{-} \bigr), \end{aligned}$$
(2)

where \(n_{k}^{+}\) and \(n_{k}^{-}\) denote the number of individuals at level k starting with a positive and negative attitude, respectively. Since u>e, the assumption that \(n_{i}^{+}>n_{i}^{-}\) for all i ensures that h il >0 for all i and l. Thus, no matter what dynamics is used, the number of positive attitudes increases until all individuals assume a positive attitude. □

Proof of Corollary 4.1

Follows from the proof of Proposition 4.1. □

Proof of Corollary 4.2

Follows from the proof of Proposition 4.1. □

Proof of Proposition 4.2

For K>2, from (2), it follows that h il <0 under the assumed conditions, since \(n_{k}^{+}-n_{k}^{-}<0\) for all k>1. For l=2, the field reads \(h_{i2}^{n}=(u-es_{i2})+e(n_{2}^{+}-n_{2}^{-})+d \sum_{k>2}^{K}(n_{k}^{+}-n_{k}^{-})\) and, if u<(K−2)d, the result follows no matter what dynamics is used. □

Proof of Proposition 4.3

Under the hierarchy

$$ h_{il}^{h}(t)=us_{p,l-1}(t)+d\sum _{k=\eta _{il}+1}^{\eta _{il}+q_{il}}s_{k,l+1}(t) $$
(3)

with \(\eta _{il}=\sum_{k=1}^{i-1}q_{kl}\) and p denoting the superior of i. Notice that, by construction, for l>2, we have \(h_{il}^{h}(t)>-u-dq_{il}\). For l=2, we have \(h_{i2}^{h}(t)>u-dq_{i2}\). Under the network, we have from expression (2)

$$ h_{il}^{n}=u\sum_{k<l} \bigl(n_{k}^{+}-n_{k}^{-}\bigr)+e \bigl(n_{l}^{+}-n_{l}^{-}-s_{il} \bigr)+d\sum_{k>l}\bigl(n_{k}^{+}-n_{k}^{-} \bigr). $$

For l>2, we have \(\sum_{k<l}(n_{k}^{+}-n_{k}^{-})\leq 3-l,(n_{l}^{+}-n_{l}^{-}-s_{il})\leq 0\) and \(\sum_{k>l}(n_{k}^{+}-n_{k}^{-})\leq - ( K-l ) \). Hence \(h_{il}^{n}\leq -d(K-3)-(u-d) ( l-3 ) \). For l=2, the field reads

$$ h_{i2}^{n}=(u-es_{i2})+e\bigl(n_{2}^{+}-n_{2}^{-} \bigr)+d\sum_{k>2}^{K} \bigl(n_{k}^{+}-n_{k}^{-}\bigr)\leq u-d ( K-2 ) . $$

A sufficient condition for \(h_{il}^{h}(t)\geq h_{il}^{n}\) is that

$$ q_{i2}\leq K-2, $$

and for l>2,

$$ q_{il}\leq -\frac{u}{d}+(K-3). $$

If both conditions above are satisfied, the hierarchy is preferred to the network. From the result above, we know that a sufficient condition is that \(q_{il}\leq \max [K-2,-\frac{u}{d}+(K-3)]\). Since \(K-2>-\frac{u}{d}+(K-3) >K-4\), it follows that the hierarchy is preferred to the network if K>q +4, where q denotes the maximum number of subordinates of any agent in the organization. □

Proof of Lemma 4.1

In the network, we know from (2) that

$$ h_{il}^{n}=(u-es_{il}) +u\sum_{1<k<l}\bigl(n_{k}^{+}-n_{k}^{-} \bigr)+e\bigl(n_{l}^{+}-n_{l}^{-} \bigr)+d\sum_{i>l}\bigl(n_{k}^{+}-n_{k}^{-} \bigr). $$

Since in this case \(n_{k}^{+}=0\) and \(n_{k}^{-}=n\) and s il =−1 for all i>1, we have

$$\begin{aligned} h_{il}^{n} =&(u+e)-u\sum_{1<k<l}n_{k}-en_{l}-d \sum_{k>l}n_{k}\\ =&u \biggl( 1-\sum _{1<k<l}n_{k} \biggr) +e ( 1-n_{l} ) -d\sum _{k>l}n_{k}. \end{aligned}$$

Since 1−∑1<k<l n k <0 for l>2 and n l ≥1, we have h il <0 for all l>2. For l=2,

$$ h_{i2}^{n}=u+e ( 1-n_{2} ) -d\sum _{k>2}n_{k}. $$

Let \(q_{l}^{m}=\min_{i}q_{il}\) denote the minimum number of subordinates of any agent in level l. Since \(\sum_{k>2}n_{k}\geq q_{2}^{m}\)and n 2>1, we have \(h_{i2}^{n}<0\), leading to our result. In the hierarchy, (3) and \(u\leq q_{2}^{m}d\) imply negative fields for all individuals. A hybrid organization can be seen as a hierarchy plus some informal links. For the case of an arbitrary individual in the second level, the initial field under the hierarchy is \(h_{i2}^{h}(0)=u-dq_{i2}\). Under a hybrid structure it is \(h_{i2}^{hyb}(0)=u-dq_{i2}-dn_{i2}\), where n i2≥0 denotes the number of informal links associated with that particular individual. Thus, for \(u\leq q_{2}^{m}d\), no individual in the second level changes attitude. Similarly, no individual in lower levels will change attitude, since

$$ h_{il}^{hyb}(0)=-u-dq_{il}-dn_{il}^{d}-en_{il}^{e}-un_{il}^{u}+u \delta _{il}, $$

where \(n_{il}^{d}\geq 0\) denotes the number of informal links with individuals in lower levels, \(n_{il}^{e}\geq 0\) denotes the number of informal links with individuals in the same level, \(n_{il}^{u}\geq 0\) denotes the number of informal links with individuals in higher levels (except the top manager), and δ il is equal to 1 if there is a direct link to the top manager and zero otherwise. This concludes the proof. □

Proof of Proposition 4.4

Define \(q^{\ast }=\max_{l}q_{l}^{\ast }\) as the maximum number of subordinates of any individual in the organization. Under the hierarchy, the field felt by any individual is given by (3). Notice that, by construction,

$$ \sum_{k=\eta _{il}+1}^{\eta _{il}+q_{il}}s_{k,l+1}(t)\geq -q_{il}\geq -q^{\ast }\quad \Longrightarrow\quad h_{il}(t)\geq us_{p,l-1}(t)-q^{\ast }d. $$

Since s 11=+1, u>q d implies h i2(t)>0 for all i=1,…,n 2 and for all t≥0. Consider first the sequential dynamics starting at t=0. For tn 2, we have s i2(t)=+1,∀i, leading to h i3(t)>0 for all i=1,…,n 3. In general, if \(t\geq \sum_{k=2}^{\nu }n_{k}\), we have s ik (t)=+1,∀i, for all kν. Hence, at t=N the system attains the fixed point configuration where all individuals have positive attitudes. In the simultaneous dynamics, all individuals in l=2 become positive at t=1. For the same reason, in the next step of the dynamics all individuals in l=3 become positive. The process goes on until all individuals become positive at t=K. □

Proof of Proposition 4.5

If u>q d, from Proposition 4.4 the hierarchy is optimal under both considered dynamics, leading to an equilibrium where all individuals have a positive attitude. If \(u\leq q_{2}^{m}d\), we know from Lemma 4.1 that, under both dynamics, the equilibrium configuration will be the one where the leader is isolated, independently of the organizational structure. Since by assumption \(q_{2}^{m}=q^{\ast }\), this concludes the proof of our statement for all values of u. □

Proof of Corollary 4.4

Follows from Lemma 4.1 and from Proposition 4.4. □

Proof of Lemma 4.2

Under the network, the field felt by individual i in level l is given by

$$\begin{aligned} h_{il} ( t ) =&u\sum_{k<l} \bigl[ n_{k}^{+} ( t ) -n_{k}^{-} ( t ) \bigr] +e \bigl[ n_{l}^{+} ( t ) -n_{l}^{-} ( t ) -s_{il} ( t ) \bigr] \\ &{}+d\sum_{k>l} \bigl[ n_{k}^{+} ( t ) -n_{k}^{-} ( t ) \bigr] . \end{aligned}$$

Let \(\Delta _{k} ( t ) =n_{k}^{+} ( t ) -n_{k}^{-} ( t ) \) and note that

$$\begin{aligned} h_{il} ( t ) >&0\quad \Longleftrightarrow\quad s_{il}<\frac{u}{e} \sum_{k<l}\Delta _{k} ( t ) +\Delta _{l} ( t ) +\frac{d}{e}\sum_{k>l} \Delta _{k} ( t ), \\ h_{il} ( t ) <&0\quad \Longleftrightarrow\quad s_{il}>\frac{u}{e} \sum_{k<l}\Delta _{k} ( t ) +\Delta _{l} ( t ) +\frac{d}{e}\sum_{k>l} \Delta _{k} ( t ) . \end{aligned}$$

For simplicity, we introduce the following notation

$$ \Delta ( t ) =\sum_{k}\Delta _{k} ( t );\qquad\hat{\Delta} _{l} ( t ) =\sum _{k<l}\Delta _{k} ( t ) ;\qquad \check{\Delta} _{l} ( t ) =\sum_{k>l}\Delta _{k} ( t ) . $$

Furthermore, assume αu/e and, for simplicity, γd/e≥1. Under this specification, the two conditions above simplify to

$$\begin{aligned} h_{il} ( t ) \geq &0\quad \Longleftrightarrow\quad s_{il} ( t ) \leq H_{l} ( t ), \\ h_{il} ( t ) \leq &0\quad \Longleftrightarrow\quad s_{il} ( t ) \geq H_{l} ( t ) , \end{aligned}$$

where

$$ H_{l} ( t ) = ( \alpha -1 ) \hat{\Delta}_{l} ( t ) + ( \gamma -1 ) \check{\Delta}_{l} ( t ) +\Delta ( t ) , $$

and by construction does not depend on i. □

Proof of Proposition 4.6

The sufficient conditions in this proposition are

$$ H_{l} ( t ) \geq +1,\quad \forall l>1 $$
(4)

for the first part and

$$ H_{l} ( t ) \leq -1,\quad \forall l>1 $$
(5)

for the second part. Since s il (t)∈{−1,+1}, we have that if H l (t)≥+1 for a given level l, then h il ≥0 for all individuals in that level. Similarly, if H l (t)≤−1 for a given level l, then h il ≤0 for all individuals in that level. Under condition (4) it follows that h il ≥0,∀l, and any dynamics implies Δ(t+1)≥Δ(t), leading to an equilibrium where all individuals have a positive attitude. From condition (5) it follows that h il ≤0,∀l, and any dynamics implies Δ(t+1)≤Δ(t), leading to an equilibrium where the leader is isolated. □

Proof of Proposition 4.7

Consider an arbitrary level l. Under the simultaneous dynamics, we have the following three possibilities

$$\begin{aligned} H_{l} \geq &+1\quad \Longrightarrow\quad n_{l}^{+} ( t+1 ) =n_{l}^{+} ( t ) +n_{l}^{-} ( t ) \equiv n_{l}, \\ H_{l} \in & ] {-}1,1 [ \quad \Longrightarrow\quad n_{l}^{+} ( t+1 ) =n_{l}^{-} ( t ), \\ H_{l} \leq &-1\quad \Longrightarrow\quad n_{l}^{+} ( t+1 ) =0. \end{aligned}$$

Under the sequential dynamics, starting with an individual with a positive attitude, we get for n l τ>0

$$\begin{aligned} H_{l} \geq &+1\quad \Longrightarrow\quad n_{l}^{+} ( t+\tau ) =n_{l}^{+} ( t ) +\max \bigl\{ \tau -n_{l}^{+} ( t ) ,0 \bigr\} \rightarrow n_{l}, \\ H_{l} \in & ] {-}1,1 [\quad \Longrightarrow\quad n_{l}^{+} ( t+ \tau ) =\max \bigl\{ 0,n_{l}^{+} ( t+\tau -1 ) -1 \bigr\} \rightarrow 0, \\ H_{l} \leq &-1\quad \Longrightarrow\quad n_{l}^{+} ( t+\tau ) =\max \bigl\{ 0,n_{l}^{+} ( t+\tau -1 ) -1 \bigr\} \rightarrow 0. \end{aligned}$$

Under the sequential dynamics, starting with an individual with a negative attitude, we get for n l τ>0

$$\begin{aligned} H_{l} \geq &+1\quad \Longrightarrow\quad n_{l}^{+} ( t+\tau ) =n_{l}^{+} ( t ) +\max \bigl\{ \tau -n_{l}^{+} ( t ) ,0 \bigr\} \rightarrow n_{l}, \\ H_{l} \in & ] {-}1,1 [ \quad \Longrightarrow\quad n_{l}^{+} ( t+ \tau ) =\min \bigl\{ n_{l},n_{l}^{+} ( t+\tau -1 ) +1 \bigr\} \rightarrow n_{l}, \\ H_{l} \leq &-1\quad \Longrightarrow\quad n_{l}^{+} ( t+\tau ) =\max \bigl\{ 0,n_{l}^{+} ( t+\tau -1 ) -1 \bigr\} \rightarrow 0. \end{aligned}$$

Therefore, due to the levels where H l ∈ ]−1,1[, the sequential dynamics starting with an individual with a negative attitude is preferred to the simultaneous dynamics, which, in turn, is preferred to the sequential dynamics starting with an individual with a positive attitude. □

Proof of Proposition 4.8

It is convenient to start by characterizing what happens with the attitude of an arbitrary individual under both dynamics.

The field felt by individual i in level l, given by (3), can be rewritten as

$$ h_{il}(t)=us_{p,l-1}(t)+d\sum^{i,l}s_{k,l+1}(t), $$

where ∑i,l denotes the sum over all subordinates of individual i at level l, and p denotes his/her superior. Let q il denote the total number of direct subordinates, \(n_{il}^{+} ( t ) \) denote the number of subordinates with a positive attitude at time t, and \(n_{il}^{-} ( t ) \) denote the number of subordinates with a negative attitude. Then,

$$ n_{il}^{+} ( t ) +n_{il}^{-} ( t ) =q_{il} $$

and the field above can be rewritten as

$$ h_{il}(t)=us_{p,l-1}(t)+d\bigl[n_{il}^{+} ( t ) -n_{il}^{-} ( t ) \bigr]. $$

Also, let q denote the maximum number of subordinates that any agent has in the organization, i.e.,

$$ q^{\ast }=\max_{l}q_{l}^{\ast }= \max_{l}\Bigl\{\max_{i}q_{il} \Bigr\}. $$

We first characterize what happens with an arbitrary individual in a hierarchy under the sequential dynamics. Since there are no same-level peer interactions in a hierarchy, we assume, without loss of generality, a dynamics that revises the attitudes of all individuals in each level at the same time. Assume that individual i in level l revises his/her attitude at t+1. Then,

$$\begin{aligned} s_{il}^{seq}(t+2) =&\text{sign} \biggl[ us_{p,l-1}(t)+d \sum^{i,l}s_{k,l+1}(t) \biggr] \\ =& \text{sign} \bigl\{ us_{p,l-1}(t)+d\bigl[n_{il}^{+}(t)-n_{il}^{-}(t) \bigr] \bigr\} . \end{aligned}$$
(6)

Consider now the simultaneous dynamics. For an arbitrary individual i in an arbitrary level l, we have in the second step of the dynamics

$$\begin{aligned} s_{il}^{sim}(t+2) =&\text{sign } \biggl[ us_{pl-1}(t+1)+d \sum^{i,l}s_{kl+1}(t+1) \biggr] \\ =& \text{sign } \biggl[ u\text{ sign }h_{pl-1}(t)+d\sum ^{i,l}\text{sign } h_{kl+1}(t) \biggr]. \end{aligned}$$
(7)

The result of the proposition follows from the comparison of these two equations.

Consider an individual i in level l, whose superior has a positive attitude and does not change it, i.e., s p,l−1(t)=s p,l−1(t+1)=+1. Under the assumption that q il q <u/d, it follows from (6) and (7) that s i,l (t+2)=+1 under both dynamics. Each of its subordinates will then have a superior with a positive attitude in the next step of the dynamics and the argument applies again until all agents under the initial superior attains a positive attitude. Since the head of the organization (the individual in l=1) has a positive attitude at time t=0 that does not change by design, the argument may apply initially to each of the individuals in level l=2 and then for all other levels. Since the argument does not depend on the dynamics, the result follows.

In the case where q =u/d, the above argument follows obviously for every individual i in level l such that q il <q . Let us focus on the first individual such that q il =q =u/d. Knowing that his/her superior has attained a positive attitude at some point under either dynamics, we consider three cases.

  • If at least one of his subordinates has a positive attitude. It follows that \(n_{il}^{+}(t)>0\Rightarrow h_{il}(t)>0\) and the above argument still holds for both dynamics, leading to s il (t+1)=+1. This clearly implies that s il (t+2)=+1 under both dynamics, since the worst that may happen is that in the simultaneous dynamics all his/her subordinates have changed into negative attitudes at t+1, leading to a null resulting field and \(s_{il}^{sim}(t+2)=s_{il}^{seq}(t+2)=+1\). However, we are left to show that if \(n_{il}^{+}=0\), his/her final attitude does not depend on the dynamics. This leads to the two following cases.

  • If the focal individual has a positive attitude s i,l (t)=+1 and all his/her subordinates have a negative attitude, then \(n_{il}^{+}(t)=0\Rightarrow h_{il}(t)=0\Rightarrow s_{il}(t+1)=+1\) under both dynamics, by the argument just described.

  • If the focal individual has a negative attitude s i,l (t)=−1 and all his/her subordinates have a negative attitude, then for the sequential dynamics \(n_{il}^{+}(t)=0\Rightarrow s_{il}^{seq}(t+2)=-1\). Notice that in the sequential dynamics all the subordinates of the focal individual will remain with negative attitudes in subsequent times since they have at most q subordinates themselves (by definition of q ) and even if all these have a positive attitude, the fact that u=q d constrains change. This same argument applies in the case of the simultaneous dynamics. Here, either signh kl+1(t)=s k,l+1(t+1)=−1 for all the subordinates of the focal individual and \(s_{il}^{sim}(t+2)=s_{il}^{seq}(t+2)\) from (7) and (6), or h kl+1(t)=0 for some subordinate, leaving its attitude s k,l+1(t+1)=−1 negative and sustaining \(s_{il}^{sim}(t+2)=s_{il}^{seq}(t+2)\).

This concludes our proof. □

Proof of Proposition 4.9

We show that under the hierarchy if the number of subordinates of any agent is larger than u/d and the maximum span q is strictly larger than u/d, a sufficient condition for the dynamics to be irrelevant is that, for all agents, the number of subordinates with a negative attitude is larger than half of the sum of u/d with the number of subordinates, i.e., \(n_{il}^{-}>\frac{1}{2} ( q_{i,l}+u/d ) \) for all i,l. The condition q il u/d ensures that \(\frac{1}{2} ( q_{i,l}+u/d ) \leq q_{i,l}\). Under the condition \(n_{il}^{-}>\frac{1}{2}( q_{i,l}+u/d ) \) we then have

$$\begin{aligned} h_{il} =&us_{p,l-1}(t)+d\bigl[n_{il}^{+}(t)-n_{il}^{-}(t) \bigr] \\ =&us_{p,l-1}(t)+d\bigl[q_{il}(t)-2n_{il}^{-}(t) \bigr] \\ <&us_{p,l-1}(t)-u\leq 0. \end{aligned}$$

From (6), we have that under the sequential dynamics

$$ s_{il}^{seq}(t+2)=-1. $$

Under the simultaneous dynamics, the argument above holds for l>2 leading to

$$ s_{pl-1}(t+1)=\text{ sign }h_{pl-1}(t)=-1 $$

and

$$ s_{kl+1}(t+1)=\text{ sign }h_{kl+1}(t)=-1. $$

From (7) we conclude that for l≥2

$$ s_{il}^{sim}(t+2)=-1 $$

thus concluding our proof. □

Proof of Proposition 4.10

We show that under the hierarchy if the number of subordinates of any agent is larger than u/d and the maximum span q is strictly larger than u/d, a sufficient condition for the dynamics to be relevant is that, for all agents, the number of subordinates with a negative attitude is less than half of the sum of u/d with the number of subordinates, i.e., \(n_{il}^{-}<\frac{1}{2} ( q_{i,l}+u/d ) \) for all i,l. Under the condition \(n_{il}^{-}<\frac{1}{2} ( q_{i,l}+u/d ) \) we then have

$$\begin{aligned} h_{il} =&us_{p,l-1}(t)+d\bigl[n_{il}^{+}(t)-n_{il}^{-}(t) \bigr] \\ =&us_{p,l-1}(t)+d\bigl[q_{il}(t)-2n_{il}^{-}(t) \bigr] \\ >&us_{p,l-1}(t)-u\leq 0. \end{aligned}$$

If the superior of a given agent i in level l has a positive attitude, the agent’s field will be positive. From (6), we have that under the sequential dynamics

$$ s_{il}^{seq}(t+2)=+1. $$

Since the top manager has a positive attitude, we conclude that the above conditions ensure the diffusion of that positive attitude, under the sequential dynamics, throughout the whole organization. Under the simultaneous dynamics, individuals whose initial attitude are positive and whose superiors do not have initially a positive attitude, will feel a field h il >−2u that may be negative, changing in that case their attitudes into negative, and increasing the number of negative subordinates (and superiors) in the system, possibly invalidating the condition \(n_{il}^{-}<\frac{1}{2} ( q_{i,l}+u/d ) \). In the second step of the dynamics a subordinate of one such individual will feel a field that may be negative for the very same reason, increasing the number of negative subordinates in the system. This shows that, under the conditions of this proposition, the sequential dynamics is preferred, thus concluding our proof. □

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Costa, L.A., de Matos, J.A. Attitude change in arbitrarily large organizations. Comput Math Organ Theory 20, 219–251 (2014). https://doi.org/10.1007/s10588-013-9160-3

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