Strategic Connections in a Hierarchical Society: Wedge Between Observed and Fundamental Valuations

Abstract

In an interconnected society, social networks grow through formation of strategic connections based on the hierarchy within the social network. Often, the hierarchy becomes self-reinforcing and the observed valuations of the individuals in the hierarchy become disconnected from the corresponding fundamentals. We propose a network model to characterize the disconnect between the observed and fundamental valuations of entities, where the difference is a function of the linkages across the entities. In a growing social network, new entrants come at every point of time and offer connections to the incumbents based on the observed valuations. Individuals care only about their ranks in the hierarchy of observed valuation. With myopic individuals, network grows in equilibrium, but the associated hierarchy becomes unstable. However, with farsighted individuals, the network growth process is hierarchy-preserving and depending on the structure of seed network, the process may be completely halted by individuals who have incentives to preserve hierarchy. These two mechanisms taken together provide a comprehensive characterization of valuation in a growing inter-connected, hierarchical society. We illustrate an application of the model by analyzing the Indian board interlocking network. Our model enables us to find the hierarchy of the board members’ network and to identify the dispersion in magnitude of network externalities across directors.

Appendix

1.1 Intra-period Setup: Linear-Quadratic Payoff Functions

To model the stage game or the intra-period game, we posit a standard linear-quadratic game. The ith individual has a fundamental valuation $\pi _i$ and they have strategic complementarities. The utility function is given by [5, 31]

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where is the action profile of the agents. Solving the problem in equilibrium (maximizing with respect to $V_{it}$) given the connections $\{\gamma _{ijt}\}$, we get a linear network model¹⁸ of observed valuation V:

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where $\omega$ denotes the network coefficient and is the set of entities at time t. Following Ballester et al. [5], we can substitute the first order conditions back in the utility functions to show that at the equilibrium,

$$\begin{aligned} U_i^*=\frac{1}{2}(V_i^*)^2. \end{aligned}$$

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Therefore, in equilibrium utility is monotonically increasing in the action profile. This is a very useful result for us because given this result, the hierarchy defined over the equilibrium action profile $V^*$ is the same as the one defined over utility $U^*$ in equilibrium.

1.2 Proof of Proposition 1

Proof

Consider a static network ${\mathbb {N}}$. From the spectral theory of graphs, we know that (see, e.g., [34])

$$\begin{aligned} \max \{E(d),\sqrt{d_\mathrm{max}} \} \le \lambda _\mathrm{max} \le d_\mathrm{max}, \end{aligned}$$

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where $\lambda _\mathrm{max}$ is the dominant eigenvalue, E(d) is the average degree and $d_\mathrm{max}$ is the maximum degree. Note that for a connected network, $E(d) \ge 1$. Assuming $\theta$ is the constant of proportionality and $\theta \in (0,1)$, we get

$$\begin{aligned} \omega= & {} \theta /\lambda _\mathrm{max} \nonumber \\\le & {} 1. \end{aligned}$$

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Thus for a generic network ${\mathbb {N}}$, $\omega \in [0,1]$ and its value depends on the adjacency matrix $\varGamma$ of ${\mathbb {N}}$. Therefore, it satisfies Assumption 1.

Next, we show that this choice of $\omega$ also satisfies Assumption 2. We have to analyze how $\omega$ changes as the network size increases. We use a result from spectral graph theory [44]. $\square$

Theorem 4

Let A be a symmetric matrix with largest eigenvalue $A_{\lambda _\mathrm{max}}$. Let B be the matrix obtained by removing the last row and column from A, and let $B_{\lambda _\mathrm{max}}$ be the largest eigenvalue of B. Then,

$$\begin{aligned} A_{\lambda _\mathrm{max}}\ge B_{\lambda _\mathrm{max}}. \end{aligned}$$

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Therefore by increasing size of the graph with symmetric connections, $\lambda _\mathrm{max}$ monotonically (weakly) increases. This implies $\omega$ decreases as n increases. Therefore, it satisfies Assumption 2 as well. $\square$

1.3 Proofs of Sect. 3.1

1.3.1 Proof of Theorem 2

Proof

The adjacency matrix for a graph of n nodes be $\varGamma _{0}~= ~[\gamma _{ij}]_{i,j\in n}$. Let us assume that a new node m comes up and offers an edge to the top node in hierarchy. Let us denote that node by node 1. The entrant has no outside option and hence, formation of an edge is always rank-improving. Interestingly, we will show that node 1 always accepts that edge as it can never decrease node 1’s rank. Note that the new adjacency matrix is

$$\begin{aligned}\varGamma _{1}= \begin{bmatrix} \gamma _{11} &{}\quad \gamma _{12} &{}\quad \gamma _{13} &{}\quad \dots &{}\quad \gamma _{1n} &{}\quad 1\\ \gamma _{21} &{}\quad \gamma _{22} &{}\quad \gamma _{23} &{}\quad \dots &{}\quad \gamma _{2n} &{}\quad 0\\ \gamma _{31} &{}\quad \gamma _{32} &{}\quad \gamma _{33} &{}\quad \dots &{} \quad \gamma _{3n} &{}\quad 0\\ \vdots &{}\quad \vdots &{}\quad \vdots &{}\quad \ddots &{}\quad \vdots &{}\quad \vdots \\ \gamma _{n1} &{}\quad \gamma _{n2} &{}\quad \gamma _{n3} &{}\quad \dots &{}\quad \gamma _{nn} &{}\quad 0\\ 1 &{}\quad 0 &{}\quad 0 &{}\quad \dots &{}\quad 0 &{}\quad 0 \end{bmatrix}. \end{aligned}$$

Therefore, the new eigenvector centrality $V'$ solves the equation

$$\begin{aligned} \varGamma _{1} V' = \lambda '_\mathrm{max}V' \end{aligned}$$

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where $V'=[V'_1,~V_2',\ldots ,~V'_n,~V'_m]^\mathrm{Tr}$ is a column vector. Therefore, Eq. 26 gives us $V^{'}_{1}=\lambda ^{'}_\mathrm{max}V^{'}_{m}$. Note that

$$\begin{aligned} \lambda '_\mathrm{max}>1 \end{aligned}$$

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for a graph which has average degree more than 1 [34]; we have also used the same argument in Proposition 1). Therefore, we conclude

$$\begin{aligned} V_1'>V'_m \end{aligned}$$

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i.e., node 1 does not lose its rank with respect to the new entrant m. Node 1 cannot lose its rank with respect to any of the other incumbents by having a new connection. Therefore, node 1 will connect to node m. $\square$

1.3.2 Proof of Theorem 3

Proof

We know from Theorem 2 that node 1 accepts the incoming individual m. We have to show that after forming that connection, m would like to connect to 2 and 2 would also like to connect to him. By induction, everyone forms connections with the new individual m. Intuitively, Ballester et al. [5] showed that V is increasing in the number of links. Thus all individuals have incentives to accept offers to form links. But we have to also show that their ranks are not negatively impacted by the link formation mechanism.

First, we prove that forming a connection between a pair of nodes cannot reduce ranks of that pair of nodes. We note that by forming an edge between any generic pair $\{i,j\}$, the edge weights $\gamma _{ij}$ and $\gamma _{ji}$ increases from 0 to 1. Now we appeal to the Theorem 1 in Roy et al. [41] (in particular, see corollary 1) which shows that by adding a row vector $\{a_{i}\}_{i\in n}$ (where $a(i)>0$ and $a(j)=0$ for $j\in n{\setminus } i$) to an $n\times n$ real irreducible nonnegative matrix $\varGamma$, the element $v_i$ of the dominant eigenvector strictly increases in ratio with respect to the rest of the elements $\{v_j\}_{j\in n{\setminus } i}$. Note that forming an edge between the pair $\{i,j\}$ in a symmetric matrix leads to $\varGamma _{ij}$ and $\varGamma _{ji}$ to be converted into 1 from 0. This operation can be interpreted as adding a row vector of zeros with 1 in the jth entry (0, 0, 0, $\dots$, 1 ,$\ldots$, 0) to the ith row of matrix $\varGamma$ and adding a row vector of zeros with 1 in the ith entry ($0, 0, 0, \dots , 1, \ldots , 0$) to the jth row of matrix $\varGamma$. Formally, the first row is $\{a_{j}\}_{j\in n}$ (where $a(j)>0$ and $a(k)=0$ for $k\in n{\setminus }k$) and $\{a_{i}\}_{i\in n}$ (where $a(i)>0$ and $a(k)=0$ for $k\in n{\setminus }i$). Thus by applying Theorem 1 in Roy et al. [41], both the nodes $\{i,j\}$ are better off in terms of centrality with respect to all other nodes k such that $k\in n{\setminus }{\{i,j\}}$. Therefore, forming a connection between nodes i and j will not reduce rank for i and j with respect to the rest of the nodes $n{\setminus }\{i,j\}$.

This step leaves out one scenario where node i connects to a lesser central node j (for a generic pair of nodes i and j) and the relative ranking between i and j is not preserved. We incorporate it as a condition and as stated in the theorem, if such a case appears, then the link would not be formed as node i would not form a link where it loses rank. However, in numerical experimentations we could not find a case where that happens and in all cases, ranking is preserved after formation of the link.

Next, we have to show that the new entrant (after connecting to all incumbents), would now weakly dominate all the pre-existing incumbents. In order to do that we first write down the eigenvector centrality of the entrant m as

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Since the entrant m forms a connection with all incumbents , $\gamma _{im}=1$ for all . For any incumbent , the valuation would be

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where the neighborhood of the jth node (since some links may be non-existent). Therefore, $V_m\ge V_j$ for all .

This completes the proof. $\square$

1.4 Proofs of Sect. 5.1

1.4.1 Complete Graph

Here we compute the network multiplier for a complete graph ${\mathbb {N}}$ of n nodes. The valuation equation is given by

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Therefore the total valuation is

$$\begin{aligned} \sum _{i\in n} V_{i}= & {} (1-\omega )\sum _{i} \pi _{i} + \omega \sum _{i} \sum _{j\ne i} V_{j} \nonumber \\= & {} (1-\omega )\sum _{i} \pi _{i} + \omega (n-1) \sum _{i} V_{i}. \end{aligned}$$

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Solving for the total valuation, we get

$$\begin{aligned} \sum _{i\in n} V_{i} = \frac{(1-\omega )\sum _{i} \pi _{i}}{1 - \omega (n-1)}. \end{aligned}$$

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Therefore, the network multiplier is given by

$$\begin{aligned} {\mathcal {M}}_\mathrm{complete}= & {} \frac{\sum _{i} V_{i}}{\sum _{i} \pi _{i}} \nonumber \\= & {} \frac{(1-\omega )}{1 - \omega (n-1)} \nonumber \\= & {} \frac{(1-\frac{\theta }{n-1})}{1 - \theta } ~~~~ \text{(since } \lambda _\mathrm{max}=n-1)\nonumber \\> & {} 1~~~~\text{(since } \theta <1). \end{aligned}$$

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1.4.2 Star Graph

We compute the network multiplier for a star graph with $n-1$ peripheral nodes. The valuation equation is given by

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If node 1 is at the center, then (for $i \ne 1$)

$$\begin{aligned} V_{i} = (1-\omega )\pi _{i} + \omega V_{1}. \end{aligned}$$

(36)

Combining the above two equations, we get

$$\begin{aligned} V_{1}= & {} (1-\omega ) \pi _{1} + \omega \bigg ((1-\omega )\sum _{j \ne 1} \pi _{j} + \omega (n-1)V_{1} \bigg ) \nonumber \\= & {} \frac{(1-\omega ) \pi _{1} + \omega ((1-\omega )\sum _{j \ne 1} \pi _{j} )}{1 - \omega ^{2}(n-1)}. \end{aligned}$$

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Therefore, the total valuation is given by

$$\begin{aligned} \sum _{i} V_{i} = (1-\omega )\sum _{j\ne 1} \pi _{j} + V_{1}(1 + (n-1)\omega ). \end{aligned}$$

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Note that for a star graph, $\lambda _\mathrm{max}=\sqrt{n-1}$. Hence, the network multiplier is given by

$$\begin{aligned} {\mathcal {M}}_\mathrm{star}= & {} \frac{\sum _{i} V_{i}}{\sum _{i} \pi _{i}} \nonumber \\= & {} \frac{(1-\omega )\sum _{j\ne 1} \pi _{j} + V_{1}(1 + (n-1)\omega )}{\sum _{i} \pi _{i}}\nonumber \\= & {} \frac{(n+2 \omega (n-1))(1-\omega )}{n(1-\omega ^{2}(n-1))} ~~~ (\pi _i=\pi \text{ for } \text{ all } i) \nonumber \\> & {} 1. \end{aligned}$$

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1.4.3 Linear Graph

We compute the network multiplier for a linear graph with n nodes. The valuation equation is given by

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To simplify the calculation, we solve the model in the limit. If $n~\rightarrow ~\infty$,

$$\begin{aligned} V_{i} = (1-\omega )\pi _{i} + \omega (V_{i-1} + V_{i+1})~~~~~~\forall i. \end{aligned}$$

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Summing over all i and using symmetry across all nodes, we get the total valuation as

$$\begin{aligned} \sum _{i} V_{i}= & {} (1-\omega ) \sum _{i} \pi _{i} + 2\omega \sum _{i} V_{i} \nonumber \\= & {} \frac{\bigg (\sum _{i} \pi _{i}\bigg )(1-\omega )}{1-2\omega }. \end{aligned}$$

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Hence, the network multiplier is given by

$$\begin{aligned} {\mathcal {M}}_\mathrm{linear}= & {} \frac{\sum _{i} V_{i}}{\sum _{i} \pi _{i}} \nonumber \\= & {} \frac{1-\omega }{1-2\omega } \nonumber \\> & {} 1. \end{aligned}$$

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1.5 Ranking Scheme Matters for Network Growth

Here we show the usefulness of the ranking scheme we assumed. To give an example, if we have three nodes with centralities 0.4, 0.4 and 0.2, then our ranking scheme would assign ranks 1, 1, 3 to the three nodes. One can possibly imagine a closely related ranking scheme that will assign 1, 1, 2. The difference between our preferred ranking scheme and the alternate one is that our ranking scheme is not purely ordinal, how many players are more influential than a given player matters for the rank of that given player. Below we show that the two ranking schemes can lead to very different types of network growth. In Table 3, we consider a linear network of three nodes (numbered as 2, 1, 3 from left to right for a horizontal linear network of 3 nodes) and the 4th node is the entrant. In Table 4, we consider a linear network of three nodes (numbered as 4, 2, 1, 3, 5 from left to right for a horizontal linear network of 5 nodes) and the 6th node is the entrant. The key take away is that the growth processes would be quite different under two schemes. See the captions for a complete description of the resulting growth process.

Table 3 Eigenvector centrality along with ranks (within parentheses; the first one represents the ranking assumed in the paper, second one represents an alternate ranking) of the nodes

Table 4 Eigenvector centrality along with ranks (within parentheses; the first one represents the ranking assumed in the paper, second one represents an alternate ranking) of the nodes