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Published in:

01-01-2021

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

Authors: Anindya S. Chakrabarti, Sanjay Moorjani

Published in: Dynamic Games and Applications | Issue 3/2021

## 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.

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Appendix
Available only for authorised users
Footnotes
1
Autor et al. [3] for example, shows that the unanticipated elimination of rent controls in Cambridge, MA in 1995, led to a sharp price appreciation in the decontrolled housing units. Interestingly, the effect also spilled over to never-controlled units as well, which were in geographic proximity. In fact, the authors had shown that a larger portion of the total property valuation appreciation comes from the indirect effect on the never-controlled units and the appreciation was far more than can be explained by the observed increase in residential investment.

2
A simple example is that presidents are chosen from pools of vice presidents in corporate board rooms. In such cases, compensation increases by a significant margin overnight, but productivity does not. There is a large literature on efficiency of the relevant compensation schemes, starting with a very influential paper by Lazear and Rosen [32].

3
In Sect. 2 we will provide a formal definition of centrality. Here, we can imagine centrality to represent the degree of influence of nodes in the network.

4
In this paper, we utilize the term Katz–Bonacich centrality to denote the same, following the textbook definition given in Newman [38].

5
There is a vast literature on the statistical analysis of such large-scale social network (see for example an analysis of the US and Italian firms by Battiston and Catanzaro [7], exclusively Italian firms by Bargigli and Giannetti [6], German firms by Raddant et al. [40]) and their interplay with financial decision-making (see, for example,, [43]).

6
We do not pursue the link with market equilibrium here. Ghiglino and Goyal [23] for example provides a link between centrality and equilibrium prices and consumption in an exchange economy. Interested readers can refer to Goyal [24] for a review on the network description various economic (both micro and macro) and financial phenomena.

7
Therefore, this network is unweighted. A connection either exists or not. Also, there is no self-loop, i.e., $$\gamma _{iit}=0$$ for all and t. Finally, one can have an alternative representation of the edges in terms of pairs of nodes it connects. However, here we will explicitly utilize the description through adjacency matrix as that will help us to economize on notations.

8
We will denote the cardinality of set by $$n_t$$.

9
In “Appendix 8.1” we provide a standard game with linear quadratic payoff functions that give rise to such interdependent valuation [5, 31]. This framework constitutes the stage game and we use it to motivate the intra-period actions where given the network structure, players optimize their action profiles. The main influential result from Ballester et al. [5] is that given a network structure, the action profile in the Nash equilibrium in a linear quadratic setup is the same as the Katz–Bonacich centrality. Our focus in the main text of the paper is on the inter-period game where the players decide on their connectivities, which leads to network formation.

10
If no one in $${\mathbb {N}}_{t}$$ accepts the offers, then no connections are made and the potential entrant cannot enter.

11
We note from Proposition 2 that asymptotically the observed valuation and eigenvector centrality give rise to the same hierarchy.

12
We discuss in Sect. 4 what kind of connections can be formed in equilibrium. As we will see, not all possible connections will materialize if the players are farsighted.

13
The weight parameter $$\omega$$ works as attenuation factor in case of Katz centrality.

14
Another way to think about it is that Proposition 4 requires us to show that Katz centrality is well defined. Dequiedt and Zenou [17] analyzes this issue in further details.

15
The giant component refers to the largest connected component in the network.

16
We have assumed $$\theta =0.99$$ for numerical calculations; the estimates will change for different values of the parameter. The goal of the exercise is to show how the network multiplier changes as the size of the network changes for a given $$\theta$$.

17
In a personal communication with social networks researchers from a Japanese business card company, we came to know that indeed there is a large number of meetings with two participants with extreme differences in eigenvector centrality. However, the data is not available in the public domain.

18
Elliott et al. [20] have used a static variant of the linear valuation equations to describe firm to firm connections via cross-holding in assets.

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Title
Strategic Connections in a Hierarchical Society: Wedge Between Observed and Fundamental Valuations
Authors
Anindya S. Chakrabarti
Sanjay Moorjani
Publication date
01-01-2021
Publisher
Springer US
Published in
Dynamic Games and Applications / Issue 3/2021
Print ISSN: 2153-0785
Electronic ISSN: 2153-0793
DOI
https://doi.org/10.1007/s13235-020-00374-9

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