Critical dimensions in the empirical measurement of common shareholding

Rosati, Nicoletta; Bomprezzi, Pietro; Martinez Cillero, Maria Article — Published Version Critical dimensions in the empirical measurement of common shareholding Research in International Business and Finance Provided in Cooperation with: Kiel Institute for the World Economy – Leibniz Center for Research on Global Economic Challenges Suggested Citation: Rosati, Nicoletta; Bomprezzi, Pietro; Martinez Cillero, Maria (2024) : Critical dimensions in the empirical measurement of common shareholding, Research in International Business and Finance, ISSN 1878-3384, Elsevier, Amsterdam, Vol. 70, Iss. Part A, https://doi.org/10.1016/j.ribaf.2024.102315 This Version is available at: https://hdl.handle.net/10419/306566 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. 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Contents lists available at ScienceDirect Research in International Business and Finance journal homepage: www.elsevier.com/locate/ribaf Full length article Critical dimensions in the empirical measurement of common shareholding✩ Nicoletta Rosati a,b,∗, Pietro Bomprezzi c, Maria Martinez Cilleroa aEuropean Commission, Joint Research Centre (JRC), Via Enrico Fermi 2749, 21027 Ispra (VA), Italy bCEMAPRE, Rua do Quelhas 6, 1200 – 781, Lisbon, Portugal cUniversity of Milan-Bicocca, Piazza dell’Ateneo Nuovo 1, 20126 Milano, Italy ARTICLE INFO JEL classification: C18 D21 D22 G11 G32 L40 Keywords: Common ownership Corporate governance Networks Mobile network operators Anti-competitive practices ABSTRACT The debate on common shareholding and its potential antitrust effects is currently on the agenda of major institutions worldwide. Discussions point to the need for improved empirical quantification of this phenomena. This work presents a flexible, multifaceted statistical framework for a set of new common shareholding indicators, covering both firm and investor perspectives, which can be adopted under different economic models. Many indices currently used in the literature fall within this framework as special cases. Aggregation at market level yields suitable industrylevel indicators, providing policymakers with tools to evaluate the extent of common ownership in strategic markets. The indices are tested using firm-level data for European Mobile Network Operators in 2007–2021, showing a sector with concentrated ownership under large corporate groups, but also the presence of institutional investors with extensive ownership across the major firms. 1. Introduction In December 2017, the OECD organised in Paris a Competition Policy Roundtable to discuss the rise of the relatively new phenomenon of common ownership.1According to this report, the previous ten years had been characterised by a ‘‘rapid growth in passively-managed investment funds, [which] has had a significant impact on the ownership structure of large firms in several industries’’. Common ownership, or the simultaneous ownership of shares in many firms active in the same market, was also on the radar of other industry watchdogs and stakeholders.2Subsequently in May 2018, the European Corporate Governance Institute (ECGI) dedicated a focus panel of its Annual Members’ Meeting to ‘‘Common Ownership: Antitrust Meets Corporate Governance’’. The ECGI Event Report3noted attendants raised concerns regarding ‘‘potential collusion between competing firms having the same shareholders’’. The ✩Disclaimer: The views expressed are purely those of the authors and may not in any circumstances be regarded as stating an official position of the European Commission, or of any other affiliation. ∗Correspondence to: LEAR, Via di Monserrato 48, 00186 Roma, Italy. E-mail addresses: [email protected] (N. Rosati), [email protected] (P. Bomprezzi), [email protected] (M. Martinez Cillero). 1The report is titled ‘‘Common ownership by institutional investors and its impact on competition’’ and is accessible at http://www.oecd.org/daf/competition/ common-ownership-and-its-impact-on-competition.htm. This Roundtable built on a previous one held in 2008 entitled ‘‘Minority Shareholdings and Interlocking Directorates’’. 2In the literature, common shareholders are mostly known as ‘‘common owners’’. The term ‘‘common owners’’ can be somehow misleading, as these investors do not actually own companies, they rather own (usually small) participations in many companies. The two terms will be used interchangeably in this paper. 3https://ecgi.global/sites/default/files/events/2018_annual_members_meeting.pdf https://doi.org/10.1016/j.ribaf.2024.102315 Received 12 December 2022; Received in revised form 28 November 2023; Accepted 4 March 2024 Research in International Business and Finance 70 (2024) 102315 2 N. Rosati et al. same year, the ‘‘Viewpoint’’ of the International Corporate Governance Network issued in October (see ICGN,2018), and a public hearing organised by the Federal Trade Commission (FTC) in December in the US4concluded that the impact of common ownership on competition required further understanding and analysis. These kind of debates are now becoming more common among policymakers. Although traditionally common ownership has not been seen as an antitrust issue, in recent years researchers and policy makers have started to consider its potential anticompetitive effects. Following seminal work on anticompetitive effects of common ownership among U.S airlines (Azar et al.,2018), numerous empirical studies have analysed the impact in specific sectors, among which agrifood (Clapp,2019;Backus et al.,2021a;Torshizi and Clapp,2021), airlines (Kennedy et al.,2017;Azar et al.,2018;Schmalz,2018;Dennis et al.,2022), banking (Schmalz,2018; Azar et al.,2021), energy (Argentesi et al.,2021) and pharmaceuticals (Newham et al.,2018); Banal-Estañol et al. 2021; (Xie, 2021)). These various studies underline the relevance of common ownership, but also highlight empirical challenges and differing evidence regarding the effects of common shareholding on competition. In 2021, the Antitrust Bulletin dedicated its March special issue to the topic of common ownership and its anticompetitive effects. In this issue, Schmalz (2021) presents a comprehensive survey of recent studies on this topic, concluding that empirical evidence has confirmed anticompetitive effects and that the study of the economic channels implementing anticompetitive incentives has advanced. Along this line, Tzanaki (2022) looks from a corporate governance perspective at varieties and mechanisms of common shareholding, and at the plausibility of common owners’ anticompetitive strategies, discussing possible policy implications. Debate on the relevant methodological topics is ongoing. At the 2020 Association of Competition Economics special panel session on common ownership, top scholars debated at length the question of empirical measurement. They highlighted that research dealing with the empirical quantification of common shareholding is at an advanced stage, but a framework for quantifying common ownership remains a key objective. Few recent scholarly articles propose tentative solutions to the measurement challenge, such as profit weights (Backus et al.,2019,2021a,b,c); (Antón et al.,2023), model-based measures (Gilje et al.,2020), or data-derived measures (He and Huang,2017).5Schmalz (2021) highlights that any proposed measure has an intrinsic synthetic nature, which will capture in turn different aspects of an industry or market, and that therefore one single best measure of common ownership does not exist, encouraging researchers to consider the relevant economic context when adopting a certain measure. This paper contributes to the quantification of common ownership with a flexible, multidimensional framework that can be adopted within different economics models for studying the impact on market outcomes. Many indices currently used in the literature fall within this framework as special cases. This new framework exploits solely the ownership links between market actors through indices based on sparse matrix theory and network analysis, avoiding the shortcomings of other common ownership measures, such as subjective assumptions about control weights or the computation of market shares.6The indices explore both the firm’s and the investor’s perspective, considering interactions between the two but also within peer groups. There are a number of useful applications for this proposed framework, which are illustrated here through a real-data example for the Mobile Telecoms market in the EU.7 In the first part of this paper (Section 2), we review the current knowledge, together with its growing critiques, and identify the main measurement issues to be tackled. In Sections 3and 4a series of new indices of common shareholding are proposed, based on balance sheet and ownership firm-level data, under a unifying statistical framework with detailed mathematical properties. Many of the indices currently used in the literature can be identified as special cases falling within this framework. The new indices cover both the firm and the investor’s perspectives, and are then aggregated to obtain suitable industry-level measurements of common ownership. Section 5goes over the application of these new indices in the context of common ownership. Finally, Section 6presents an empirical application of the proposed measures using firm-level financial and ownership data for Mobile Network Operators (MNOs) active in Europe over the period 2007–2021. The last Section concludes. 2. Measuring common ownership All debates mentioned earlier point to the increased need for developing sound measures of common ownership and of its potential impacts. This Section reviews some of the measurement approaches used in past literature, together with their main drawbacks, and recent developments in this area. Table 1 summarises the main measures outlined below, together with their main limitations. The most popular tool used to assess the effects of common shareholding was, until recently, the so-called Modified Herfindahl– Hirschman Index (MHHI), a market-level indicator that captures the distortion introduced in market competition by the presence of common shareholders. It does so by correcting the Herfindahl–Hirschman Index (HHI) of competition according to the ownership and control shares of common shareholders in competing companies. The MHHI however presents several drawbacks. Some relate to the computation of the MHHI itself — it requires computing the market shares of the firms, and also control weights of shareholders in firms which are difficult to determine in practice. In addition, the equations generally used to compute the index in the empirical 4US FTC public hearing on ‘‘Common ownership’’, 6 December 2018, https://www.ftc.gov/news-events/events-calendar/ftc-hearing-8-competition-consumer- protection-21st-century. 5Section 2below discusses more in detail these measures. 6The statistical framework was initially developed in the European Commission report ‘‘Common Shareholding in Europe’’ (Rosati et al.,2020), where it was illustrated through a simplified market example. The present work largely draws from Rosati et al. (2020). 7Rosati et al. (2022a) present an application of the new indices proposed here to test the possible effects of common ownership on competitiveness in the EU beverages industry. Research in International Business and Finance 70 (2024) 102315 3 N. Rosati et al. Table 1 Summary of common ownership (CO) measures. Index Type of measure Description Limitations MHHI market-level concentration measure competition measure; direct policy interpretation market and not firm-level; needs the calculation of market shares and control weights; not a measure of CO; misspecification and endogeneity issues in empirical applications Profit weights (Backus et al.) model-based; firm-level measure define weight a firm puts on competitors’ profits need assumptions on control weights GGL (Gilje et al.) model-based; firm-level measure measures managers’ incentives shifts due to CO rules out strategic interactions of firms; not suitable for some industries Descriptive measures (He and Huang) data-driven; market- and firm-level firm-level explanatory variables used to model effects of CO not model-based Our approach data-driven; market-, firm- and investor-level multidimensional framework; includes most above measures as special cases; measures distorsions and links created by CO at firm–firm, investor–investor, firm–investor levels; measures market structures due to CO; measures network effects not model-based literature suffer from a misspecification problem, which may generate a (positive) correlation between price and the measure of common shareholding, even in the absence of a causal effect of common shareholding on price (O’Brien and Waehrer,2017). Finally, some of the factors that drive prices may also affect institutional investors’ stock purchasing decisions, and consequently the financial shares of investors, which then become endogenous (Kennedy et al.,2017;O’Brien and Waehrer,2017). More importantly, it fails to measure directly the extent of common shareholding itself. More recently, Backus, Conlon and Sinkinson introduced the ‘‘profit weights’’ (see Backus et al.,2019,2021a,b,c), a measure representing the weight a firm puts on its competitors’ profits. These weights arise within a firm’s objective function under common ownership, where the firm maximises a combination of its own profits together with its competitors’ profits, duly weighted by the proposed weights. In the authors’ own words the ‘‘profit weights [...] are the channels through which common ownership [...] affects firm behaviour’’. Similarly to the MHHI, the profit weights need for their calculation the choice of control weights (Pareto weights) of shareholders on firms, representing the influence of the investors on firm decisions. The typical choice throughout the literature has been one of proportional control (where the control weights equal the ownership shares), but other alternatives are possible. The profit weights have been used in some recent empirical studies such as Antón et al. (2023) and Boller and Scott Morton (2020). Gilje et al. (2020) derive a bi-directional, pair-level measure of common ownership aiming at capturing the extent to which common ownership shifts managers’ incentives to internalise externalities. The index accounts for the shares held by common investors in two competing companies, as well as for the relative weight of each firm in the investors’ portfolios. The measure is based on a model where assumptions are made on how managers deal with externalities imposed on one another by commonlyowned firms, specifying a function capturing how attention is allocated across portfolio companies. However, as pointed out by Schmalz (2021), the measure is not suitable for capturing the competitive effects of common ownership in certain industries, given that the model underlying this measure rules out strategic interactions of firms. As alternatives, other studies have generally limited the measurement of common shareholding to a small set of descriptive measures. Examples are the proportion of common shareholders among all the investors present in a market; the proportion of firms that are cross-held by a common holder, at a certain level of ownership; the number of competitors linked through a common shareholder; the proportion of a firm’s shares held by common shareholders, or still the shares held by common shareholders in a firm’s competitors, and so on (see for instance, He and Huang,2017). Such descriptive measures have been used as firm-level explanatory variables in models trying to capture the effect of common shareholding on markets, together with other measures capturing the corporate ownership structure, such as the proportion of atomistic shareholders of a firm. However, several other aspects of investors’ behaviour and of portfolios’ composition can help draw a more precise picture of the phenomenon. The same applies to the analysis of the firms’ shareholding structures, which can reveal interesting patterns of overlap in a given market. Focusing on the behaviour of individual investors, we consider several dimensions of interest. A general overview of the degree of connectedness of a market due to the presence of common shareholders is a starting point, but the investment decisions are driven by a variety of objectives, which determine not only how many and which firms to include in an investor’s portfolio, but also the amount to be held in each of the chosen companies. The distribution of investments within a portfolio can also vary, being more or less concentrated around few players rather than equally spread across all chosen firms, revealing different shareholders’ strategies. Another aspect of interest is the comparison of portfolios of concurrent investors. This allows for the analysis of possible market-level structures, in particular considering whether a market is split into segments allotted to different stakeholders or – on the opposite side – total access to any company is available to all potential investors. Finally, the Research in International Business and Finance 70 (2024) 102315 4 N. Rosati et al. consideration of the shareholders’ type (such as industrial company, financial company, public authority, individual, etc.) is also of interest to investigate possible differentiation of investments across certain groups. A complementary perspective of the one proposed above regards the presence of common shareholders across individual firms’ ownership structure. Here, the objective becomes the study of the shareholder structure of a given firm, and the assessment of the degree of overlap with other competitors’ ownership information. The stronger the similarity of the shareholding structures of competitors, the stronger the potential distortion in competition due to common investors. It is crucial then to assess the strength of the firm–firm links induced by common owners. Furthermore, the degree of overlap will be contingent on the shareholders considered for a given firm, and will vary based on the threshold of equity considered in the analysis. We consider in the following some methodological strategies to construct indices of the extent common ownership, which capture the various aspects mentioned above. 2.1. Sparsity and networks methods in the context of common ownership A starting point for the analysis of common ownership is the matrix representation of a given market. The utility of this representation is evident when the analysis shifts into a more formal framework where the properties of these matrices can be leveraged to compute the relevant indices. A simplified representation of the ownership structure of a market can be obtained through a table, where – for instance – each row corresponds to a shareholder and each column to a firm: F1 F2 ⋯ SH1 𝑒11 𝑒12 ⋯ SH2 𝑒21 𝑒22 ⋯ ⋯ ⋯ ⋯ ⋯ The elements 𝑒𝑖𝑗 of the table can either report the corresponding ownership share, in which case we can name it ownership matrix (OM), or simply report a value of one if a link exists between a firm and an owner, zero otherwise (relation matrix - RM). The empirical structure of this matrix in the ownership context plays a relevant role in the choice of appropriate statistical techniques for its analysis. In fact, the number of investors is typically way larger than the number of firms, with an average shareholding structure easily presenting dozens of owners; on the other hand, the large majority of shareholders only invest in one firm, hence displaying no link to the remaining competitors.8This gives rise to a very large matrix where the majority of the elements are zero. Several different statistical techniques that extract patterns from given matrices are available, both for the case of numerical and of binary (relational) matrices. Such techniques allow for the identification of matrices’ characteristics, as well as for the calculation of indices quantifying specific aspects of the relationships represented in the matrix, and are therefore a valuable starting point for the analysis of CO. Given the multiplicity of possible matrix aspects to be considered, we shall analyse in this work the measures related to the concept of sparsity of a matrix, which is more directly linked to the CO problem. In fact, the concept of sparsity has to do with the representation of a phenomenon where only a small number of coefficients contain a large proportion of the total information, the remaining elements of the representation being negligible, in most cases considered just noise. In matrix language, a sparse matrix or vector is such that most of its elements are zero, just like the typical empirical structure of a market represented though the ownership and relation matrices — as just noted above. The distance or similarity between matrices will also be analysed. The similarity between matrices is defined according to a specific metric used to determine the distance between two given matrices. If one of the two matrices is a benchmark – for example the most sparse matrix in a specific context – the distance or similarity measure can be used to identify the degree of a certain phenomenon with respect to the given benchmark. In the study of CO, a specific benchmark matrix can be easily defined, representing for instance absence of CO, rather than total interconnection between owners and firms, or any other market structure of interest. Finally, network methods will also be considered, applied separately to the network of investors and to the network of firms. The assessment of the strength of the links existing in these two networks will be performed applying the standard network indices, but also using some of the matrix methods mentioned above, applied to the matrix representation of the network links. In the following sections, the aforementioned concepts and statistical methods will be reviewed, and their relevance in the context of CO measurement analysed. 3. Sparsity The concept of sparsity is often linked to definitions of inequality or diversity of the distribution of a phenomenon in a population of size 𝑁, say (𝑐1,…, 𝑐𝑁). Although the literature presents different interpretations and measures of sparsity,9a common agreement is that a distribution with all its information concentrated in one coefficient, and all other zero, is the most sparse. On the other 8For example, in the EU Mobile Telecoms industry studied in Section 6, more than two-thirds of the investors are ‘‘single owners’’ i.e. hold stakes only of one of the firms active in this industry, consistently throughout the period of observation. This is in line with what observed in other EU industries, such as Oil&Gas and Electricity (Rosati et al.,2022b), or Beverages (Rosati et al.,2022a), but also with empirical findings for the US listed firms (He and Huang,2017). 9A comprehensive review can be found in Hurley and Rickard (2009). Research in International Business and Finance 70 (2024) 102315 5 N. Rosati et al. Table 2 Some common sparsity measures and their properties. No. Measure Definition RH Sc RT Cl BG Ba 1. 𝓁0∕𝑁#{𝑘∶𝑐𝑘= 0}∕𝑁✓(✓)✓ 2. 𝓁0 𝜖∕𝑁#{𝑘∶𝑐𝑘≤𝜖}∕𝑁(✓)✓ 3. −𝓁1∕𝑁−1 𝑁∑𝑘𝑐𝑘✓(✓) (✓) 4. −𝓁𝑝∕𝑁−1 𝑁(∑𝑘𝑐𝑝 𝑘)1∕𝑝,0< 𝑝 < 1✓ ✓ (✓) 5. 𝓁2∕𝓁1√∑𝑘𝑐2 𝑘∕(∑𝑘𝑐𝑘)✓ ✓ ✓ 6. − log∕𝑁−1 𝑁∑𝑘log(1 + 𝑐2 𝑘)✓(✓) (✓) 7. 𝑁𝜅4𝑁∑𝑘𝑐4 𝑘∕(∑𝑘𝑐2 𝑘)2✓ ✓ (✓)✓(✓) 8. Hoyer 1 √𝑁−1 (√𝑁−∑𝑘𝑐𝑘 √∑𝑘𝑐2 𝑘)✓ ✓ ✓ ✓ ✓ 9. 𝑝𝑞-mean −(1 𝑁∑𝑘𝑐𝑝 𝑘)1∕𝑝 ∕(1 𝑁∑𝑘𝑐𝑞 𝑘)1∕𝑞 𝑝≤1, 𝑞 > 1✓ ✓ ✓ ✓ ✓ 𝑝 < 𝑞 10. Gini 1 − 2 𝑁∑𝑐𝑘∑𝑁 𝑘=1(𝑁−𝑘+1 2)𝑐(𝑘)✓ ✓ ✓ ✓ ✓ ✓ Notes: Properties as presented earlier: RH = Robin Hood; Sc = Scaling; RT = Rising Tide; Cl = Cloning; BG = Bill Gates; Ba = Babies. Properties in brackets are only valid for the normalised version of the measures. hand, there is agreement that the least sparse distribution is found when the information is evenly spread over all coefficients. In the following, this will be the reference definition of sparsity. In the case of corporate ownership, a sparse investment behaviour would correspond generically to a portfolio with high concentration of ownership in few firms, or, from a firm’s perspective, a shareholding structure with few shareholders owning large stakes. Low sparsity would be observed, instead, in case of more widespread investments across the market, typically with minority participation.10 3.1. Some common sparsity indices There is a large set of sparsity measures in the literature, coming mainly from the fields of signal processing and information theory. The most common are the Kurtosis, the Gini Index, the Hoyer measure, the 𝑝𝑞-means, the 𝓁𝑝norms and their combinations. These and other measures are discussed, for instance, in Karvanen and Cichocki (2003), Rickard and Fallon (2004), Hurley and Rickard (2009), Zonoobi et al. (2011), Pastor et al. (2013,2015). The choice between alternative measures can be motivated by their mathematical properties, which represent minimum criteria a ‘‘good’’ measure of sparsity should satisfy. Six main properties are recognised to be desirable for a sparsity measure11: the Dalton’s Laws (‘‘Robin Hood’’, scaling, ‘‘rising tide’’ and ‘‘cloning’’), the ‘‘Bill Gates’’ and the ‘‘Babies’’ properties. Their aim is to guarantee that a sparsity index goes in the right direction when a change occurs in the underlying distribution. The properties are described in detail in Table A.1 in Appendix A, where their meaning and relevance in the context of common ownership is also discussed. Table 2 presents some among the most common sparsity measures with their properties. Again, the coefficients of the distribution under analysis are denoted by 𝑐𝑘,𝑘= 1 … , 𝑁, while their ordered set is indicated by 𝑐(𝑘)(in increasing order). Measures 1–4 and 6–7 are presented in their normalised version, which accounts for the length 𝑁of the vector representing the distribution. The measures in Table 2 do not present a unified notation in the literature; we follow here the suggestions of Hurley and Rickard (2009), where some measures have been modified – either with a minus sign or by changing the direction of some inequality – in order to obtain an homogeneous interpretation in the sense that an increase in sparsity yields a positive increase in the corresponding measure. In particular, notice that measure 1 is usually defined in the literature as the count of non-zero values, but this would go in the opposite direction, increasing when sparsity decreases. For this reason here the definition is reversed, counting the number of zeros instead. For all these measures, the less sparse the distribution, the smaller the value of the index, the value increasing with sparsity. As we can see from Table 2, most measures fulfil many of the presented properties, if not all. Karvanen and Cichocki (2003) compare measures 1–4 and 6–7. Quéré and Frélicot (2012) compare 7 and 8 through a set of simulations on binary (0–1) distributions, testing their performance in the context of fuzzy partitions. Measure 10 is considered, among others, in Rickard and Fallon (2004) and Zonoobi et al. (2011). Hurley and Rickard (2009) and Pastor et al. (2015) propose more comprehensive accounts of original and normalised measures, other additional measures and properties not discussed here, as well as proofs of the fulfilment of the respective properties. 10 A more detailed analysis of the application of sparsity concepts to corporate ownership is discussed in Appendix A. 11 See for example Hurley and Rickard (2009). Research in International Business and Finance 70 (2024) 102315 6 N. Rosati et al. Measures 1 and 2 simply compute the proportion of zero or negligible elements of the distribution; the higher the proportion, the more concentrated the distribution, i.e. the higher the sparsity. In measure 6, any zero coefficient gives a zero log value in the summation, contributing towards a smaller total; the minus sign reverts the direction of this effect, so that the more the null coefficients in the sum, the higher the sparsity measure. Measure 7 is based on the Kurtosis coefficient 𝜅4– named by analogy to the measure of peakedness of a probability distribution – while measure 10 is the well-known Gini Coefficient of inequality. Measures 3–5 and 8–9 are based on 𝓁𝑝-type norms, which are sums of the coefficients each raised to a certain power 𝑝, sum that in turn is raised to the power 1∕𝑝in order to go back to the original scale of the coefficients. The 𝓁𝑝-type norms do not have in general an intuitive interpretation, except for the 𝓁1norm which is simply the average. However, a special note should be devoted to two measures, namely 𝓁0and 𝓁2, from which several other measures in the table are derived. The 𝓁0index is the base for the popular density measure, giving the proportion of non-zero elements of a vector. The density concept is complementary to sparseness, and has several applications both to matrices and to networks, which will be discussed later. Measure 2 restricts the attention to relevant coefficients only, thus ignoring all those of a negligible size. A corrected density index can be proposed according to this more restrictive exclusion criterion, only considering values above a certain threshold. The 𝓁2norm is also known as Euclidean norm, and for vector 𝐶= (𝑐1,…, 𝑐𝑁)of coefficients is given by: 𝓁2(𝐶) = √ √ √ √𝑁 ∑ 𝑘=1 𝑐2 𝑘 This is one of the most popular norms in several fields of application; it is commonly used to compute the ‘‘length’’ of vectors of size 𝑁, since it corresponds to the distance from the origin to the point 𝐶in an 𝑁-dimensional space. The vectors presenting larger coefficients will have a larger 𝓁2norm, showing that they are further away from the origin (the null vector). Based on this norm, several distance measures have been developed, especially in error minimisation contexts, such as the least squares criterion (which minimises the square of the 𝓁2norm), or the mean square error measurement (divides by 𝑁the square of the 𝓁2norm). Notice that the 𝓁2norm gives higher weight to larger coefficients, by squaring their values, contrary to the simple average 𝓁1. For this reason, the derived measure number 5, given by the ratio of 𝓁2to 𝓁1, can be seen as assessing the relative weight of larger coefficients over the coefficients’ total, hence showing larger values for more concentrated distributions, i.e. for higher sparsity. For example, considering three distributions with same total value and increasing concentration – say (2, 2, 2), (4, 1, 1) and (5, 1, 0) – measure 𝓁2∕𝓁1takes values 0.58, 0.71 and 0.85 respectively. In the sparsest case, i.e. (6, 0, 0), we get 𝓁2∕𝓁1= 1. 3.1.1. Extension of sparsity measures to ownership and relation matrices Sparsity measures can be of particular interest in the framework of CO when considering a given investor and the relationship with all the firms in its portfolio. Referring to the matrix representation of a market introduced in Section 2.1, the dimension of interest is the so called row-wise sparsity,12 given that each row in the matrix corresponds to an investor’s portfolio, reporting the full shareholder-firm relationships. If we look at a row, the sparsest case occurs when an owner only holds shares of one firm in the market (whatever the percentage), the minimum sparsity being achieved when the owner owns shares in all firms in the market, at a constant percentage, i.e. does not show preference for any firm in particular. This also represents the most extreme form of common ownership, an investor reaching all competitors active in a market. Consider now the application of the sparsity measures to either the OM or RM matrix, say 𝑋, whose elements shall be denoted by 𝑥𝑖𝑗 , where again index 𝑖= 1 … , 𝐼 spans the rows i.e. the owners, and index 𝑗= 1 … , 𝐽 the firms in the columns. Recall that the elements represent, respectively, either the ownership share or the presence/absence of a owner–firm link. Any of the sparsity measures of Table 2 can be applied to each row 𝑖of 𝑋, and then aggregated across rows according to some criterion. Denoting generically by 𝑆(𝑥𝑖)a sparsity measure applied to row 𝑖(i.e. to elements 𝑥𝑖𝑗 , with fixed 𝑖), an overall measure of row-wise sparsity for matrix 𝑋is given by 𝑆(𝑋)defined as follows: 𝑆(𝑋) = 𝐼 ∑ 𝑖=1 𝑆(𝑥𝑖) This type of aggregation is mentioned in Rickard and Fallon (2004) as a common measure of matrix sparsity, however other ways of summarising rows information can be considered. Table 3 proposes some alternative aggregation methods of row sparsity indices 𝑆(𝑥𝑖), discussing the interpretation of the resulting matrix measures. Besides the sum of row indices, the average across rows is considered, as well as the median and other relevant percentiles; as alternatives, the maximum or minimum row sparsity are also of interest, representing, respectively, the value of sparsity corresponding to the owner with most concentrated investments (in the limit not a common owner), and to the most ‘‘democratic’’ owner, investing more equally across firms in the market (in principle a common owner). In general, low values of 𝑆(𝑋)raise concerns in the CO context, showing more evenly spread investments of owners across firms in the market, going in the direction of CO. Among the possible row-sparsity measures that can be used to construct the matrix indices, measure 5 from Table 2 will be used to discuss the application to the OM and RM matrices, the remaining measures presenting in general analogous interpretations. The special case of measure 1, given its links with other fields in the literature, will be discussed separately below. 12 See more details in Appendix A. Research in International Business and Finance 70 (2024) 102315 7 N. Rosati et al. Table 3 Matrix sparsity measures 𝑆(𝑋)constructed applying different row aggregation criteria. No. Criterion 𝑆(𝑋)Interpretation 1. Sum 𝐼 ∑ 𝑖=1 𝑆(𝑥𝑖)Total owners’ sparsity. A high value denotes high concentration in the investment behaviour, i.e. owners tend to hold shares of few firms. A low value denotes tendency of owners to distribute investments across firms. 2. Average 1 𝐼 𝐼 ∑ 𝑖=1 𝑆(𝑥𝑖)Average owners’ sparsity. Same as above, but normalised by 𝐼, the number of owners in the market. 3. Median Med 𝑆(𝑥𝑖)Median sparsity. 50% of owners have an investment behaviour with sparsity lower than 𝑆(𝑋). If 𝑆(𝑋)is high, then owners do not diversify much investments; if low, there is stronger tendency for CO. 4. 𝑝th percentile 𝑄𝑝𝑆(𝑥𝑖)Same as above, with now 𝑝% of owners having investments with sparsity lower than 𝑆(𝑋). Threshold that determines the degree of sparsity of the 𝑝% most ‘‘democratic’’ owners. 5. Minimum min𝑖𝑆(𝑥𝑖)Sparsity of most ‘‘democratic’’ market owner, holding a very similar proportion of shares in all market firms. 6. Maximum max𝑖𝑆(𝑥𝑖)Sparsity of most ‘‘unequal’’ market owner, holding very different proportion of shares across market firms, in the limit having invested only in one firm. Measure 5 is the 𝓁2∕𝓁1sparsity index; the expression applied to one row of the OM or RM is given by: 𝑆(𝑥𝑖) = 𝓁2(𝑥𝑖) 𝓁1(𝑥𝑖)=√∑𝐽 𝑗=1 𝑥2 𝑖𝑗 ∑𝐽 𝑗=1 𝑥𝑖𝑗 The general meaning of this measure was discussed in Section 3.1; in the CO application, the 𝓁2measure considers one owner 𝑖at a time, sums the squares of the shares the owner holds in each subsidiary 𝑗in the market – giving more weight to larger shares – and finally takes the square root of the total. This measures the (Euclidean) ‘‘distance’’ of the owner’s investment behaviour from the ‘‘null’’ owner, i.e. an owner that holds zero (or negligible) shares in all firms in the market. The ratio of 𝓁2to the 𝓁1measure (simple sum of the shares held by the owner in all its subsidiaries), rescales such distance according to the owner’s total investment, giving a relative measure of how concentrated is the owner’s investment behaviour across the market. Such individual behaviour can then aggregated according to any method proposed in Table 3. A little more attention should be devoted to measure 1, whose link to the vector density measure was discussed in Section 3.1. If we choose 𝑆(𝑥𝑖) = 𝓁0∕𝐽i.e. the proportion of null elements of row 𝑖, then the average aggregation criterion would give: 𝑆(𝑋) = 1 𝐼 𝐼 ∑ 𝑖=1 𝑆(𝑥𝑖) = 1 𝐼 𝐼 ∑ 𝑖=1 #{𝑗∶𝑥𝑖𝑗 = 0} 𝐽=#{(𝑖, 𝑗) ∶ 𝑥𝑖𝑗 = 0} 𝐼𝐽 This is the overall proportion of null elements of the 𝑋matrix, the complement of the well-known matrix density index – an index very widely used in matrix analysis and also in the networks literature – i.e. the proportion of non-zero elements of the matrix: density(𝐶) = #{(𝑖, 𝑗) ∶ 𝑥𝑖𝑗 ≠0} 𝐼𝐽 A thorough discussion of this index in the analysis of networks will follow in Section 4.2. In the ownership application, however, all lines of the OM or RM must have at least one non-zero element, i.e. each owner is included in the matrix if and only if it owns at least one firm in the market. Therefore, the sparsity index for each row should compute the proportion of null row elements excluding those that are structurally non-zero. i.e. should take the expression 𝑆(𝑥𝑖) = 𝓁0∕(𝐽−1), otherwise the row index would never reach the maximum of one in the case of maximum sparsity (absence of CO). This yields the following CO-corrected matrix sparsity measure: 𝑆CO(𝑋) = #{(𝑖, 𝑗) ∶ 𝑥𝑖𝑗 = 0} 𝐼(𝐽− 1) For the same reason, a CO variation of the matrix density index should also be considered as follows: densityCO(𝑋) = #{(𝑖, 𝑗) ∶ 𝑥𝑖𝑗 ≠0} − 𝐼 𝐼(𝐽− 1) Research in International Business and Finance 70 (2024) 102315 8 N. Rosati et al. Table 4 Some common matrix norms. No. Norm Definition — Single matrix Definition — 𝑑(𝐴, 𝐵) 1. 𝓁𝑝-type (∑𝑖∑𝑗|𝑐𝑖𝑗 |𝑝)1∕𝑝(∑𝑖∑𝑗|𝑎𝑖𝑗 −𝑏𝑖𝑗 |𝑝)1∕𝑝 2. 𝐿2,1∑𝑖√∑𝑗|𝑐𝑖𝑗 |2∑𝑖√∑𝑗|𝑎𝑖𝑗 −𝑏𝑖𝑗 |2 3. Frobenius √∑𝑖∑𝑗|𝑐𝑖𝑗 |2√∑𝑖∑𝑗|𝑎𝑖𝑗 −𝑏𝑖𝑗 |2 4. 𝐿𝑝,𝑞 (∑𝑖(∑𝑗|𝑐𝑖𝑗 |𝑝)𝑞∕𝑝)1∕𝑞(∑𝑖(∑𝑗|𝑎𝑖𝑗 −𝑏𝑖𝑗 |𝑝)𝑞∕𝑝)1∕𝑞 Notes: The elements of the reference matrix are denoted by 𝑝𝑖𝑗 ,𝑖= 1 … , 𝐼,𝑗= 1 … , 𝐽 , were 𝑝=𝑎, 𝑏 or 𝑐, according to the case. In the third column norms applied to a pair of matrices 𝐴and 𝐵in order to determine their distance 𝑑(𝐴, 𝐵). which computes the proportion of non-zero elements only among the entries in the matrix which can actually be zero, thus excluding the cases that are bound to be non-zero. This modified density index will reach the minimum value of zero in case of absence of common owners, hence allowing to have a zero-density benchmark for a market with only single owners. 3.2. Similarity measures for comparison to benchmark scenarios A possible alternative approach to the measurement of the extent of CO, as mentioned earlier, is the comparison of the observed market structure with an hypothetical benchmark scenario of interest. As long as it is possible to express the desired benchmark through a specific matrix, then the comparison with the benchmark can be performed through the computation of a similarity measure between the benchmark matrix and the matrix representing the actual market. Matrix similarity measures are intended to assess the ‘‘distance’’ 𝑑(𝐴, 𝐵)between two matrices 𝐴and 𝐵, with identical number of rows and columns, not necessarily square. Similarity measures are usually calculated entry-wise, that is the elements of the two matrices in the same position are compared (usually calculating differences), and these values are then aggregated through a matrix norm, i.e. a matrix measure similar to the vector measures considered earlier. In order to compute a matrix norm, the matrix is treated as a long vector, where the columns (or rows) have been stacked together; the norms treat a 𝐼×𝐽matrix as a vector of size 𝐼𝐽, and apply any vector measure to it. Among the most popular, we have the 𝓁𝑝-type norms, the Frobenius norm and the more general family of 𝐿𝑝,𝑞 -norms, whose formulas are given in Table 4. The structure of these norms is very similar to the sparsity measures presented earlier, the main differences being that summation is now performed over the two row and column indices, 𝑖and 𝑗respectively, to span all elements of the matrix. Both the 𝐿2,1and the Frobenius norm are particular cases of the family of 𝐿𝑝,𝑞 norms, since the Frobenius is actually an 𝐿2,2 norm. The 𝐿2,1norm is a popular error function used in robust data analysis and in sparse coding, given that the error for each data point (the matrix row in this case) is not raised to any power, but simply summed over all points (rows). The Frobenius norm is also rather popular, as being the Euclidean norm on the 𝐼×𝐽space of the matrix elements. Both norms are invariant under rotations of the row and/or of the columns, i.e. the order in which the firms and the owners are arranged into the matrix is irrelevant. For a fuller account of matrix theory and applications, see for instance Zhang (2017), or Boyd and Vandenberghe (2018). For the calculation of the distance between matrices, the chosen norm is not applied to the original matrices, but to the transformed matrix (of differences). In the right column of Table 4, the norms presented above are used for the measurement of matrices similarity, based on the matrix of differences 𝐴−𝐵. If we wish to use the density index (or its variation presented earlier) in the case of comparison of matrices, we can either compute the densities of the two matrices and then compare them, or compute directly the density of the matrix of differences. Since the density is not a linear operator, in general Density(𝐴) − Density(𝐵)≠Density(𝐴−𝐵)(unless 𝐴=𝐵, in which case they are both null). In the first computation on the left-hand side, we assess the difference in sparsity between 𝐴and 𝐵; if the difference is positive, then matrix 𝐴is more dense i.e. has more non-zero elements than 𝐵, the reverse applying in case of a negative value. On the other hand, the density of 𝐴−𝐵will always be a positive value, computing the proportion of non-zero values of the difference matrix, i.e. the proportion of cases for which the elements of 𝐴and 𝐵are not equal. A small value indicates that the two matrices coincide in most entries, while a value close to one means that most of the elements of the two matrices are different, hence producing non-zero entries in the difference matrix. 4. Network methods Social network analysis studies the empirical structure of social relations and associations that may be expressed in network form. It can therefore be applied to the analysis of the corporate ownership structure of a market, which can be easily represented through a network of relationships between owners and firms. A light introduction to social networks can be found in Borgatti et al. (2009), while König and Battiston (2009) present the main features of economic networks. For a comprehensive account of social networks analysis see, for instance, Scott (2017)orBorgatti et al. (2018). A social network is constituted by a set of nodes (or actors), and a set of ties (or links) that connect pairs of nodes. The nodes can be all of the same type, for example when studying relationships between individuals, giving rise to a so-called one-mode network. Research in International Business and Finance 70 (2024) 102315 15 N. Rosati et al. Table 5 Summary statistics for firms active in the EU28 in the MNOs sector between 2007 and 2021. Year Number Number (%) firms Number Number (%) Number (%) Firms Cross-held by BH of SH Common SH Single SH 2007 84 61 (72.62) 640 188 (29.38) 452 (70.63) 2008 87 64 (73.56) 652 193 (29.60) 459 (70.40) 2009 90 63 (70.00) 662 187 (28.25) 475 (71.75) 2010 91 67 (73.63) 717 194 (27.06) 523 (72.94) 2011 94 65 (69.15) 639 191 (29.89) 448 (70.11) 2012 95 63 (66.32) 657 169 (25.72) 488 (74.28) 2013 94 67 (71.28) 645 185 (28.68) 460 (71.32) 2014 94 67 (71.28) 617 177 (28.69) 440 (71.31) 2015 96 72 (75.00) 642 205 (31.93) 437 (68.07) 2016 100 76 (76.00) 616 209 (33.93) 407 (66.07) 2017 100 78 (78.00) 593 188 (31.70) 405 (68.30) 2018 100 79 (79.00) 595 179 (30.08) 416 (69.92) 2019 96 75 (78.13) 712 185 (25.98) 527 (74.02) 2020 95 71 (74.74) 692 164 (23.70) 528 (76.30) 2021 95 73 (76.84) 500 139 (27.80) 361 (72.20) Notes: Percentage of firms cross-held by block-holders (BH) at minimum 5%. Percentage of common shareholders = Percentage of shareholders with more than one firm in their portfolio. Single owners = Those holding shares only of one firm. 2007 and 2021.18 All of these operators are based in Europe, 97% being registered in the EU28, while the remaining 3% (i.e. 3 firms) are registered in Norway. Table 5 presents the summary figures for the shareholder structure of the industry over the period of observation. The number of firms in this industry has been increasing slowly, stabilising just under 100 in 2021. The sample includes an average of 642 shareholders per year. The proportion of common owners has been rather steady around a mean of 29% throughout the period. The percentage of MNOs that share block holders with competitors has increased between 2007 and 2021 to almost 77% of firms. By definition, two firms are cross-held by a block-holder if the common investor holds at least 5% in both firms. The empirical evidence for our sample implies that the vast majority of firms are linked to at least one other company in the market through a substantial amount of shares in the hand of a common owner. This is mainly due to the peculiar structure of this market, where most companies belong to large corporate groups such as Vodafone or Orange, where all subsidiaries share a common parent. In fact, these figures are markedly higher than EU and US indicators calculated on a variety of markets for the same period in Rosati et al. (2020) and He and Huang (2017), respectively, reporting proportions of block cross-held firms well below 70%. Notice that institutional investors hold very large portfolios in this market, as will be detailed later, and hence are responsible for linking a substantial number of competitors; however, the shares held are usually below the block threshold of 5% considered here, so their holdings do not influence this indicator. 6.1. Main players Since most of the MNOs included in our sample are registered in the EU28, there are only few large players from outside (Norway). Therefore, in the analysis of the main firms active in this industry ten EU28-based companies and only two non-EU28 corporations will be considered. Table 6 displays summary information for the top 10 largest enterprises in EU28 and top 2 outside EU28, by country of registration. The firms are ranked according to the value of their total assets (TOAS, in Bln e) in 2021. Note that some of the top firms are controlled subsidiaries; in these cases, the ownership information refers to the mother companies (see ‘‘Controlled by’’ column). As a single investor, BlackRock dominates the MNOs industry, with rather large shares in some of the top players. This behaviour is in line with other empirical studies of institutional investors in strategic markets, which has in turn been associated to anticompetitive behaviour.19 Vanguard and Norway lag behind, with relatively small participations, while State Street plays only a minor role. France is also present with small shares in only a handful of top competitors. To the best of our knowledge, this paper also represents the first instance where the role of sovereign investments funds in common ownership is methodically quantified. The additional association to anticompetitive behaviour of sovereign funds could be an interesting topic for further research. 6.2. Density and uniformity Table 7 now presents summary statistics for the density-based and uniformity indicators calculated on 2021 data for MNOs.20 These statistics represent industry-level empirical estimates for the presence of common ownership. The density index indicates that 18 Most of the companies (82%) classify their core activity within the NACE Division ‘‘61: Telecommunications’’. The other 18% of firms are scattered across different activities (e.g. Retail, Postal and Courier Activities and Head Offices). 19 See for example Azar et al. (2018), Banal-Estañol et al. (2021), and Azar et al. (2021) which identify Blackrock as one of the key common shareholders in the airlines, pharmaceuticals, and banking sectors, respectively. 20 Tables displaying the full set of matrix indices calculated for all years between 2007 and 2021 are available upon request from the corresponding author. Research in International Business and Finance 70 (2024) 102315 16 N. Rosati et al. Table 6 Largest firms of MNOs sector and selected ownership data on their main common owners. Country TOAS, Black France Norway State Vanguard 2021 rock street (Bln e) Registered in EU28 Controlled by Shares held in 2021 (%) VODAFONE GROUP UK 153.95 9.11 0.13 3.09 2.40 3.77 ORANGE FR 99.46 4.87 29.41 1.65 0.27 2.00 TELECOM ITALIA IT 57.48 1.55 0.75 0.21 1.34 TELIA COMPANY SE 20.25 3.00 1.12 0.18 1.90 WIND TRE IT 16.41 CK HUTCHISON HOLD. (KY) 4.90 1.07 0.91 2.04 EE LIMITED UK 13.60 BT GROUP PLC (UK) 4.55 2.99 2.16 3.31 KONINKLIJKE NL 12.74 4.84 2.80 0.38 2.20 O2 HOLDINGS LTD UK 12.58 (50%) LIBERTY GLOBAL (UK) 3.49 0.92 1.50 2.02 (50%) TELEFONICA (ES) 2.90 1.89 2.49 TELENET GROUP BE 9.76 TELENET GROUP HOLD. (BE) 0.64 0.99 0.14 1.13 PROXIMUS SA BE 9.23 4.88 1.14 0.15 1.17 TELEFONICA DE DE 7.59 1.78 0.90 0.10 0.64 Registered outside EU28 Controlled by Shares held in 2021 (%) TELENOR ASA NO 20.03 1.60 53.97 0.80 1.25 TELIA NORGE AS NO 3.76 TELIA COMPANY AB (SE) 3.00 1.12 0.18 1.90 Notes: Direct or indirect participation shares (%) held in 2021 are displayed. Table 7 Summary statistics for MNOs sector common shareholding indices (year 2021). Index Mean 75th percentile 99th percentile Maximum Density 2.05 2.11 12.11 17.89 Uniformity 11.45 19.19 74.78 76.54 TOAS density 12.67 26.40 67.25 68.48 TOAS weighted density 0.25 0.15 3.68 5.73 MKT CAP density 16.35 28.04 96.61 98.98 MKT CAP weighted density 0.20 0.08 4.22 8.77 Notes: Percentage points displayed, index values between 0 and 100. the MNOs investors tend to have small portfolios, including at most 18% of the firms in the market. The top 1% of largest portfolios are actually rather smaller, holding shares in just above 12% of the market firms. As noted earlier, this is due to the presence, in this industry, of large groups mothered by Telecom giants such as Vodafone, Orange or Deutsche Telekom, which have little if no overlap in their portfolios, and have basically partitioned the set of firms active in this industry into independent domains. From this perspective of the shareholders then, common ownership in 2021 remained a relatively contained phenomenon. This is also confirmed by overall lower levels of uniformity indices,21 where the maximum did not reach 77% in 2021, showing a higher concentration of investment strategy. This suggests that investors are less ‘‘democratic’’ and tend to target higher participations in a smaller set of firms, rather than widespread low investment across all market. In terms of value-based indices, there are evident differences in level between total assets and market capitalisation results, given that the set of listed firms in this industry is rather reduced, and therefore the second set of indices is computed on a very restricted part of the market. We see that the top 1% of investors hold stakes in firms that represent almost 68% of the Total Assets of the industry, showing that such top investors tend to privilege the largest enterprises. Given the high participation shares in this market, the weighted TOAS index reaches high values, with the top investor holding 5.73% of the market total assets through shares. A similar picture is found for listed firms, where the largest portfolios hold stakes in firms representing basically the entire market capitalisation (almost 99%), and through their participations control about 9% of the market value. Unlike the unweighted indices, these industry measures imply a degree of common ownership that could present anticompetitive concerns, as a few top investors exert influence over a large portion of the market assets. 6.3. Common ownership indices for top investors Table 8 presents the density indices for the top investors engaged in common ownership in the MNO sector in 2021. These are mainly large Telecom groups, with several subsidiaries in different countries. For example, Orange holds 2.38% of the market Total Assets through virtually full ownership of its subsidiaries in Belgium, Poland, Romania, Slovakia or Spain. Similarly, Deutsche Telekom holds 1.89% of the Total Asset of the industry through control of firms in Austria, Check Republic, Croatia, Hungary, Poland and Slovakia. 21 Compared to other markets. See for example Rosati et al. (2020). Research in International Business and Finance 70 (2024) 102315 17 N. Rosati et al. Table 8 Top investors in MNOs sector (year 2021). SH name SH No. Density TOAS W TOAS Largest subsidiaries and shares held (%) country Subs. density den. FRANCE FR 9 9.47 46.39 5.73 ORANGE (29.41); VODAFONE GROUP (0.13); ORANGE ESPAGNE (25.01); ORANGE POLSKA (25.01); ORANGE BELGIUM (25.01); ORANGE ROMANIA (25.01); ORANGE SLOVENSKO (25.01) DEUTSCHE BANK DE 12 12.63 67.46 3.82 VODAFONE GROUP (13.68); ORANGE (0.48); TELECOM ITALIA (0.17); TELIA COMP (0.16); TELENOR (1.62); KONINKLIJKE (0.64); PROXIMUS (0.63) CK-HUTCHISON HOLDING KY 6 6.32 3.89 3.82 WIND TRE (98); HUTCHISON DREI AUSTRIA (100); THREE IRELAND (98); CK HUTCHISON UK (98); HI3G ACCESS AB (98); HI3G DENMARK APS (98) BLACKROCK US 12 12.63 67.46 3.79 VODAFONE GROUP (9.11); ORANGE (4.87); TELECOM ITALIA (1.55); TELIA COMP (3.00); TELENOR (1.6); KONINKLIJKE (4.84); PROXIMUS (4.88) TELEFONICA ES 3 3.16 4.14 3.74 O2 HOLDINGS LTD (100); TELEFONICA DEUTSCHLAND (69.93); TELEFONICA MOVILES (100) NORWAY NO 17 17.89 68.48 3.35 TELENOR (53.97); VODAFONE GROUP (3.09); KONINKLIJKE (2.80); ORANGE (1.65); TELECOM ITALIA (0.75); TELIA COMP (1.12); PROXIMUS (1.14) ORANGE FR 7 7.37 2.94 2.38 ORANGE ESPAGNE (100); ORANGE POLSKA (50.67); ORANGE BELGIUM (78.32); ORANGE ROMANIA (99.20); ORANGE SLOVENSKO (100); ORANGE ROMANIA COMM. (53.58); ORANGE COMM. LUX. (78.32) VODAFONE GROUP UK 9 9.47 2.41 2.22 VODAFONE ITALIA (100); VODAFONE PORTUGAL (38.62); VODAFONE MAGYARORSZAG (100); VODAFONE CZECH REPUBLIC (100); VODAFONE ROMANIA (100); VODAFONE LIBERTEL (50); VODAFONE ENTERPRISE GERMANY (100) SWEDEN SE 15 15.79 58.73 1.97 TELIA COMP AB (39.50); VODAFONE GROUP (0.18); ORANGE (0.27); TELENOR (0.13); KONINKLIJKE (0.23); PROXIMUS (0.19); TELEFONICA DEUTSCHLAND HOLDING (0.12) TELIA COMP SE 6 6.32 1.94 1.90 TELIA SVERIGE (100); TELIA FINLAND (100); TELIA NORGE (100); TELIA LIETUVA (88.2); TELIA EESTI (100); LATVIJAS MOBILAIS TELEFONS (50.01) DEUTSCHE TELEKOM DE 8 8.42 2.40 1.89 MAGYAR TELEKOM (40.77); T-MOBILE POLSKA (100); T-MOBILE AUSTRIA (100); HT-HRVATSKI TELEKOMUNIKACIJE (52.17); SLOVAK TELEKOM (100); T-MOBILE CZECH REP. (100); TELEKOM DEUTSCHLAND (100) VANGUARD US 12 12.63 67.46 1.68 VODAFONE GROUP (3.77); ORANGE (2.00); TELECOM ITALIA (1.34); TELIA COMP (1.90); TELENOR (1.25); KONINKLIJKE (2.20); PROXIMUS (1.17) Notes: SH country: Country of shareholder. Last column displays the firms with largest total assets in portfolio and respective quantity of shares held. Rather important are also the States, with France and Norway ranking amongst the top investors, holding the largest percentages of the market Total Assets, with 5.73% and 3.35% respectively. As for the Funds, only BlackRock has large enough participation (i.e. 3.79% of the market Total Assets) to appear in the top positions, while Vanguard is left behind with a lower, yet still sizeable, share of Total Assets held (about 1.68%). While our indices do document a degree of common ownership among state investors (such as France, Norway and Sweden), there is still no evidence in the literature of governance related channels which tie the presence of common ownership to anticompetitive behaviour. 6.4. Network indices The results for shareholders’ and firms’ networks in the MNOs sector are reported in Tables 9 and 10, respectively. There are strong interconnections between actors of the same type. The proportion of pairs of portfolios that are linked through commonly held firms (i.e. proportion of non-zero correlations in Table 9) has been oscillating over time between 4 and 5%, with a slight decline in the last few years. On the other hand, among the large22 connected portfolios the links are rather strong, having observed 22 Portfolios in this sector are considered large if they hold stakes in at least 10% of the market’s firms. Research in International Business and Finance 70 (2024) 102315 18 N. Rosati et al. Table 9 Shareholders’ network indices for MNOs sector. Year Total No. Prop. Non-zero No. High Corr. Prop. High Corr. Common SH Correlations High Dens. High Dens. 2007 188 4.57 21 6.46 2008 193 5.02 35 6.34 2009 187 4.37 38 7.25 2010 194 4.24 14 4.07 2011 191 5.27 19 6.33 2012 169 4.23 12 4.08 2013 185 4.51 9 4.46 2014 177 4.72 14 5.74 2015 205 5.05 22 7.97 2016 209 5.28 20 6.67 2017 188 4.73 14 6.67 2018 179 4.30 12 7.84 2019 185 3.26 12 7.02 2020 164 2.72 19 13.97 2021 139 3.06 21 23.08 Notes: High-density portfolios are considered as such if the densities are higher than 10%, i.e. if they hold stakes in at least 10% of the market’s firms. Table 10 Firms’ network indices for MNOs sector. Year Total No. firms Prop. Non-zero No. of high Prop. of high cross-held by BH correlations correlations correlations 2007 61 3.96 73 52.90 2008 64 4.49 99 58.93 2009 63 4.39 99 56.25 2010 67 4.37 98 54.75 2011 65 3.84 88 52.38 2012 63 4.32 105 54.40 2013 67 4.12 97 53.89 2014 67 4.37 87 45.55 2015 72 3.84 82 46.86 2016 76 3.31 88 53.66 2017 78 3.25 95 59.01 2018 79 3.41 101 59.76 2019 75 3.55 86 53.09 2020 71 3.85 85 49.42 2021 73 3.54 83 52.53 Notes: Block-holders (BH) hold at least 5% of shares. an increasing proportion of highly correlated investors, on average about 7% of owners showing almost coincident investment strategies, with portfolio overlaps of 80% or more. The firms’ network is also highly interconnected through block holders, with between 3.5% and 4% of pairs of firms connected through some common shareholder. More importantly, steadily over half of these linked portfolios present a correlation of more than 80%. Regardless, this very high fraction of highly correlated ownership structures is probably the result of the presence of clusters of firms belonging to the large Telecom groups, which are almost all wholly owned or at least controlled by the parent, showing therefore very similar shareholders’ structures. Again, compared to other sectors like Oli&Gas, Electricity or Beverages (see Rosati et al.,2020), the MNOs show stronger network interconnections both between firms and between shareholders. As mentioned earlier, the source of the quantified links between firms is relevant for competition concerns. If the large share of highly correlated ownership among firms derives from corporate groups, it would not necessarily indicate anticompetitive behaviour stemming from common ownership as the empirical literature typically finds. The indices proposed provide insight into these different dimensions of industry links. 7. Conclusions The existence of common shareholders among competing companies in a given industry has raised the concern of academics and policy makers worldwide, due to its possible effects on market efficiency and competition. The main empirical works investigating this issue cover only a limited number of industries, and a more general effect on the economy has yet to be analysed. This is in part because the measures used so far to assess the effects of common ownership in a given market have been subject to criticism from several scholars, raising the need for the development of a sound measurement framework. Besides specific technical shortcomings of the known metrics, critics underline that each measure will capture different aspects of an industry or market, and encourage researchers to consider the relevant economic context when adopting a certain measure. The absence of a single one-size-fit-all Research in International Business and Finance 70 (2024) 102315 19 N. Rosati et al. ‘‘best’’ measure of common ownership points towards a plural approach, where a set of indices are able, together, to capture the complexity of the phenomenon. This work considered some novel multifaceted methodological strategy to be applied to the measurement of common ownership. The methods are taken in part from the theory of sparse matrices and in part from social network analysis, and are adapted to reflect the need for proper measurement of the phenomenon of common shareholding. From sparsity theory, a series of concentration indicators are derived for the analysis of a shareholders’ portfolio, describing its investment behaviour both at industry level and at investors’ level. Value-based indices are also proposed, where the size of the firms held in each portfolio is taken into account. The strength of the relationships existing within the groups of firms and of investors is also studied, based on purpose-built network indices. The network analysis looks, in turn, at similarities between portfolios of pairs of shareholders, or at overlaps of shareholders structures for pairs of firms. The proposed methodologies can be applied both to the set of owner–firm relations induced by corporate ownership, as well as to the full information about the amount of participation shares. Comparison with given benchmarks such as the case of absence of common ownerships are also considered. Because the framework proposed here abstracts from previous, theory-based approaches, there are limitations to what the measurements can say about the transmission mechanisms of common ownership trends to anticompetitive behaviour. We argue however that their model-free nature makes them more objective and compatible with a range of different economic models, as they do not require a number of the assumptions made by previous metrics. As a matter of fact, many of the indicators currently used in the literature are identified as special cases falling within this framework. The measures are tested using firm-level data for European Mobile Network Operators between 2007–2019, with special attention dedicated to the role of the largest investors. Results show that investors’ portfolios on average in this industry are rather specialised, due to the presence of large corporate groups controlling country-specific subsidiaries, such as Vodafone. However, certain institutional investors typically associated with common-ownership-related anticompetitive concerns, such as Blackrock, present a share of firms held and a total asset share comparable to these corporate groups. Moreover, the subsample of large investors with extended interests (again generally corresponding to institutional investors), shows a rather strong network connection with highly-correlated portfolios. This is not the case, instead, for the corporate groups, which present non-overlapping portfolios, creating clusters of ownership segmenting the market. This highlights the importance of a holistic approach to common shareholding as proposed here. The framework proposed in this paper, based on directly observed linkages between firms and investors provides a data-driven measure for common ownership. Its multidimensional approach helped disentangle the differential behaviour of common owners of different nature, such as the parents of corporate groups as opposed to institutional investors. The policy implications in terms of possible anti-competitive behaviour of these investors are distinct, so the approach provides policy makers with a more complete picture of the market distortions introduced by different varieties of common owners. The empirical application proved useful also in highlighting new relevant trends for research. In particular, our findings underscored the growing role of states as common shareholders, something which merits closer attention in future research. CRediT authorship contribution statement Nicoletta Rosati: Conceptualization, Data curation, Formal analysis, Investigation, Methodology, Project administration, Software, Supervision, Validation, Writing – original draft, Writing – review & editing. Pietro Bomprezzi: Data curation, Formal analysis, Investigation, Software, Validation, Writing – original draft, Writing – review & editing. Maria Martinez Cillero: Data curation, Formal analysis, Investigation, Software, Validation, Writing – original draft, Writing – review & editing. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Data availability The authors do not have permission to share data. Acknowledgements This paper benefited from valuable contributions from Jean Bergevin, Maria Elena Despott, José Enrique Elías Cabrera, Sean Greenaway, Issam Hallak, Cyril Hariton, Igor Jelinski, Michela Nardo, Cornelius Schmidt, Joanna Sikora-Wittnebel and Anthea Wingender. The authors also thank valuable comments from seminar participants at the Essex Finance Centre and from conference participants at European Economics and Finance Society Annual Conference and at the International Conference on European Studies. Author Nicoletta Rosati was partially supported by the Project CEMAPRE/REM - UIDB/05069/2020 financed by FCT/MCTES through national funds. The other authors did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. Appendix A. Application of the sparsity framework to common ownership In order to apply the sparsity concept to the CO problem, and in particular to the two matrices defined in Section 2.1 (the ownership matrix OM and the relation matrix RM), it is necessary to identify the most and least sparse scenarios and their meaning. Research in International Business and Finance 70 (2024) 102315 20 N. Rosati et al. The main issue to be considered is the applicability of the sparsity concept to a matrix, since the concept itself, as well as the measures, were initially developed in relation to a vector of coefficients — representing the distribution of wealth. The extension of a vector measurement to a matrix can be done in three different ways, which in turn give rise to different benchmark scenarios, as discussed below. Let index 𝑖= 1,…, 𝐼 denote the owners, and 𝑗= 1,…, 𝐽 the firms. The sparsity of a given ownership structure can be studied looking at the following dimensions in the OM and RM: (1) column dimension:sparsity of the shareholders’ distribution for a given firm. In the analysis of a column, the maximum sparsity is achieved when a given firm 𝑗presents only one owner who holds 100% of the shares, the least sparse distribution corresponding instead to having 𝑛𝑗shareholders, each with a proportion of 1∕𝑛𝑗of the shares. For the RM, these cases correspond, respectively, to a vector with one unit element and all the rest zero, or to a vector of all ones. (2) row dimension:sparsity of the investment distribution of a given owner into the firms constituting the market. If we look at a row, the sparsest case occurs when an owner only holds shares of one firm in the market (whatever the percentage), the minimum sparsity being achieved when the owner owns shares in all firms in the market, at a constant percentage (does not show preference for any firm in particular). (3) overall dimension:sparsity of the firm–owner links present in the market. In the overall analysis of the matrices we can define maximum sparsity either of all the rows (one firm only per each owner, for all owners; one firm can have more owners, 𝐼≥𝐽), or of all the columns (one owner only per each firm, for all firms; the owner can be common to other firms, 𝐼≤𝐽), or of both at the same time (square diagonal matrix, one owner only per each firm and one firm only per each owner, 𝐼=𝐽). Defining maximum sparsity of a matrix looking at the column dimension would consider only the shareholders’ structure of a given firm, and not the inter-linking between firms caused by common owners, so does not correspond to the primary objective of our study. However, this approach can be used to assess other market characteristics in a later stage. The last approach (joint maximum sparsity of both rows and columns) again imposes the column maximum sparsity, which is not of central interest at this stage, so it will be left for later analyses. It follows that the row-wise approach for the definition of overall maximum sparsity of a matrix seems more appropriate for our case, as being directly connected to the study of CO. In fact, in this approach the most sparse matrix corresponds to the absence of common owners, i.e. each row only presents one non-zero element, while the opposite happens when each row is completely filled with positive and equal values, that is each owner being linked to all firms in the economy with equal shares (least sparse scenario). Notice that this last case, being repeated for all owners, implies that the shareholder structure of all firms is identical. The relative weight of each shareholder in a firm is not constrained to a specific value, however it will depend on its financial capacity. Properties of sparsity measures Table A.1 presents the definition and interpretation of the most common properties of sparsity indices. Desirable criteria that a sparsity measure should fulfil are, among others, the so-called ‘‘Robin Hood’’ property, the scaling or homogeneity, the ‘‘rising tide’’ property, and the ‘‘cloning’’ property. These four criteria were initially proposed by Dalton (1920) in the financial setting of inequality measurement of wealth distribution, but are nowadays generally recognised to be minimum criteria a ‘‘good’’ measure of sparsity should satisfy, being referred to as ‘‘Dalton’s Laws’’. More recently, Rickard and Fallon (2004) add two extra properties, named ‘‘Bill Gates’’ and ‘‘Babies’’, respectively. Additional axioms and attributes, whose analysis goes beyond the scope of this note, can be found in Pastor et al. (2013,2015). More details about the above properties can be found in Hurley and Rickard (2009); the paper also presents sixteen sparsity measures and prove which of them satisfy which properties. The main result is that only two measures satisfy all six properties, namely the Gini Index and the 𝑝𝑞-means (see definitions in Table 2), while all remaining measures satisfy some, but not all of them. Properties revisited in the context of common ownership Given that the properties were initially considered in a welfare inequality context, they need to be critically reassessed in the CO framework, specifically in the sense discussed above, i.e. looking at row-wise sparsity (increase in sparsity = decrease of the extent of CO; decrease in sparsity = stronger presence of CO). Each property will be analysed initially looking at a change in behaviour of one single owner, subsequently extending the same behaviour to all owners in the market (if meaningful) in order to assess the overall effect on the OM and RM. This exercise will also help exclude sparsity measures that are inappropriate for our purpose. The revised properties are those presented in Table A.1. Summarising, all six properties of sparseness measures are overall still meaningful in the context of CO, although the scaling and cloning properties show some limitations when applied to the OM, being less problematic in the RM case. Robin Hood: in case an owner holding a large share in a firm decides to divest, redirecting the investment towards other firms where it holds smaller (or zero) shares, the measure captures the change as a decrease in sparsity, i.e. a movement from absence or low level of CO towards a higher level of CO. The same applies if all owners change their behaviour in an analogous way: the measure would detect a reduction in sparsity. This is a desirable property as it goes in the direction of the effect we would like to measure. Therefore sparsity measures that fulfil the Robin Hood property are appropriate in our context. Research in International Business and Finance 70 (2024) 102315 21 N. Rosati et al. Table A.1 Properties of sparsity measures. No. Property Definition Description 1. Robin Hood A ‘‘fair’’ readjustment of coefficients decreases sparsity Taking some amount from the larger coefficients and ‘‘redistributing’’ towards smaller coefficients yields a less concentrated i.e. more equal distribution. 2. Scaling A change in scale of the coefficients does not change sparsity Multiplying wealth of all units by an equal factor does not change the level of (in)equality of the distribution. 3. Rising Tide An identical increase in all coefficients decreases sparsity Adding a fixed amount to each coefficient reduces the relative difference between large and small values, i.e. yields a more equal distribution. 4. Cloning Doubling the number of coefficients by cloning the same values does not change sparsity If a second population is cloned, reproducing the same values as the first one, then the level of (in)equality of the merged populations equals that of the initial population. 5. Bill Gates A significant increase in one coefficient increases sparsity When one coefficient becomes very large, the level of inequality increases. 6. Babies The addition of extra zero coefficients increases sparsity Adding individuals with zero wealth to a population increases inequality, by concentrating the (positive) total wealth in the hands of a smaller share of individuals. Scaling: If an owner, say, doubles the ownership shares held in all its subsidiaries, the measure does not detect a change in sparsity, i.e. the extent of CO is considered unchanged. For a single row, this change in investment does not alter the presence of CO, since the number of existing links is unchanged. However, given that the strength of the links increased, and that the column sum of the shares is constrained to be ≤100%, this implies a decrease of the strength of the links other owners have with the same subsidiaries. Such decrease in the limit could even lead to a zero share, therefore eliminating an existing link, and altering the sparsity. It follows that this property is somewhat controversial, especially in the case of the OM, which might suffer an (undetected) readjustment of all rows when one owner changes its investments. The property seems more acceptable for the RM, which in general will be unchanged under this scenario, although in an extreme case the actual number of links might be affected, as mentioned earlier. Therefore, a measure fulfilling this property raises some concerns, and should be tested further. The situation where all owners would double their shares is an impossible event given the above constraint on the column totals, therefore is not analysed. Rising Tide: If an owner increases its ownership shares by 𝑘 > 0points in all firms in the market (even in those where it had previously no shares), then the sparsity decreases. In the limit, an owner that only owned shares in one firm, becomes common owner of all companies in the economy, so there is a change towards an increased level of CO. This property is in line with the dynamics of CO, therefore is acceptable for our study. If all owners made a similar change in investments, the RM would immediately become the sparsest one proposed earlier, therefore going again in the direction of this property; however, in the case of the OM we cannot add indefinitely extra shares to all elements, due to the column total constraint, so this case will not be contemplated. Cloning: If the number of firms in the market doubles, and an owner invests in the new firms exactly the same amounts of shares held in the original set of firms, sparsity does not change. Duplicating (‘‘cloning’’) the initial vector of investments does not change their relative concentration, since the proportion of firms held by the owner is unchanged. However, the absolute number of firms linked by the owner doubled, introducing more inter-connections between firms. In the most extreme case, an owner who had invested only in one firm (and therefore who was not a common owner) will introduce a connection between two firms, and become a common owner. Therefore, measures with this property are acceptable if we seek to measure the relative extent of CO, but are less suitable for absolute measurements. Bill Gates: If one owner increases largely its investment in one firm, sparsity increases. The owner will have to divest with respect to other firms held, in order to move funds massively to that specific firm. Therefore all remaining existing links of that owner will decrease in strength, some even reaching a zero value, i.e. some links may disappear. The same will happen at column level, as a larger share held by the owner under consideration makes all other shares of that specific firm decrease largely, again possibly causing some links of that firm with other owners to vanish. In both cases this implies an increase in sparsity, therefore the property is in line with the market dynamics and is acceptable for our analysis. Babies: Adding an extra firm in the market whose owners are not common to any other firm increases sparsity. This amounts to adding a new column with the shareholders’ structure of the added firm, and some new lines corresponding to the new added owners, which were not present before in the market, since they are not common to any other firms. Therefore, the column will be filled with zeros, except for the last few elements containing the shares held by the new owners. This implies that each line corresponding to an existing owner will have an extra zero, hence increasing the sparsity of that line. On the other hand, since the new owners are not common to other firms, the degree of CO decreases. It follows that the market dynamics in this scenario goes in the direction predicted by the property, which therefore is admissible in the CO framework. Appendix B. New measures of CO summary tables See Tables B.1 and B.2. Research in International Business and Finance 70 (2024) 102315 22 N. Rosati et al. Table B.1 Common shareholding indicators for shareholders’ portfolios, and market level counterparts. Indicator Definition Interpretation Market level Density Nsubs/Nfirms Number of firms in a shareholder’s portfolio over total number of firms in the market. Represents the share of the market to which an investor has access through shareholding. Average, median, maximum, percentiles TOAS density Total TOAS subs/Total market TOAS Sum of the TOAS of all firms in a shareholder’s portfolio over sum of TOAS of all firms in the market. Represents the relative weight of the firms chosen by a specific investor over the whole market. Average, median, maximum, percentiles TOAS weighted density Sum weighted TOAS subs/Total market TOAS Sum of the TOAS of firms in portfolio, each weighted by the actual ownership share of investor, over sum of TOAS of all firms in the market. Represents the actual share of TOAS of the market in the hand of an investor through the specific ownership shares held. Average, median, maximum, percentiles, sum for ‘Big 3’ MKT CAP density Total MKT CAP subs/Total market MKT CAP Sum of the MKT CAP of all listed firms in a shareholder’s portfolio over sum of MKT CAP of all listed firms in the market. Interpretation is same as for TOAS density, but refers to market capitalisation. Only listed firms in portfolio. Average, median, maximum, percentiles MKT CAP weighted density Sum weighted MKT CAP subs/Total market MKT CAP Sum of the MKT CAP of listed firms in portfolio, each weighted by the actual ownership share of investor, over sum of MKT CAP of all listed firms in the market. Interpretation is same as for TOAS weighted density, but refers to market capitalisation. Only listed firms in portfolio. Average, median, maximum, percentiles, sum for ‘Big 3’ Uniformity 1 - √∑𝐬𝐡𝐚𝐫𝐞𝐬𝟐 ∑𝐬𝐡𝐚𝐫𝐞𝐬 One minus the following ratio: (Square root of the) Sum of the squares of the shares in portfolio, over sum of all shares in portfolio. Index assesses the relative weight of larger participation shares over the shares total, showing smaller values for more concentrated distributions. Average, median, maximum, percentiles Number of Block-holdings Number of holdings > 𝑝% Number of participations in portfolios with share value above a certain percentage 𝑝. Represents the number of more intensive investments of portfolio. Computed for 𝑝= 3,5,10. Sum for ‘Big 3’, sum for all SH Table B.2 Common shareholding indicators for shareholders’ and firms’ networks. Shareholders’ network Indicator Definition Interpretation Correlation Pearson’s 𝜌Pearson’s correlation coefficient 𝜌between pairs of portfolios. Reveals the level of overlap between two shareholders’ investments. Proportion of non-zero correlations 𝐍𝐨.𝐧𝐨𝐧−𝐳𝐞𝐫𝐨 𝜌 𝐧(𝐧−𝟏)∕𝟐Number of non-zero correlations over total number of possible connections between pairs of portfolios. 𝑛is the total number of portfolios. Represents the proportion of existing links in the shareholders’ network, i.e. the network’s density. Number of highly correlated high-density portfolios No. of 𝜌 > 80% Number of correlations higher than 80% between pairs of large (high density) portfolios (holding more than 10% of the market’s firms). Counts the number of very strong links among large shareholders’ portfolios. Proportion of highly correlated high-density portfolios 𝐍𝐨. 𝜌>𝟖𝟎% 𝐤(𝐤−𝟏)∕𝟐Number of correlations higher than 80%, over total number of possible connections between pairs of large (high density) portfolios (holding more than 10% of the market’s firms). 𝑘is the total number of high density portfolios. Represents the proportion of very strong links among large shareholders’ portfolios, i.e. the degree of similarity of their investments. Firms’ network Indicator Definition Interpretation Correlation Pearson’s 𝜌Pearson’s correlation coefficient 𝜌between pairs of SH structures of firms. Reveals the level of overlap between two firms’ SH structures. Proportion of non-zero correlations 𝐍𝐨.𝐧𝐨𝐧−𝐳𝐞𝐫𝐨 𝜌 𝐧(𝐧−𝟏)∕𝟐Number of non-zero correlations over total number of possible connections between pairs of firms. 𝑛is the total number of firms. Represents the proportion of firms that are connected through some common shareholder, i.e. the firms’ network density. Number of highly correlated SH structures No. of 𝜌 > 80% Number of correlations higher than 80% between pairs of firms. Counts the number of very strong links among firms’ SH structures. Proportion of highly correlated SH structures 𝐍𝐨. 𝜌>𝟖𝟎% 𝐤(𝐤−𝟏)∕𝟐Number of correlations higher than 80%, over total number of non-zero connections between pairs of firms. 𝑘is the total number of connected firms. Represents the proportion of very strong links among connected firms, i.e. the degree of similarity of SH structure. Research in International Business and Finance 70 (2024) 102315 23 N. Rosati et al. References Antón, M., Ederer, F., Giné, M., Schmalz, M.C., 2023. Common ownership, competition, and top management incentives. J. Polit. Econ. 131 (5), 1294–1355. Antoniou, I.E., Tsompa, E.T., 2008. Statistical analysis of weighted networks. Discrete Dyn. Nat. Soc. 2008, 375452, 16pages. Argentesi, E., Banal-Estañol, A., Seldeslachts, J., 2021. A retrospective evaluation of the GDF/Suez merger: Effects on the Belgian gas hub. Energy J. 42. Azar, J., 2012. A New Look at Oligopoly: Implicit Collusion Through Portfolio Diversification (Ph.D. Dissertation). Princeton University. Azar, J., Raina, S., Schmalz, M.C., 2021. Ultimate ownership and bank competition. Financ. Manage. 1–43. Azar, J., Schmalz, M.C., Tecu, I., 2018. Anti-competitive effects of common ownership. J. Finance 73 (4), 1513–1565. Backus, M., Conlon, C., Sinkinson, M., 2019. The common ownership hypothesis: Theory and evidence. Econ. Stud. Brook. 1980–2017. Backus, M., Conlon, C., Sinkinson, M., 2021a. Common Ownership and Competition in the Ready-to-Eat Cereal Industry. NBER Working Paper No. w28350. Backus, M., Conlon, C., Sinkinson, M., 2021b. Theory and measurement of common ownership. In: AEA Papers and Proceedings. vol. 110, pp. 557–560. Backus, M., Conlon, C., Sinkinson, M., 2021c. Common ownership in America: 1980–2017. Am. Econ. J. Microecon. 13 (3), 273–308. Banal-Estañol, A., Newham, M., Seldeslachts, J., 2021. Common ownership in the U.S. pharmaceutical industry: A network analysis. Antitrust Bull. 66 (1), 68–99. Bernard, H.R., 2012. Social Research Methods: Qualitative and Quantitative Approaches. Sage. Boller, L., Scott Morton, F., 2020. Testing the Theory of Common Stock Ownership. NBER Working Paper No. w27515. Bonaccorsi, A., Giuri, P., 2001. Network structure and industrial dynamics. The long-term evolution of the aircraft-engine industry. Struct. Change Econ. Dyn. 12 (2), 201–233. Borgatti, S.P., 2012. Social network analysis, two-mode concepts. In: Computational Complexity. Springer, New York, pp. 2912–2924. Borgatti, S.P., Everett, M.G., Johnson, J.C., 2018. Analyzing Social Networks. Sage. Borgatti, S.P., Mehra, A., Brass, D.J., Labianca, G., 2009. Network analysis in the social sciences. Science 323, 892–895. Boyd, S., Vandenberghe, L., 2018. Introduction to Applied Linear Algebra - Vectors, Matrices, and Least Squares. Cambridge University Press. Bureau van Dijk, 2023a. Orbis Brochure, Bureau van Dijk. Available from https://www.bvdinfo.com/en-gb/our-products/data/international/orbis. Bureau van Dijk, 2023b. Bureau van Dijk historical ownership databases - User Guide - 2022 Q4, Bureau van Dijk. Available with the purchase of the Orbis Ownership database. Clapp, J., 2019. The rise of financial investment and common ownership in global agrifood firms. Rev. Int. Political Econ. 26 (4), 604–629. Dalton, H., 1920. The measurement of the inequality of incomes. Econ. J. 30 (119), 348–361. Demsetz, H., Villalonga, B., 2001. Ownership structure and corporate performance. J. Corp. Finance 7 (3), 209–233. Dennis, P., Gerardi, K., Schenone, C., 2022. Common ownership does not have anti-competitive effects in the airline industry. J. Finance 77 (5), 2765–2798. Egghe, L., Rousseau, R., 2003. A measure for the cohesion of weighted networks. J. Assoc. Inf. Sci. Technol. 54 (3), 193–202. Fichtner, J., Heemskerk, E.M., Garcia-Bernardo, J., 2017. Hidden power of the Big Three? Passive index funds, re-concentration of corporate ownership, and new financial risk. Bus. Politics 19 (2), 298–326. Gilje, E., Gormley, T.A., Levit, D., 2020. Who’s paying attention? Measuring common ownership and its impact on managerial incentives. J. Financ. Econ. 137 (1), 152–178. He, J., Huang, J., 2017. Product market competition in a world of cross-ownership: Evidence from institutional blockholdings. Rev. Financ. Stud. 30 (8), 2674–2718. Hurley, N., Rickard, S., 2009. Comparing measures of sparsity. IEEE Trans. Inform. Theory 55 (10), 4723–4741. ICGN, 2018. Common ownership: do institutional investors really promote anti-competitive behaviour? In: ICGN Viewpoint, 2018. International Corporate Governance Network. Karvanen, J., Cichocki, A., 2003. Measuring sparseness of noisy signals. In: 4th International Symposium on Independent Component Analysis and Blind Signal Separation. pp. 125–130. Kennedy, P., O’Brien, D.P., Song, M., Waehrer, K., 2017. The Competitive Effects of Common Ownership: Economic Foundations and Empirical Evidence, Manuscript. Available at SSRN 3008331. König, M.D., Battiston, S., 2009. From graph theory to models of economic networks. A tutorial. In: Networks, Topology and Dynamics. Springer, Berlin, Heidelberg, pp. 23–63. Latapy, M., Magnien, C., Del Vecchio, N., 2008. Basic notions for the analysis of large two-mode networks. Soc. Netw. 30 (1), 31–48. Newham, M., Seldeslachts, J., Banal-Estañol, A., 2018. Common Ownership and Market Entry: Evidence from the Pharmaceutical Industry. Department of Economics, Discussion Paper Series, City, University of London, No. 18/03. Newman, M.E., 2003. The structure and function of complex networks. SIAM Rev. 45 (2), 167–256. O’Brien, D.P., Waehrer, K., 2017. The competitive effects of common ownership: We know less than we think. Antitrust Law J. 81 (3), 729–776. Opsahl, T., Agneessens, F., Skvoretz, J., 2010. Node centrality in weighted networks: Generalizing degree and shortest paths. Soc. Netw. 32 (3), 245–251. Pastor, G., Mora-Jiménez, I., Jäntti, R., Caamano, A.J., 2013. Sparsity-based criteria for entropy measures, wireless communication systems (ISWCS 2013). In: Proceedings of the Tenth International Symposium. pp. 1–5. Pastor, G., Mora-Jiménez, I., Jäntti, R., Caamano, A.J., 2015. Mathematics of sparsity and entropy: Axioms core functions and sparse recovery. Manuscript, arXiv preprint arXiv:1501.05126. Quéré, R., Frélicot, C., 2012. A new index based on sparsity measures for comparing fuzzy partitions. In: Joint IAPR International Workshops on Statistical Techniques in Pattern Recognition (SPR) and Structural and Syntactic Pattern Recognition. SSPR, Springer, Berlin, Heidelberg, pp. 291–300. Rickard, S., Fallon, M., 2004. The gini index of speech. In: Proceedings of the 38th Conference on Information Science and Systems. CISS’04. Robins, G., Alexander, M., 2004. Small worlds among interlocking directors: Network structure and distance in bipartite graphs. Comput. Math. Organ. Theory 10 (1), 69–94. Rosati, N., Bomprezzi, P., Ferraresi, M., Frigo, A., 2022a. Firm ownership and competitiveness in the European beverages industry. Manuscript. Available at SSRN 4046403. Rosati, N., Bomprezzi, P., Ferraresi, M., Frigo, A., Nardo, M., 2020. Common Shareholding in Europe, EUR 30312 EN. Publications Office of the European Union, Luxembourg, ISBN: 978-92-76-20876-1, http://dx.doi.org/10.2760/734264, JRC121476. Rosati, N., Bomprezzi, P., Martinez Cillero, M., 2022b. Institutional investors and common ownership in the European energy sector. Manuscript. Available at SSRN 4046563. Schmalz, M.C., 2018. Common ownership, concentration and corporate conduct. Annu. Rev. Finan. Econ. 10, 413–448. Schmalz, M.C., 2021. Recent studies on common ownership, firm behavior, and market outcomes. Antitrust Bull. 66 (1), 12–38. Scott, J., 2017. Social Network Analysis. Sage. Seldeslachts, J., Newham, M., Banal-Estañol, A., 2017. Changes in common ownership of German companies. DIW Econ. Bull. 7 (30), 303–311. Torshizi, M., Clapp, J., 2021. Price effects of common ownership in the seed sector. Antitrust Bull. 66 (1), 39–67. Tzanaki, A., 2022. Varieties and mechanisms of common ownership: A calibration exercise for competition policy. J. Compet. Law Econ. 18 (1), 168–254. Xie, J., 2021. Horizontal shareholdings and paragraph IV generic entry in the US pharmaceutical industry. Antitrust Bull. 66 (1), 100–112. Zhang, X.D., 2017. Matrix Analysis and Applications. Cambridge University Press. Zonoobi, D., Kassim, A.A., Venkatesh, Y.V., 2011. Gini index as sparsity measure for signal reconstruction from compressive samples. IEEE J. Sel. Top. Sign. Proces. 5 (5), 927–932.