What is Control?
The term ‘control’ is multifaceted, and to truly pin down the meaning of ‘control’ we need to understand its context. This means that control is used and interpreted differently depending on context, and there is not one, all-encompassing meaning. For instance, ‘control’ might signify a degree of influence over a company or, conversely, the authority to dictate outcomes.
In the EU Regulation 2024/1624 (31 May 2024) on the prevention of the use of the financial system for the purposes of money laundering or terrorist financing, the following definition of control is found in Article 53(2)(a):
‘control of the legal entity’ means the possibility to exercise, directly or indirectly,
significant influence and impose relevant decisions within the legal entity;
The challenge, then, is that the definition above is just a set of words. The definition has to be converted into a mathematical model that operationalizes the definition into a measure that can be used, for instance, in daily AML-work. Making a mathematical interpretation can be done in many ways. In the following we will discuss three mathematical models of control that will illustrate different aspects of financial control, and most importantly; how they differ.
We will review the following control definitions:
The Threshold Model
The threshold model of control follows directly from the fact that having more than 50% ownership of a company, gives you the necessary power to dictate most decisions (note; modifications of the articles of association, may require a higher threshold). There are situations where the threshold is set to lower values, for example within AML legislation where the threshold is often set at 25% ownership.
The threshold model is by far the simplest in terms of algorithmic complexity. The main problem of the threshold model is coverage; meaning that its measure is not applicable for all ownership configurations. For thresholds below 50%, there is also a problem that influence is not proportional to the share of the voting rights – as we will see in examples below. Simplicity comes at the cost of analytical capability, and considering the ownership structure in some way or another, is required.
The Banzhaf Power Index
Power indices is a concept from game theory that measures the power of a voter in a weighted voting system. These approaches quantify the “power” of an individual or company where voting rights are not necessarily equal. Among the power indices, Banzhaf power index is probably the most widely used. Instead of simply counting votes, it focuses on how often a voter can “swing” the outcome of a vote by changing their decision. Essentially, it measures the probability that a voter will be critical in a winning coalition and equating this with influence and control.
The algorithm for calculating the Banzhaf power index for a shareholder is simple to state:
The Banzhaf power index can be used to analyze voting power in many different situations, such as shareholder voting, corporate board decisions, and political structures. It can also uncover unfair voting power distribution, even when it appears to be fair. The Banzhaf power index calculation can be extended to any company, regardless of shareholder structure or ownership differences. However, the calculation becomes complex in systems with many voters, cross-ownership, or circular ownership structures. Although the Banzhaf index calculation can be complex, there are solutions to these issues.
Network Control
The concept of network control originates from work on analysis of control within complex networks, particularly in economic and financial systems. The work containing the full algorithm for calculating network control was published in 2011 as The Network of Global Corporate Control and further popularized in the TED talk Who controls the world?
In essence, the definition of network control measures the capacity to influence economic activity through a network of ownership relationships. The focus is on identifying how control is distributed within these networks, and this includes determining which nodes have the most influence and how they can shape the behavior of the entire system. Therefore, rather than a standard dictionary definition of control, the paper provides a methodology to quantify control within a network context.
Loosely speaking, calculating network control resembles mathematically the calculation of indirect ownership, because one quantifies control through direct ownership and their indirect effects. An important preprocessing step of the methodology is the following: ownership data is truncated according to the Threshold model with a predefined threshold of 50%. This means that the network control metric is calculated based on a subset of the full ownership dataset. In quantifying the indirect control effect one introduces a flow analogy that matches the concept of indirect ownership. There are similar challenges in the computations of both network control and indirect ownership, for example through having to deal with the occurrence of cross-ownership and circles in the network. A differing aspect though, is the addition of a type of value measure (operating revenue, total assets, or market capitalization) into the network control calculation. This adds a dimension to the network control definition that can come across as mixing up different concepts in an ad-hoc fashion.
Example 1
This simple example illustrates an inconsistency of the Threshold model for control. Assuming a threshold of 50%, the method will conclude that it is ‘Ebony Purdy’ owning 70% that has control over the company ‘Connelly, Nikolaus and Fritsch’. However, reducing the threshold to 25%, and the model produces the result that both owners have control over the company, and this is inconsistent given that it is ‘Ebony Purdy’ that controls everything and the other owner has no influence at all.
Example 2
Imagine a situation where there are 17 shareholders of a company: one shareholder owns 20%, and the remaining 80% is equally distributed among the 16 shareholders. In this situation the threshold model with a threshold of 25% will not produce any shareholders with control; even less being able to differentiate between the shareholders. On the other hand, the Power index will produce the following result: ‘Adrien Jacobson’, the 20%-owner, has a 66.77% probability of being part of a critical coalition. This is then a disproportionate boost in control, compared to the 20% ownership in the company. For Network control, the preprocessing step renders the data void, and the method is not able to construct any viable result.
Example 3
This is a more complex example. With a threshold of 25%, the Threshold model renders both persons in control and does not differentiate between the two owners. Considering the indirect ownership, it is ‘Mireya Senger’ that has the greatest indirect ownership, more than 13 times that of the other person, ‘Dereck Moen’. However, The Power index nominates ‘Dereck Moen’ to have complete control over the company ‘Trantow-VonRueden’. It is important to realize that there is a distinction between ownership and control over a company, and seemingly small differences in ownership translates into big differences in actual control.
Example 4
This example illustrates the real strength of the Banzhaf power index – since this is the only control model that is able to produce interpretable and good results. Glancing only at ownership, this can come across as being dominated by ‘Velma Koepp’, ‘Kiley Bins’, and ‘Orie Boyer’; and totally disregarding ‘Kailyn Kling’. However, in a voting game ‘Kailyn Kling’ will have a probability of 75% of being part of a critical coalition, and despite its marginal ownership this means that ‘Kailyn Kling’ will be able to influence greatly. Hence, the learning from this example is that if one wants to measure high influence, and not only total control, in principle there exists no threshold able to cover all situations.
Example 5
Be aware of the existence of circular ownership. Circular ownership needs to be handled by the choice of control model, and modifications to the algorithms are expected in order to handle these cases.
Example 6
The flow of control. Setting the threshold to 50%, both the Threshold and Network models will produce no viable values to assess the control over the company ‘Franecki Group’. This example illustrates the fact that control flows downwards (from persons to companies), and that care needs to be taken in order to preserve this. It is not trivial to modify the Banzhaf power index to preserve the flow of control, but it is possible and care must be taken.
Summing Up
The threshold model is straightforward but lacks nuance. It fails to accurately represent control in complex ownership structures with numerous shareholders, frequently leaving control undefined. Its “all-or-nothing” approach overlooks the varying degrees of influence smaller shareholders may wield.
The Banzhaf power index, derived from game theory, offers a more sophisticated measure of voting power in weighted systems. However, its computational complexity increases significantly with larger networks. T-rank has technology that overcomes all these challenges and complexities and can calculate the Power index for any shareholder/company in the World. This contributes to making the Banzhaf power index tractable and applicable in real-world scenarios with intricate ownership structures.
Network control quantifies influence within interconnected ownership networks by considering both direct and indirect control. Though it provides a comprehensive view of economic influence, its calculation mirrors that of indirect ownership calculations, including the challenges of cross-ownership and cyclical relationships. Additionally, it requires the integration of value measures, adding another layer of complexity.
Stay tuned for more fascinating insights related to shareholder networks, control and power!
Previous: The Tangled Web of Ownership
Written by Kenth Engø-Monsen – kenth (at) trank (dot) no