In an earlier blog post: Different Approaches to Control, we looked at the T-rank Power Index (Banzhaf type), the Threshold Method, and Network Control. In this blog post we are going to look at some newly discovered alternative measures for control, and see an example of how they compare to the T-rank Power Index.
We have reviewed the literature and checked on new and recent developments within the area of Weighted Voting Games. There are two developments in the past five years that we want to highlight and review, and also compare the T-rank Power Index up against: Firstly, the Helmholtz-Hodge decomposition (HHD) method found in the article: “Stock Ownership Structure in Japan”, 2021. Secondly, there is the Network Power Flow method published in “The flow of corporate control in the global ownership network”, 2023.
The Helmholtz-Hodge Decomposition
In order to get off to a good start, we need to take a deep breath: there is a very, very fascinating result in mathematics and differential geometry called the Helmholtz-Hodge decomposition! In order to understand what this gives us, we need to comprehend what a “vector field” is. A good analogy is wind, and imagine wind blowing: In order to describe the blowing wind you’ll need to know: i) speed of the wind, and ii) direction of the wind. Voila! And there you’ve got your vector field!
So, what is it that the HHD result has been hiding from us? The result claims that any “vector field” can be separated into two parts: i) a purely directional part originating from a scalar potential, and ii) a purely circular part. Adding these two parts together (which each is a vector field on their own) will then restore the original “vector field“.
So, what does this have to do with ownership networks, you ask? It might take some convincing, as is often the case with mathematical concepts, but the key lies in understanding that ownership networks can be viewed as a discrete “vector field”. Recall how we quantified blowing wind: “speed” of the wind, together with “direction” of the wind. Analogous, for an ownership network, we have: i) “amount” of ownership, and ii) in that “specific company”! This allows us then to apply the Helmholtz-Hodge decomposition to this ownership “field”!
The decomposition of the ownership network will produce two parts, and the part that we want to compare against below is the generated potential score for each node in the ownership network. This potential can be used in identifying ultimate owners and also to rank the nodes, where higher potential is considered better.
The Network Power Flow
Network Power Flow (NPF) is a method that quantifies the influence of all shareholders, including intermediate ones, within a global corporate ownership network. It builds upon the Network Power Index (NPI), a prior method that focused exclusively on measuring the power of ultimate owners to control companies based on their voting rights and coalition-building capabilities. NPF extends NPI by allocating power to all intermediate shareholders in the ownership network.
The NPF methodology is based on the Stanley-Shubik method, which calculates a player’s power as the probability of being the pivotal voter across all possible voting permutations. This is different from the Banzhaf index by the fact that in the Stanley-Shubik methodology the order of the voting is essential, and being pivotal is defined accordingly. For the Banzhaf index the order of the voting is not essential in defining winning coalitions. Check out Different Approaches to Control, for more details on how to calculate the Banzhaf index.
A Detailed Example
In the ‘Green-Sipes’ example above, we can read out the T-rank Power Index scores and Integrated Ownership from the figure. Why does ‘Kailyn Kling’ end up with such a high T-rank score? Well, the thinking goes like this: ‘Kailyn Kling’ will gain control in ‘Beahan, Schiller and Reichert’ through coordinated voting with either ‘Velma Koepp’ or ‘Kiley Bins’. Given this control in ‘Beahan, Schiller and Reichert’, adding the direct votes of ‘Kling’ renders control in ‘Green-Slipes’. If ‘*Kailyn Kling’ coordinates her voting with ‘Orie Boyer’ this will also give control in ‘Green-Sipes’. ‘Kailyn Kling’ only needs to agree with one of the other three owners, while on the other hand, all three other owners needs to coordinate their voting to go against ‘Kailyn Kling’.
In the below table we have summarized the scores generated by the different methods: T-rank Power Index, Helmholtz-Hodge Decomposition, and Network Power Flow/Index. In addition, we have included; ‘Ownership’ – the integrated ownership share; and the ‘Threshold method’ based on the quota 50%.
| Boyer | Bins | Koepp | Kling | Reichert | Green-Sipes | |
| Ownership | 49.5 | 23.76 | 24.255 | 2.485 | 49.5 | 0 |
| T-rank Power Index | 25 | 25 | 25 | 75 | 50 | 0 |
| Threshold (50) | 0 | 0 | 0 | 0 | 0 | 0 |
| HHD Potential | 12.997 | 25.8 | 25.8 | 22.599 | 12.803 | 0 |
| Network Power Index (NPI) | 20.442 | 24.175 | 23.978 | 31.405 | 0 | 0 |
| Network Power Flow (NPF) | 14.628 | 17.299 | 17.159 | 22.473 | 16.513 | 11.927 |
We make the following observations from the table:
- The Threshold method with quota of 50% comes across as meaningless, and a valid question is whether this is at all useful even with lowering the threshold?
- T-rank Power Index: The T-rank Power Index assigns an equal 25% power to all three owners with the higher ownership share. ‘Kailyn Kling’ is assigned 75% despite its significantly lower ownership. This suggests that even small shareholders can have substantial influence in a voting game context.
- HHD Potential: The HHD Potential displays higher correlation with NPI/NPF than with T-rank Power Index or Ownership. There is no clear understanding of what the potential value is meant to be in terms of interpretation. Along with the decomposition itself there is the risk that this comes across as a different version to the original ownership network (recall that the potential is obtained from on half of the decomposition) . This all obfuscates the measure in terms of direct interpretation.
- Network Power Flow/Index: As mentioned above, the NPI index suffers from focusing only on the ultimate owners and does not assign values to the intermediate owners–in this case; ‘Beahan, Schiller and Reichert‘. The Network Power Flow fills in missing NPI values, but it is a bit confusing why ‘Green-Sipes’ is assigned a score and how to interpret this. NPF has a fairly high correlation with the T-rank Power Index, which is expected given that they are both based on a similar mechanism at core. However, they do not come across as equal.
Summing Up
The T-rank Power Index offers a unique and granular perspective on the phenomena it quantifies. Its inherent properties are not fully captured by simpler metrics, providing a more profound understanding due to its distinct construction.
While other measures might offer a superficial overview, they would inevitably miss the subtle yet crucial distinctions captured by the T-rank Power Index. This highlights the importance of choosing the right tool for the job, even if that tool is more or less sophisticated than its alternatives.
The value of a measurement lies not just in its ease of interpretation, but in its ability to accurately and comprehensively reflect the reality it seeks to explain.
Previous: Different Approaches to Control
Written by Kenth Engø-Monsen – kenth (at) trank (dot) no