Shapley values for channel attribution equal to linear attribution

I am looking into Shapley values for online marketing attribution. In recent time many articles seem to have been made on this particular approach to attribution (there are more):

• 1. https://medium.com/analytics-vidhya/the-shapley-value-approach-to-multi-touch-attribution-marketing-model-e345b35f3359

And i.e.: https://blog.dataiku.com/step-up-your-marketing-attribution-with-game-theory

It seems that, at least in certain cases, the result will be identical to linear attribution, so I am trying to get more information regarding whether this is to be expected / correct.

The problem:

The Shapley value approach for online marketing attribution in these articles seems to be as follows:

This runs into issues in certain cases though, due to a large number of possible coalitions - which is why it was reformulated for online marketing specifically in 2. https://arxiv.org/ftp/arxiv/papers/1804/1804.05327.pdf

At link 2, on page 7, the following example is shown:

The example in formula 13 in this case seems to be a formulation which is identical to linear marketing attribution (where all touchpoints get equal credit). A quick test seems to show the same result:

1. Fake data with 3 channels:

2. I am using the Shapley value functions from link 1.

3. My implementation of the simplification from link 2 (I am not showing linear attribution also, as the implementation would be the same - right?)

So I guess my questions boil down to:

• is Shapley value attribution really identical to linear attribution in this case? If it is different, which input will make the output between the methods differ? If not, why the focus on Shapley values instead of the simpler method?
• Is something wrong with these two definitions/ implementations of Shapley values causing the results to be identical to linear attribution?