🔗 Erik’s post: theintrinsicperspective.com/p/i-figured-...

🔗 The paper: arxiv.org/abs/2510.02649

I know you like showing pictures of lenses but this seems a little excessive

There's so much more in the paper, largely thanks to Patrick who really pushed this to the next level. We're already working on applications: SVs are often used for XAI, but now we can do this for vector-valued functions--the kind implemented by transformers... Stay tuned!

To summarise:
⬆️Möbius inversions construct higher-order structure.
⬇️Shapley values project this down again, in the 'right' way.

We derive generalisations of both, to directed acyclic multigraphs, and group-valued functions.

This shows how intimately related Shapley values and Möbius inversions are: we derive an expression that expresses Shapley values *purely in terms of the incidence algebra*!

Doing so required also generalising the Möbius inversion theorem to this setting (prev. only defined for ring-valued functions). We show that it's a natural theorem in the *path algebra* of the graph:

But we go further.
Classical Shapley values only work for real-valued functions on power sets of players (or lattices).

We generalise them even beyond posets to
✅vector/group-valued fns
✅weighted directed acyclic multigraphs
, and prove uniqueness!

That’s exactly what we do.
We reinterpret Shapley values as projection operators: a recursive re-attribution of higher-order synergy to lower-order parts.

This turns Shapley values into a general projection framework for hierarchical structure, valid far beyond game theory.

Möbius inversions are a way to derive higher-order interactions ion a system's mereology. I wrote a blog post about this here 👉https://abeljansma.nl/2025/01/28/mereoPhysics.html

If Shapley values are truly general, we should be able to express them for any Möbius inversion/higher-order structure.

But Shapley values (SVs) aren’t just about fairness.
They're really a projection operator: the right way to push higher-order structure back down to lower levels.
So… can we do this more generally? 🤔

Enter Möbius inversions...

If a group of people earn a payoff together, how should it be fairly distributed?
Shapley values are weighted sums of sub-coalition "synergies", and provably the fairest possible distribution.
It earned Shapley the Nobel Prize. 🧮

Hi! sorry I'm not on often on this site. They're now online: www.youtube.com/@dutchinstit...

Yes!

The link doesn’t seem to work for me