An Interactive Shapley Value Explainer

Thanks Austin, yes—the weeks I've spent trying to really understand why Shapley uses such a complicated method to calculate the possible coalitions, has left me feeling that it is actually prohibitively cumbersome for most applications. It has been popular in machine learning algorithms, but faces the problem that it is computationally expensive.

I created a comparison calculator to show Shapley next to my own method that simply weights by dividing all the explicit marginal values by the total of all the explicit marginal values and multiplying that by the grand coalition value. I found that, for realistic values being entered, it yields very similar results to Shapley, and yet is easy to calculate on a spare napkin. It also satisfies Shapley's 4 axioms, and seems more intuitive to me at least.

There might be an issue with mine that you need the total of all marginal values (which is involved in the weighting) to find any one weighted value, whereas Shapley can be used to calculate each weighted marginal value in isolation.

Regardless... who am I to argue with a Nobel Prize winning economist? But I can't be accused of not trying to get on board :) I like the look of Quadratic Funding, perhaps for a future post.

Adding such virtual players is an interesting idea. I guess it should be possible to properly add probability weighting, without virtual players, once one has fully understood the reason for Shapley's permutation method of weighting. (Which I have not.)

Another thing I noticed: It seems one can generalize a "coalition" of "players" as a conjunction of propositions, where the propositions of the players not participating in the coalition are negated. So for players/propositions A, B, C, the coalition {A, C} would correspond to the proposition $A\wedge \neg B\wedge C$. The Shapley value of a player would simply be the value of a proposition, like A. Since coalitions are just propositions as well, this has the (intuitive?) consequence that coalitions can also be players.

Furthermore, instead of talking about "values" one could talk about "utility", as in expected utility theory. There is actually a formalization of utility theory, by Richard Jeffrey, which applies to a Boolean algebra of propositions (or "events", as they are called in probability theory). So theoretically it should be possible to determine which restrictions placed on a Jeffrey utility function are equivalent to Shapley's formula. Which could, presumably, aid understanding the assumptions behind it, and help with things like integrating probabilities into the Shapley formalism, since those are already integrated into Jeffrey's theory.

Of course this would probably be a lot of work...