# Coalitional games and Shapley value used for assessing explanatory variable contribution

I have a problem that at the first sight seems not as coalitional game, but rather could be described as logistic regression with all dichotomous variables: 1 response variable Y (I would call it later as a feature/violet star) and 5 explanatory variables A, B, C, D and E (also binary).

I'm trying to infer which explanatory variables contribute the most to Y and preferably somehow assess quantitatively their severity (something like ranking predictors, variable selection, etc.) and I find strong similarities with Shapley value.

It's very similar to coalitional games, because of it's strong cooperative nature, but I'm not sure whether I can use it in this particular case. The main problem with this approach is stated at the very bottom in bold.

Could you have a look on the situation below and recommend some methods how using Shapley value to infer which explanatory and to what extent contribute the most to response Y (feature/violet star).

Let's say, we have five different objects denoted as labelled balls: A, B, C, D, E creating some subsets and the single special feature of interest marked as violet star. If the star is filled, the feature is present in a specific set, whereas hollow star denotes its absence. Any subset of these five objects (there're $2^5 - 1$ possible types of such nonempty subsets) can appear many times, sometimes with the feature being present and sometimes without it, as depicted below, where there're six instances of the full set and $\frac{2}{3}$ of them have the feature.

We can assume (if it's needed) there's always at least one instance of the full set with the feature present (for example the one marked with thicker red envelope).

For every of the $2^5 - 1$ possible nonempty subsets we have three values, e.g. for (A, C, E):

• $F_{ACE}$ - number of all occurrences of subset (A, C, E) \textbf{with} the feature.
• $N_{ACE}$ - number of all occurrences of subset (A, C, E) \textbf{without} the feature.
• $T_{ACE} = F_{ACE} + N_{ACE}$ - simply total number of all appearances of subset (A, C, E).

The cardinalities of instances $T_X$ of every subset type $X$ can be different, but there's a very obvious and rather trivial rule that if $X \subset Y$ then $T_X \geq T_Y$. For example, $(A, C, E) \subset (A, B, C, D, E)$ thus $T_{ACE} \geq T_{ABCDE}$. That's because any instance of subset (A, B, C, D, E) is at the same time an instance of (A, C, E), but for narrower (A, C, E) we can have more instances.

In this description I use six instances for each subset type just bacause of limited space for pictures and those six instances are intended to reflect the overall proportion between instances having and lacking the feature.

Among all these $2^5 - 1$ types, some of them are more likeli to have the feature (for this particular subset/type, there're more instances with the feature than without it), whereas other types are more prone to lack it. Please look at two figures below and the following numbers:  $$\frac{F_{ACE}}{T_{ACE}} = \frac{5}{6}, ~~ \frac{F_{BD}}{T_{BD}} = \frac{1}{6}, ~~ \frac{F_{ABCDE}}{T_{ABCDE}} = \frac{4}{6} ~\text{(the situation of the very first picture)}$$ A blatantly obvious property is that in my model the ratio $\frac{F}{T}$ is not monotonic (in neither direction) with respect to inclusion relationship.

And finally the most crucial property I would like to make my model had, is some kind of cooperative interaction between objects in subset, which is the last thing I'm going to describe beneath. A moment before, we've spotted that the subset (A, C, D) is exceptionally favorable for the feature appearance, so we would pay a special attention to its constituent singletons and their corresponding ratios: $$\frac{F_A}{T_A} = \frac{2}{6} ~~ \text{(2nd & 4th)}, ~~ \frac{F_C}{T_C} = \frac{2}{6} ~~ \text{(2nd & 3rd)}, ~~ \frac{F_E}{T_E} = \frac{1}{6} ~~ \text{(2nd only)}$$   Even now, we can clearly see a sort of cooperative, since those three singletons (A), (C) and (E) all together cover only three subsets {2nd, 3rd, 4th} = {2nd, 4th} $\cup$ {2nd, 3rd} $\cup$ {2nd}, whereas subset (A, C, E) triggers off the feature in most subsets except the 1st one.

Moreover, merging A with E or C with E gives none additional coalitional effect, so E itself contributes poorly to overall effect.

However, merging A with C gives a significant interaction resulting with the feature apperance in more instances than it follows from simply summing instances: $\{2nd, 3rd, 4th, 6th\}$ vs. $\{2nd, 3rd, 4th\}$. The occurrence in the 6th instance is this coalitional effect. Nonetheless, the E is necessary to trigger off the full potential and reveal the feature also in the 5th instance, as depicted in the second figure.

To sum up, I would like to infer from the data which single object are favourable for the feature appearance. In our example we can clearly see that A and C are very conducive (in cooperation with other), E is not so useful, but required to reach the full potential, whereas B and D seem to have no cooperative effect on the feature.

The main problem with incorporating the Shapley value: I would like to use ratio $\frac{F_X}{T_X}$ as characteristic function $v(X)$ for the particular subset $X$ ($v:2^{5} \mapsto \mathbb{R}$), but this ratio isn't monotonic with the respect to inclusion relation and the total coalition $(A, B, C, D, E)$ would get smaller worth than narrower $(A,C,E)$ (compare 1st and 2nd figure). I don't think that B or C should bring negative contribution, rather neutral.

Do you see how can I infere such contribution?