Suppose I have a game in which $n$ players which can form coalitions of size $k < n$. I then have a characteristic function $v$ which assigns a worth to each of the $n^{k}$ possible $k$-player combinations. Assume further that the order of the players matters, e.g. the coalition {1,2,3} may not have the same worth as {2,1,3}.
Is it possible to calculate Shapley values for each player in this problem?
Any help would be appreciated!
Edit:
One approach I wondered is the following: Denote by $N = \lbrace 1, \cdots , n \rbrace$ the set of players and K the set of size $k$ coalitions made up from players in N in which order matters i.e $|K| = n^k$. Let $v:K \rightarrow \mathbb{R}$ be the characteristic function.
For $S \subseteq K$, let $S_m$ denote the player in position m. Additionally, denote by $S[S_m = k]$ the original coalition S with the $m$-th player swapped for player $k \in N$.
Define the value assigned to player $i$ as
$$ \phi_{i}(v) = \frac{1}{C} \sum\limits_{m=1}^{3} \quad \sum\limits_{S \subseteq K \text{ : } S_m \quad \neq i} v(S[S_m = i]) - v(S) $$ where $C$ is some appropriate normalization. Im not sure how to prove it satisfies the axioms or what C should be but intuitively I am averaging the effects of swapping players in a coalition with the player of interest. Could this be a fruitful avenue?