I have a specific problem I have been set, I'm asking here because I can't really find an answer anywhere else.

Consider the scenario where a company offers some service to its users. The company has an enterprise value which is a function of the number, $n$, of users it has, $f(n)$.

When people use the service provided by the company, each user $i$ gets a non-negative utility $u_i \geq 0$.

Let $N$ denote the population of potential users, and $F$ denote the company.

We can view this as a cooperative game, $(N \cup \{F\}, v)$ played by the company and its potential users.

The value of a coalition formed is the sum of the enterprise value of the company and the utilities of its users, if the company is in the coalition. If the company is not included in the coalition, its value is $0$.


for any $C \subseteq N \cup \{F\}$, we have

$$v(C) = \begin{cases} f(|C| - 1) + \sum_{i \in C \cap N} u_i & \text{if $F \in C$} \\ 0 & \text{otherwise}\end{cases}$$

We are given that the game is superadditive, so we assume the grand coalition forms. We are asked to compute the Shapley value for the company $F$ and each user $i \in N$.

The only thing I am sure of is that, due to the efficiency property of Shapley values, the sum of values for all users and the company must be $v(N \cup \{F\}) = f(|N|) + \sum_{i \in N} u_i$. After attempting the problem, I ended up with a large equation which I don't believe to be the answer that they were looking for.

Is there a simple way to compute these values, possibly based on some of the properties of the Shapley value? Or does it have to be a long convoluted equation?


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