I am facing problem to compute the ShapleyValues. Suppose there are 5 people in a city. First person has 10 dollars, 2nd person has 6 dollars, 3rd person has 11 dollars, 4th person has 12 dollars and 5th person has 1 dollar. To make them co-operative how many dollars to each of them means what will be their ShapleyValues.
Can anyone please compute their ShapleyValue by using ShapleyValues formula and show the steps of that calculations.
Thanks in advance.
The Shapely values determine the average marginal contributions across all coalitions. A coalition is to encourage players to combine resources in order to create a larger resource. A contemporary example of this is the Euro.
For example if player $A$ is worth $A_v$ and player $B$ is worth $B_v$, then their combined worth would be $A_v+B_v$. To encourage coalitions, a function $v(s):2^{|S|}\to\mathbb{R}$ is created to represent the worth of the coalition. So we might write $v(\{A,B\})=A_v+B_v$ but to make the coalition more worthwhile, we might define $v(\{A,B\})=A_v+B_v+A_vB_v$ or other similar ideas.
If $N$ is the set of players, then $S=\mathcal{P}(N)$ is the set of coalitions, and $v_S(s)=W_s$ is the worth function with $s\in S$. The Shapely values aim to help a player decide which coalition they should join.
As there are $2^{|S|}$ possibilities, I have trimmed your example to $3$ players. Note the worth function can be completely arbitary - it doesn't have to be 'realistic'.
$N$ is players $A$, $B$ and $C$ with $v(A)=10$, $v(B)=11$ and $v(C)=12$. The remaining values of $v$ are in the table.
First we list the coalitions and the corresponding values of the worth function:
$$\begin{array} {c|c|c|c|c} \text{coalition}&\text{worth}&A&B&C\\ \hline \emptyset&0\\ A&10\\ B&11\\ C&12\\ AB&24\\ AC&27\\ BC&30\\ ABC&40 \end{array}$$
Next we calculate the marginal contributions. These are, for example with player $A$, the contributions they make to the coalitions $A$,$AB$,$AC$ and $ABC$. We can start off easily enough as we know the marginal contribution each player makes to the coalition consisting of only them, $A$ makes to $A$ for example. To calculate marginal contributions for larger coalitions, we remove the contribution made by the coalition WITHOUT the player in question in it, for example the marginal contribution of $A$ to $ABC$ is $v(ABC)-v(BC)$.
$$\begin{array} {c|c|c|c|c} \text{coalition}&\text{worth}&A&B&C\\ \hline \emptyset&0\\ A&10&10\\ B&11&&11\\ C&12&&&12\\ AB&24&13&14\\ AC&27&15&&17\\ BC&30&&18&19\\ ABC&40&10&13&16 \end{array}$$
Now we can calculate the Shapely values. This involves a little bit more work, we take the average of the averages for coalitions of size $k$. In this example $A$ appears twice in coalitions of size $2$, in this case $(13+15)/2=14$. So the Shapely value $\phi_A=(10+14+10)/3=11.33$. So $\phi_B=(11+(14+18)/2+13)/3=13.33$ and $\phi_C=(12+(17+19)/2+16)/3=15.33$.
See BIS for another example.