Calculate the Shapley value of the Airport Game in the case of six pilots whose aircraft, all different, require progressively longer runaways costing 40, 50, 60, 70, 80, 90 units respectively
$v(I) = 40, v(II) = 50, v(III) = 60, v(IV) = 70, v(V)=80, v(VI)=90$
We need to define the cost vector $c = (40, 50, 60, 70, 80, 90)$ through $$ C(S) = \max_{i\in S}{c_i}, \forall S \subseteq N $$ possible games are (I), (II), (III), (IV), (V), (VI), (I, II), (I, III) and so on. There will be $2^N -1$ games. In this case, there are 63 games,
then the associated cost game is $$ cv = (40, 50, 60, 50, 60, 70, 80, 90, ...) $$ 1 40's, 2 50s, 4 60's, 8 70's, 16 80's, 32 90's.
I am having trouble determine what the Shapley value is? How would I derive the Shapley value?
For airport games, there is a particular characterization of the Shapley value that can be found in the following article.
Shapley value for airport games
By this article one should be able to compute the Shapley value of the associated cost game, which is given by
$sh_{cv}=(40,52,67,87,117,177)/6.$
Alternatively, transferring the cost game into a savings game, and the Shapley value is given by
$sh_{av}=(200,248,293,333,363,363)/6.$
Hope this helps.