Accept my apology in advance if my question sounds stupid as i am early phase of exploration. Can someone give answer or point out the literature that gives answer of my two questions related to Shapley value:
Q1: Does Shapley value also works for the non-linear nature of utility within coalition's member? Q2: What is the maximum possible number of agents can Shapley value compute?
Thanks in advance
On
To get a quick overview I recommend the following Wikipedia article
http://en.wikipedia.org/wiki/Double_precision
from there you can find some additional literature. Have also a look in the book by Won Y. Yang, Applied Numerical Methods Using Matlab, Wiley (2004). However, if my calculation is correct, you need for a 51-player game at least 16384 TB physical memory. I don't know if a present computer system has so much memory. Moreover with a quadruple-precision floating-point format (128-bit), you can extend this limit up to 111, but this imposes on a 64-bit system some extra costs on the computational speed. For more information related to this topic you should consult the documentation of the FORTRAN language.
Probably, my game theory Matlab toolbox might be also of interest for you, which can be found here
http://www.mathworks.com/matlabcentral/fileexchange/35933-mattugames
See in this respect the manual. The main functions are tested up to 35 players. The computation time of the Shapley value for a 21-person game is in average about 4 seconds in serial mode. Larger games should run in parallel mode.
To answer both questions at once: the Shapley value is well-defined for each agent in a standard co-operative game, regardless of the number of agents and the possible non-linearity of the utility values.