I find Combinatorial Game Theory very interesting as my primary interest is mathematics. My question is why do Computer Scientists (who tend to have a more practical approach) study it as well? Are there some applications of it? This question is similar to this one but I am asking about combinatorial games such as nim. Could someone provide me with a reference?
I think Kaveh's comment is the correct answer: applications? We don't need no applications.
But despite all that, combinatorial game theory does appear to have some applications in error correcting codes. See Conway and Sloane, "Lexicographic codes: Error-correcting codes from game theory", IEEE Trans. Inf. Th. 1986.
More simply, if you are willing to think of nim-addition (bitwise exclusive or) as something that came out of combinatorial game theory, then there are plenty of applications, for instance in tabulation hashing, or as one of the components of Schieber and Vishkin's lowest common ancestors data structure, or as one of the components of some block cipher systems.
I was able to find an application of combinatorial game theory in cryptography. See the link here
Combinatorial game theory and topics thereof has received a lot of attention in modelling and verification of reactive systems. I would draw your attention to this nice reference for more information.