The best (IMHO) derivation of the $\sum p \log p$ formula from basic postulates is the one given originally by Shannon:

Shannon (1948) A Mathematical Theory of Communication. Bell System Technical Journal. http://cm.bell-labs.com/cm/ms/what/shannonday/shannon1948.pdf

However, Shannon was concerned not with physics but with telegraphy, so his proof appears in the context of information transmission rather than statistical mechanics.  To see the relevance of Shannon's work to physics, the best references are papers by Edwin Jaynes. He wrote dozens of papers on the subject.  My favorite is the admittedly rather long

Jaynes, E. T., 1979, `Where do we Stand on Maximum Entropy?' in The Maximum Entropy Formalism, R. D. Levine and M. Tribus (eds.), M. I. T. Press, Cambridge, MA, p. 15; http://bayes.wustl.edu/etj/articles/stand.on.entropy.pdf

How do you prove $S=-\sum p\ln p$?
Just curious for games going forward. 

Because the expansion allows you to see how people have played on missions, can reveal people's cards, etc, it seems like these are all an advantage mostly to the resistance. 

In people's experience - does it make it easier for the resistance?
