How much Shannon entropy is there in people voting for Trump? What I mean is: a voter choosing Trump or Clinton had a simple binary choice (I'm simplifying by 'forgetting' other candidates). Yet, would it not be unfair to state that all voters' votes contained the same, small, amount of entropy?
After all, much information has gone into forming the final choice for each of them. 
ALthough Shannon's theorem may not (from my understanding) take this into account, is there a way to measure this? 
An extension of these could be: voters pick delegates, who pick the president, who picks the UN ambassador (or whatever) who makes several yes/no binary decisions.
Yet, would it not be unfair, from the point of view of how much information it contains, to measure each step as containing equal amount of information. Is there a way to discriminate these?
 A: Based on what I know about Shannon entropy, it is not really a calculatable quantity in the situation you have described.  I suppose it could be calculated in principle, but certainly not in practice.
The question you have posited does involve a sequence of "choices", and the Shannon entropy of a series of "choices" (or bits) is generally a thing that can be calculated.  However, in order to calculate the Shannon entropy of a series of choices, you have to know the probabilities of making these choices.
It's a bit of a stretch to compare binary "choices" with bit strings, but I think I see the overall gist of your question.  To explain a little further, we can calculate the Shannon entropy of a bit string such as:

1011 0011 0111 1001 1000 0010 1000

provided that we know the probabilities of 1s and 0s appearing in each position.
However, in the situation you have described, we don't know the probabilities of the binary choices that are being made.  I suppose we could make wild conjecture about "what is the probability of a delegate choosing such-and-such", but any Shannon entropy you calculate would only be as good as your wild conjecture.  
I've heard of "Econophysics" - a branch of physics that tries to use physics principles to predict financial markets.  However, this is a rather awesome attempt (and the first I've seen) at "Political Physics".  Awesome!
