Decisions are made on the basis of expected value, not probability.

Yes, that's the point here!

your analysis of the first bet ignores the value of the information gained from it in executing your options for further play thereafter.

By "the first bet" I take it that you mean "your first opportunity to put a coin in a green box" (rather than meaning "brown box").

My analysis of that was "you should put some coins in the box", exactly because of the information gain.

This statement indicates a lack of understanding of Jaynes, or at least an adherence to his foundations.

This post was based closely on the Chapter 18 of Jaynes' book, where he writes:

Suppose you have a penny and you are allowed to examine it carefully, and convince yourself that it is an honest coin; i.e. accurately round, with head and tail, and a center of gravity where it ought to be. Then you’re asked to assign a probability that this coin will come up heads on the first toss. I’m sure you’ll say 1/2. Now, suppose you are asked to assign a probability to the proposition that there was once life on Mars. Well, I don’t know what your opinion is there, but on the basis of all the things that I have read on the subject, I would again say about 1/2 for the probability. But, even though I have assigned the same ‘external’ probabilities to them, I have a very different ‘internal’ state of knowledge about those propositions.

Do you think he's saying something different from me here?