The three wise men problem

I suppose I went the wrong way from the outset. I should have started with the original puzzle and gone backwards.

The original:
Three wise men are summoned by the king. He says he wants to test their wisdom and the test will be equally fair for all three men.
The king tells them each will be wearing a hat which is either red or white, and they won't be able to see their own hat.
Then he tells the three that at least one of the hats worn will be a red hat.
The first wise man to say he knows the color of the hat he's wearing, wins the chocolate fish (sayeth their ruler).

So the king tells them several things. I tried it without the "at least one" information.
In this case the wise man with his eyes closed reasons he will have to open them to see the color of the other two hats, which must be the same if the test is "equally" fair.

What if the king also leaves out the part about the chocolate fish? He doesn't really need to tell three wise men they will win some kind of prize, a royal title perhaps? Or that the first to figure out their hat color can say so? etc.
Suppose the bare bones version excludes the king saying anything, each man has to figure out why they've been summoned, the nature of the test, the whole enchilada?

How wise do you need to be?
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