Head Up Theory Question

[M.B.E.]

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Ok, I have read the thread and understand why something around 20 is much more likely to be right than the obvious 499,998 or so.

However, what I don't understand is whether the "correct" answer assumes that our opponent never overvalues his hand, or whether that doesn't matter.
...
Are we to amend the original question to assume an opponent who won't overvalue his hand and plays correctly?

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Original question said we are playing against an expert. I take that to mean there are no exploitable weaknesses in his/her strategy so we are trying to calculate the game-theoretically optimal strategy.