I'm currently reading about how to value an american option and I have a few questions about it. Would be very grateful if anyone can spare the time and answer them.
$1)$ Since an american put and a european put both have the same payoff and they have identical cash flow, don't they have the same value at any point by the Law of One Price?
$2)$ The source I'm using says that you need the condition of "holder chooses the early exercise strategy in order to maximize the option’s value" in order to price it. What does this condition mean? From what I've read is it that once you reach some "optimal" value, you the exercise the option if the value decreases? If yes, then how is this "optimal" value determined?