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Why are American-style options worth more than European-style options?
An option is an instrument that gives you the "right" (but not the obligation) to do something (if you are long). An American option gives you more "rights" (to exercise on more days) than a European option. The more "rights," the greater the (theoretical) value of the option, all other things being equal, of course. That's just how options work.  You could point to an ex post result, and and say that's not the case. But it is true ex ante.