Without reading below, I'm sure others have already argued this point:
I can think of a few situations where a player could chose the lower sum and be much happier. The chance to live and play in a city you love for a team you love near your family and still have a better chance to win? If the player is already rich for whatever reason (comes from money, previous contract, etc.), and $180MM won't make the man? I wouldn't call that man an idiot for choosing $20MM.