So the simplest answer here is that the image of the duel at high noon is mostly a work of fiction. Stuff like that didn't happen. The West was certainly violent in many ways, but if you wanted to kill someone, you probably were going to shoot them in the back. I've written about this before here. That said, in a real duel, there was an advantage in letting the other person shoot first. Depending on the rules dictating the exchange, it could mean that the first person to shoot is more rushed and hurried, and has more chance of missing, thus giving the second shooter time to be slower and more deliberate. This, for instance, happened with Andrew Jackson, in a duel I wrote about here.