Would you want to give up?

If anything for most guys its going to up the ante on how hard they try to win you over, if its really you they want. When you say really confident type of guy the immediate thing that pops into my head as cocky, which is quickly followed by competitive. Being competitive in nature, its very unlikely that this guy is just going to fold his hand when he realizes there's other people interested in you.

If you meet someone who is your equal he won't be really competitive (think of a dominant lion, he doesn't work hard because he doesn't have to), unless he gets a kick out of that. Really it depends on the guy because some guys play by their own rules, and really are different/have a different mentality than the average joe.
There is a common myth that guys like the chase, (depending on the guy) for the most part they don't. Girls enjoy the variety of emotions that come from a chase more often than guys. If he dates a lot of girls (or has decent self esteem and is very secure) he will only care about how you treat him, and how you two interact. Other guys won't be important.

Oh yeah, and if he is the type of guy who likes to date girls one at a time and expects the same from girls, he may give up and look for a girl who will do the same rather than compete with a bunch of other guys (because he wouldn't do the same to you). It really depends how you are and how he is in your dating styles.

Most confident guys would still try to ask you out because in many guys' minds, other guys mean nothing unless they are going out with you. So the only time they would stop asking you out is if you say yes to someone else.

A really confident guy probably won't be deterred from pursuing you if he really wants you. In some cases seeing the girl get asked out by lots of guys makes him realize that he should ask you out. And sometimes it brings out the competitive side of the guy and he sees it as a competition to "win" you from the other guys who are asking you out. Good luck.

A really confident probably won't care because he thinks he's all that and he can get anyone he wants. On the other hand he might be a little defensive because your getting all this attention.

Hopefully this guy is right for you and he's not too cocky or anything because if that's the case he's probably not worth your time. If you've got guys wanting to date you all the time than just make sure you choose the right guy, okay? Good luck.