When it comes to social intelligence (with women or otherwise) being competitive is, like most positive leadership traits, great to an extent but self-defeating if you overdo it it. For every 5 "alpha" guys puffing the chest out at the club and showing their stuff in one way or another, there's a silent guy flying under the radar, just being chill, and picking off beautiful girls quietly while the louder guys make a spectacle of themselves constantly trying to one-up everyone.
Of course I know "competition" refers to a lot more than that, but it just goes to show how a competitive nature needs to be tempered a bit as well. That's because having the presence of a leader isn't about winning all the time. I can't remember exactly where I read this, but there are studies out there that even show how true leaders often allow men who are weaker than them (and part of their tribe) to put on a show of one-upping them from time to time (maybe Chase could chime in on the science here).
For example, I'm a much bigger and more experienced fighter then my younger brother and quite a few other guys I hang out with (not to humble brag too much lol). But at times they like to put on a show of making an aggressive joke towards me in front of other people, giving me a little flex when I walk by them, or roughhousing a bit (typical guy stuff, at least if you come from the blue collar crowds).
When I was younger this really used to irk me - as in, "How dare they underestimate me, don't they realize that I..." - but as I've gotten older, I've realized that it's a form of supplication and display of our bond. In a way, it's more about them than me - as if they want to say to everyone, "Look how I can poke this big bastard and he allows me to do it. Ain't I something?" If they were clearly bigger than me, instead, them doing this would just make them look like jerks.
And how much of a prick would I be if every time this happened, I grabbed my brother or buddy by the throat and choke-slammed him to remind him how easily I could physically dominate him - or even just rose my voice to bully him back into submission? How much longer would he follow me around, and how would women who saw this feel about me being so quick to intimidate or hurt someone in my "pack"?
This is a bit of a tangent and a very simplified example, but bottom line is nobody wants to follow a leader who always has to be the top dog and trample on those less solid or less skilled than him.
If you've read Jordan Peterson's book, he talks about a time when he worked on a railroad crew and a new guy showed up that didn't want to accept the good-natured ribbing any new guy got as kind of an "initiation." This guy was set on being taken seriously and showing he wasn't to be messed with - and by doing so, he missed the fact that groups of men, too, are as much about cooperation as they are about competition.
In the end, his lack of social intelligence and fierce pushback only made the other men antagonize him ten times more...until he was forced to storm off the site to save face and the potential of physical violence. Regardless of what you might think about Peterson, he's a keen observer; this dynamic is very real among masculine groups of any size.
You dominate groups, especially, by being accepted first, and that usually requires being willing to submit as much as compete.
A friend who is genuinely always trying to compete with you gets really annoying after a while. I don't want to go to the gym with someone who is always trying to get me into a contest - I know who I am and what I can do, and I don't need to prove myself. In fact, I prefer to work out alone.
Maybe it's biased to think so, but it has long been my opinion that the buddy trying to outdo himself in an effort to show his stuff (whether in the gym, around girls, with various skills, etc) reveals himself to be needy and insecure - while the guy who just cooly goes about his biz and let's his actions and accomplishments speak for him, is and is perceived as the actual leader.
Here's another example.
Have you ever noticed that legitamely tough guys don't go out constantly looking for fights in bars (execpt in cases where they are directly provoked or have something else dysfunctional going on that they need to make up for)? They don't have anything to prove and they have enough experience to know that being over aggressive isn't going to magically win them the "star of the night" award at the bar. And just that look of cool, collected confidence is generally enough to send a very powerful message.
My advice - sure, a competitive streak is a healthy masculine trait in doses, but know when it's time to compete, know when it's time to draw a line in the sand (i.e. when being blatantly disrespected or pushed around), and know when it's time to just kick back and chill while everyone else tries to "win" some imaginary contest with the room. If you aren't sure when is time for what, observe other guys who are relaxed without sacrificing their dominant aura and watch how they behave in crowds.
As far as competing for women, I can think of few things more revealing of a scarcity mentality than a guy who feels he needs to work against his friends for a single girl. And even with enemies/non-friends, what puts a woman on a higher pedestal than outwardly competing for her attention?
Let the other guys compete and play your cards right - she'll often come to you. You can silently relish your achievement when you're out with her and catch those same guys sneaking envious looks.