When you chase after someone, that means they're running. And just because you decided to chase them doesn't mean they're going to stop running. In fact, they usually speed up.
If you catch them, you may experience a moment's joy at the victory, but then you'll have to decide if winning them was worth the fatigue and exhaustion. If it's not, resentment instantly builds and the relationship equity is now in a hole that it's impossible to escape from.
Maybe it's fun. Maybe that's a fine reason to do it. And in that case, go nuts - just don't be surprised or hurt when the thrill of that chase is far more exhilarating than the victory.
To me, there's no upside. So don't chase.