If you are not in a relationship yet and there is competition, such a situation is much more complicated. In such a case you feel very often that you can lose this person at any time, hence automatically you start to care about him/her much more, and thus you are willing to show great interest towards him/her.
For many people such a situation causes a panic attack, and they want to start the relationship as soon as possible and in this way defeat their competition - which actually is a big mistake, because they cease to be a challenge and only make things worse.
Therefore, the most important rule is always to stop the first impulses driven by our emotions and approach the situation in the most rational way possible. The difficult truth you have to face is that you don't have an influence on everything. There is no magical sentence or indisputable argument that you can use to make the person you are dating be with you. You have to accept that you can do everything exactly as should be done, show up your best self, and still not win over that person.
The second fact that is worth realizing is that your influence on what this person feels for your competitor is low. You may think that discrediting your competition, emphasizing his/her flaws, or even ridiculing him/her may lead to something - but in practice you only harm yourself. You express your interest but you do it in a negative way. If you didn't care, you wouldn't say such things.
Similarly, if your competitor tries to say something bad about you, it will not have any significant impact on what the person you’re dating feels for you, because, first of all, the credibility of your competitor is very low, and secondly, what your love interest feels for you depends primarily on your behavior.
For all these reasons it is not worth to directly or indirectly attack your competitors. The best reaction is to focus on building your own attractiveness and not to assume that you have to win someone over. The best approach, which will have an incredibly positive effect on your attractiveness, is constant thinking: "Do I really want to be with this person?", "Is it a good match for me?", "Will it be possible to create a happy relationship with this person? This is a very attractive approach that will transfer to all your behavior and significantly increase your chances of actually being able to choose this person - as long as it really meets your expectations. It's incomparably better than thinking about what I can do to beat the competition.
Therefore, the best way to beat the competition is not to compete at all. Do absolutely nothing more than you would do if there was no third person around. The best attitude that will allow you to win over the person you care about is to be ready to lose them at any time. The worst way is to try to stop them at all costs and to act under time pressure - because it may be too late otherwise. This way you can shoot yourself in the foot.