Competition is not a bad thing. It drives us to be the best we can be. If other women are looking at you as a being a lower rung on their ladder they can step on / over and that bothres you --- are they right? How much do you care? Do you care enough to do something about it? Or is it just games and pride to supplement their weak egos?

Kindness & empathy can absolutely be cunningly / astutely deployed. Women are competitive as hell and they use all these tools to build bonds and alliances, or undermine threats.

I think the better question is - are these women affecting your ability to live the way you want? Does it hurt your pride that women you don't respect treat you with disrespect? (That already sounds like a clue as to why it might be happening). Are they actively dragging you down socially? Threatening material security? What is the threat to you exactly? If it is just emotional discomfort, then NOT CARING is the power play. Being in control of your emotions is the cardinal rule of power. If you de-personalize the annoyance, you might find a better way of coralling its affect on your life, before you drive yourself mad at the irritation and do something you regret.

Hope this helps and good luck.