I'm going to suggest that this is not the most helpful distinction. It's important to take care of yourself AND to take care of your partner, but you can kill both of those birds with one stone by making strong choices. What we need here is a deeper understanding of the word "support."
In Improvise, Mick Napier puts it this way:
If the first thought in your head when you approach an improv scene is "Support your partner" ... [w]hat are you supporting them with?So, what's the best way to take care of myself? To make a strong choice. No brainer.
Are you supporting them with thoughts about supporting them? That's very nice but not very supportive. ... Do you say nice things to them, do you uber-agree, do you pat them on the head, offer them a chair, rub their shoulders? No, the most supportive thing you can do is get over your pasty self and selfishly make a strong choice in the scene. Then you are supporting your partner with your power, and not your fear.
If you want to support your partner in an improv scene, give them the gift of your choice.
And what's the best way to take care of my partner? Also, to make a strong choice. Not deferring to them, saying "yes" a lot, and keeping your own ideas to yourself.
For me, the latter concept was difficult, because I confused 'support' with 'submission' for my first couple of years of improv. I'm sure there are guys who deal with this, too, though I haven't met many. I have seen this over and over with evangelical women.
Conservative evangelical gals grow up being told that good Christian girls are polite and deferential. We're told, for instance, that the only reason Deborah and Jael were allowed to lead is that Barak and the rest of the Israelite men were too wimpy to step up. A woman could only be strong if all nearby men had abdicated their manhood.* Even if you don't consciously buy into these ideas, they're in the water, and they need to be fought.
Being polite will not serve you or anyone else. Being generous will. It means giving of yourself, not abdicating yourself. Generosity means making strong choices.
It's not as though strength is a single cake, and for one woman to have more of the cake, it means a man or another woman has to have less.
Strength is NOT a cake.** It's more like the widow of Zarephath's oil, which never dried up during the famine; she always had enough to give some food to Elijah.
Or like the other widow's oil, which Elisha told her to divide into other jars. She took all the jars in the neighborhood, and no matter how many jars she poured her oil into, there was always enough to fill another jar.***
In God's upside-down economy, giving things away doesn't necessarily mean you have less for yourself. Grace isn't a zero-sum game. The more I give of myself, the more I have. That's how we're supposed to live, and good improv is a small, concrete example of how it can play out.
Making strong choices yourself doesn't mean your scene partner can't. My strong choices should make it easier for you to make strong choices, which will make it easier for me to make strong choices, in an endless loop of strength and support.
*Here is a more reasonable interpretation of that story, preached earlier this summer by Rev. Karen Miller. I highly recommend investing 20 minutes of your day listening to this.
**THIS IS A WAY IN WHICH IMPROV IS NOT LIKE CAKE. My world may collapse.
***Elijah and Elisha had a thing for widows and oil, I guess?