You can add stuff to paint to slow down the growth of mildew, but it's not very effective.
Since the last administration changed the laws in how paints are allowed to be made, only Valspar and Sherwin Williams seem to be any good. Prices have gone crazy because of this too. Valspar at Lowes is the better bargain, but if you want the VERY BEST, go to Sherwin Williams. I think their best of the best is around $85 a gallon.
For bathrooms, and especially for rental properties, I always use exterior paint. It's designed to deal with the weather, and it offers a lot more protection. I'm not a fan of semi gloss on the walls. It reflects so much light that you see every imperfection in the wall. Studs are wood, and with all things nature, they move and change shape over time. All you need is one stud to bow a little and that's all you will see on a semi gloss wall. This is why flat paint is used on ceilings, it hides the sag from the joists.
Even more important then the paint used is the primer. My personal favorite to the point that I won't use any other brand regardless of how far I have to drive, or how long it takes to get it, is Zinsser. The 123 is an excellent primer that soaks into new sheetrock and gives you a very solid finish to apply your paint. After the law was changed on how paint had to be made, the paint companies came out with a sales gimmick saying the primer was built into the paint. People fell for it and tried to paint without primer. It took twice as much paint, or more to do the job, so the paint companies sold more paint!!!! Nothing replaced good primer. There are other brands, but none come close to what you get out of Zinsser.
