I have four cats. Three of them catch and kill mice, but Todd my twenty lb tabby, kills rats, I've also once saw the laying hens I had two years ago kill a young rat that ran through their outdoor pen. I've heard that if you have just one cat who isn't allowed in the house (Todd lives in the garage because he pees on walls otherwise) it's a lot more likely that they will hunt effectively, as opposed to a housecat. I think this is true. If I get mad at Todd he tends to find a rat and leave it on the front walk where I can see it as if too remind me he's earning his keep.
I only have bantams at the moment and when they are out in the yard they boss the cats, but that's probably because I had large hens first and the cats got used to chickens and seem to recognize that they are part of the family and not toys.