This board has SO many kind and generous soles on it, and trying to equally make/treat multiple animals within a household/Pack, seems so fair. However, in a natural Pack environment there is a Pecking Order, that is bread into our animals, and they are quite ok with it.
So many times, when we introduce a new Dog into the Pack, there is a struggle for dominance. This can be controlled so easily, if the owner/provider understands the mentality, and supports the resident Alpha. That tenure in the residence, is what should determine that, even though most think it takes a Male to hold that position.
This is not true with the Yorkshire Breed, the Females are excellent pack Alphas, and the ones I've raised, hold and control the pack with grace and a very cool head when there's conflict.
So often, we think that when there is conflict in the Pack, that we Humans must intervene. Most times that it totally unnecessary, IF we have supported the Pecking Order in our Pack, and shown our "authorization" of who's in charge, besides us.
When there is the introduction of a Male, who comes under the authority of an existing Alpha Female, as time passes She will let him have the Alpha position. BUT, that is in appearance only. Subtly she will still be in charge, and from my experience, that's just fine with the Male.
A natural application of "the Power behind the Throne".
One can have a favorite Dog, but if you intend to create a stable environment within the Pack, you shouldn't show that preference openly to the Pack, unless that animal is the Pack Alpha. This is so difficult for some of us.