English Mastiffs are great with other pets, especially if they are brought into the home at a young age. They will follow your lead in terms of expectation (and your expectation should be that they follow the pack order of the house, which is you and your partner, the kids, and then the other pets (generally in order of pet’s age, but not always). When you bring in the new pup, the other pet(s) assume their role is let the new kid know the pack order, as they see it. For this reason, you want to carefully consider your other pets’ personalities when choosing a puppy, because there definitely is a pack adjustment with any new addition!
There are 3 basic personality types in pets: alpha, beta, and omega.
Alpha: These are the leaders and they require strong leadership, expectations, and socialization from their humans! There are very few true alphas. "True" alphas rule not just by strength of body, but by strength of personality. Everything they do is meant to keep the pack safe and functioning smoothly. From a human perspective, you could call it integrity.
Some "wannabe" alphas will incessantly try to work their way up the pack order, which can cause problems with other pets (and perhaps your children and your partner!). Their goal is to be number one and they tend to dominate rather than lead the rest of the pack.
Beta: These are the most adaptable of the group. They are spunky and playful, but will finally settle down for some petting, training, and socialization. They are the ones most inclined to want to please you and the others, and for the most part could care less what order they are in the pack, as long as you pay attention to them!
Omega: These are the exact opposite of the alphas. They also require strong, but GENTLE leadership, socialization, and training so they do not evolve into fearful submissives.
If you currently have a pet that can’t stand to be messed with (a wannabe alpha), you will want to choose a submissive puppy. In this case a beta is probably your best bet.
If your current pet already knows he/she is the boss, but is still playful and tolerant, you may consider a beta or omega as your best bet. Depending on the older pet’s personality, you may also be able to bring in an alpha pup-especially if it is of the opposite sex. If you have several pets, a beta is definitely your best bet.
Over the years, we have brought 4 Mastiffs and a friend’s English Lab into a home with an 11 year old Rottweiler, a mixed breed, and 2 cats. (The cat’s weren’t quite so accommodating; it took them at least a month (per additional dog) to forgive us enough to make their presence known again).

EGOR
To give you an idea of this works; our Rottweiler (Egor) is a big brute, but very nurturing. He knows he is the boss and is perfectly confident in saying “stop” to a pup without snapping on them. In the dog pack, he would be considered a “true alpha”. At that time, we also had a 2 year old mix (Dexter). Dexter was submissive to Egor, a “beta” in personality; and he loved to please his humans more than anything!

DEXTER
When we first brought “omega” Billie home, Egor ignored her completely, no matter how much she tried to kiss up to him. Egor remained “alpha” and Billie was assigned a lower role by him. Dexter saw that Egor wanted nothing to do with Billie, so Dexter felt confident in his role as second in command to Egor.
However, Dexter was a little threatened by the new addition (just in case Billie might want to move her way up to 2nd in command). Dexter never snapped on Billie or took her down because she never disrespected his authority. Instead, Dexter would steal Billie's toys and run circles around her to tease her with them (passive-aggressive type behavior). In effect, Dexter was letting her know that he was “alpha” over her, even though he was submissive to Egor/us. After a month or so, Dexter slowly started sharing the toys with Billie.
Another month or so later, once Egor saw Dexter had sealed Billie's role as 3rd in the doggie pack order, Egor began to acknowledge Billie's presence by letting her kiss him (although he would growl at her if she got too mushy with it-and she understood and backed off).
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DEXTER, BILLIE & EGOR
Our next pup was Sadie-a definite “ALPHA”. Whew! She required immediate puppy classes! She wanted to rule not only the dogs, but the humans in the home to boot! Billie did her best to nurture her as a pup, but Sadie was more interested in Dexter’s and Egor’s roles. Again, Egor ignored her as unworthy, so Sadie tried to scrap her way into Dexter’s “second in command” role. Dexter took her right down to the ground (without snapping on her). Since getting over on Dexter didn't work, Sadie did her best to dominate Billie. Billie, as an omega, allowed her to do this while Sadie was a pup (Sadie would climb on her neck and try to mount her at just 2 months of age!!)

SADIE & BILLIE
Our next mastiff was Rogue, another “beta”. I don’t think there is much of anything that bothers Rogue. He could care less where he is in the pack order as long as he gets his loving! Rogue wakes up every morning and bestows wet, slobbery kisses on everyone in the pack. He is the ultimate in "laid-back".
ROGUE
Sadie was ruthless with Rogue when he first arrived. She was forever trying to find new ways to pin him. Rogue didn’t care-he liked the attention. Once he outweighed her by a good 50 lbs., it was tougher for Sadie to take him down, so she had to be trickier (like ramming him at full-speed when he was going potty). As a "beta", even that didn’t bother Rogue (although he did seem to enjoy his new trick of rolling over her and smashing her into the ground, pretending he was just scratching his back). Rogue loved Sadie and Sadie loved that she could be alpha over him! It worked for them.

SADIE PINNING ROGUE
Then Voltaire arrived, another “beta”. Much like Rogue, Voltaire wants to please everyone in the pack. He is forever passing out kisses, and all he asks in return is a little attention!

VOLTAIRE LOVES SARAH!
Egor, Dexter and Billie pretty much ignored Voltaire. Sadie loved having someone new to be “alpha” over and wrestled him to the ground every chance she got. Rogue smothered him in wet kisses.

SADIE & VOLTAIRE

ROGUE (6 months) & VOLTAIRE (4 months)
So all was well until….
Dexter’s hips went bad…and when he could no longer use them we had to put him down at just 4 years of age. Billie was devastated and slept in his spot for a month. Egor was also depressed, and he was forced to come out of "retirement" to help train the new kids (a job he had pretty much delegated to Dexter). Sadie, true to her "wannabe alpha” personality, thought she would try for the 2nd in command role. That’s where Billie drew the line. It was the first time Billie ever took Sadie down, and she did it so fast and hard (without snapping) that Sadie sat up next to the house for hours, afraid to move. The new roles were sealed, and those roles were in age order.
At Dexter's death, Billie was 2 years old and Sadie was 9 months old. Billie had unwillingly become "2nd in Command" in terms of the dog pack order, but it was a role she took seriously and it was a role she showed integrity in at all times. Billie continued to show deference to Egor and, for the time being, Sadie showed deference to both Egor and Billie. As time went on, however, this changed. An extremely heartbreaking choice would need to be made, and it was. For more info on Sadie, click here.

BILLIE & SADIE Checking the Stream for Muskrats