I too had heard that letting your dog win makes him try to be dominant.
However, I don’t find it on any serious dog training website so far, and my dogs have been tugging to a win/lose/draw for some years now without any attempt to be the boss, so I’m going to file that under “yet another plausible-sounding urban myth without a shred of basis in reality.”
In the thread above you might have noticed that although many dogs love to tug, the bulldogs and their kin (including Boston Terriers and American Pit Bull Terriers) are super-enthusiastic about it. That’s because the “bully breeds” have been selected for around 2,000 years for their willingness to grab and hang on; another term for this class of working dog is “grip dogs.”
My little APBT, Simone, just LOVES tug games. You can see in her eyes that when she’s got her teeth on an approved toy and her feet braced, she KNOWS she’s doing what she was literally bred to do; she’s visibly fulfilled.
When she gets worked up, I can pick her entirely up off the floor and she holds on with her teeth (I try to make sure she’s got a solid grip on a substantial toy before doing this). I usually can’t resist calling my wife and saying, “Where did you want this?” or “Who asked for one of these?” while holding the dog in the air like a prize. My wife usually forgives me my repetitiveness.
I have always emphasized to Simone that she needs to control her enthusiasm and aim her teeth carefully; if her catches my hand in her mouth the game is over and I walk away. Now if her teeth graze my skin she stops the game and sits down hard, her eyes wide, hoping she hasn’t ruined everything.