At first glance, the use of bows in play may appear random. They do not occur every N actions, or every N seconds. But it turns out that the bow has a very important function: the bow is regularly used before and/or after other actions that could be misinterpreted by the other dog, and could disrupt the social play. For example, bows were used either immediately before or after bites during play 74% of the time in infant and adult domesticated dogs, 79% in infant wolves, and 92% in infant coyotes.
In addition to communicating "I want to play," bows performed during the play sequence itself seem to mean "I still want to play despite what I am going to do or just did." It's like what your Mom used to tell you - it's always fun until someone gets hurt. The dogs seem to have internalized that rule, and continually make it clear that any damage done was all done in good fun. No hard feelings. (What I learn from this is: if my brother had simply bowed to me when we were kids, there wouldn't have been any fights in the first place. Right?)
Observations show that coyotes use the bow more often than domestic dogs, probably because a coyote fight is serious business for the pack. Link