The Booker we play as predicate the fate of the other Bookers in the multiple universes where Booker exists?
If the multi-universe theory is true, wouldn't it stand true that at least in one universe, Booker decides not to take his life, thus keeping the vicious cycle to continue?
Jaime Lannister sends his perspective...
http://tinyurl.com/Jonsnowface
You're right he doesn't change other Bookers in the multiverse. He only stops himself from continuing his own loop.
This is evidenced in Burial at Sea (spoilers) where Elizabeth finds herself in a universe that was never "attached" to hers, or part of her loop. It's easy to mistake that Comstock in Burial is 'her' Comstock but he isn't. Remember the point of of Booker's sacrifice is to end his loop - which he does. So in order for Elizabeth to have gone and interacted with the kidnapping again she'd have to find a new universe/loop that is not her own. Remember the tears allow crossing between universes - not time, not really.
The merging of Elizabeths at the end of Bioshock isn't showing he's fixing the other universes, it's supposed to be symbolic that he will never turn another Ana into elizabeth because he will never kidnap her. It's symbolic we know because the place Booker goes to die is similar to the Paris Elizabeth goes to in Burial, an untethered universe where the laws of reality are not the same