Winning battles is a question that doesn't have a flat answer because it's based upon how badly the other guy wants the kid. The other guy may have done just enough to get listed, see that he's not alone, and then decide to go searching for another option, in which case winning the "battle" might be as simple as promising 10 minutes or bringing the kid to campus for a visit. On the other hand, the other coach might want the kid REALLY BAD, in which case you're either going to have to follow suit and spend through the wazoo to stay in contention or find yourself knocked off or getting a "you're too late coach" email.
Same thing for when I guy comes in on you late, but in those cases, the guy probably is serious about trying to get the recruit since he made the conscious decision to try and grab the kid from another human coach. The essence boils down to the same question though -- how bad do both people want the player? At that stage it's pretty much a game of chicken as both coaches spend cash and throw influence at the kid until one person decides the time has come to veer away.