Gandalf says the winner would emerge stronger and free of doubt. Stronger because the winner would have one less rival and free of doubt because the winner knows the loser did not have the Ring.
But the above point by Ddesque was based on the premise of what would happen if Saruman did gain the ring. And I don't see what is particularly narrow minded about that. Though we can't be sure (luckily for the free peoples of middle earth), I believe that if Saruman gained the ring he would have likely defeated sauron if he could learn to wield the ring quickly enough. That would be the key, as sauron's forces were infinitely greater, and if Saruman was slow in learning how to wield it properly the power of mordor could defeat him without much difficulty.
Yeah, Middle earth wouldn't even be a nice place to visit.Not that this would have helped the Free Peoples very much.