[lathspell]

I always believed Gandalf thought he was to weary to fight the Balrog single-handedly and therefore made his final blast on the bridge, knowing he did it when the Balrog would fall in. Why else would one of the most powerful wizards break his staff knowingly?

And Gandalf was certainly not chasing the Balrog up the Endless Stairs. He tells Aragorn, Legolas and Gimli that he didn't know where he was or how to get away and had to depend on the Balrog to get into regions he knew. The sentence: 'Too well he knew them all!' is quite famous. It is said by Gandalf, referring to the Balrog and the paths underneath moria.
They fought and chased eachother up the tower, and I don't believe Gandalf actually thought while on these stairs that he was going to beat the Balrog. More kinda despair to win the battle of the peak, without a staff ready.

greetings,
lathspell