In the books, it is very heavily and even directly implied that the breaking of another Wizard's staff is both a show of another Wizard's authority and a symbol of the 'bad' Wizard's expulsion from both the order and the Council.
In The Two Towers, Gandalf says to Saruman (Houghton Mifflin, paperback, p. 569):
...'Behold, I am not Gandalf the Grey, whom you betrayed. I am Gandalf the White, who has returned from death. You have no color now, and I cast you from your order and from the Council.'
He raised his hand, and spoke in a clear cold voice. 'Saruman, your staff is broken.'
Bearing this in mind, Gandalf was specifically given power and duty in the book as part and parcel to his resurrection to cast Saruman out of his order and the Council and part of that power included the hand-in-hand destruction of Saruman's staff.
However, if one recalls back to The Fellowship of the Ring where Gandalf led the Fellowship across the Bridge of Khazad-dûm, Gandalf's staff broke when he slammed it into the bridge to break the bridge... which coincided with his later death as Gandalf the Grey. As such, the breaking of his staff was his own doing - an immense show of his power and, given what happened to him later, perhaps an 'exhaustion' of that power similar to the 'take away' of Saruman's power by Gandalf the White later on.