Sometimes bribing an AI to go to war with another civ can be quite cheap.
In a recent quick-speed 2 player multiplayer game with 5 AI's, I bribed Oda, to my SE, with 25 gold and 2 horses to go to war with Hiawatha, his neighbor and rival. This was pittance for me and performed its role (of redirecting Oda's NW-moving swarm of catapults and swordsmen to someone other than me) admirably. Of course they made peace 10 turns later and then Oda DOW'ed me (as was his original plan, I think), but by then I had a number of Camel Archers... the point is that for the cost of 25 gold and horses for several turns, I caused two enemy civs to lose hundreds of gold worth of units and ensure my safety for long enough to build a defensive force.
I think that the cost of getting an AI to attack another civ is related to how well the bribed AI likes you, and how much it hates the target civ, so a civ that is 'friendly' with you and 'guarded' towards another will be easier to bribe than a civ that is 'guarded' towards you.
In the end, always remember: Strategic bribery is your friend!