It depends on how I'm doing. If I'm keeping up and have a well-equipped military, I usually support an underdog about to go to war with a bigger civ. I'd much rather be in the Modern Age with Mahatma Ghandi tucked away on some island that gained three cities during a war, than Gengis Khan, great warmonger on the mainland who just conquered India with my help. If I'm losing horribly, I'll be nice to the biggest civ when they go to war with the weakest (or when I play, second weakest.)
I live by the rule, Either be your master's sword or let your slave be your shield. For swords steal the glory and shields take the blow! -My quote