Eager to curb the bickering, we started something called the “In Charge” system where each child has a day that she gets first pick on whatever issue arises.
The child “in charge” gets first pick that day. We often use it for TV or movie-watching, but the options are endless. A child "in charge" could have first priority on whatever comes up - first to be pushed on the swing, first to talk to Grandma on the phone, or first to choose a book to read at bedtime. The other child (or children) are ok with letting the “in charge” child choose because they are certain their turn is coming.
This system is so easy - so peaceful! - and I get to step out of the picture. The girls frequently ask, "Whose day is it?" and all I have to say is, "Let's check the calendar!" Having a child "in charge" eliminates many of the little arguments that start the fire. The flame is instantly extinguished when the kids know whose "in charge."
Read more about this idea on Rudey's Blog.