I have two daughters, 12 and eight. Both girls argue with each other several times a day. There has been competition between both of them for many years. My oldest tries to tell me that her sister is to blame for the arguments and vice versa. When I am not around and they fight, the younger calls me to tell me the whole story. It usually ends up that the older will call the younger "stupid" or another hurtful name which my youngest becomes very sensitive to. When I try to talk with both, of course they blame the other and think the other should change. Neither will own up to starting the argument. I don't know where to go from here. This has gone on for at least two years and it just seems to be getting worse.
You can find some very clear guidelines about relating to arguments between siblings in a book called Siblings Without Rivalry by Adele Faber and Elaine Mazlish.
In short, the first thing to do is to acknowledge the feelings of each girl. "Wow! You two sound really angry with each other!" Then, hear each side and reflect back to each girl what you heard. "So you were watching your favorite TV show and your sister changed the channel."
To the other sister: "You've been waiting all week to see this movie and you wanted to check to see when it was coming on. This really is a difficult situation."
You can express your confidence that the girls can find a solution that will be fair to both of them and they can let you know what solution they came up with. Sometimes when the quarrel continues incessantly, the parent needs to step in and decide what the solution will be this time. When everyone is calm and has time, a problem-solving meeting can be held during which the parent can describe the problem again to the children and ask them to come up with solutions that can be implemented the next time a similar incident arises.
The goal of parents is to model and teach skills to children so that the children will learn effective and respectful ways of dealing with their differences and conflicts on their own, without needing our intervention.