When he is in the majority decision, he decides who gets to write the majority, which is a huge advantage. Two people can agree on one end but how they reached that end decision can be hugely different. The one being written down will be the one setting precedence. He can also put appellate cases on a "no hear list", meaning that it won't come to the Supreme Court unless a majority of the other justices want to hear the case. Otherwise, there is the rule of four for hearing a case, although sometimes if they get three, a fourth will join just to get the case heard. I may be forgetting more, I should take notes in class