For groups on both sides, legal experts say the biggest shift comes from the new position of Chief Justice John Roberts, who is now perceived as the most moderate of the five conservative justices and the court’s swing vote. Chief Justice Roberts is viewed as considerably more conservative than Justice Anthony Kennedy, the previous swing vote, who had a libertarian streak and could at times be sympathetic to the left on social issues including gay rights and the death penalty.

Practically, this means pitching to Chief Justice Roberts’s particular legal perspective, including what observers see as his hesitancy to overturn precedent. Some lawyers said this could mean, in cases where a precedent might cut against one’s arguments, emphasizing to the chief justice that those prior cases are slightly different and therefore don’t apply.

Others said they would address the issue of precedent head on. “If you want Chief Justice Roberts to overrule precedent you would have to, one, pitch to him the lack of constitutional basis for the original decision, and two, address the consequences of overruling that precedent,” said Mat Staver, chairman of Liberty Counsel, a Christian legal organization that represented two Kentucky counties in a Supreme Court case about public displays of the Ten Commandments.