https://en.wikipedia.org/wiki/Chief_Justice_of_the_United_States
Excerpt:
The chief justice's formal prerogative—when in the majority—to assign which justice will write the court's opinion is perhaps his most influential power, as this enables him to influence the historical record. He may assign this task to the individual justice best able to hold together a fragile coalition, to an ideologically amenable colleague, or to himself. Opinion authors can have a large influence on the content of an opinion; two justices in the same majority, given the opportunity, might write very different majority opinions. A chief justice who knows the associate justices well can therefore do much—by the simple act of selecting the justice who writes the opinion of the court—to affect the general character or tone of an opinion, which in turn can affect the interpretation of that opinion in cases before lower courts in the years to come.
The chief justice chairs the conferences where cases are discussed and tentatively voted on by the justices. He normally speaks first and so has influence in framing the discussion. Although the chief justice votes first—the court votes in order of seniority—he may strategically pass in order to ensure membership in the majority if desired. It is reported that:
Chief Justice Warren Burger was renowned, and even vilified in some quarters, for voting strategically during conference discussions on the Supreme Court in order to control the Court's agenda through opinion assignment. Indeed, Burger is said to have often changed votes to join the majority coalition, cast "phony votes" by voting against his preferred position, and declined to express a position at conference.[
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