What Does a C.E.O. Actually Do?

They're paid a fortune — but for what, exactly? What makes a good C.E.O. — and how can you even tell? Is "leadership science" a real thing — or just airport-bookstore mumbo jumbo? We put these questions to Mark Zuckerberg, Richard Branson, Indra Nooyi, Satya Nadella, Jack Welch, Ray Dalio, Carol Bartz, David Rubenstein, and Ellen Pao. (Part 1 of a special series, "The Secret Life of C.E.O.'s.")