On a dreary night in April 2014, Phil Jackson, the president of the New York Knicks, sat down to dinner with Carmelo Anthony at a downtown Manhattan steakhouse and presented his star player with an ambitious request.

His proposal was for Anthony to delay his impending free agency by one year. By opting in to the final season of his contract with the Knicks, Jackson explained, Anthony wouldn’t become a free agent until 2015, a time when the Knicks would have greater salary-cap space to improve the roster around him. And by...