Two years ago, Art Basel directors Marc Spiegler and Annette Schönholzer came to this city on a mission: to start a new contemporary art fair in Asia. But after scoping out the competition, they decided to change tack. The Hong Kong International Art Fair, also known as Art HK, “was not only a popular show, but a show whose quality was rising from year to year,” said Mr. Spiegler. After that, he said, “this idea of working with the existing team made a lot of sense.” Art Basel’s owner, MCH Group, bought a majority stake in the Hong Kong fair last year.