Just started thinking about this today: who's run a game that later wrote a book?
(Note: the list includes everybody who wrote a book; whether or not the book had anything to do with game shows is not a consideration. And "author" in some cases is a reach -- I doubt Pat Sajak had a lot to do with the crossword books that bear his name.)
Feel free to add the names I've missed or forgotten.
Allen, Steve
Barker, Bob
Barris, Chuck
Bergeron, Tom
Carey, Drew
Carson, Johnny
Clooney, Nick
Collyer, Bud
Downs, Hugh
Enberg, Dick
Eubanks, Bob
Garagiola, Joe
Griffin, Merv
Hall, Monty
Ludden, Allen
Marshall, Peter
Martindale, Wink
Myerson, Bess
Pennington, Janice
Perry, Jim
Sajak, Pat
Trebek, Alex
White, Betty
White, Vanna