"The Running man" by Stephen King was published in 1982, the movie starring Schwarzenegger was released in 1987 (another adaptation will be released this year in 2025)
The story is similar to a Robert Sheckley's short story from 1952,"The Prize of Peril" adapted into a movie in Germany in 1970 "Das Millionenspiel" and in France in 1982 "Le prix du danger" (same year as King's Running Man novel)
I would say "The Prize of Peril" is the grandfather of these books, movies and series, as far as I know. Battle Royale is the start of another branch, though : it's not one vs many anymore, it's many vs many.
He also has his protagonists hail from the fictional town of Shiroiwa-cho 白岩町, which is a direct translation of King's frequently-seen "Castle Rock". I wonder how many Japanese readers of the original spotted that.