What event initiates the rising action in the plot in "The Most Dangerous Game"?
In any story that demonstrates the the classic organization of plot, an event has to occur (called the inciting incident) that sets the story in motion. If there isn't an initial problem, then there is nothing to complicate or to try to solve.
Had Rainsford not fallen off the boat, this would not have been possible. His departure from the boat is a turning point in the action of the story that helps also point to an eventual climax. The climax has to demonstrate a significant turning point as well. Usually that turning point is an answer to the inciting incident. If Rainsford hadn't fallen off the boat, he wouldn't have had to find an island, climb up to a place where someone lives, learn about that person, complicate a relationship with that person, and survive that relationship.
In my opinion, the event that starts the rising action is when Rainsford falls off the boat into the ocean. I think that the rising action pretty much begins when he gets to Zaroff's place.
If that is the rising action, then it is started by the fact that Rainsford is on the island in the first place. And why is he on the island? He is on the island because he fell off the boat.
So that is why I think that him falling off the boat is the event that starts the rising action.