The Most Dangerous Game is a short story, and so must get to the point quickly to avoid losing the reader. Setting up the character of Rainsford, master game hunter, we find him on a ship with a friend named Whitney who is not himself important to the story.
Most summaries would point to Rainsford's falling off the ship as the inciting incident. However, before he falls, he is napping on the deck and hears gunshots where there should be no human habitation:
An abrupt sound startled him. Off to the right he heard it, and his ears, expert in such matters, could not be mistaken. Again he heard the sound, and again. Somewhere, off in the blackness, someone had fired a gun three times.
Rainsford sprang up and moved quickly to the rail, mystified.
This is the true inciting incident. Rainsford would have continued to nap on the deck and never moved if not for his acute hearing and his curiousity. In attempting to pinpoint the source of the gunshots, Rainsford climbs the railing and falls overboard, setting the plot into motion.