There are two main conflicts in The Most Dangerous Game. The first is Man Vs. Man and the second is Man Vs. Nature.
The conflict of Man Vs. Man is obvious; protagonist Rainsford is forced to fight for his life against the sociopathic Zaroff. Both men are expert hunters, with mastery of their skills, and as Rainsford sets traps and uses his knowledge to evade pursuit, Zaroff negates the advantage by predicting Rainsford's decisions and path. Ultimately, Rainsford is willing to risk a possible death to avoid a certain one, and is victorious over Zaroff.
Two slight annoyances kept [Zaroff] from perfect enjoyment. One was the thought that it would be difficult to replace Ivan; the other was that his quarry had escaped him; of course, the American hadn't played the game -- so thought the general as he tasted his after-dinner liqueur.
The conflict of Man Vs. Nature is more subtle. Rainsford hears pistol shots and falls overboard; the salt sea keeps him from crying out, and the ship leaves him behind. After swimming in the direction of the shots, Rainsford finds himself on an island, shored by jagged rocks and filled with a dense jungle. When he sets out as prey for the hunt, he realizes that the island is too small to simply run, and he uses his knowledge of woodcraft to set false trails. However, Zaroff is equally skilled. After wounding Zaroff with a trap, Rainsford finds himself blocked by a quicksand swamp; instead of giving up, he digs a pit in the soft dirt and plants another trap. This fails as well, and as a last ditch effort, Rainsford ties his knife to one of the many saplings that had been slowing his pace. The springy tree fails to kill Zaroff, and in final desperation Rainsford gives himself to the very ocean that brought him to the island.
"Nerve, nerve, nerve!" he panted, as he dashed along. A blue gap showed between the trees dead ahead. Ever nearer drew the hounds. Rainsford forced himself on toward that gap. He reached it. It was the shore of the sea. Across a cove he could see the gloomy gray stone of the chateau. Twenty feet below him the sea rumbled and hissed. Rainsford hesitated. He heard the hounds. Then he leaped far out into the sea....
Ultimately, by using the harsh environment of the island to his advantage, Rainsford is triumphant.