"The Most Dangerous Game" is largely a story that emphasizes man versus man and man versus nature conflicts, though there are elements of a man versus self conflict within Rainsford.

Rainsford's conflict with nature occurs early on, when the ship begins to sink. Not only does the water threaten him, but the dark of night makes it harder for him to see where he's going or what's happening:

Rainsford sprang up and moved quickly to the rail, mystified. He strained his eyes in the direction from which the reports had come, but it was like trying to see through a blanket. He leaped upon the rail and balanced himself there, to get greater elevation; his pipe, striking a rope, was knocked from his mouth. He lunged for it; a short, hoarse cry came from his lips as he realized he had reached too far and had lost his balance. The cry was pinched off short as the blood-warm waters of the Caribbean Sea dosed over his head.

This is the first time the mighty hunter Rainsford is rendered vulnerable by powers out of his control.

By and large, most of the conflict is between people. The moment Rainsford first encounters another person on Shiptrap Island is immediately fraught with mortal danger:

The first thing Rainsford's eyes discerned was the largest man Rainsford had ever seen—a gigantic creature, solidly made and black bearded to the waist. In his hand the man held a long-barreled revolver, and he was pointing it straight at Rainsford's heart.

The servant Ivan prefigures the more potent threat represented by Zaroff. When Zaroff first appears, Rainsford regards him as handsome and aristocratic in appearance, but within moments, he shows his "red lips and pointed teeth," foreshadowing his brutality and thirst for violence.

While much of the conflict between Zaroff and Rainsford is physical, earlier on, the two clash over philosophies of hunting. Zaroff has no qualms killing humans he considers lower than himself (which appears to include almost everyone), while Rainsford still has moral scruples, even if he does enjoy the thrill of the hunt:

"Thank you, I'm a hunter, not a murderer."

"Dear me," said the general, quite unruffled, "again that unpleasant word. But I think I can show you that your scruples are quite ill founded."

"Yes?"

"Life is for the strong, to be lived by the strong, and, if needs be, taken by the strong. The weak of the world were put here to give the strong pleasure. I am strong. Why should I not use my gift? If I wish to hunt, why should I not? I hunt the scum of the earth: sailors from tramp ships—lassars, blacks, Chinese, whites, mongrels—a thoroughbred horse or hound is worth more than a score of them."

"But they are men," said Rainsford hotly.

"Precisely," said the general. "That is why I use them. It gives me pleasure. They can reason, after a fashion. So they are dangerous."

This goes to show that the conflict between these two men is as psychological as it is physical.

Lastly, while man versus self is not prominent, it does appear in Rainsford's own fears of losing his life. The sound of the general's laughter in particular chills him:

He stood there, rubbing his injured shoulder, and Rainsford, with fear again gripping his heart, heard the general's mocking laugh ring through the jungle.

Eventually, Rainsford overcomes all three of these obstacles: his hunting prowess allows him to survive the elements and Zaroff, but his own courage allows him to overcome inner anxieties about his ability to survive.

Man vs. Man:

After General Zaroff and Rainsford finish their meal, the general elaborates on the "game." His comment describes the man vs. man conflict throughout the story when he tells Rainsford,

Your brain against mine. Your woodcraft against mine. Your strength and stamina against mine. Outdoor chess! (Connell 11).

At the end of the story, General Zaroff congratulates Rainsford for surviving the game. However, Rainsford still feels like an animal being hunted and challenges the general by saying,

I am still a beast at bay...Get ready, General Zaroff (Connell 15).

Man vs. Self:

After spending his first night resting on the limbs of a tree, Rainsford watches as the general calmly approaches the tree before casually walking away. Rainsford realizes that General Zaroff is saving him for another day and begins to feel anxious and afraid. However, Rainsford calms himself by saying,

I will not lose my nerve. I will not (Connell 12).

Man vs. Nature:

At the beginning of the story, Rainsford falls off the yacht into the tumultuous sea. He struggles to regain his composure and swims to the surface against the waves. Rainsford's struggle to survive the rough open waters is considered a man vs. nature conflict. Connell writes,

He struggled up to the surface and tried to cry out, but the wash from the speeding yacht slapped him in the face and the salt water in his open mouth made him gag and strangle. Desperately he struck out with strong strokes after the receding lights of the yacht, but he stopped before he had swum fifty feet (2).

While Rainsford is attempting to flee General Zaroff, he encounters numerous obstacles throughout Ship-Trap Island. Connell once again depicts a man vs. nature conflict through Rainsford's struggles to avoid and survive the island's Death Swamp by writing,

Then, as he stepped forward, his foot sank into the ooze. He tried to wrench it back, but the muck sucked viciously at his foot as if it were a giant leech. With a violent effort, he tore his feet loose. He knew where he was now. Death Swamp and its quicksand (13).

"The Most Dangerous Game" contains various themes of conflict, and acts as a story about a man facing death, adversity, and struggle in the most primal ways. The barbarity of Zaroff is contrasted with the beautiful island and Rainsford's pragmatism. Here is one quote for each of the three queried conflicts:

Man versus Self:

Rainsford had fought his way through the bush for two hours. "I must keep my nerve. I must keep my nerve," he said through tight teeth.

Rainsford, forced into being hunted, is at first unsure how to act. He has always been used to hunting animals and can't figure out what to do at first. However, he admonishes himself not to lose his nerve; his final act demonstrates just how strong his nerve truly is.

Man versus Man:

"...let me congratulate you. Not many men know how to make a Malay mancatcher. Luckily for me I, too, have hunted in Malacca. You are proving interesting, Mr. Rainsford."

Rainsford and Zaroff are both accomplished hunters, and initially they have similar beliefs about natural law. Both believe that all living things are either hunters or prey, and so the strong are meant to survive. The are similar enough in their skills that Rainsford cannot shake Zaroff from his trail, and only  Rainsford's personal determination allows his victory.

Man versus Nature:

Dusk came, then darkness, and still he pressed on. The ground grew softer under his moccasins; the vegetation grew ranker, denser; insects bit him savagely.
(Connell, "The Most Dangerous Game," classicreader.com)

the island is well-suited for a hunting reserve; it has cliffs, swamps, and jungles. Rainsford is trapped inside the island's boundaries, but uses the thick jungle, the soft dirt of the swamp, and finally the dangerous cliffs all to his advantage. Rather than being constrained by nature, Rainsford embraces it as a tool of survival.