What is the ending of "The Most Dangerous Game"?

It is strongly implied, although not outright specified, that at the end of "The Most Dangerous Game," Rainsford recognizes that the only way to deal with General Zaroff is to attempt to beat him at his own game, of treating humans as if they were animals to be baited and used as prey.

Zaroff, thinking that Rainsford has escaped him as a quarry, goes home, thinking morosely about how he is going to have to replace his servant, Ivan, whom Rainsford has killed. He listens to opera and then goes to bed. He believes that Rainsford has been one of the most interesting beasts he has ever attempted to pursue. This opinion is borne out when, upon going to bed, he realizes that Rainsford is in his bedroom, waiting behind a curtain for him to arrive.

Zaroff tells Rainsford that he has won the game, but Rainsford parries, saying that actually he is still "a beast at bay," or still part of the game. He will not give up so easily, and he will not let Zaroff off the hook. Zaroff enjoys this and says that whichever of them gets the better of the other in their fight will sleep in the bed that night, while the other will be fed to the dogs.

Although there are no details of the fight, the fact that Rainsford is the one who ends up sleeping in the bed is a sure indication that he has bested Zaroff.

In the plot, or sequence of happenings in a literary work, the exposition, or beginning of the story, is part of the rising action, during which there is often a complication, or problem. This is a point at which the protagonist, or central character, meets some opposition, or conflict.

Therefore, in "The Most Dangerous Game," during the exposition Whitney and Rainsford discuss hunting; Whitney feels some sympathy for the prey, contending that they understand fear, "[T]he fear of pain and the fear of death," an observation that foreshadows the experiences of Rainsford to come. Rainsford, however, disagrees and bids his friend goodnight while he remains on deck. When he hears three shots, Rainsford rushes to the rail of the boat, but loses his pipe. As he struggles to catch it, he falls overboard.

In the rising action, Rainsford swims ashore and collapses. The next day he discovered a trail made by hunting boots and follows it where he finally sees the "shadowy outlines of a palatial chateau...set on a high bluff." He introduces himself to the occupants, one of whom has menacing eyes and is dressed in a black uniform. The other man, General Zaroff, is handsome but possesses "an original, almost bizarre quality" about his face.

At dinner, Rainsford converses with the general about hunting; the general informs him,

"I hunt more dangerous game....I live for danger, Mr. Rainsford.....I have done a rare thing. I have invented a new sensation."

Rainsford senses that there is something bizarre about the general. This recognition is the beginning of the complication. For, he and General Zaroff have a strong disagreement about "the ideal quarry." For, Zaroff hunts men, and Rainsford finds this action "cold-blooded murder." Thus begins the opposition of Zaroff as the antagonist and Rainsford as the protagonist. This conflict between Rainsford as the prey and Zaroff as the hunter continues until it reaches the climax of the duel between the two men. Therefore, the rising action of "The Most Dangerous Game" continues through most of the narrative.

In a literary work, the denouement is the final section in which all the strands of the plot are tied together and resolution is achieved; it is the end result, the culmination of the action.  The denouement of "The Most Dangerous Game" occurs after the climactic scene in which Zaroff discovers Rainsford hiding in his bedroom following the general's hunting expedition in which Rainsford is the prey.  Rainsford refuses to end the game at this point and kills Zaroff.  Following his successful encouter with General Zaroff, Rainsford enjoys a sound night's sleep in Zaroff's bed.  The denouement of the story, then, is the final segment of the plot: Rainsford's encounter with Zaroff in the general's bedroom, the murder of the general, and Rainsford's assuming of Zaroff's place in the estate.

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.

The question is correct to suggest that there is more than one conflict present in this story. The main conflict is the man versus man conflict that exists between Rainsford and Zaroff. Rainsford falls over the side of the ship that he was on, and he is forced to swim to the nearby island. Rainsford discovers that the island is inhabited by a fellow gentleman that also happens to be an avid hunter. Unfortunately, Zaroff's favorite prey is humans. He forces Rainsford to be his prey. Rainsford must survive for 72 hours. If he is successful in doing that, then Zaroff claims that he will release Rainsford from the island.

"And if I win—" began Rainsford huskily.

"I'll cheerfully acknowledge myself defeated if I do not find you by midnight of the third day," said General Zaroff. "My sloop will place you on the mainland near a town."

The bulk of the story details how Rainsford tries to outwit Zaroff. Zaroff is a good hunter, and Rainsford works very hard to throw him off of his trail. The conflict is a dire conflict because if Rainsford fails . . . he dies.

In Richard Connell's suspenseful adventure tale of a deadly hunt, plot is the essential story element.  With an exposition that includes a discussion between Sanger Rainsford and his friend and fellow hunter, Whitney, Connell subtlely suggests the dangerous game to come that will alter Rainsford's initial attitude about the feelings of the prey that is hunted:

"We should have some good hunting up the Amazon. Great sport, hunting."
"The best sport in the world," agreed Rainsford.
"For the hunter," amended Whitney.  "Not for the jaguar."
"Don't talk rot, Whitney,...You're a big game hunter, not a philosopher. Who cares how a jaguar feels?"
"Perhaps the jaguar does..."
"Bah! They've no understanding."

Ironically, then, it is Rainsford who comes to have the deepest of understandings about how the jaguar feels as he himself becomes "a beast at bay." And, it is the first major event of the plot, the complication of the rising action, that initiates this change in feeling for Rainsford; namely, the evening when Rainsford leaps upon the rail of his ship and balances to ascertain from which direction shots have been fired, and he loses his balance and falls overboard,

The cry was pinched off short as the blood-warm water of the Caribbean Sea closed over his head.

Rainsford is able to swim toward the sound of a "screaming animal" and reaches a shore. Exhausted, he falls into a deep sleep.

Richard Connell's "The Most Dangerous Game" is one of those short stories which rather breaks the conventional plot structure of a short story, and there could certainly be some disagreement about what event qualifies as the climax/crisis/turning point of the story.

The inciting action is when the action begins, and the rising action is everything that happens after that which leads to the climax of the story. After the climax is falling action, followed by the resolution or denouement. If we were to follow this traditional form, the climax would have to be when General Zaroff announces that he intends to hunt Rainsford or perhaps when the actual hunt begins. That reading would allow for plenty of falling action and at least some resolution. This story does not really have a long resolution, though, so a more useful reading is as follows. 

The inciting action is either when Rainsford drops his pipe and falls off the boat or when he lands on the island. The rising action goes on for a very long time, until it peaks with Rainsford jumping off the cliff to the rocks and water below and then reappearing in Zaroff's bedroom. Very little story is left after that, but it is enough to resolve the key issues: Rainsford lives and Zaroff dies. 

The climax of a story is 

the moment in a play, novel, short story, or narrative poem at which the crisis comes to its point of greatest intensity and is resolved. It is also the peak of emotional response from a reader or spectator, and it usually represents the turning point in the action. 

Given this definition, a case could be made that Rainsford jumping off the cliff is the climax; the case could also be made that the moment of "greatest intensity" is when Rainsford appears from behind the curtain in Zaroff's room. It is certainly a great surprise to us--and an even greater surprise to Zaroff. The resolution and denouement are only a line or two in either case, but that is enough to ensure that we know who wins the battle. 

Either reading seems defensible and depends on what the reader views as the "peak of emotional response." The direction of the story turns when Rainsford escapes from Zaroff, but everything really turns when Rainsford wins the game and Zaroff becomes the hunted. 

The general sucked in his breath and smiled. "I congratulate you," he said. You have won the game."

For me, the climax of this story is when Rainsford appears from behind the curtain and wins the game. 

Climax is best defined as the "turning point in a storyline," the moment that the rising action turns into the falling action (Literary Devices, "Climax"). At this moment, the conflict in the narrative reaches its greatest intensity, bringing the resolution into sight. Although some literary critics use the terms crisis and climax interchangeably, "the climax usually follows" but can sometimes overlap with the crisis (Dr. Wheeler, "Literary Terms and Definitions: C"). At the point of crisis, the reader is still unsure what the resolution will be, if the protagonist will succeed or fail in battling the conflict. Hence, to find the climax, we first want to determine the resolution.

In Richard Edward Connell's short story "The Most Dangerous Game," we know the story resolves with Rainsford deciding "he had never slept in a better bed" than General Zaroff's, which means that Rainsford won the battle between he and Zaroff in Zaroff's bedroom. Hence, we also know that the climax comes just prior to this moment.

Before Zaroff has dinner that night and locks himself in his bedroom, the last thing we witness Rainsford doing is standing on the beach, staring into the sea, and hearing the dogs drawing closer. The narration in the paragraph ends with, "Then he leaped far out into the sea ...." At this point, the reader is not completely sure what happened to Rainsford. It seems as though he may have committed suicide in order to escape being caught by the dogs and Zaroff. Since the resolution of the story is uncertain at this point, we know that this is the moment of crisis in the story and that the climax is soon to follow.

Hence, the climax is the moment that Zaroff locks himself in his bedroom, goes to his bedroom window to admire the moonlight on the courtyard, and turns around to see Rainsford, "who had been hiding in the curtains of the bed," standing there. At that moment, we know that Rainsford has outwitted Zaroff and won the game; all that's needed next is the final battle. Since at this moment the resolution is already in sight, we know this moment is the climax.

In addition, The conversation in which Zaroff declares Rainsford has won and in which Zaroff realizes Rainsford intends to fight Zaroff to the death counts as the falling action that leads to the resolution.

In "The Most Dangerous Game," how is the ending ambiguous?

Ambiguous means that somthing is open to or has several possible meanings or interpretations.  The end of this story is ambiguous because the author does not really tell us the Rainford kills Zaroff.  Zaroff finds Rainsford back at the house.  Zaroff states that one of them will be food for the dogs, meaning that one will be dead, and one will be sleeping in an "excellent bed."  The story ends with Rainsford in the bed so we can assume that Zaroff is dead. The author does not state this specifically, but leaves the ending in an ambiguous manner.

"In contrast, Connell takes a different approach at the end of the story. Having stretched out intense moments throughout the story, including the involved description of General Zaroff's return, Connell quickly describes the final confrontation. He grants it only a few paragraphs of sparse dialogue before ending the scene abruptly with "He had never slept in a better bed." By describing none of the final battle, Connell stretches the suspense as far as he can. He waits until the last two words of the story to reveal the survivor with: "Rainsford decided."

As the main characters in fiction are mostly dynamic characters, the reader expects some type of change in Rainford.  So, your supposition is correct. 

When Zaroff and Rainsford dine the first night after his capture, the general describes the type of hunting in which he engages and seeks the approval of Rainsford as he says, "Surely your experiences in the war--"  Horrified, Rainsford stiffly finishes the general's sentence: "DId not make me condone cold-blooded murder..."  Later, as he is hunted by Zaroff, Rainsford experiences the "fear of pain and the fear of death" of the animal at bay, a condition he scoffed at when talking with Whitney on the ship before his accident.  While hiding in the tree he sees Zaroff leave: "Rainsford's second thought was even more terrible." He knows the fear of the animal at bay. 

As the general finishes his after-dinner liqueur, he reflects that Rainsford "hadn't played the game." As Zaroff retires for the night, Rainsford comes from behind the curtains and tells Zaroff, "I am still a beast at bay...Get ready, General Zaroff."  Rainsford means for the general to prepare himself to fight as he is ready to kill a man.   

After killing Zaroff, Rainsford has "won the game" as the general has told him and will "sleep in this very excellent bed."  Since Rainsford sleeps in the bed and enjoys it, he is, therefore, no longer the hunted. He has done what once horrified him.  He is changed, for he has killed a man.  There is little ambiguity here except that the author does not directly state that R. kills Zaroff.

What is the climax of "The Most Dangerous Game"?

The climax of a story is often the most exiting point in the action, but since the entire story of "The Most Dangerous Game" is pretty exciting, locating the climax isn't a simple as it may be with other works of fiction.

For this story, you need to examine the events carefully to determine the main turning point, which is another indicator of the climax. Where in the story does the action or a character change?

I think the most obvious answer is near the end of the story when Rainsford confronts Zaroff in his bedroom. Zaroff tells Rainsford that he is free to go, since he won the game. At this point Rainsford has the opportunity to show his moral superiority to Zaroff, but instead, he chooses to continue the game. Interestingly enough, after having killed Zaroff, Rainsford does NOT free the "prey" that Zarroff has been holding for future hunts. This inaction further supports the idea that Rainsford's confrontation with Zaroff is the climax of the story, since the possibility of his replacing Zaroff (symbolized by a comfortable night's sleep in Zaroff's bed) definitely indicates a change, or turning point, that takes place in Rainsford's character.

Are all loose ends resolved in "The Most Dangerous Game"?

The conclusion of "The Most Dangerous Game" is suggested rather than stated overtly. The story ends with Rainsford sleeping comfortably in General Zaroff's bed, having presumably killed the General in single combat. The author does not reveal what exactly has happened to Zaroff or whether Rainsford has fed his corpse to the hounds, as Zaroff himself suggested.

Several matters are unexplained in the story. We are never told how General Zaroff manages to maintain such a luxurious lifestyle in this deserted place. He does not appear to have servants, with the exception of Ivan, who presumably does not cook his meals. In any case, when he enjoys his final excellent meal, Ivan is dead. It is also not clear how Zaroff has all the comforts Connell describes delivered to the island.

The nearest thing to a true loose end, however, is the fate of the prisoners in the institution Zaroff calls his "training school." He tells Rainsford that he has about a dozen pupils "from the Spanish bark San Lucar" who are poor specimens and will not provide him with any sport. Now that Zaroff has such a celebrated hunter as Rainsford in his sights, he gives up any idea of hunting these wretched sailors, and the reader has no idea what happens to them.