"The Most Dangerous Game" is often cited in anthologies for the structure or plot, not to mention its great suspense. To identify a symbol is a bit of a stretch in this work, but there are items worthy of use.
In the beginning Whitney and Rainsford discuss the jaguar. As they talk, Whitney points out that even an animal can contain fear, while Rainsford laughs the thought away. Although this is a great moment of foreshadowing and as it is later fulfilled we see the irony of an animal's capacity to fear, I think Connell uses this animal to represent the innocence of all animals. This story was written as a reflection of the big game hunting that was done for sport in the early 1900s. Connell uses this animal to later illustrate what it would feel like if a human had to endure the fear of being hunted for days on end.
Similarly, the cry that Rainsford hears while swimming close to shore after hearing small caliber gun shots, symbolizes the torment and agony animals go through when hunters aren't responsible about their sport. A kill shot is a necessity for many moral hunters because they actually care about wildlife game management. They don't want to put animals through pain, but I think Connell uses this cry to demonstrate that hunting creates pain, and that death is painful, no matter how it is done.
The term game is used to have a double meaning, thus it is often a play on words when used in the story to mean either or both the animal and the sport. I think you could use game as a symbol to demonstrate that their is a win and a loss. Connell would like his audience to understand that hunting may be a loss for us in the long run.
These may be some stretchers, but the work has many other devices that it is regularly recognized for. I hope you can use some of these.