Why does Rainsford agree to play General Zaroff's game?
Rainsford agrees to play General Zaroff's game because he wanted to see if General Zaroff would keep his word and let him go if he survived three days of the hunt. It is always important to follow the word of a "gentleman and sportsman." I also think that Rainsford wanted to see if he could match wits with Zaroff. Zaroff says it is like a game of "Outdoor Chess." It all depends on the moves and how you outmaneuver your opponent, which Rainsford definitely made all the right moves. Rainsford knew if he didn't agree to the hunt that Ivan would enjoy diposing of him, but I think he felt he had a better chance using his wits to win.
Reference: The Language and Literature Book by McDougal Littell.
Rainsford is trapped on this island. He is, at this point, a prisoner of General Zaroff. On one level, he has no choice but to agree to play, unless he wants to be killed immediately. By playing, he has a fighting chance.
Yes, Rainsford does not have much of a choice. He must either compete in the most dangerous game or face torture at the hands of Zaroff's assistant.