who wins the game ? howdo you know he won ?

Rainsford won the game. We know this because the story ends with him sleeping in Zaroff's bed.

"That night, Zaroff eats a good dinner, all the while lamenting the death of his faithful servant Ivan. After drinking a nightcap, he heads towards his bedroom. Immediately after switching on the light a man appears from behind the bed’s curtains. It is Rainsford. Zaroff congratulates Rainsford on his win. Rainsford tells Zaroff that the game is not over yet. Zaroff agrees—whoever wins the last battle would sleep in his bed, the loser be fed to the dogs. The story ends with Rainsford remarking on the quality of the bed he slept in that night."