How did the game end?

The game ended with Rainsford sleeping in Zaroff's bed.

The general made one of his deepest bows. "I see," he said. "Splendid! One of us is to furnish a repast for the
hounds. The other will sleep in this very excellent bed. On guard, Rainsford." . . .
He had never slept in a better bed, Rainsford decided.

Zaroff believed that Rainsford hadn't actually played the game because he jumped off the cliff.... killing himself rather than be killed. Unfortunately for Zaroff, he was wrong.

Two slight annoyances kept him from perfect enjoyment. One was the thought that it would be difficult to replace Ivan; the other was that his quarry had escaped him; of course, the American hadn't played the game--so thought the general as he tasted his after dinner liqueur.