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Confronting Zaroff allowed Rainsford to not only kill him, but to let him know he'd won as well. Rainsford wanted Zaroff to know he'd been beaten at his own game.
The Most Dangerous Game
R was like a trapped animal with his back to the wall. It was kill or be killed. He had to kill Z in order to survive as Z would never have given up and it was unlikely R could survive trapped on the island with Z. Z would have eventually figured out R was still alive and continued the hunt.