This is about the actual club. You may be looking for the story, The Nimrodian Club.

The Nimrodian Club was an organization founded in 2314 by Robin Maxwell, an incredibly rich man who was bored with life. Other founding members included Carl Hinton and Tristram Keeler-Finch. The rules of the club were that each year a computer would randomly select two members of the club. One week later, they would face off in a duel in a secluded woody area, and be charged with hunting the other down. The victor would be the one who killed the other; if one ran away, then the entire club would be charged with hunting down and killing him. The only exception would be for members who were gravely wounded.

The first lot ws drawn on June 22nd, 2314, placing Hinton against Maxwell one week later. Hinton won, cutting Maxwell in half with a laser.