Possibly Shade's Children (1997) by Garth Nix?
In a futuristic urban wasteland, evil Overlords have decreed that no child shall live a day past his fourteenth birthday. On that Sad Birthday, the child is the object of an obscene harvest resulting in the construction of a machine like creature whose sole purpose is to kill.
The mysterious Shade — once a man, but now more like the machines he fights — recruits the few children fortunate enough to escape. With luck, cunning, and skill, four of Shade's children come closer than any to discovering the source of the Overlords' power — and the key to their downfall. But the closer the children get, the more ruthless Shade seems to become ...
This Amazon review mentions a game of chess and Ferrets:
Puberty is rough, but in Garth Nix's science fiction thriller, it's deadly. You see, the "Change" has taken place in this once-familiar world and children who reach the "Sad Birthday" (that's 14 to you) are harvested for their brains, which in turn are placed in creatures used for war games by Overlords. These unlucky children go off to the Meat Factory before having their brains implanted in either a Screamer, a Tracker, a Ferret, a Myrmidon, or a Winger, the beasts used by Overlords in their sprawling and violent battles. For these seven Overlords (four men, three women), it's like a giant chess game, and the pawns are the children their creatures hunt down.
Found by searching this site for
[story-identification] ferrets which returned, among others, Story Identification: Human Resistance to Conquering Aliens, the description of which had battles fought by proxy and robots:
This is a story I read back in the late 1990's or early 2000's. It was a sci-fi story and the plot was centered in a city. I believe it was on Earth. There were different tribes of Aliens that fought each other through proxy. Each tribe had augmented humans in different ways and used the humans to fight each other.
I believe there were also robots or cyborgs of some sort.