This would be a nice, knock down drag out brawl that would probably push both men to their limits.
I'm not a huge gamer or anything but based upon what I've read in various character biographies regarding Kratos, his powers have often fluccuated. He's defeated some extremely powerful foes throughout the game franchise but he's often done so due to tools & help that've been given to him by the gods. In the first game, for example, he was given a sliver of Poseidon's rage, he had Poseidon's Trident, he had Zeus' Thunderbolt and many other special weapons that he would eventually lose. I think a similar pattern emerged in the sequel games in which Kratos is able to temporarily gain additional weapons & powers before ultimately losing them.
If we're talking about Kratos as he was in the very beginning of the GOW series in which he has only his physical attributes and the Blades of Chaos, then I see Wolverine eventually beating him. Wolverine's healing factor coupled with the fact that his bones are infused with a virtually indestructible metal with claws that can cut almost any material and nearly unrivaled combat experience is enough against most opponents.
In terms of strength, Kratos has the edge but that's nothing new for Wolverine as he's frequently gone one on one with powerhouses like the Hulk, Thing, Namor and Colossus and fought to either a standstill or victory in most cases. In terms of stamina, I'd probably put it pretty even as Kratos is able to drain the lifeforce of others after wounding them with the Blades of Chaos. That would be a hurdle for Wolverine to overcome but he's defeated Omega Red at various times in the comics, who's able to do the same thing with his Carbonadium coils.
Agility and reflexes I'd give to Wolverine. Neither he or Kratos are on the same level as someone like Spider-Man in these areas, but they're both at superhuman levels but I think Wolverine has a slight edge.
If Wolverine faces off against Kratos when Kratos is in possession of all the various other powers and weapons he's had throughout the series, then it'll be a different story I think. Wolverine would give him a tough go before Kratos' physical powers along with his energy powers and godly weaponry would give him the win.
Since I didn't see any of those listed in Kratos' character biography for this tourney, I'm guessing that it's mostly just him and the Blades of Chaos so I'm sticking with Wolverine.
It'd eventually break down into a bloody brawl because neither man is known for being especially graceful in fights. While both highly skilled and do often display such skill, it's not at all uncommon for them to just throw technique out the window and go at their opponents like rabid animals.