The circumstances of an average pet dog and average pet cat weigh heavily in favor of the dog, based on what seems like a universal estimation of the dog being much larger. But that would be an unfair fight. Pound for pound the question is tougher. Someone already mentioned that felis and canis varieties of similar size avoid each other in the wild. Most predators prefer smaller prey. Wolves, as a pack, might go after a solitary puma, with the advantage in numbers, but even then the wolves would probably be cautious about getting close to another animal with large teeth.
So in the hypothetical cage match between a dog and cat of the same size what would happen? Let’s assume we have picked a dog breed that has some instincts and ability for a one on one fight. All cats seem to have approximately the same methods and resources for killing, but let’s keep it close and chose a leopard, solitary, in the size range of dogs. The leopard is known for its intelligence, but giving IQ tests to cats is difficult, and it would only determine how well the cats are at taking IQ tests, but leopard’s certainly wouldn’t be categorized as the dumbest cat. Let’s assume also that the animals have equal experience at fighting, but not against the other specie specifically.
The cat has teeth and claws for weapons. I suspect the cat is more agile, but maybe not enough to get the upper hand. The dog has a more powerful jaw and neck, and a body designed to take advantage of that. The dog may have an edge intelligence, but in the heat of battle it probably doesn’t help much.
I call it for the dog. If the dog gets his jaws on any part of the cat, he can hang on, shake and swing the cat around, and kill the cat. The cat’s teeth and claws aren’t strong enough to control the dog, or deliver a kill in one shot. Big cats kill comparable sized and larger prey by strangulation via the snout or throat, a slow process. The fight is close though. If the cat can under the dog, it might get its jaws into the dogs throat in a manner that leaves the dog without any more offense. So if a series of fights were held, the outcome wouldn’t always be the same.
I mentioned before that the animals had no experience fighting the other’s kind. If either were well trained in dealing with the other’s tactics, it may become a dead heat. Dog’s are easily fooled and distracted. A cat that can learn to say ‘Look at the curls on that French Poodle’ might have an obvious advantage. On the other hand, skillful manipulation of a dog’s tail can mesmerize a cat. If either species can imitate the sound of a can opener, it will win every time.