Well, you can look at this logically, also. There may be a level advantage, or disadvantage! But supposing all these Pokemon are at level 100, we'll take a look at their attacks and stats, supposing these are wild Pokemon.
Houndoom
Attacks:
Odor Sleuth (Learned at level 35)
Faint Attack (Learned at level 43)
Flamethrower (Learned at Level 51)
Crunch (Learned at Level 59)
Stats (Discluding Nature)
HP: 290
Attack: 215
Defense: 135
Speed: 225
S.Attack: 255
S.Defense: 195
Arcanine
Attacks:
Helping Hand (Learned at Level 37 [Growlithe])
Agility (Learned at Level 43 [Growlithe])
Flamethrower (Learned at Level 49 [Growlithe])
Extremespeed (Learned at Level 49)
Stats (Discluding Nature)
HP: 320
Attack: 255
Defense: 195
Speed: 225
S.Attack: 235
S.Defense: 195
Ninetales
Attacks:
Flamethrower (Learned at Level 29 [Vulpix])
Safeguard (Learned at Level 33 [Vulpix])
Grudge (Learned at Level 37 [Vulpix])
Fire Spin (Learned at Level 41 [Vulpix])
Stats (Discluding Nature)
HP: 286
Attack: 187
Defense: 185
Speed: 235
S.Attack: 197
S.Defense: 235
My Thoughts
Well, Houndoom has the advantage of two Dark attacks, although both Pokemon have high S.Defense. And everyone here thinks Arcanine has the Speed advantage over Ninetales, well, you're all wrong.(According to my findings) Arcanine actually has a Defense and S.Defense disadvantage. And I must say, so does Houndoom. Ninetales has the highest Defense, S.Defense, and Speed of the both of them. But Ninetales has to learn all of it's attacks from Vulpix. Ninetales also has the Ability of Flash Fire, as does Houndoom, but Houndoom as two which is Early Bird and Flash Fire. Here, I see that Arcanine doesn't stand a chance, due to it's stat disadvantage. Because, you see, Ninetales powers up when hit by fire attacks.(So may Houndoom.) I see Ninetales the winner of this big battle, due to the fact of stats and special ability. But then again, Houndoom may win if it has the Flash Fire ability. If it does not have this ability, thye Ninetales would win. (According to my findings.)