A narrower tire will help your traction in winter, but not much in the mud.

 

from my experience with a locker it makes a huge difference in traction, more traction when both tires are spinning, we dont have much snow in hawaii but, in mud I have used lots and lots of brands of tires, Ive had mud terrains, all terrains, Good year MTR's, all kinds of swampers. I didnt really see a difference in the mud with all terrains and mud terrains, swampers made a huge differnce. But in your case I would get the good year MTR's, they have stronger sidewalls and they grip alot more.