From what I've heard, here in Minnesota, almost all off-road winter riding is done on Fat Bikes, because riding on snow (even packed snow) is difficult with any skinnier of a tire. Fat bikes aren't cheap either, though they're cheaper if used.

In my opinion with winter riding there's almost only 2 sizes of tires that work - 30c-40c, for riding on pavement, where if there's fresh snow the tire cuts through the snow to the pavement. Or a fat bike which is as good as you can get on snow where it's either to deep to cut through, or just not possible (offroad). I'd be hesitant to spend money on a bike that only takes 2" tires or anything close to it for winter riding, myself.