This is one of those things that took 25 years to accurately quantify in the road riding community. In that arena, many thought 20c tires (about 1" wide) inflated to 130psi gave the lowest rolling resistance. It was the Belgians who really promoted 25c tires at 90psi because their roads suck.
The bottom line - this is about tire deformation. Using that road tire example: If you are riding on a polished wood velodrome with almost no surface imperfections at all, you can get away with 18c tires inflated to 160psi. But, as the surface gets worse, so too will the ride and that tire will begin to ride like a wagon wheel and deflect off those imperfections and not roll over them. The imperfections need to dimple into the tire as it rolls along. Too much air and that can't happen. So, lowering psi would allow the tire to roll OVER those imperfections and not glance off them. You can only go so low before you have to offset that psi with added volume...tire width. Width and PSI go hand in hand as it relates to resistance. You can't discuss rolling resistance and tire width without including air pressure.
Back to the dirt - The road tire analogy still translates to the dirt. A thin tire will have lower resistance as long as the surface is pretty tame. Maybe a 1.9 tire would be best at 40psi on a buff fire road. Should the terrain get more ugly, that optimal combo might be 30psi on a 2.3. Like all things, the extremes are seldom optimal. 1.9 tires won't always fit many situations, nor will 2.4 tires. This is why most mountain bike tires fit in that 2.1 range. Same for road tires. 23c tires on the pavement are pretty standard.
Probably the worst two scenarios are small tires under inflated - flat city. Or, big fatties over inflated - big weight, high resistance.
