The single biggest factor in determining the power to incapacitate is the damage dealt to the BG per second. This has two major considerations; controllability and depth/diameter of holes produced.
Any bullet - ANY bullet - in the right place will instantly incapacitate. A .22 in the eye socket will penetrate the brain and short circuit a few vital neurons. However, a .22 in the wrong place is practically useless; lodged in the arm, you'd be lucky to break bone. You therefore need a LOT of .22 bullets in the wrong places to make a guy bleed to death, even though each one if it only contacts soft tissue would probably go right through. Therefore, the issue is having a bullet big enough that you can just slightly miss and still contact vital tissue like heart muscle, blood vessels or nerve trunks, which cause rapid incapacitation. The bullet must also penetrate deeply enough to get to said tissue.
By that consideration alone, a .50AE +P+ HydraShok is the cartridge of choice. It makes a big hole (it would probably expand to 3/4" or more) and penetrates deep, so you're virtually guaranteed to damage something important and do a lot of damage per shot. However, .50AE is a VERY powerful round needing a VERY big gun and producing a LOT of recoil. All references to the Matrix aside, it is not a practical cartridge based on the availability of weapons to fire that round that are of a practical size for concealed carry.
By contrast, a smaller-diameter round may not make as big a hole, but it also requires less energy to propel through a BG (thereby doing maximum damage), resulting in less recoil. You can then have a smaller, easier-to-manipulate, easier-to-control weapon, allowing you to fire faster. Say you can fire a 9mm accurately twice as fast as a .50AE. The .50AE doesn't do twice as much damage; therefore even though the Desert Eagle wins shot-for-shot, a 9mm does more damage over the same time spent shooting.
Therefore, the final answer to the $1,000,000 question is that the best cartridge is the biggest and highest-power load that you can fire accurately at a good rate of speed. "Good" is subjective, but if you can rapidly unload at 5 yards and achieve a 6" grouping I think you can handle your average BG. It doesn't matter what that caliber is; if 9mm is your most comfortable caliber, go for it. If you can shoot a .40S&W at least as fast, that's an advantage. If you can fire a .45ACP cartridge that fast as well, more power to ya. And yes, if you can shoot a Desert Eagle rapidly and accurately, and it's practical for daily life
then by all means carry one.