Here is how I would play each faction in certain matchups, which may shed a little more light on my rankings.
1. Swadia.
Against Rhodoks. This is potentially your toughest battle. The Rhodok AI (at least with Diplomacy installed) will usually refuse to charge, unlike other factions. The cheese to use here is to keep your troops back and repeatedly circle their formation by yourself, picking off troops at the edges. Works with a polearm. Probably works a lot better with a bow, but I don't do horse archery personally. You just keep doing this until they finally charge. Then you grab your cavalry and circle around them so that they spread out. Clumped Rhodok Sergeants equals death, even to Swadian Knights. Spread them out, position cavalry near Sharpshooters, never Sergeants, and charge. If you choose to not use cheese to make them charge, you will suffer casualties. However, as long as you have enough "I win" units called Swadian Knights, you should still be able to beat Rhodok armies that outnumber your own.
Against Sarranids. As I said in the rankings, these guys are about your equals. You'll win because you're human controlled and you likely built more heavy cavalry than they did. It's really that simple.
Against Vaegirs and Khergits. These two literally have nothing on you. Their cavalry is worse and their armor is light. Just charge and win.
Against Nords. Just be smart and circle to spread. They might stay in formation like the Rhodoks. Just use the same tactics. If the army outnumbers yours and they have enough Huscarls, you will take casualties.
2. Sarranids.
These guys are so similar to Swadia in how they play that the matchups are pretty much all the same.
3. Vaegirs.
These guys are a little more interesting to play than the Swadians or Sarranids because their cavalry isn't as good but they have great archers. This calls for more of a combined arms strategy. You want to give your archers the chance to inflict casualties before the melee rather than getting to a cavalry charge as quickly as possible as with the other two. If the enemy wants to approach you slowly, then that is great. Just rain arrows the whole time they approach. When they finally charge, crash into their flank with your cavalry. How you play this faction depends a lot on what units you decide to build, though. If you go the expensive route and just build knights, you will pretty much play like Swadia or Sarranid. But then that would beg the question why did you pick Vaegir.
Against Nords. These are your biggest enemies in terms of map position and AI tendency, and they are a pretty difficult matchup for you. They have huge shields to negate your archers and their heavy weaponry is a serious threat to your lighter cavalry. You have to get them in open terrain and spread them out. You need a cavalry heavy army for this matchup. Your archers won't be worth much. If all knights is too expensive, guards can do okay. Just make sure you focus on suppressing their archers with your cavalry.
4. Rhodoks.
Make your army about 1/3 Sergeants and 2/3 Sharpshooters. Make even your companions foot soldiers. There's not much use in having 8 isolated horsemen that don't fit into the tactical concept. You can make super archers with bow, arrow, arrow, great long bardiche. Better ranged units than Sharpshooters and better cav killers than Sergeants. Anyway, set your Sergeants up as a screen in front of your Sharpshooters, preferably on a hill and let any enemy come to you. Have Sergeants clump together if the enemy is cav heavy. Use above mentioned cheese to make them come to you if necessary. You will win every battle with minimal casualties doing this. The Sharpshooters will just annihilate anything but Nords before they even reach the line. Clumped Sergeants on a hill will own Swadian Knights. That's why I say with some basic tactics they can be the best faction in the game. The only downside is it takes some patience. You can't just auto-charge battles. You can be the one guy on horseback to circle and distract. That definitely helps. However, I like to be on foot myself because when the cavalry gets stopped against my wall of Sergeants, I like to be there with a huge weapon like the Great Long Axe to butcher the enemy heavy cavalry. I can sometimes butcher a huge number of them. That helps the cause even more than being on horseback.
Against Nords. Even though Nords aren't that great overall, they are the toughest matchup for Rhodoks. Their shields will allow them to get to your line relatively intact, and once they do you'll regret having a 2/3 ranged army. This is the one faction that requires you to adjust your ratio to more like 50/50. It's tough, but you will win anyway because you're the player.