I'm gonna say Shogun 2.
I don't know what everyone's talking about when they say "Rock, paper, scissors." I didn't get any of that, and I crushed it on legendary--couple times. Yeah, some unit types fare better against other unit types, but playing smart is always superior to whatever bonuses the units get against other units.
Samurai are strong, yes, which is why you don't fight armies full of samurai on even footing on the higher difficulties. You always make sure you have the advantage; i.e. more units, or you're defending from a fort. And if you're playing on the lower difficulties, you can do basically whatever you want, so again, "Rock, paper, scissors"--moot. All of the TW games have had some sort of "Unit X beats unit Y" mechanic such as yari > cavalry, and it's not a bad thing. I don't know why Shogun 2 is getting singled out.
There are are some issues with it, but they're few & far between.
-Sea battles are, honestly, pretty lame. There's just not much to 'em, and they're extremely boring. UNLESS you go Nanban... Nanban is super fun!
-Enemy army compositions are sometimes a little whacky. Like occasionally you'll go up against a squad of, say, 2 samurai, 1 general, and 17 bow samurai. They have no chance of winning, but you've no chance of not sustaining extremely heavy losses. It's like a kamikaze attack, but with no follow up. This comes back to the hit-or-miss AI that's been present in, well, every TW game.
-The Realm Divide mechanic is a little silly. If you've been allied with someone--or someone's been your vassal--since the beginning of the game, it doesn't really make sense for them to turn on you, especially if you're much stronger. But they still do it, although sometimes they'll give you a few extra turns.
-Choosing how to level up your generals pretty much always goes the same way. Leveling them up as melee combatants just doesn't work on anything but the lowest difficulty settings, which is kind of a shame. If any TW game was begging for kick ass, Dynasty Warriors-esque generals, it's Shogun 2.
-Diplomacy is kind of stupid at times. There were some memorable diplomatic moments in one of my legendary campaigns... towards the end of the game, I had a treasury of about 50k and about 40 provinces. Otomo, who owned 6 provinces, had virtually no income and was at war with pretty much everyone (including me), continually asked (talking almost every turn) for the ENTIRETY of my treasury in exchange for a trade agreement that would've made me about 300 income per turn, and a peace treaty...
That's more or less all that really frustrated me about it, but I may be forgetting some stuff. Now onto the good stuff.
+"Hey, everybody, I'm a ninja! I'm doing ninja stuff now!" PLUS! CGI assassination cinematics.
+Samurai. Samurai everywhere.
+Graphics scale well, and you can even scale how many units are in each squad.
+Zooming in and watching the action unfold is not unlike watching a movie. As long you have the blood pack. Though, you probably won't be doing this much on higher difficulties.
Honestly, I'm realizing, everything that's good about it is more or less what's good about other TW games, but Shogun 2 just does it real well, and in my opinion, has the fewest flaws. The learning curve is solid. You'll suck at everything at first, but you'll get better. The tutorial is solid, too.