Duke is one of those cards that seems pretty innocent at first, and you think, most of the time you just ignore it. But then you play a couple interesting games and... you realize it has the potential to be extremely game-warping.
Part of the problem with evaluating duke, as is the case with basically all your alternate VP strategies, is that your clocks get all messed up. Lots of people have a good feel for how long the game is going to be given the kind of deck they're building. Oh, it takes BM/Smithy about 15 turns to complete the game, say. Well, that's based on rushing to half of the provinces, then slowing down a little for duchies toward the end. In a game which your opponent isn't helping you end the game by buying provinces himself, you tend to choke a bit harder on green cards, and it can go significantly longer.
So that's the first lesson of Duke: in non-mirrors, you're in for a very long game. Prepare accordingly.
Okay, next thing to notice is the math on Dukes specifically. How many of them are you going to need? When do you want to buy duchies, and when do you go for dukes? Well, after a little thought, you'll see that the first three should definitely be duchies, and then with 3 duchies and 0 dukes, a duchy is worth as much as a duke, and thereafter, if you want to maximize your immediate points, you should alternate. However, if you're playing to have duchies/dukes as your main victory points, you're eventually going to want at least 6 duchies, right? Because in order to beat all the provinces, you need at least 11 duchy/dukes, and it's optimal if you've got 7 duchies and 4 dukes, but 6 and 5 will get you a tie. As a rule of thumb, you just buy duchy duchy duchy until the duchies are gone (well, you might want to consider going duke before grabbing the last duchy), with the big factor being denial. If your opponent isn't going for dukes as well, getting the duchies denies them a lot more points than getting the dukes does. And even if they are mirroring you, the duchies are what give the dukes their potency anyway, so they're what you need to focus on.
So, you want to get all this stuff - how do you do it? There are actually a lot of ways to do it. Because there's a higher ceiling on points from duchy/duke than from provinces (technically higher even than colonies, but you need like ALL of them), I've actually seen some engine or combo kind of decks that go for duchy and duke. It's pretty interesting, and it's something you need to keep aware of. But if you're going that route, you need to make sure that your deck is capable of getting lots and lots of 5 buys (or gains) without the engine gumming up too much with the green and the big deck. Most engines do gum up like that, and so aren't so good with Duke. Still, there's some nice combo potential in a lot of ways - Tactician can work really well, bridge or highway and some of your gainers is cute, heck bridge by itself is pretty nice. Generally though, your traditional +Cards/+Actions engines don't work so well (though they still can), so you need to be a little more creative to go with an engine or combo and dukes. Plain old money goes very nicely with dukes, on the other hand - even copper.
Because all your key cards cost $5, copper brings your $$ average closer to being there. And because you can actually get above that $1 per card you want, the extra coppers smoothing out your money distribution is important. Moreover, the coppers stop each successive green card from hurting as much. Okay, copper isn't something you should really be buying that often anyway - you should be getting something better most of the time. But if you have an extra buy, and you're playing money for dukes, grab those coppers, even from turn 1. This suggests you actually want those buys. And you do; +Buy cards help BM for duke a lot more than traditional BM precisely because those coppers are very nice to have. And so those cards are nice to have. Of course, cards that can get you lots of silvers or golds are magnificent as well.
Sifters (e.g. warehouse) are also going to be pretty nice for your BM for dukes deck (actually, engines too for that matter), as no matter what, you're going to have pretty high variation in money production from card to card, as such a high number of your cards produce $0. I want to pay special attention to oasis here. It produces a little money for you as well as sifting, and in these kinds of decks will most often be better than silver, as it tends to drive you to that magical $5 that you want. Ultimately you'll probably want a mix between oases and silver, but lots of oases are going to be your friend here.
In addition to sifters, you've got similar cards which help you manipulate your shuffles. I'm looking squarely at mandarin and courtyard here - interestingly enough, one's pretty weak at going provinces (though, I'm convinced, quite underrated), whilst the other is really strong - but even stronger for duke strategies. The idea's simple - you need exactly $5, and you can save the rest, as much as possible, to hit $5 again next turn.
You've got all kinds of cards that are always good with green cards, things like Hoard, crossroads, and scout. Okay, that's only half-joking on scout. No, seriously, it can help. Those things are nice here, too, of course.
Okay, so that's the strategy - what about the tactics?
First, playing against a duchy/duke opponent. You want to know that duchy/duke is a little better than going for provinces on a straight-up big money board. So if you're not going for the dukes yourself, you must have a reason (this is very often the case).
Usually that reason is because you think there's a deck to be built that will get you ALL the provinces before your opponent gets to 11 duchy/dukes. If that's the case, you should NOT, I repeat NOT try to contest duchies until very late (assuming that you've identified that your opponent is going duchy/duke, hold off on duchies yourself until later than you normally would), when you've got the vast majority of your points, as they'll only clog you up, probably much more so than your opponent.
Sometimes, though, you think you can get to a large number of provinces pretty quickly in some kind of deck that can deal with having some duchies. Here, you want to contest early, keying on 2-3 duchies, and then leaving alone. Of course, if you can get 4+ duchies, you can potentially even just head duke yourself. The thing here is, if you can get that 3rd duchy, they need ALL of the other duchies AND ALL the dukes to have more points than you do, if you can get all the provinces. And then they have 1 more point than you. With 2 duchies, they'll need the rest of the duchies plus 6 of the dukes to equal you. The thing is though, if they can sneak a province, you can be in trouble, not to mention that 2-3 extra early green cards will probably seriously hinder your chances of getting all 8 provinces fast enough. There just aren't many I'm-going-provinces-and-have-time-to-contest-duchies decks out there. And if you can't get 2 duchies, it's really not worth it (until the game is on the verge of ending).
If you are playing your duchy/duke strategy against a province-seeking opponent, something very important is to stick to your strategy. You're in it for the long haul. You're in it for a long game. You need the game to go long, so assume it will (and make it happen, if you can). Don't buy your first duchies too early, and find yourself having to rebuild your economy halfway through the game. It's better to build a bit of an economy first. Simulations show that around $15 money is where you want to be, which is only a few coin less than when you're playing straight big money for provinces. Support cards will change that a bit, of course. But stay your course, gobble up all those duchies, resist the temptation for the quick points of going dukes early - don't switch until the duchies are out, or at least until you've got your 7. Whether or not to go for a province on a lucky $8? Well, generally, you don't want to. Your dukes will end up being worth as much by the end of the game, and, more importantly, you're speeding up the end of the game for your provincing opponent, which gives them real chances. After a certain point, though, once you're into your duke run, and you have a lead, particularly if your opponent is going for a big mega-turn, but basically any time you're in the duke stretch with a lead, you can go for it. Just do the math to make sure that it's more province denial on your opponent than it is speeding up the end of a game they have a good chance to win.
Finally, the mirror matchup. The mirror is probably going to be determined by who gets more duchies. Obviously, there are a lot of different ways to go for duke, so you could both be going for dukes while playing totally different decks, and ultimately, you want to follow your strategy for that particular board. But in general, the only reason a player who wins a duchy split in the mirror will lose is if they are totally out of economic gas, and being crippled, have to spend a lot of time rebuilding before they can get very many duke at all. Because, if you win the duchy split 5-3 even, even if you lose the duke split 5-3, you have a 6 point lead (heck, if you lose it 6-2 you're only at a 2 point deficit). The upshot of this is that the mirror matchup often turns into a game of chicken. You're both sitting there, building up your deck, and one of you makes the move to dive for the duchies first. The other one probably has to follow pretty soon. Take stock of what's gone on when the duchies run out. In a 4-4 split, you keep the pedal to the metal on those dukes. Same thing, more or less, if you win the split. But if you've lost the split, you may well want to give serious consideration to spending several turns re-tooling your economy to go for some provinces, only contesting dukes later on. This mixed strategy will often hold a better chance of success for you in the long run, as you can still pick up a couple dukes later on. 3 duchies and 2 dukes with 5 provinces matches up against the rest of the dukes, and it's probably easier than trying to stick with the full mirror, because once those dukes run, the duchy/duke player needs to find a 3rd pile.
Finally, there are two specific combos that I've got to mention here. The first is pretty well known, Horse Traders/Duke. There have been a few articles about this combo already, I suggest reading them if you aren't familiar.
The second is even stronger - Feast/Duke. This is actually a rush strategy, as you can start basically right away pounding out those greens. Silver on 3, Feast on 4, Duchy (later duke) on 5 is pretty fast, and because getting to 4 is actually a heckuvalot easier than getting to 5, especially in these decks, you can do it much sooner. Is it still gonna be a long game? Yup. But not much longer, actually, than your average province vs province affair.
Works With:
Duchy
Duchess
Money
+buy cards - your woodcutters, bridges, and especially horse traders
Filterers or sifters, like cellar, warehouse, oasis...
Gainers, particularly of silver - even workshop! - but most notably bureaucrat, particularly in the mirror
Cache
Hoard
Feast
Duchy
Did I mention duchy?
Doesn't work with:
Colonies
Particularly fast engines
Opponents that are resilient to greening (a la chancellor/stash, a nightmare matchup for this deck)
Hunting Party
Opponents' bishops (you actually want those coppers!)
Trash-for-benefit cards (most normal trashers, too)
Embargo - mostly (i.e., embargoing the provinces in reply can be very strong as well, and it totally changes the complexion of the game).
Swindler
