Gardens: An Introduction and Dominion Theory

Ah, Gardens. For a long time, you were my favorite card.

At some point, I determined to not lose to this card - which is probably how I got the itch for all the alternate VP that came later. It's true, though, that mastery over it is not a simple thing. I have learned an awful lot, and I think there's still a chunk out there that I have yet to bite off. But let's dive in.

Running in head-long
The most naive strategy in "going for Gardens" is to just buy as many as possible as soon as possible. People don't stay on that for very long, because it doesn't take much to figure out that your deck gets so bad, you actually have to build more to be able to afford all the Gardens. But it's not too uncommon to see players who do only a few turns of preparation before starting their charge into Gardens, buying every card possible as soon as possible (i.e. maximally jamming coppers into their decks), and trying to get lots of Gardens and lots of cards more or less ASAP. Let's take a look at a couple of examples to demonstrate this kind of approach.

Candlestick Maker
I have recently seen it floating around that Candlestick Maker is one of the better "Gardens enablers", with the implication that a headlong strategy here could make real strides. The idea was to end the game around turn 17, by which point it was supposed to be not uncommon to have 6 point Gardens. Memories can apparently be faulty: In order to get to 6 point Gardens, you need to add 50 cards to your starting deck. Turn 17 means 17 natural buys, which means you need 33 additional buys from CSMs. Work out the example yourself, and you'll see that even if you stack your deck, you can't get that many - indeed, you max out at 55 if my memory serves right. However, having a best-case-scenario mentality isn't very useful anyway, so what I'm going to do here is look more at an average case. Examples like the following can show the power of theory-crafting in planning out many kinds of strategies, broadly.

Turns 1-2 you are buying CSM. Turns 3-4 you are getting, most likely 3 more. How did I come up with 3? Well, the maximum possible is 4, but to get that, you need a CSM turn 3 with at least $4, another one turn 4 with at least $4. And you can float a token, potentially. However your deck is only capable of producing $9, and 2 cards miss the shuffle. If either of those is a CSM, you're done. If they're both coppers, you once again can't get there. If both CSMs fall in the same turn, you're once again hung out to dry. There are a lot of fail cases, and indeed it's not crazy unlucky to only get 2 more (though it is fairly unlucky). So 3 is a reasonable guess. Then turns 5,6,7 will be a single shuffle, in which you are pretty likely to get the remaining 5 CSMs. At this point, you are likely to have played CSM about 6 times. Assuming you're using all your spare buys for coppers, this means you're looking at somewhere around a 23 card deck. That means you're looking at a 4 turn shuffle next, and 1-2 CSMs, on average, missing that shuffle, with it being more likely for more to miss if you have been luckier earlier and have more cards now. 4 natural buys plus 9 CSM buys gets you 13 more cards, taking you after turn 11 to around a 36 card deck. This gives you a 7 turn shuffle, in which 0-1 CSMs will miss, most likely. Thus, after turn 18, you're looking at 53 cards.

Here is where we need to start looking at what exactly we're buying on these turns. Surely lots of the buys are heading to coppers. But we're also obviously going to want to buy actual things, too. And especially with the coin tokens to smooth us out, we don't want to waste any of the 'real' money we are producing. If we're playing this headlong rush, that means we will be buying Gardens. If we go back to the point at which we're anticipating the CSMs running out, at the end of a shuffle on turn 7, we see a 23 card deck with 3 Estates, 10 CSMs, and 10 Coppers. The money density of this deck is 20/23 or about 4.35 per hand. We can also anticipate having 1-2 tokens left when we enter this stage. Given that, it seems very likely that we'll be able to buy Gardens on each of these 4 turns, along with our 9 coppers. We're likely to pick up another couple tokens as well. This means that at the start of the shuffle on turn 12, our money density will be 29/36, which is a hair over 4 per turn. Thus, we can still Gardens-per-turn for the next 4. Following this, though, we're dropping off a bit, so by the time we go for Estates, we won't be able to double consistently. Overall, then, we're looking at 20-21 turns to be able to empty our 3 piles. If we look back, this is most likely going to leave us just shy of hitting 60 cards (which would take 1-2 additional turns, average case).

So what we end up with is: 20-21 turns to empty piles and end the game, 8 5-point Gardens, and 11 Estates, for a total of 51 points. This equals the number of points from a player who is able to empty all the Provinces without trashing their Estates or touching Duchies. Of course, the problem is that this simply isn't good enough. I'm not sure if this is even good enough to be straight-up Big Money. Probably yes, but only barely. Of course, they are almost always going to have some improvement on straight money, so you'd better have an improvement on straight CSM/Gardens if this is what you want to do.

Workshop
Workshop plays out in some ways very similarly to CSM. Gain a bunch of them, buy out of the Gardens, empty the Estates. It's the most classic rush strategy in the game. In this case, we are even Workshopping the Workshops (which is not great if you're contested, but as I often say, you want to consider non-mirrors when picking strategy, and only consider mirrors when implementing). Of course, Workshop is also terminal, which means you are going to be a little bit slower to empty them, but also means you want to stop 'shopping 'shops a bit before you actually empty them (buying the last 1 or 2 when you have exactly $3, or if it gets late enough, gaining them later on). The terminal-ness of the Workshop is a pretty big downside in comparison, but this is more than compensated for by the increased economic upside. When I gain Gardens, Workshop made $4. Even when I gain Workshop, it made $3, and $2 for Estate is still better than the $1 from a CSM. Of course, collisions happen, and without tokens, you probably aren't going to be able to use all of your $$ efficiently, providing some waste. This means you won't see the full benefit of the $4 economic potential.

Still, the games will start out playing more or less the same. Then, because you're worried about collision, you start gaining Gardens a turn or two faster. The key point is that throughout the period you are gaining Gardens, you're also buying other cards which actually help you end the game (mostly Estates), as opposed to just picking up coppers. 7 turns before you start greening, into about 9 turns of gaining Gardens+Estate, plus an extra turn to finish off stragglers means you are going to be finishing somewhere around turn 7. Of course, you're going to have gained way fewer cards in that time-span - we're looking at barely 4 point Gardens as opposed to almost 6 - but the increase in speed is a big deal. 43 points is enough to outlast 6 Provinces, which in all honesty is still not all that special, but it's quite a lot better.


Some Fundamental Concepts
There are two main approaches you can take with this kind of head-first charge for Gardens. One is to try to rush down and end the game. This is great if you can actually end things fast enough, but most often, that's just not going to be possible (the major exception, of course, is the card Ironworks). The other is to try to score incredibly large numbers of points. This also tends to not be great most of the time (Beggar is, once again, a very large exception).

Does that mean that the card isn't good? No, not at all. Does it mean it's doomed to be merely an alternative to Duchy as a means for getting a few extra points at the end of a game? Sometimes, but not always. It can still be the star of the show. How? Well, headlong rushes can still happen sometimes with multiple enablers. More to the point, if you build just long enough to be getting 1-2 Provinces and THEN dive for the Gardens, things can turn out well. But most often there are one of two strategies with which this card can get you there:

Aggro Pile
If you haven't checked it out, I might suggest my previous blog post to familiarize yourself with what this deck is all about. But how Gardens comes in is, if you add some kind of attack, particularly a junker, to the Headlong rush mix, then suddenly you are in business. You're emptying piles slower, but only slightly slower. More importantly, you are slowing your opponent down quite a lot, and while you certainly aren't shrugging the junk off yourself, it doesn't hurt you nearly as badly. This thus gives you enough time to finish up piling out without getting horribly outscored by an opponent going after the big cards.

Engine
I know, it might seem crazy. But in all honesty, engines tend to be the ways to maximize Gardens. Engines love Gardens. After I made that determination to not lose to Gardens, the way I most often did was when I went head-long, and my opponent opened Chapel... and then went for Gardens.

The thing to understand here is that engines are better suited to gain more cards over the long term than virtually any other deck. Once you get a running engine, it's very common to be gaining 3,4,5, even more cards every turn, as you add a little bit of payload and maintain your draw apparatus. Going straight for enablers doesn't permit that, and over a time-span of almost any length (look at the CSM as an example), you really aren't going to be doing better than 2 cards per turn that way. The fundamental problem is that as your deck gets bigger, your shuffles get longer, and you see your enabling cards less often. In contrast, as engines get bigger, they continue to see every enabling card nearly every turn - so even though they have fewer overall, they see them much more often.

Because of this, Engine decks routinely end up to be nearly the size of your more 'traditional' Gardens decks, and especially with the help of wasting a bunch of extra final-turn buys, they can often actually end up largely, sometimes by quite a substantial margin. Chapel-ing away ten cards is insignificant against an extra card or three per turn across the remainder of the game.




As an example of this, I would like to point to this game. Of course, this isn't a comparison to a head-long rush - my opponent didn't go for Gardens, and indeed this would have been a clobbering regardless. But the point is that, in 13 turns, I was able to gain 8 Gardens and have them all be worth 5 points, and you don't do that by rushing them, you do it by building an engine. The reason why this is so powerful in an engine is that it gives you a lot of flexibility, because it's a massive well of points which you can 'float'. If your opponent goes straight for Gardens, you clean up by just taking down all the Provinces. If, though, they go for Provinces, then you have a massive reserve of points waiting for you in the form of Gardens, which means you don't really need to worry about losing a Province split by any margin other than 8-0.


Of course, there's a lot more to the card than this, but I hope this introduction has been helpful.

Dominion: Treasure Map

Treasure Map is one of the most misunderstood cards in all Dominion. Often cited as the prime example of a card which going for can sometimes bring a weaker player a luck-based win even when it's not the optimal strategy, it is actually a card which is fairly high-skill, and which by its presence will lead the better player to win more often.

The Level Zero Strategy

The most basic thing people do with Treasure Map is buy two as fast as they can, maybe get a third to help them collide, buy some treasures, and then use the golds off the collision to rumble in for Provinces.

This is a bad strategy. A really bad strategy.

In fact, the optimal play with Treasure Map and no other cards on board barely defeats not getting Maps at all. If this were all that was going on for a given board, then it really would come down to luck in if the Map player collides fast enough. (On the other hand, that's not really much different than straight Big Money, where it's all about how well that money clumps together).

Fortunately, there is almost always something better available on the board. And the big point here is that even when the Map player collides immediately, on the vast majority of boards, the other strategy is STILL going to beat this naive one. There is just so much else you can be doing, and four Golds is not really enough to finish the game out without support.

The Level One Strategy

The next step is to add in cards to help your Maps line up with consistency, greatly accelerating when they collide and give you the pay-off. The classical combos here are Haven, Warehouse, and Chapel (though to be fair, Chapel tends to help other things more). This is way stronger than the level zero strategy, and has enough raw power to be more or less correct on a non-trivial, if still small, number of boards.

Four golds probably still isn't enough, but a key point for this strategy is that the helper card helps you to smooth out later draws as well - this is the classical purpose of Haven, if you notice, and one of the main roles of sifters - at least in money-based or "good stuff" decks.

Still, this strategy tends to not be terribly strong, because though it has some punch, it isn't anything special - just a decent version of Big Money, and not even a great one.

The Best Usage

All this might lead you to think that Treasure Map just isn't a very good card and that you can pretty much ignore it. For the most part, you would be right. But there really are some cases in which it can be key. I am taking about cost-caring trash-for-benefit cards a la Salvager and most especially, Bishop. In this case, Maps are basically a way to use 2 buys and $8 to get 4 6-cost cards of value to trash.

In order for this to work, you need a very potent engine - you're gaining 4 golds with this every turn, and what's more, you're putting them on top of your deck. You need a lot of drawing power in order to break through all those stop cards. This tends to be pretty slow to set up. But if you can get it going, this really is a lot of fuel. For this reason, Bishop really is the best combination - the Golds provide you with a huge source of points to overcome even a big deficit of green cards, and the player who is trying to buy victory is probably going to stall, trying to grab them all.

Theory: Anti-Engine Slogs

So we've talked about engines, what their strengths and weaknesses are, how to build them, etc. The next couple of articles, I want to talk about my favorite ways to beat them. This time: by playing a slog.

One of the great strengths of engines in general is that they are going to win the long game, and they can twist things to try to make the game go long. They have inevitability. Playing a slog seeks to flip this on its head. Classically, the slog player is looking to get a matrix of victory points so large that the engine opponent won't be able to overcome it, potentially because it's over 50%, but often actually because the engine can't support having the number of green cards necessary while maintaining its draw-ability. Sometimes, though, the slog player can leverage pressure on piles to squeeze the engine player's options.

What You Need:

Generally you need some kind of VP pile that will work well for you (and less well for an engine). The best here is Duke, since that makes you need to get lots of Duchies, and it's worth a zillion points. Silk Road is also good, since you need other green cards to make it work, and engines often can't afford to get lots of those. Gardens is decent, but it's not as good as the others, since an Engine often ends up with LOTS of cards just from getting lots of components.

More importantly, what you really need is for your engine opponent to actually care about the bloat that will happen in their deck. This means you need them to not be able to get to all the points in too efficient a timespace. Things like strong trashing and megaturn ability are not your friends here. Against a megaturn, you need your own strategy to be very fast in order to get enough points before they can fire off their megaturn. Another thing to watch out for is their ability to go back over the top of you - if there's VP chips available, in most cases Vineyards or Colonies, and sometimes Fairgrounds, they will be able to have inevitability, and the onus flips back on you to be able to end it, which isn't really where you want to be.

A good point to look at is what form of +Buy is available. If it's something terminal and dead, like Woodcutter, that is going to be very favorable for the slog player, who can just get a few and not worry too much about getting other cards to support. If it's something like Market on the other hand, it's going to be a lot more helpful to the engine player - he can build pretty easily towards one or two very big turns, because he doesn't have to add tons of extra draw power and reliability to get his payload off the ground. Ultimately, this is what the engine player is trying to do - build build build and go for VP at a spurt.

Theory: Building Engines

Engines are great in lots of circumstances, as is well know, but how do you go about building them? Is it just a coincidence that the top players end up doing better in engine games than other players? (Hint: No, it isn't). A lot depends specifically on the exact configuration of cards available, but there are some things which are generally true that will hopefully be helpful.

Build Your Draw Up First
The first mistake that a lot of players make is to get too much payload too fast, without building their draw sufficiently. This is an easy trap to fall into - it looks good, since it helps your deck have better turns economically in the (extreme) short term. You get to buy more/better stuff right away. The problem is, it ends up actually being slower in the end. The reason goes back to how engine economies can explode once you get to drawing your deck. Once you get to that point, you can pick up all those 'missed' economic pluses over the course of only a couple turns. At the end of all that, it doesn't matter that more of your turns did 'worse things', because your final product is either 'there' faster, or with extra draw stability to boot, and at that point, it doesn't matter how you got there - the resulting deck is just better.

The Development Phase
At the beginning of the game, you're pretty far from drawing your deck - you have ten quite bad cards, you're only drawing five, and you need to add more cards in order to do anything. The top priority is almost always to thin down by getting rid of your (bad) starting cards. It's the quickest way to get you to drawing your deck, and it gives you good consistency. If you can't thin, it's usually a good idea to get some sifting cards - they will cycle you back to your key buys faster.

Beyond thinning, you'll want to get the minimum possible economy in order to get the components you need. How much this is depends on a lot of things - principally what you're looking to get and what your other things currently are - but a good rule of thumb is that you need very very little to hit $4 consistently, 1-2 silvers to hit $5 consistently, and 2-3 silvers to hit $6 consistently. Be on the lower end of this when you can - extra economy is typically extra dead cards which you'll have to fight through. If nothing else, every silver slows down the rate at which you see your trasher (or whatever other good card) one slot every other shuffle. This sounds like very little, but it adds up pretty quickly, especially in your deck that wants to shuffle every single turn, and especially with that effect compounding itself.

In terms of what order to buy components in, during this stage you want 0-1 more terminals than your villages can support - in other words, terminals before villages. The reasoning here is pretty simple: there's a good chance the terminals aren't colliding yet anyway, and their effects are more important than the villages'. There are lots of exceptions to this rule though - usually when your village has some significant effect such that it's a good bit better than a straight cantrip.

The Economy Phase
One you get to the point that you can draw your deck - even if you can't do it reliably yet - things start to change. You really can't ignore building the draw aspect of your engine, as you will be adding more payload cards and/or green cards that you need to draw, and you also need to increase your reliability. At the same time, you do need to start looking to getting your economy going.

At this phase, you're looking to add a mix of cards such that you're increasing your payload whilst simultaneously reinforcing your draw and adding reliability. It's at this stage that cards like Scheme really start hitting their potency in terms of reliability (I should note that there are other applications for the card).

Specifically in this situation, the script flips from the first phase, and you want villages before Smithies. Obviously, you want to keep enough draw to be able to scoop everything up. But when it comes to over-building, you're much better served in getting extra villages compared to superfluous draw. This is because the biggest chance for you dudding out is from finding insufficient villages to get going - it's worth noting that extra villages don't hurt your chances of finding a Smithy on time, but extra Smithies do hurt your chances of making it to that Village. Of course, this isn't true when your villages don't draw you cards.


Exceptions
Exceptions are so many and varied that I am definitely not going to go over all of them. But I will mention the two biggest ones here: in mirrors, when you are pinched for some particular resource, you want to put a bit of an extra priority on that resource, because winning the split can be a big deal. But the biggest exception of all comes with game-end considerations, where you need to maneuver pile depletion in your favour.

Engine Theory: Consistency and What You Need

Following on from these two posts, let's continue our look at engines. We know what the payoff is, and we know the difficulties of getting them set up reliably. So when do you go for them? Well, as with most questions about Dominion, this is going to vary a good bit based on what's available. You always have to ask yourself, for any strategy, "what's the alternative?". Nevertheless, there are some things we can cover in general, and as long as we know there are going to be exceptions, and remember that context is king, we can learn some things.


Ok, so the biggest problem we were left facing at the end of the reliability article is that it's difficult to get to the point where we're consistently drawing our deck. We have basically four main ways of dealing with this.

1. Buy lots and lots of components for redundancy
   This one is tempting, because it's going to be possible every single time you have a draw engine that's at all possible, and getting stuff is what you want to do anyway. Unfortunately, if you check the math on things, it basically just doesn't work - to even get 3/4 reliable, you basically have to empty two stacks, and that's with pretty small payload. For sure this strategy is something you want to look to in order to complement the others, but in terms of getting it to work just on its own, you need to be gaining something like 3 components per turn every turn somehow, all while not emptying too many piles - and even then, it isn't great.
2. Don't Care About Consistency
   This isn't really something you can do terribly often - you are just eschewing the big benefit of engine economies exploding in on themselves like breeder reactors. Nevertheless, it can really be a viable thing on some boards. Much as I think people tend to overestimate their chances of drawing their deck every turn when it's 75-90% (i.e. you SHOULD dud out every 4-10 turns on average), I think people feel cheated when the other person 'gets lucky' and strings things together. Well, if they're 25% to do so on any given turn and they have 5 shots... they're going to get there over 3/4 of the time. In order for this to be at all viable, of course, you need some REALLY powerful payload. Traditional Megaturns are the order of the day here - 7 Bridges or Horns of Plenty, and it won't really matter that you won't get them together again. And this can lean your engine a bit more towards a 'combo' feel.
3. Thinning
   ...is, as my friend Adam Horton would tell you, whinning. =D Seriously, though, the consistency numbers get a LOT better when you have fewer dead cards to draw. Good thinning also speeds you up quite a bit, because it's just a lot faster to get rid of cards from your deck than to add cards to be able to draw it, considering that whilst adding cards, you have to draw all of those, too! Now, it might surprise a lot of you that I'm listing this here, and not last, because you would think this is the biggest solution to the consistency problem. But while it does give you a very good amount of speed, and it helps the consistency some, it doesn't really solve the problem. You're still going to want payload at some point, and it's almost always going to be more than the 5-6 cards for which you can be very confident to draw it all. That means you are going to be somewhat inconsistent unless you're significantly over-drawing. And well, you can usually just overdraw a bit, but that leads to inefficiency, which costs you time. And if the thinning isn't fast, you can have some real problems - even something pretty quick like Steward will take a couple of shuffles before you're reaping real benefits, and if it's slower like Trade Route? Yeah, good luck with that. So it turns out, even with trashing, you want something more....
4. Sifting
   Really? I know that's what you're asking. Yes. Sifting is amazing. It doesn't help you get to the point where you can draw your deck - in fact, a lot of what is traditionally considered sifting hurts you there - but once you are there, it starts becoming very, very valuable. The thing is, how are you dudding out? You draw all of your dead cards together, without having the cards you need to kick off. Sifting cards let you see LOTS more cards to get to your key starters. And when you are engine building, to get your consistency up, you would otherwise need to buy lots of extra draw cards anyway, for redundancy (a la point 1). This would lead to you overdrawing anyway, at which point it's just much more efficient to get some sifters instead of pure draw-more. The big thing to note here is that we are traditionally thinking of cards like Warehouse, Cellar, and Dungeon, but something like Cartographer, Journeyman, or Catacombs can fill this role as well. It's just worth noting that for these, they don't really help you find that first village. Even more important, cards that let you mess with the random draw of 5 cards to start your hand work here, too - Scheme, Guide, and even top-deckers like Watchtower and Royal Seal.

Of course, the real thing that you need in an engine is a good payload and enough time to effectively deploy it. Consistency helps you here, because it speeds you up through the late game. Thinning also speeds you up through the early game. In general, though, this won't be enough. Gainers which are cost-limited (e.g. Workshop) are going to help you through the midgame, because they let you grab more components to get your draw going or stabilize it somewhat (a la point 1).

But in general, +buy is going to be superior, and quite necessary. Why? You will need to be able to string together multiple big things at once - usually victory cards, really, but frequently some more expensive economic bonuses and/or engine components as well. Let's look a little more in depth at this. Getting one Province per turn is going to be too slow might be your first thought. And that's generally true. But even when you can get to this point pretty quickly, you are still going to have a problem: if you are only buying one card a turn, your engine is going to be consistently degrading as you get that Province - and as we saw, this is actually even true if you can add a Smithy and a Village with the Province. So unless your deck will be inconsistent as you continue down that green path. And really, who is going to be duchy-dancing better, your engine, or the opponent's money deck? If you are doing this in a protracted fashion, it's the money deck; it just degrades a lot less. So, instead, you want to plan to have the game end before this breakdown happens. This leads to a guiding principle of engine play which I'll cover more in a later article: Don't green until you have a way to end the game in sight. There are a number of exceptions to this, of course, with the biggest one being (unless you need the points to not immediately lose).

The concept of time is the most important thing here. It's a shifting scale with multiple axes. In general, in a non-mirror, the Engine will be playing the role of Chaser, whilst its opponent will be the Pace-Setter. But that will be in another article!

Engines' Economic Explosiveness

A point that I think a lot of players get to – and I know I certainly did – is to look at the world of Dominion in a Big Money Paradigm. What I’m talking about, in my case at least, is a concept I call “money density”. Basically, a rough measure of quality for a BM deck is something like “take your average coin production per card, multiply that by your average hand-size, and that gives you your average economic capacity per turn.”

This is perhaps a bit simplistic, but it really does give you a rough assessment of the capabilities of a big money deck. I will note that even when you are playing big money, there are lots of imprecisions to modelling things this way: card efficacy doesn’t, in general, scale linearly, unused money has no value, turn-to-turn variations are quite important, ancillary benefits of cards beyond coin-value are real, etc. Even within this Big Money Paradigm, there are lots of nuanced decisions to make.

When you are stuck in this mindset, it can be hard to think of why you would add “do-nothing” cards – a la Village – to your deck. Sure, the cantrip makes it not hurt so much, but it doesn’t actually help your money density, and you are potentially missing out on getting a better card – silver, at least, will do more for you. And in a Big Money deck, the extra action usually doesn’t do a lot for you – if you get to the point you’re needing one very often, you probably should have just bought fewer terminal actions. Or not play Big Money at all.

This is where the Engine Paradigm (or draw-your-deck if you want slightly less pithy but arguably more descriptive names) comes in. If you can get to the point where you are drawing every card in your deck every turn, the way you look at your economic output on a turn changes completely. Now, you don’t need to look at average coin per card; you can actually just add up the sum total of all economic production in your whole deck.

To illustrate this, let’s look at an example of adding a Gold (which is not what you want to do terribly often in engines, but is always available and gets the point across). Adding the gold in a Big Money Paradigm will increase your money density by the difference between $3 and your old average value, divided by the number of cards in your deck. So if you had a starting deck plus five silvers, your old density was 17/15 = 1.133, and you increase by (3-1.133)/16 = 0.117. Multiply that by 5 cards per hand, and you get .583 coin per turn on average. This is pretty good for a Big Money deck. It’s worth noting here that each successive gold will do less though – the next one only adds .103 per card, or .515 per hand. This is because the difference between where you were and the $3 of the Gold continues to shrink, and the impact of each card is less as your deck gets bigger.

If you’re drawing your deck, on the other hand, you simply get to add the full $3 to your ever-turn spending power. The next gold just adds $3 again, too. This is a lot more, and it gets to be a progressively bigger gap over time. So you can see that even treasures can get leveraged more if you are drawing your deck.

Now, there are reasons why it’s not quite so rosy for engines as the above might make it sound. As it relates to the discussion above, the most notable thing is that if you are adding payload cards which don’t help you to draw (like Gold), you will have to get more pieces that do so you can continue to draw your deck every turn (terminal payload also requires getting more villages). This diminishes the ability to really add as much economy as you might otherwise be able to. On the other hand, being able to add to your per-turn payload so quickly self-synergizes, exploding in on itself in a chain reaction – getting that extra $3 now means I have more that I’m able to spend next turn to keep increasing my economic capabilities without falling behind on draw. This ramping effect virtually always more than compensates for the need to get extra pieces to keep drawing, at least if you have the capability to get extra buys – otherwise making $30 on a turn doesn’t do much for me. On top of that, there are ancillary benefits – if there are cards which are much better in combination or in multiples, you get to reliably do that, and you get to hit them with your attacks every turn. Engines also give you better control of ending the game just when you want.

The real downside of engines, which might make you not want to go for one, is that they can be slow to set up. This, along with increasing the reliability of the engine, is why trashing and/or sifting is such a boon to the engine. It’s all about getting to that point where you are drawing your deck as quickly as possible, because once you are there, even if it takes a long time, even if you are forced in to buying victory points in less efficient chunks, the raw power of an engine’s chaining buildup, if one is possible on the board, is usually enough to overcome the potential speed deficit.

Theory: Engine Draw Basics

I have written elsewhere about the benefits of Engine Economies (and I will probably port that over here at some point). But in order for all that to work, you actually need to draw your deck consistently. Given this, a good question to ask is, how consistent is it?

Fortunately, we have mathematics. The first thing to note are some very basic statistics. How many cards can your deck draw? Seriously, this is as easy as totaling up the number of +cards on all your cards, and adding five for your initial hand. Obviously you need to subtract out cards you discard (e.g. Embassy counts for 2, not 5), and cards like Journeyman which don't technically draw cards should still count for their obvious number (in this case, 3). Obviously variable-drawers like Scrying Pool will make things trickier, but for now I'm going to focus on your more basic engines. You can compare your total draw to the number of cards in your deck, and this will tell you whether it's even possible to draw your deck. Consistency, however, is another question, and it requires math that's a bit more advanced.

The specific math we're going to be leaning on here is a mainstay of card games, the Hypergeometric Distribution. There are a number of Hypergeometric Calculators available on line, such as this one. I do most of my work in Excel, which has this stuff built in, of course. The biggest reasons for this are that it's easy to generalize doing a lot of similar calculations across a spreadsheet, and I almost always have at least one Excel tab open anyways.

I'm not going to walk through all the mechanics here, but I will outline you through how to calculate the basics for the first scenario I'm looking at. That scenario is 12 "dead" cards (anything that doesn't draw, from Curse to Goons), which is going to be reasonably common, as you start with 10 and often have to add about 2 payload cards to get your economy going, along with 4 Villages and 4 Smithies. Ok, this means 20 cards in our deck total, and we start with 5. The Villages each draw a card, and the Smithies each draw 3, so 5 + 4*1 + 4*3 = 21, or actually one more card than we have in our deck. Great. So we should be able to draw our deck pretty reliably, right? Let's see.

The first thing we need to get going on our turn is a village. So we use our hypergeometric calculator to come up with the chance of getting zero of those out of 5 cards. You should come up with 28.17%. Then, we need a Smithy. Of course, we can play any number of Villages before that Smithy. So the way to model this is, we need to see the chance of drawing none of 4 Smithies in a hand of 5 cards, from a total of 16 cards, where the 16 is the size of our deck minus the Villages, since we can just cycle through those. We should come up with 18.13% here. Okay, so now we've hit at least one Village and a Smithy. To continue, we need another Village. The key thing here is that it doesn't actually matter whether we found one before the Smithy or after, the overall chances of getting at least one remain the same - we just need to check for having none in our hand, which is now 7 cards, from our deck, which now has 3 Villages left out of 18 total cards. We get a 20.22% chance of missing now. Same thing for hitting the next Smithy - 3 left out of a deck of effectively 15 cards, and a 7-card hand. That's a 12.31% chance to whiff.

Continue that pattern down the line until the deck is drawn. Remember that what we've been calculating are the chances to miss. To get from this to the chance to draw our deck, we first need to turn them into chances to succeed by subtracting them from 1 (=100%). This gets us the chance of e.g. drawing AT LEAST one Village. Then, because we need ALL of them to succeed, we need to multiply those chances together to get the chance we draw our deck. When we do that, for our example, we get... a whopping 20.08% chance to draw the deck on any given turn, starting from a fresh shuffle. It's worth noting that if we only hit 3 Smithies, we will have drawn all but 1 payload card - the chance of doing this with an action live is not much better, though, at only around 30%.

As we move to bigger decks and adding more cards, it's important to note that, as we get to the point we're over-drawing our decks, we don't necessarily need to hit EVERY village and smithy - at a certain point, the deck is drawn, and the superfluous drawing components do nothing (except, perhaps, give us enough actions to complete all our payload).



I'll wrap up this post by noting some general trends. With 12 dead cards, you don't reach a 50% chance of drawing your deck until you are at 6 Villages and 5 Smithies. Adding in even a single extra dead card starts hurting your chances pretty significantly. If we're limited to a single pile of Villages and Smithies to draw with, we can't do better than 77.64% to draw our deck with 12 dead cards, which we reach at 10 Village and 8 Smithies. After a point, adding extra superfluous Villages is the best way to get your deck drawn. Non-drawing Villages are way worse for our consistency, even if we compensate by making our drawing cards better - Festival + Hunting Grounds is quite a bit less consistent than Village + Smithy on the same number of each pair, generally leading to deck drawing itself between one-third and two-thirds as often as the traditional Village-Smithy roles. It's also worth noting that the "more villages after a point" advice from the Village-Smithy framework does NOT translate to the Festival-Hunting Grounds one - you want a balance here.


As for the strategic implications of what all this means, I'm going to leave that for a different post. For now, let's just say to not be too surprised or upset when you get those dud hands - they're most likely more common than you think.

Instructional Game #6: Empty the Treasure Maps

Game Log

...with Shelters.

My opponent opens Minstrel/Silver, gets a pair of Maps on turns 3 and 4, and collides them pretty quickly. I go for Silver/Silver into a Count and start trashing down. After turn 9, our decks look as follows:

The battle lines are drawn: he's attempting to use his quick map activations to power through Provinces and lock the game up. I'm trying to build more of an engine and... wait and see what to do. I have some options: just get some treasures and be able to single Province more consistently; use Stonemason to grab tons of components, probably to Graverob lots of things over a couple turns; use Bishop to score tons of points. I feel that probably any of these would actually work, but as the game state progressed in this particular game, I like the options progressively more as you got through that list. The reason for this is that the later ones give me increasingly more time before the game ends, and extra time should help me with my superior deck, as what I really need is to survive long enough for that quality to shine through. The key point here is that I expect my opponent's deck to break down if I give it long enough to do so.

At the time of the game, though, I was a bit wishy-washy, and my natural instincts of taking the points lead when it seems to be enough caught on for a moment, leading to my buying a Province on turn 14. This is a pretty bad play, as it just speeds up his win condition (empty the Provinces), while not really advancing my own thing at all. Soon after this, I strike on the optimal plan: Bishop golds for points, Stonemason overpay for more components, Stonemason Golds/Catacombs for components as well, loading up on more Maps to refuel. And it works like a thing of beauty. The other nice thing about this deck (besides being a strong, resilient way to gain a zillion points) is that it gives me a lot of control over when the game ends. I can always buy another Province to lock the game out, or I can also smash lots of things and make a pile ending happen, but either way, it will be on my terms.


As we finally round the bend on turn 18, I see an opening, using the on-trash ability of Catacombs, and pile out on Stonemason, Wandering Minstrel, and... Treasure Map!

FPS: KISS (Fancy-Play Syndrome: Keep It Simple, Stupid)

Game Log

I go for a big engine here - Lookout to trash down, Armory gains things, Throne Room for Shenanigans. It's a strong deck.

My opponent goes for Scavenger+HP.

On his 9th turn (he's first player here), he gets the final Hunting Party, his 6th. At this point, our decks look like this:

At this point, we're off to the races. By the middle of turn 13, our decks look like this:

Unfortunately for me, I have a dud here, and I can only Mine up a Silver to Platinum and buy a Duchy. But the thing is, this just ends the game. Yup.

Let's break this down a little more. On that turn 9, I have a little more power than him, I can gain a bit more stuff, thanks to Throne Room and Armory. Island can mitigate the deleterious deck-bloat aspect I run into to some extent. In exchange for this, my opponent's deck is nigh-unbreakable. The key thing to note is, as long as he can get to $11 without drawing ALL of his Hunting Parties, he can Scavenger right then to seed his next hand with a Hunting Party. This gives him tremendous reliability. His deck will not break - if mine does, I'm dead. And though I am a pretty good favorite to not dud out on any given turn, I'm not terribly likely to make it through ALL of them, especially given that I need to make some magic happen with my Islands, and I'm somewhat limited on actions.

Now, part of me wants to cry "first player advantage", and indeed, though I lost by 20, I did have to turn for Province a bit early, and more importantly, that 1st-turn lets him get the 6-4 HP split here, which is pretty big. The problem with taking this mindset, though, is that even though I may be favored were I first player here, I would always be in a *better* position if I just play the simpler strategy. It doesn't blow my bigger engine entirely out of the water, and first-player might be a bigger deal, but it does have some advantage everywhere.