Tuesday, April 12, 2011

Duel Theory: Part 2 - Deck Design

After a much shorter Part 1 which was mostly self-explanatory, I'm afraid break time is over. I have to warn you here that there is some math involved in this next part. Thankfully, it's still mostly simple. For the most part, we've been using simple arithmetic as a framework to keep count of the game and adjust our Tempo similarly to how Blackjack card counters operate. But now you need to know how to make a deck. Before you can do this, you will need to know a little bit about probability. I won't do a whole lecture on things like how the Law of Averages works or the Gambler's Fallacy. If you don't know those things, I suggest you search them out now. We're going to be talking about Hypergeometric Probability Distribution. As the name seems to imply, this is a rather complex breed of probability calculation, however you will not need to know the formula or even how it works.

Please see exhibit A; a calculator!

And now to exhibit B; a handy guide!
  • Population size - Is the total amount of cards in a stack you're testing from. For most cases, this is the size of your deck. This experiment will show you why 40 cards is a must for just about every deck.
  • Sample size - Is the number of cards you wish to draw and sample from. For deckbuilding this is usually the 6-card opening hand.
  • No. Successes in population - This is the number of chances you have to draw a specific card in a deck. Duelmonsters has a limit of 3, 2, or 1 on copies of a card. However you can effectively increase this as I will explain later.
  • No. Successes in sample - This is what we're actually trying to find. How many copies of this card do you need to draw? Usually this is just 1.
  • Cumulative Proability: P(X >= x) - This last box is your answer. It will give you the probability versus "1". This means that you can simply show it as a percent.
Alright, now that you know how to use the calculator, we should jump into designing the deck.
For the probabilities I will be giving you, they are based on a deck size of 40 cards and drawing in the opening 6 hands with only looking for 1 success.
You ought to know by now, in this game there are limitations on the number of copies of a card you can have; either 3, 2, or 1 depending on what format and month you're in. The following will show you the constant probabilities of drawing those numbers in the first hand:
  1. 15% chance to draw
  2. ~28% chance to draw
  3. ~39% chance to draw
  4. ~49% chance to draw
"Oh wow, she included 4 copies! You can't have that!" In fact, you can effectively make your deck contain 4 copies of a card. Take Reinforcement of the Army for example. This makes all of your Lv4 and lower Warrior-Type monsters searchable if you draw that card in the opening hand. Thus, when calculating the probability of drawing, say, Marauding Captain in the opening hand (with 3 copies), if you have 1 Reinforcement of the Army, you actually have a 50% chance of being able to get that card from your deck! This is the reason you usually see searchers Forbidden or Limited. But not all of them are. If you run 3 Necrovalley, 3 Terraforming, and 3 Gravekeeper's Commandant, any 1 of those 9 cards will get you the Necrovalley. Therefore you have an ~81% chance of getting it in the opening hand! That's only 2 out of 10 duels that you don't get this card immediately.

Cards that search give you an immense advantage of options. When designing a deck, you should keep in mind what you absolutely must draw in the opening hand consistently, what you'd like to get, and what's totally optional. Identifying those properties takes some keen intuition and experience that will come with time and repetition. What I suggest is taking a deck that's topped well and studying it carefully, calculate the probability for drawing any given card in the deck, and seeing if you can improve upon it by making the meat of the deck more consistent.

Let's use Gravekeeper's as an example. So you want to build this deck. You're going to want Necrovalley of course, so let's ignore that 3 Terraforming can clog you down for now and just go for the highest draw probability. Add in the 9 cards you need,
-3 Necrovalley
-3 Terraforming
-3 Gravekeeper's Commandant

Now you want to see about getting out your Gravekeeper's. No point in adding them into your deck if you're just relying on blind luck to draw them. Well, looking at Gravekeeper's Spy, he's a good utility card that can act as another copy of each GK you have. Let's put him in. Since most of them have < 1500 ATK, we can also use Sangan.
-3 Gravekeeper's Spy
-1 Sangan

Next, look at the other effects and decide what you'd like to center the deck around. We have spy, so we need lots of fodder or his effect might turn useless mid-game. Howabout 2 Descendent, 2 Guard, and 3 Assailant to start off. We can re-adjust them later if need be. Let's center the deck around getting out--and keeping--Visionary. He's a beefy hitter that only needs one tribute, easily gets even beefier, and can stop himself from dying. For his second effect we'll need lots of smaller fodder to discard just in case. We don't want 3 of him because that wastes precious space for card that keep him out. So let's go with 2 Visionary. So far our monsters look like this:
-3 GK Spy
-3 GK Assailant
-3 GK Commandant
-2 GK Descendant
-2 GK Guard
-2 GK Visionary
-1 Sangan

Well, the deck is shaping up quite nicely. We have a lot of DARK monsters, so this could help us in our Spell/Trap selection. In fact, it can help us now. We can add in Gorz, Caius, Dark Armed Dragon, and Tragoedia with ease. Let's pop in 2 Caius, 1 Dark Armed Dragon, and 1 Gorz. We'll leave out Trag for now since his discard effect is counter-intuitive to keeping Visionary out.
-2 Caius
-1 Dark Armed Dragon
-1 Gorz the Emissary of Darkness

At this time he have exactly 20 monsters. This means that 50% of our deck is monsters. A very nice 66% chance to draw half our opening hand in monsters. And the same for Spell/Traps, conversely. And a mere 1% chance to clog the opening hand with all monsters. So let's stop here before that number rises.

Our Spell/Traps so far just account for our field spell, Necrovalley. Let's go ahead and get some staples out of the way and add in 1 Mirror Force, 1 Dark Hole, 1 Book of Moon, 2 Mystical Space Typhoon, 1 Giant Trunade (as of 4-12-2011, all these are at their max). Since we have so many ridiculous chances to have Necrovalley, and we want to beef up Visionary for a winning hit, why not put in Royal Tributes? 2 of them will give us the chance to absolutely devastate an opponent's hand provided they aren't using Dark World monsters. We might want to get some easy +1's and get back the GKs we discard too, so let's add in 2 Gravekeeper's Stele. Because of Necrovalley, we aren't using Call of the Haunted or Monster Reborn. We have something better for this deck, Rite of Spirit. It's 3 free GK revivals. So let's put the max number of them in for massive swarming and getting our discarded GK's back to re-use them. We're kind of lax on Spell/Trap countering. We have the MSTs, no doubt, but we can't stop getting Dimensional Prision'd, Royal Opression'd, or Solemn Warning'd. Let's put in 2 Royal Decree to stop those shenanigans. But since we're playing 2, let's remove our Mirror Force to keep our trap count low since the Decree will be preventing its use. We have Dark Hole for nuking.

So far we're looking pretty good! We don't have much drawing power. Just immense deck-thinning. Allure of Darkness will serve us well, as will Pot of Duality if played correctly. Since each one is either a 2:2 or a 1:1 that speeds through our deck, let's fill it up with those. 1 Allure and 2 Duality. But now we have 23 Spell/Traps, over the 40 card limit. Let's remove the 3 Terraforming since they will be useless draws once we get Necrovalleys. With that, we still have a ~65% chance to draw it in the opening hand which increases every turn.
-3 Necrovalley
-3 Rite of Spirit
-2 Gravekeeper's Stele
-2 Royal Tribute
-2 Royal Decree
-2 Mystical Space Typhoon
-2 Pot of Duality
-1 Book of Moon
-1 Giant Trunade
-1 Dark Hole
-1 Allure of Darkness

And now we have a playable 40-card deck which is designed with probability of the interactivity of each card in mind.  Some things you can side-in are Magician's Circle in case the opponent doesn't have Spellcasters, Terraforming in case you need to lock their graveyards quickly, Gravekeeper's Watcher in case they have discard effects, a third Stele and Pot of Duality in case you really need to speed the deck up instead of using utility so often. Another option is removing Dark Armed Dragon and putting in Gravekeeper's Recruiter in case your Dragon is unable to use his effect due to Necrovalley too often. Recruiter can do some nasty combos with Descendant.

Hopefully you've learned a little something about the design process. But science isn't content with just formulating a hypothesis! Your deck is completely guesswork at this time. You must go out and apply it to the metagame and see how it fares.  After each duel, record your winning/losing card advantage and break everything down so you can see what cards are becoming problematic. The answers may surprise you. Use that new data from experimentation to refine the deck and do it all over again. This is how winning decks are made.

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