Thursday, March 19, 2015

The Mathematics Behind Good Deck Building

I’ll start off this post with an example that telegraphs what I’ll be discussing today:

Two players choose between two decks of cards at random. The cards are blank, and each player draws 1 card at the same time, every second. The winner of the game is the player who draws all the cards in their deck first. Now, if one player has a 40 card deck and the other player has the 37 card deck, who will win? The player with the 37 card deck will always win the game, with the above specifications applied.

This example is obviously a very simplistic scenario, but gets the point across, and projects the topic of today’s discussion: Mathematics in respect to deck building. For as long as I can remember, ever since my first Regional I attended, I’ve always made it a mandatory requirement to keep my decks no greater than 40 cards. The smaller your deck is the greater chance you’ve given yourself to draw the cards you need. In every collectable card game (excluding games like Poker and 21) you are expected to have a deck size between a certain amounts of cards, or you are expected to use a defined amount of cards. Yu-Gi-Oh and Magic are games where you have a minimum deck size of 40 and 60, respectively. Essentially, one of the solid foundations of deck building is ensuring you draw the necessary cards to play the game properly.

The Most Basic of Mathematical Concepts

In regards to Yu-Gi-Oh, currently, a lot of players play a single copy of Book of Moon, because the card is limited to 1 copy per deck. If you choose to go second, you will draw 6 cards for your opening (I’ll re-visit this segment shortly). The odds of drawing Book of Moon, using the most basic mathematical concepts, in a 6 card hand is 6/40 (15%). If you play a 60 card deck, and decide to go second and draw 6 cards, the chance of drawing Book of Moon become 6/60 (10%). Notice that there is a 5% difference between them. I could start explaining how factorials and hypergeometric distribution works, but this will not benefit you if math isn’t your strong suit. You don’t need to be a rocket scientist to understand you’ll draw Book of Moon more in a 40 card deck than a 60 card deck.

As well, there’s a mathematical difference between going first and going second. If you win the die roll and choose to go first, you’re only drawing 5 cards. In a 40 card deck, that’s 12.5% of the deck. I suppose some people will look at it and consider “it’s only one less card; it’s not a big deal”. Clearly, there are some decks that prefer to go first, but most should choose to go second. Keep in mind that we are in a format where most of the tier 1 decks are combo-based decks: Nekroz, Burning Abyss and Dragon Rulers in a few days when Dark Matter Dragon is released. Combo decks critically rely on having enough combo pieces to generate a play. Going second and drawing that extra card is an extra card that can enhance your combos.

That being said, there are particular decks that benefit from going first. Usually, these are more control-based decks that rely on grinding the opponent’s resources and/or stopping their plays all together. If you decide to go first, and open with multiple cards that auto-defeat your opponent, you are obviously in a winning position. Does that mean going first is necessarily better? Not entirely. Here’s a situation for you to review:

You’re playing Qliphorts and the first 5 cards of your deck are Scout, Raigeki, Skill Drain, Emptiness and Saqlifice.

Say you chose to go first and you drew these cards. Obviously, you’d play Scout, search a monster and summon it, equip Saqlifice and set your floodgates. It’s pretty amazing, but it’s rather underwhelming. You might stop your opponent from playing the game, and maybe they won’t draw an answer to your field. However, you will lose if they have an answer to the floodgates. It’s also relevant to mention that the opponent will just be saving their resources until they have an answer, and if they draw it, there will be a devastating outcome. But of course, you might win anyway if the answer to the floodgates doesn’t arrive: Advantage only matters if you can play the game.

Say you chose to go second instead. You have these 5 cards in your hand, and draw another card (the other card doesn’t matter in this example). If they wasted a lot of resources to establish a field, Raigeki will punish the opponent immediately, and then you’ll build a similar field to the first scenario, and actually deal damage. Keep in mind there’s no reason to waste Raigeki if you can clear their field courtesy of a Scout search. Now you have an extra card to work with because you chose to go second, and you’re in a good position once your turn ends; possibly better if you disestablished their board.

Is the best deck the smallest deck?

Currently, Nekroz is the best deck in the format. One of the reasons why this is the case is because the deck is so thin. So many cards thin out the deck for specific combo pieces, then other cards allow you to replenish those resources to thin out your deck next turn. Burning Abyss also has sufficient search power, however, it is quite different than Nekroz. Burning Abyss focuses on milling cards to accelerate your plays. If you mill more monsters, you are getting the most out of the deck thinning. If you mill all spells/traps, you might lose the game. Qliphorts have cards that can thin the deck for specific cards (Scout, Summoner’s Art, Saqlifice and Disk), but there’s not much more than that, unless you’re play drawing cards that let you draw directly, like Upstart Goblin. Dragon Rulers will have some solid draw and thinning power due to cards like Dragon Shrine and Cards of Consonance. The point of these examples is that Nekroz is just so consistent and the deck constantly sheds itself with useful cards, so your draw phase has a higher chance of being something amazing. Really, how lucky is drawing one of 3 Manjus if the deck has 10 cards by that point?

Heart of the Cards

I know I’ve stated this before, but the key to become a powerful duelist starts by building your deck right. The thinner you can make your deck, the more likely you’ll draw the cards you need to win the game. The Draw Phase is the most important phase during your turn, as it will dictate the events that occur during your turn. Why squander this phase by unnecessarily adding cards that will dilute your deck, and take away from the most brilliant top-decks? Most players call it “sacky” when you topdeck that Snatch Steal for game. However, it’s easier to topdeck that amazing card if your deck is smaller.

It’s during that deck-building phase where problems start to occur. People will squeeze their deck to 45 cards, justifying their actions with a thought similar to “this is a good card to play right now, why shouldn’t I play it”. Before you know it, you’ve done this a dozen times with random good cards like Mirror Force, Dark Hole, Mind Crush, and Maxx “C”, and then that deck is now over 40 cards. My response to thoughts like this can be summarized in a simple sentence: Suck it up. There are a lot of good, powerful cards in this game, and if you could play all of them, Konami would have to expand the maximum deck size allotted in this game.

Picking these amazing cards is the hardest part of deck building. Ideally, the key to building the best deck is playing the best cards. Picking cards that will crush the majority of your matchups will provide you a greater base to cover your deck’s weaknesses. You may also decide to play cards that are complete blowouts in certain matchups but ineffective in others. For example, topdecking Book of Eclipse is amazing against Nekroz, but no so much against Qliphorts. However, if your deck is lacking powerful cards to deal with Nekroz, and you need to ensure you have answers to the Djinn lock, you may need to consider maining cards that will assist you, like Book of Eclipse.

Once you decide which blowout cards to play, you need to make sure you draw them. In a deck like Nekroz, you’ll have the greatest chances of drawing them since the deck is good at thinning itself, usually. If you need to draw that Book of Eclipse for the mirror match, you’ll be more likely to draw it in a 37 card deck than a 40 card deck.

Now, I can’t use a heading “Heart of the Cards” without make some sort of Yugi reference. In the Battle City finals, there’s a scene where he is building his deck. He knows one of his win conditions is summoning Slifer the Sky Dragon. He had to ensure his deck would be able to generate three creatures that can be tribute summoned for the Egyptian God. If he knew what he was doing, he would’ve made his deck 40 cards to maximize the chances of summoning Slifer. Using the Pharaoh isn’t the best example for this, but the Heart of the Cards was always on his side. In real life, I don’t believe this exists, but you need to construct your deck to maximize those “Heart of the Card” moments.

Here’s the simple point of this discussion: Building the thinnest deck possible is mathematically the smartest decision you can make. It will reduce bricking in the long run and allow you to “sack” a little more often. You’ll still have those games where you draw unplayable. Keep in mind that an opening hand of Book of Moon, Raigeki, Snatch Steal, MST and Shurit is full of amazing cards, but it isn’t necessarily an optimal opening hand. You won’t be able to do much until you draw something that will allow you to create a strong play, and the odds of doing that in a 45 card deck aren’t as strong as a 40 card deck. This is math, and math is absolute.

Let me know what you guys think, and thanks for reading!

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