Marginal Yu-Gi-Oh

patrickGood evening everybody! The April 2014 format kicks off in a couple days and I’m sure you’re all busy working hard perfecting your decks for the upcoming format. Today I’d like to present to you a revolutionary deck building concept that I’ve been working on for quite some time. It applies economic concepts to deck building by allowing you to make decisions at the margin. It has a wide range of applications and can be applied in just about every area of deck building. I’m going to start out by giving you some background on the topic to show you my thought process, then I’m going to explain the basic economic concept that goes with it, and finally I’m going to apply that concept to deck building and give several examples of its applications.

I started out this previous format by playing Hieratic Rulers after I discovered the card [ccProd]Hieratic Seal From the Ashes[/ccProd]. It essentially generated a plus one every turn that it stayed on the field by giving me the dragons that I needed to summon a Dragon Ruler back from the graveyard. The first event of the format came, Nashville, and I did not do nearly as well as I had hoped I would do. At the time, I was very unsure why the deck had not performed as well as I might have hoped. The entire concept of the deck seemed very strong and I couldn’t place my finger on why it simply wasn’t working. I reexamined the games I was losing and one of the most common things that were happening was that I would draw a card like Ashes or [ccProd]Reckless Greed[/ccProd] when they had already established a field. I would then have to set the card and since I could not use it the turn that I drew it, it was essentially a minus one for the turn. Often times by the time I could activate one of them, it was too late. Both Hieratic Seal From the Ashes and Reckless Greed were bad when drawn to established fields. When constructing the deck I thought that this would easily be countered by the advantage that each card generated once it had resolved. What I was neglecting at the time was that having too many of them in my deck could hurt more than they could help.
If I have a total of six cards that are bad to established fields, I’m going to top deck one of them to an established field more often than I would if I were to have only five cards that were bad to established fields. If I have a total of six cards that are bad to established fields, I’m going to have multiples in my hand at the same time more often than I would if I only had five cards that were bad to established fields. In these situations I would like to have as few of these cards as possible.
When the idea of cards that are bad against established fields is first presented to you, your first instinct might be to cut those cards altogether, but that is not necessarily correct either. Let’s take a look at Reckless Greed by itself. Reckless did a great deal for both Hieratics and Mermails in the last format as they were both combo decks and it could correct for missing combo pieces. Here, you can see how running Reckless Greed might be beneficial, even with the risk of drawing it when your opponent already has established a field. This begs the question, is there some ideal number of cards greater than zero that are bad to established fields that we can play that will maximize utility?
Around this same time I was studying for the first exam of the semester in my Political Economy class. One of the things I was studying was called the Laffer Curve. Here it is:

laffer curve

Arthur Laffer is an American economist who served as member of Ronald Reagan’s Economic Policy Advisory Board. His curve explains the relationship between possible rates of taxation and the amount of tax revenue the government receives.
The curve is pretty easy to understand and is the basis for the deck building concept. It says that there is a specific tax rate (t) that the government can tax the people that will bring in the most tax dollars. This makes sense because if you have a 0% tax you’re going to bring in 0 dollars. Similarly, if you have a 100% tax you’re also going to bring in 0 dollars because no one is going to have incentive to work if they have to give it all to the government. It’s not hard to see that every rate between 0% and 100% will not bring in the same amount of tax revenue. That means that at some point in between 0% and 100% you’re going to bring in more tax dollars than you would at any other taxation rate. This is the point T and is the highest point on the curve. Any tax rate higher or lower than T won’t bring in as many tax dollars as T would. The curve goes down after point T because every dollar people earn will be worth less since it is more heavily taxed. This will make working less worth it to some and they will cut back hours or stop working altogether which results in less tax revenue.
Are you beginning to see how similar the Laffer Curve is to the problem of having too many cards that are bad against established fields? There is some benefit to having some of those cards, but if your entire deck is bad against established fields you’re going to be in bad shape as you’ll effectively lose as soon as your opponent summons a monster.

Since we’re not interested in taxes and tax revenue, let’s go ahead and replace those to make our own curve. For Laffer, he’s trying to maximize tax revenue. For us, we want to maximize utility. Because we’re looking at the utility of each additional card, we’re going to replace tax revenue on the Y axis with marginal utility. Then we’re going to replace the tax rate with whatever kind of cards we’re trying to analyze. In this case, it’s going to be cards that are bad against established fields. Laffer measured this in percent. Since we can’t run 3/5 of a card, we’re going to measure it in whole numbers only. Lastly let’s replace Laffer’s optimal point, T, with U for the point that will give us the highest utility.
Now we’ve got a basic model that we can work with for various deck building concepts. This will allow us to find the maximum utility for what we’re looking for. Here is the curve to finish out the example of cards that are bad against established fields.

Alright, there’s a curve that can be used to see the correct number of cards that are bad against established fields to give us maximum marginal utility. After realizing this, I switched from Hieratics to Mermails and dropped the Ashes, but kept the Recklesses. This resulted in going undefeated in swiss at Atlanta and winning in Charlotte the following weekend. It’s pretty safe to say that three was the number of cards that are bad against established fields that gave us the most utility. So we’re done, right?

Unfortunately, that’s not the case. If you think about it, there’s still a certain degree of utility in your deck even if you have no cards that are bad against established fields. If we take the above curve to be true, it says that having zero cards that are bad against established fields results in your deck having no utility. The above curve would also suggest that having six cards that are bad against established fields results in your deck having no utility. Now, three may be the point that will give us the highest utility, but that does not mean that you have no utility at zero or at six. In actuality, the curve probably looks something more like this.


Here, you can see that playing zero cards that are bad against established fields has some utility. Each additional card you add up to three has increasing marginal utility. Every card after that has decreasing marginal utility. Marginal utility will continue decreasing with each card that is bad against established fields that you add until you get to 40 (your whole deck), which will have utility that is effectively zero.
Where Does the Ideal Point Come From?
So how do we know that three is better than every other number? While I believe that theory is vastly more important than play testing when it comes to deck building, this is the area that play testing is most important in. This process is going to be trial and error and it’s going to be a completely separate issue to train you to be able to identify what is or isn’t working. Did you see that combination of cards enough? Did you see them too often? Under what situations were they bad? How many of them together are bad? These are important questions to ask yourself when looking at groups of cards that have the same potential drawbacks such as being bad to established fields.
Are there any Assumptions We Should Make?
Specific Groups - You have to be very specific when you group something together. Let’s say you are looking for the ideal number of monsters in your deck. That category is too broad. Technically, [ccProd]Maxx “C”[/ccProd] is a monster. Should we count it towards our ideal number of monsters? If the ideal is 18, is the ideal really 16 with two Maxx “C”? Realistically, we shouldn’t be counting Maxx “C” for this total. Because of this, the ideal number of monsters might be too broad. It might be more advantageous to look at a more specific group such as monsters that we would normal summon.
Assume the Most Effective Cards – X card that is bad against an established field does not give us the same utility as Y card that is bad against established field. Let’s look at Ashes and Reckless again. One of these cards is going to give us more utility than the other one. That means that if play testing determines the ideal number of cards to play that are bad against established fields is three, either Reckless Greed or Hieratic Seal From the Ashes is still going to give us more utility than the lesser card would. We should assume that the point U of maximum utility will be the best card available from that group. Using the my example from Charlotte, I determined that to be Reckless Greed instead of Ashes. This means that after you play test to find the ideal number of cards for whatever group you’re looking for, you’re going to have to play test to find the ideal cards to fill that group. You’re only going to achieve the point U when both of these conditions have been fulfilled. The result of this is that the curve is not usually a normal distribution. I’ll have more on this later.

Externalities – There are some externalities that make it difficult to determine exactly where U is on the curve. For example, there is obviously a lot greater benefit in drawing multiple Reckless Greeds than there is in drawing multiple Hieratic Seal From the Ashes. This will affect U and should not be ignored.
What Are Some Other Groups we can apply the Curve To?
Number of Normal Summons
You can also apply the idea of maximizing marginal utility to the number of normal summons you play in a deck. Above I’ve drawn what I think to be the curve for the number of normal summons you should have played in a Dragon Ruler deck during the second Dragon Ruler format. Through play testing I had determined five to be the amount that gave the highest utility. Too many normal summons means that you will draw multiple monsters that you have to normal summon more often than if you were to play a fewer number of normal summons. Since you can only conduct one normal summon per turn, the utility of any given hand goes down with each additional normal summon you have after the one you are allowed to conduct per turn. Once you’ve used that normal summon, if you’ve got a second monster that must be normal summoned, you can’t get any value out of the card until your next turn.
You have to remember that each curve gives the point U for a very specific group. For this curve, it gives the U for the number of normal summons during the second Dragon Ruler format. If you look at another format such as Chaos Return or Plant Synchro, you might have a much higher number of normal summons for your ideal value of U.
It’s also important to make the distinction that these curves are often times not a normal distribution. What I mean by this is that moving four spaces below U and moving four spaces above U do not necessarily achieve equal utility, even though they are an equal number of integers from the ideal point. In this particular case it is caused by clogging of normal summons. If you play nine normal summons, point 2, it would not be uncommon to see a six card hand with three normal summons in it. Here, once you conduct your one normal summon of the turn, you cannot use either of the other two and do not gain any value by them being in your hand for the rest of the turn. This translates to playing with a three card hand after your normal summon rather than a five card hand since you aren’t gaining anything by having those two additional cards. This type of clogging can cause utility to go down. If you only have one normal summon in your deck, you won’t encounter this problem and once you’ve conducted your one normal summon, you’ll always still have the five remaining cards in your hand to play. This results in point 1 having a higher utility than point 2, even though both points are an equal distance from the ideal.
Number of Defensive Cards


Above I have the curve for the number of defensive cards in a combo deck. Throughout last format and this format you can see that I’ve stuck with about seven defensive cards in all of my decks. There is more utility in having no defensive cards than having a bunch. This is because every card in your hand that is a defensive card cannot be a combo card. If you’ve got two defensive cards, you’ve only got four cards left to combo with in your combo deck. Each additional defensive card has even less value than the previous one did which causes the giant drop off.
Again this can change depending on the deck. Fire Fist from this format probably have a much higher number of defensive cards to get to their U than Mermails did. That is because they are based less around two card interactions and the majority of their cards are stand-alone. Since Mermails need multiple cards to do their powerful plays, they are going to be hurt more by having too many defensive cards than Fire Fists would be.
Ideal Number of Cards in Deck

I get asked all the time would I play [ccProd]Upstart Goblin[/ccProd] in my Dragon Ruler deck from the WCQ. My answer is ‘no’ and this curve will explain why that is the case. People take my Upstart theory to mean “fewer cards are better.” Hell, when I wrote it, I might have had that in mind, but I’ve since gained a greater understanding. There is always going to be an ideal number of cards for any given deck in any given format. It isn’t always the fewer the better. How are you going to win with a 15 card deck in the first Dragon Ruler format when just the Dragons took up 20 cards themselves? There is a certain point for any deck that it’s going to hurt you to cut that extra card. This is where the benefit of the extra card exceeds the cost of the extra card, simply put, where the marginal benefit is greater than the marginal cost, but you don’t go take the added benefit. Here, it would actually hurt you to cut a card. For the first Dragon Ruler format, I determined that ideal point to be 40 and not any lower.
Does that mean that no one should play Upstart Goblin and effectively make their deck 37 cards anymore? Certainly not. If anything, that format were more of an exception and the ideal point is usually a much lower number of cards. Let’s take this current format for example. I’d say that the ideal number of cards for a Mermail deck is about 33. This is the number that I think would yield the most utility. The game says that we are not allowed to play fewer than 40 cards. I say we’re not allowed to play fewer than 37. Since 37 is the lowest number of cards that you can play, it is going to be the point that gives us the highest utility from the options available to us. While 33 may be better than 37 for Mermails, 37 is going to give us the greatest utility of all the legal options as utility will further decrease as you increase the number of cards from 37. Because of this, it makes sense to play Upstart Goblin in Mermails to be as close to the ideal point as possible.
The final example I give, U3, is for Exodia. It has an ideal number of five. This is because if it were to only play five cards it would win every game. It’s utility for playing less than five cards drops to zero because it cannot win without all five pieces of Exodia. It’s utility as you increase the number of cards away from five decreases constantly, but not at the same rate it would if you would go below five. You’re always going to want to play closest to your ideal number. The closest legal number to Exodia’s ideal point is 37, thus it makes sense to play Upstart Goblin in Exodia.
It’s also important to note that these are not equal distributions. Playing 39 cards in Dragon Rulers will give you more utility than playing 41 cards. This is because with 39, you’ll see [ccProd]Super Rejuvenation[/ccProd] and [ccProd]Sacred Sword of Seven Stars[/ccProd] more often than you would at 41. Even though they are both one card off the ideal point, one of them gives you more utility than the other one would.
That wraps up my article on deck building at the margin. This concept can apply to just about everything in deck building, not just the examples put out in this article. Some things to remember are that distributions are generally not even and that two cards right of the ideal point and two cards left of the ideal point will often achieve different utilities. Determining what U is for any given group comes through play testing. It is commonly thought that you should optimize card choices when play testing. While this is true, you should also play test to determine the number of those group of cards you should play. Only through looking for both will you be able to achieve U. This is a deck building concept that I apply to every one of my decks for many different groups. I hope that you will be able to apply what you’ve read today to decks of your own. Leave a comment down below with other ways you can apply the concept of marginal deck building. The Circuit Series comes to Richmond, VA on April 26-27, 2014, click the pic below for the details! Until next time, play hard or go home!


Patrick Hoban

Patrick Hoban

Patrick Hoban

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