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The Unfair Mana Base


Welcome to NOT This Week in Legacy! This week I'll be presenting an incredibly detailed guest article from Cheng Zhi. Cheng is a Chinese Magic enthusiast and content contributor, along with game commentator and video producer. He is the owner of themeditators.org, the only Chinese Legacy forum. I'd like to thank both him and James Hsu (of Humans of Magic fame) for getting in contact and giving me the opportunity to publish this exceptional piece detailing mana base construction, fair and unfair deck categorisation and going deep in Magic deck building theory. Do note, although including Legacy, this article also includes examples incorporating the Modern format.

Enjoy!


Preface: What do you mean by fair/unfair?

People on Internet communities talk a lot about a couple of words: fair and unfair. These two words have been mentioned in Magic daily chat and articles for years, and many game strategies are based on whether one is playing against a fair or unfair deck. However, nobody has yet given a definition nor stated a clear boundary between the two.

On Japanese website MTGwiki.com, unfair decks are described as decks which tend to gain victory via “abnormal” methods. They consist of combos that try to win as quickly as possible and/or decks with high-synergy components. Reanimator is an example of a pure combo deck, while high-synergy deck include decks such as Infect and Affinity. Modern Bogles and Tron are further regarded as unfair decks. Decks trying to win through “normal” means are considered fair. They utilize individually powerful cards like Tarmogoyf, Lingering Souls, and mass removal spells. Fair decks are further categorized as beatdown/aggro, control, midrange, aggro-control (tempo), etc. and sometimes may have a small combo add-on. This categorization is via a deck’s winning plan, and meets the typical understanding of players.

However, my understanding has been: chasing high inequity on a certain game resource is all unfair decks are about. Take Legacy MUD as an example. Its advantage is that it can accumulate mana much faster than the opponent and can lock opponents’ spells by using permanents like Chalice of the Void and Trinisphere. It has gigantic creatures (Wurmcoil Engine, Platinum Emperion) and deadly planeswalkers (Karn Liberated and Ugin, the Spirit Dragon). However, its disadvantage is illustrated by the great variance in its possible draws. When you have seven or eight mana, your top deck could be a game-ending Karn or a Grim Monolith. Another problem is the deck being colorless, so it cannot obtain any advantage from the effects of the rest of the color pie. These merits and faults make MUD a balanced deck as a whole, but with high inequity. Reanimator and Sneak & Show seek to have absolute inequity of creatures at the expense of card advantage (Lotus Petal) and life (Ancient Tomb), while Storm tries to have absolute inequity on spell intensity in a single turn at the expense of card value (Dark Ritual) and life (Ad Nauseam). Elves tend to flood the board, at the expense of individual creature strength for synergy, likewise.

For the resources mentioned above, I think mana is the most crucial element. As the “transaction cost” in the process of resource transformation, mana to a large extent determines the speed of transformation, and thus it can reflect the unfair level of a deck. In this article, I attempt to find an index of unfair by analyzing various decks’ mana bases.

Analysis: Decks’ Mana Base

Mana Curve

In our impression, “unfair” stands out when we see key spells having a higher-than-usual mana cost. For instance, a deck that wins via Karn Liberated is a typical unfair deck while decks that win by Delver of Secrets and Death's Shadow are typical fair decks. We often hear that a deck’s mana curve is high or low. But what is a mana curve then? And is there any connection between mana curves and a deck’s degree of unfair?

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There are two forms of mana curves. One is a mana distribution curve (MDC) and the other mana cumulative curve (MCC). Let’s take Modern Jeskai Control as an example to examine these two forms.

The deck list is as below. (Deck lists used in this article are all League 5-0 decks, most of which are in July and August of 2017.)

After removing the 25 lands, we get a two-column sheet showing the amount of each cost.

Based on this sheet we can create a chart showing the deck’s mana distribution. The horizontal axis is mana costs and vertical axis is number of copies.

The blue curve is the MDC. We can see this Jeskai deck’s major mana costs are 1 or 2. Let’s examine Modern Eldrazi Tron in the same way.

Deck list is as below:

And the MDC:

It seems that Eldrazi Tron’s mana costs center on 0, 1 and 3. However, this conclusion is too simplistic. The converted mana cost and actual mana requirements can be so far away from each other in real games. For example, Snapcaster Mage as a spell theoretically costs 2 mana, but is primarily played when you have three lands, since one mana is spared for flashing back Lightning Bolt, Path to Exile or Serum Visions. Sphinx's Revelation’s theoretical cost is 3, but nobody would play it for X equal to 0.

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Likewise, we seldom get a chance to hard cast Dismember or play Chalice of Void without spending any mana when playing Eldrazi Tron. Furthermore, there are some functional land cards like Ghost Quarter, which we do not intend to play often as a mana-generating land when designing the deck. This might not be so obvious is Modern, but very common in Legacy, and lands such as Wasteland, The Tabernacle at Pendrell Vale, Dark Depth and so on can be seen as spells. Considering all the above, we get to fix cards’ mana cost to their actual mana requirement. Taking the above Jeskai deck once again as an example:

We have changed Snapcaster’s mana from 2 to 3 and Sphinx's Revelation from 3 to 6. Ghost Quarter is seen as a 1-mana spell, because you need to tap and crack it. It cannot be seen as a 0-mana spell since it is a land and occupies the play-one–land allowance.

Likewise, we can fix Eldrazi Tron as:

Chalice of the Void and Walking Ballista are modified from 0 to 2, and Dismember from 3 to 1. Specifically, because Eldrazi Temple and Urza lands have the effect of generating multiple mana, the mana requirements of all high cost spells can also be reduced. In an ordinary Tron deck for instance, if we have Urza's Mine, Tower and Power Plant, we can cast Karn Liberated with these 3 lands. Thinking about this another way, if we assume that each land generates 1 mana, Karn is a 3-mana spell in Tron. However, we can’t have the three Urza lands together on time in each game, so we use an empirical method to find an approximate outcome. If we can play many games, assuming the opponent doesn’t waste our lands, and find that on average we can cast Karn when we have 5 lands, then we set Karn’s mana requirement as 5 for this deck. Note that a key precondition of this model is that each land generates only one mana. Only by this can we compare different mana bases. Hence when I talk about mana costs below, I mean the corrected mana requirement such as the approximate mana requirement of Karn given above.

We can re-examine the MDC of the two decks after correcting mana costs:

In the two charts we see the two curves are in a descending form, which infers that the less spells cost, the more copies are in the deck. MDCs of this shape make much more sense than the ones drawn at the beginning. We also can understand the advantages and disadvantages of MDC. It can show us whether a deck’s mana consumption is balanced, or, in more generalized terminology, whether a mana curve is “smooth.” MDC of normal decks should be in a descending form, and it will be straighter if the mana costs are evenly distributed. MDCs can help us analyze the difference in mana consumption of the two decks, and tell us how cards of a certain cost are concentrated and what the higher cost is.

However, MDC has serious defects. Firstly, a “smooth” MDC doesn’t indicate that the deck has a good mana base. For a deck with 24 lands and 36 spells, we can hardly imagine mana costs are evenly distributed, like six spells at each cost from 1 to 6. In The Deadly Four-Mana Mark (Alexander Shearer, 2012) there is a model for the calculation on “the probability to have N mana at turn N”. I call it the “mana insurance level.” Shearer’s article also assumes that one land can only generate 1 mana. Therefore, after mana cost fixing the two decks above, Jeskai Control has 36 spells and 24 lands, while Eldrazi Tron has 38 spells and 22 lands. Their mana insurance levels are as below:

We can see, for each of the two decks, the probability of “3 mana on turn three” is over 70%, which is a high probability. If we omit extremely-high-cost spells and look at simply relatively expensive spells, we see Jeskai’s four-drop insurance level is over 63%, while Eldrazi Tron’s five-drop insurance level is less than 37%. So we conclude that Eldrazi Tron’s mana base is worse than Jeskai’s, though its MDC looks much smoother.

Secondly, we often say “high” or “low” when we talk of a mana curve. However, the height of MDCs are irrelevant. The height of a MDC only shows how mana costs are agglomerated. The peak of Jeskai’s MDC is 12, while that of Eldrazi Tron is 10. But the former’s average cost is 2.5, the latter’s average cost is 2.76. Thirdly, the smoothness of MDC indicates the evenness of a deck’s mana costs to some extent, but evenness is not the same as stability. One of the key statements of Shearer’s article is “five mana costs way more than four”. The higher spells cost, the lower the insurance level would be. If we have two decks each with 25 lands and 35 spells, one them having seven cards at each mana cost of 1-5, and the same but at costs 3-7, the former’s mana base is much more stable than the latter.

Same MDC heights, huge difference in mana base stability.

To address this issue we can use mana cumulative curves (MCC). For making these we sort spells in an ascending order according to their mana costs, and then make a continuous accumulation to draw an ascending curve.

For Jeskai Control:

Then Eldrazi Tron:

Superimposing the two charts:

We see the two decks’ MCCs overlap in low-cost region but separate after the amount cumulates to 20. In the mid- and high-cost region, the MCC of Eldrazi Tron is visibly higher than that of Jeskai. If we go back to the statistic sheets and look at the points where accumulated numbers are around 20, we can see 3-drop spells of Eldrazi Tron appear earlier than those of Jeskai. When players say “high” or “low” mana curves, actually they are referring to MCCs.

Can we find a general law of unfair decks from MCCs? For example, the higher a MCC is, the more unfair the deck shall be? Statistics on 18 Legacy decks and 14 Modern decks show:

Some MCCs in the two charts ascend sharply at the end. This effect has something to do with my model construction. When fixing the mana costs, I hypothesized that for spells which cannot be hard casted or won’t be hard casted in regular games, their mana costs are set to 9. This is because the most expensive spell in Legacy and Modern is Ugin, the Spirit Dragon (8 mana) and we can regard 9-mana spells as unplayable, like Griselbrand in SNT.

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Despite this effect, we can identify the highest and lowest MCCs in Legacy and Modern. For Legacy, the highest ones are 12-Post, Death & Taxes, 4c Loam and Eldrazi, while the lowest are UR Delver, Infect, Grixis Delver and Ad Nauseum Tendrils. For Modern, the highest are hard to find but there are MCCs that are much lower than others: Affinity, Ad Nauseam, Grixis Shadow and Naya Burn. Legacy decks’ MCC heights seem to have no relationship with unfairness level. No matter the highest ones or the lowest ones, fair and unfair decks (to our regular understanding) are mixed together. For Modern’s lowest ones, apart from Grixis Shadow, the other three decks are normally considered unfair. But other unfair decks, as Eldrazi Tron and Gifts Storm, show no uniqueness in their MCCs. So MCC is not a clear indicator for judging unfair decks.

Variance and Mana Uneven Index (MUI)

MCCs can reflect the mana consumption of a deck, but do not work well in showing its mana uneven level. To our understanding, mana costs of spells increase linearly for fair decks. That is to say, we cast 1-drop spell on turn one, and cast 2-drop spell on turn two, and so on. However, unfair decks tend to consume mana in an abnormal way during the transformation of resources. Say ANT of Legacy. Using Lotus Petal and Dark Ritual exchanges card advantage for mana. Unfair decks may chase also for high synergy, in order to exchange card advantage (or other resources) for the board. For example, Affinity uses a lot of 0-mana creatures. It has the potential to pour its entire hand onto the board on turn one and beat down on turn three. By nature it is somewhat similar to Belcher of Legacy which can make sixteen Goblin tokens on turn one and win two turns after. Hence, Affinity is broadly recognized as an unfair deck. This kind of construction idea leads to the unique mana bases of unfair decks. Therefore, intuitively, their mana costs should be distributed unevenly.

In mathematics, variance is applied to measure how far a set of numbers are spread out from their average value. I calculated 18 Legacy decks’ average value and variance of (corrected) mana costs respectively.

In this sheet, there are five decks whose variance is greater than 1.5, and all of them are unfair decks in our common impression. They are Sneak & Show, Reanimator, 12-Post and Elves. Other unfair decks, like ANT, Lands, Eldrazi, Turbo Depths and Burn, show no uniqueness on their variance value.

It infers that variance has some value as an indicator, but has defects, too. Variance shows how far a set of numbers are spread out from their average value, which could create cognitive bias. Suppose we have two decks named A and B and they each have four spells. Deck A’s spells cost 1, 2, 3 and 4, respectively, while deck B’s spells cost 3, 4, 5 and 6. Although variance of these two decks are equal, their mana bases are totally different from each other. For deck B, we can guess that we should cast spells linearly as the turns go ahead. But for Deck A, it is possible to play all spells in turn one, by casting low-cost spells being the precondition to casting the high-cost ones. Hence deck A accords more with unfair decks in our mind. Therefore, we need another indicator to describe such a mana distribution.

The indicator I have found is the Gini coefficient. It is typically used in economics to measure the inequality among values of a frequency distribution, like levels of income. A Gini coefficient of 1 expresses maximal inequity among values. A Gini coefficient of 0 shows a perfect equity, drawing a straight ascending line. I borrowed this concept and used it in mana-base analysis. I construct an index called Mana Uneven Index (MUI). Its calculation is based on MCC. In the graph below, MUI is equal to the area marked “x” divided by the sum of the areas marked “x” and “y”. That is, MUI = x/(x + y). Modern Jeskai Control is depicted below:

 

Using the deck A and B example, deck A’s spells cost 1, 2, 3 and 4, respectively. Compared to the least costly spell, the three other spells cost twice, three times and four times more, respectively. Deck B’s spells cost 3, 4, 5 and 6, respectively, so the three more costly spells cost 1.33 times, 1.67 times and twice more respectively. From this angle we see that deck A’s mana construction is more uneven. Thus, MUI must play a better role in presenting the inequity of mana bases than variance.

I calculated MUIs of 18 Legacy decks and 14 Modern decks.

It turns out those decks with MUI greater than 0.38, no matter if they are from Legacy or Modern, are mostly unfair decks to our common understanding. The only deviation is Legacy Mentor Miracles. However, considering its winning plan is resolve a Mentor and buff it in the latter two or three turns, in a high-synergy and explosive way, it may not be a material fault to classify it as an unfair deck. The decks with MUI lesser than 0.37 are mostly fair decks. In this field, Lands, Eldrazi, 12-Post and Burn from Legacy, and Eldrazi Tron from Modern are not considered unfair, since they seem to be unfair decks in our impression. Reasons may be two fold.

Number one, 12-Post and Eldrazi decks use multi-mana generating lands. We supposed that each land only generates 1 mana,and for the very special mana base as 12-Post or Eldrazi, we used an alternative way of tuning the mana cost of spells. When you tap two dual-mana lands to cast a 4-mana spell, it is turned into tapping two single-mana lands for a 2-mana spell. This flexible method guarantees the comparability of different decks, though it blurred some uniqueness of mana base.

Number two, we can re-assess whether Lands and Burn can be fair decks. At the beginning of this article, I quoted from MTGwiki.com that a control deck with a combo add-on should be considered a fair deck. For Lands, in its using of Life from the Loam, Wasteland, Maze of Ith and Punishing Fire it can be identified as controlling. It is very different from Turbo Depths which intends to create a 20/20 ASAP. But the dilemma is that, we shouldn’t deny the law we made before. If Miracles with a Mentor add-on may be unfair, why can’t Lands with a Depth-Stage add-on? This need further study. As for Burn, it has no combo and its components mainly stack direct damage cards that have low synergy, so it is reasonable to categorize it as typically fair.

Nonetheless, as far as we can see, a higher MUI is a good criterion for identifying unfair decks.

Further Questions

1. Is MUI the best criterion for judging fair/unfair decks? Due to the limits of the model, unfair decks like 12-Post and Eldrazi are obscured by a low MUI. But decks with relatively higher MUI are mostly unfair decks. So we may have confidence to say that an unfair deck must have a high MUI, however, can we say the converse that “all decks with high MUI must be unfair decks”? Is it possible to find a better indicator?

2. I said, for unfair decks, we know that they are decks trying to win by combos or components with high synergy. If we just want to make a brief judgment on fair/unfair, we don’t need to do data analysis. After we do this analysis and have MUI, do we need to get a MUI value once we meet a new deck to see if it’s an unfair, even without knowing its game plan?

3. Now that we filtered unfair from many decks, does it have any practical meaning? Research that I did is more for my own interest and “I just want to know.” So what? Shearer’s article states that “five mana costs way more than four,” which is a guiding reference for a deck builder, helping players build better decks by considering mana base and predicting when can they possibly resolve a 5-mana spell. Nevertheless, what does “an unfair deck must have a high MUI” contribute to deck building? Maybe MUI can help deck builders construct a more reasonable mana base when building an unfair deck. I am not sure.

4. Did I take the best way of fixing mana costs? I will list my fundamentals at the end of this article, for your references. Maybe the statistical results can be more persuasive if we make the model better.

Laws of Mana Fixing

1) For those spells which cannot be hard casted or won’t be hard casted in regular games, their mana costs are set to be 9. This is because the most expensive spell in Legacy and Modern is Ugin, the Spirit Dragon (8 mana). We can regard 9-mana spells as can never be played. Examples are Griselbrand, Omniscience and Emrakul in Legacy Sneak & Show or Bloodghast and Prized Amalgam in Modern Dredge.

2) Spells with alternative costs that do not consume mana have 0 as their mana requirement. Examples are Force of Will, Daze, Gitaxian Probe, etc. The same principal applies to Dismember, Vault Skirge, Gurmag Angler, with their mana requirement as 1.

3) Functional land cards like Wasteland, which we do not intend to play as a mana-generating land when we design the deck, have a mana requirement equal to their activated cost respectively. For example, Ghost Quarter is seen as a 1-mana spell; it cannot be seen as a 0-mana spell, since it is a land and occupies the play-one-land allowance so normally you can’t play several Ghost Quarters to waste lands in a single turn. Likewise, Maze of Ith is 1, and Sheltered Thicket is 2. Other lands without mana abilities at all are set to 1. For example, we regard Tabernacle and Glacial Chasm as 1-mana enchantments, because they follow the typical land playing allowance. But lands played mainly for mana generation while also having other functions are not in this category, like Rishadan Port, Horizon Canopy, as well as Eye of Ugin.

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4) For spells with X in their mana cost, like Chalice of the Void and Green Sun's Zenith, the highest probability or weights were considered. For example, most times Chalice of the Void is used to lock 1-mana spells, so its mana requirement is set to 2. In Maverick, Green Sun is always for fetching a Dryad Arbor on turn one or to get a Knight of Reliquary when we have 4 mana. So its mana requirement is set to 2.5 (= 1 x 0.5 + 4 x 0.5).

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5) Direct mana-generating cards’ mana requirement is 0, like Simian Spirit Guide.

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6) For spells that have to wait until we have more mana in order to cooperate with another spell, their mana requirement is slightly increased, like Snapcaster Mage and Monastery Mentor.

7) For some cards that have variable costs, like Batterskull (in case of Stoneforge Mystic), and components of Post, Tron and Eldrazi (because of multi-mana generating lands), their mana requirement is reduced.


Thanks for joining me once more, this time for a little different. Join me next week for a typical look at the Legacy metagame!

Some links, as always:

  • Mengucci plays Goblin Stompy in Legacy at CFB.
  • Julian plays a League with Elves at itsjulian.com.
  • The Brainstorm Show is back with episode 38! Find that here.
  • Jarvis Yu joins TheEpicStorm to talk Through the Looking Glass on Lands. Find that here.

‘Til next time.

Sean Brown

Email: sean_brown156@hotmail.com
Reddit: ChemicalBurns156
Twitter: @Sean_Brown156


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