4eak
08-11-2008, 04:39 AM
One of the most talked about cards in magic, but also one of the most difficult to evaluate. How should we value brainstorm? I’ve decided to breakdown the value of mechanics that form the card brainstorm. This should give us insight into why and when we use brainstorm.
I separate brainstorm’s effect into 3 components:
1.) [Cantrip]
2.) [Library Manipulation]
3.) [Hand+Library Manipulation].
1.) [Cantrip]–Guaranteed +1 Card in your hand, but just as important, a guarenteed -1 card in your library. Brainstorm, like many cantrips, is 1 blue mana for 1 card, which is already a fair effect (not broken, but fair). This thins your deck like a street wraith or fetch land. A cantrip replaces lower quality cards in a deck, allowing you to see the highest quality cards of a specific function.
If you run 4 brainstorm in a 60 card deck, and you go through a quarter of your deck on average before the end of a game, then you are paying, on average, a single blue mana over the course of the game to have a 56 card deck. The scaling of the cost to cycle through more of your deck per average game is linear too.
Why would you want to turn 60 cards into 56? Some cards have a higher utility, value, or relevance to your deck or specific circumstance than others, and in reality, we only want to play those instead of lower quality cards if possible. Cantrips, like brainstorm, remove lower utility cards from the equation, allowing the remaining 56 cards to have a higher average utility value than the average 60 card deck. The question then becomes: was the average cost of using cantrips worth the card-quality gains?
I’ll use as straightfoward a case as I can think of to show you what a cantrip means to card quality. This case by no means showcases the brokenness that is Brainstorm that we might find in decks that abuse it best, but the case shows the principle behind cantripping.
Let’s say you were playing a deck that had 16 Volcanic Island, 40 Lightning Bolts (120 damage value), and 4 Shocks (8 damage value). Notice that shocks, on average, are 1/3rd less valuable than a bolt. We’ll say you see 11 cards per an average game, putting you at (11/60)*4 (total mana cost of cantrips in deck), or 0.733 of a U, on average, to go from a 60 card deck to a 56 card deck. What happens when we replace the lower quality cards of a deck, shock in this case, with a cantrip?
[Total value of win conditions]/[Total mana cost of win conditions]=[Average Win condition to mana ratio per card].
Shocks– 128/44=2.909
Cantrips– 120/(40+0.733)=2.946
The gain, in part, is one of mana efficiency. The otherside of the cantrip is how it affects your average “win-condition” value met per card.
[Total value of win conditions]/[Total cards in deck]=[Average Threat value per card] (Think of DPS for you MMORPGers)
Shocks– 128/60=2.133
Cantrips– 120/56=2.143
What if we made a deck with with 40 shocks and 4 ‘Flashback-less’ lava darts, how good is a cantrip then? Presume we see 13 cards per average game, or (13/60)*4=.866. Notice that Flashback-less lava darts are only 1/2 as effect as a shock, and contribute proportionately less to the total win condition value of the deck. The Shocks are more relevant to the bolt-deck than lava-darts are to the shock deck. This difference in proportion will illustrate the rising advantages on cantrips in decks and formats that have larger card quality disparities.
[Average Win condition to mana ratio per card]
Lava Darts– 84/44=1.909
Cantrips– 80/40.866=1.958
[Average Win condition value per card]
Lava Darts– 84/60=1.4
Cantrips– 80/56=1.429
The worse your cantrip-replaced cards proportionately compare to the average mana-efficiency and win condition values of the rest of your deck the better a cantrip becomes. Here are your comparisons:
Bolts’n'Cantrips/Bolts’n'Shocks Mana efficiency ratios– 2.946/2.909=1.013
Shocks’n'Cantrips/Shocks’n'Darts Mana efficiency ratios– 1.958/1.909=1.026
Bolts’n'Cantrips/Bolts’n'Shocks Win Condition density ratios–2.133/2.143=1.005
Shocks’n'Cantrips/Shocks’n'Darts Win Condition density ratios–1.429/1.4-1.021
As the win-condition value of the least valuable cards of a deck (those to be replaced with cantrips) proportionately decreases as compared to the more valuable cards in a deck, the proportionately better a cantrip becomes.
If all your spells have fairly equal win-condition value, then the effectiveness of a cantrip decreases. So, while the greater variation in the value or relevance of your cards the better a cantrip becomes, the other side of this equation is that perfectly balanced decks with card quality equivalence would not want to use cantrips. For example, if you ran 16 Volcanic Islands and 44 Bolts, would replacing 4 bolts with 4 cantrips be worth it? Let’s say you see 11 cards per game.
[Average Win condition to mana ratio per card]
Bolts–132/44=3
Bolts/Cantrips–120/40.733=2.946
[Average Win condition value per card]
Bolts–132/60=2.2
Bolts/Cantrips–120/56=2.14
Running straight bolts is simply better than having a cantrip. Why? There is too little variation in the mana-efficiency and win-condition values of the cards in a deck with nothing but bolts and land. In cases where all things in your deck are equal in value, then cantrips are not worth it. A perfectly balanced deck would not need cantrips. Building this “perfectly balanced” deck is more complicated than many would realize though. Remember the arguments against running more than 60 cards in a deck? Usually, because there is such extreme differences in the quality of cards in older formats, we seek the smallest decks possible to abuse the few cards that are just too darn good for their mana costs. Cantrips act as the glue between the broken cards of eternal formats in these cases. However, technically, there are cases where 65-card decks could be perfectly balanced, even better than 60 card decks. Perhaps you could make a 65-card deck that had a several functions, all maximal and equal quality cards for their slot and function, and any removal of a card would imbalance the deck’s card quality. Here, you would take a 65-card deck over a 60-card deck. But, how many decks do you know are this well made? The sort of perfection in balancing card value to make it such that a 65-card deck would be preferred to a 60 card deck is the same sort of calculation and balance that a deck would need in order not to consider cantrips. If there is any imbalance in the value of the cards in your deck, then cantrips are worth considering.
Since that perfection rarely exists, often due to format card pool constrainsts, we opt for cantrips. The proportionately less valuable a card is compared to the average card quality of a deck the more likely we should replace it with a cantrip.
Take a more extreme case, say 16 Volc-Islands, 40 4-damage for 1 mana burn cards (henceforth: Uber-Bolt), and 4 Flashback-less Lava darts. Say you’ll see 10 cards in an average game. (10/60)*4=.667 mana cost to goto 56 cards.
[Average Win condition to mana ratio per card]
Lava Darts– 164/44=3.727
Cantrips– 160/40.667=3.934
[Average Win condition value per card]
Lava Darts– 164/60=2.733
Cantrips– 160/56=2.857
Cantrips give proportionately larger gains when they replace cards of proportionately lower relevance. In this case, cantrips let us not run flashback-less lava darts and stick to straight uber-bolts, giving much higher mana efficiency and average card quality.
Cantrips aren’t the end-all-be-all solution though. Take a case where we ran 16 Volcanic Islands, 24 cantrips, and 20 Uber bolts vs. 16 land and 44 Uber Bolts. Say we’ll see 20 cards per game; (20/60)*24=8. Ouch, that is 8 mana, per game, spent on just lowering the count to 36. It will take too many turns to see the cards we need to see to be mana-efficient at all.
[Average Win condition to mana ratio per card]
Cantrips– 80/32=2.5
Uber Bolts– 176/44=4
[Average Win condition value per card]
Cantrips– 80/36=2.222
Uber Bolts– 160/2.667
This is an extreme example, but it shows that there is a specific number of cantrips we wish to run in any given deck. You can easily run too many or too few cantrips in a deck.
Burn, of course, can be a more straightfoward calculation than other decks. And, you’ll notice several burn decklists use bauble-cantrips to maximize and balance average card quality..for good reason. Other decks are certainly more complicated, but the principle still remains the same though:
The higher degree of disparity found between the relevance and value of the different cards in your deck, the more useful a cantrip becomes. Eventually, if you follow this path, you’ll see the extreme in silver-bullets and tool-box decks that rely upon card-quality, cantrips and tutors to consistently find the singleton card that may be the only relevant thing in your deck against an opponent.
This is why running Yawgmoth’s Will with cards that aren’t nearly as powerful would drive us to use cantrips: by running cantrips we will receive a higher average use and benefit of Yawgmoth’s Will over the course of many games. The benefit, often enough, is worth the cost of replacing weaker cards with cantrips.
Decks that have similar components are less likely to desire cantrips. An aggro deck, for example, may have few deviations from the mean value of cards in the deck. On the other hand, a combo deck may often find themself in situations where they have 2 of 3 combo pieces in hand, but need the last one. In this case, only the missing combo piece may be relevant to our situation, and cantrips increase the likelihood of finding the relevant cards.
The cantrip component of brainstorm fulfills a major glue-mechanic in which decks can more consistently run and play with higher quality cards with different functions and values in different circumstances.
2.) [Library Manipulation]–This is a more straight foward effect to consider. Think of this as Sage Owl. How many cards do you have left in your library? To what degree do the individual cards in a deck deviate from the mean win-condition value of cards in a deck? The higher this deviation, and the lower your library count, the more effective library manipulation becomes. Bare in mind, the mean win-condition value and deviations vary per metagame, per deck, per matchup, and per specific game circumstance. This makes it incredibly difficult to calculate, but it highlights the variance we see in even the mean values across a spectrum of conditions. Library manipulation, like a cantrip, helps isolate and condense the average variance from the mean value of win-condtions across the spectrum of play-conditions.
Sage Owling into a 40-card library with nothing but Lightning Bolts isn’t going to net you anything. However, Sage Owling into a 40-card library that has only three or four relevant card in the deck (perhaps you MUST Wrath of God next turn or you lose) increases your likelihood of seeing relevant cards sooner. Like the cantrip, library manipulation benefits the deck that has higher variance in card value from the average card value.
Let’s take a basic example:
If you had 36 cards left in your library, 3 WoG’s in the library, and Wrath was the only relevant card, what does a 4-card Library manipulation effect do for you?
Without the library manipulation you have a 1 in 12 chance to draw Wrath of God next turn.
With a 4-card library manipulation you have a 1 in 9 chance to draw Wrath of God next turn.
Library manipulation is still very good even beyond looking for 1-of a specific card in your deck. It lines up your deck plays too. It could be as simple as counting your land drops drops for the next several turns and making sure relevant spells are on top with land being placed exactly where you would need to draw it so you could make a land drop for the next several turns.
Library manipulation allows you to order cards in their relevance to your current game position. If you need a counterspell before you need a land, then go ahead and put the land under the counterspell. The land may be relevant, but maybe it is less relevant than the counterspell. Library manipulation increases the quality of your future draws. A basic permutation grid of a 4-card library manipulation ensues.
Actual Card1 (in Slot1) — Value at Slot1=1, V@S2=6, V@S3=8, V@S4=5
Actual Card2 (in Slot2) — Value at Slot1=3, V@S2=5, V@S3=4, V@S4=6
Actual Card3 (in Slot3) — Value at Slot1=0, V@S2=5, V@S3=6, V@S4=2
Actual Card4 (in Slot4) — Value at Slot1=4, V@S2=2, V@S3=3, V@S4=3
Card1 moves to Slot3, Card2 moves to Slot4, Card3 moves to Slot2, Card 4 moves to Slot1.
Originally, we have a top 4-card value of 15. After a 4-card library manipulation we have a top 4-card value of 23.
The permutation grid is actually much more complicated than I’ve provided. For example, what if Card 3 only has a value of 5 in Slot2 if and only if Card 4 is in Slot1? Multiply this type of value calculation, and you see that identifying the value specifics cards, even in a very specific circumstances, can be quit complicated. These are the sorts of mental calculations that we make on the fly. It seems obvious, but drawing out the reason why we do what we are doing is more complex than we initially thought.
Shuffle effects has a specific effect right here too. You know the value of the top X cards of your library. Is that value below the average value of X cards in your library? If it is below, then a shuffle effect increases the value by [Average Value of X cards]-[Value of Current X cards].
Decks with higher variations of value per slot make the most use of library manipulation. Again, perfectly rounded decks with zero variance from the mean value per card would not want library manipulation. It must be noted that this perfect balance might not be found at 60 cards in a specific format and metagame, and thus a perfect deck without cantrips might not be possible in many circumstances.
Brainstorm has a 2-card library manipulation value. However, it’s 3rd effect is the game-breaking ability that twists library manipulation into relevant and immediate card advantage and quality.
3.) [Hand+Library Manipulation] This is a very odd effect in magic. This is the effect that makes brainstorm more than just a mere cantrip and library manipulation. This ability might be seen as an extension of library manipulation, but we must distinguish this component of brainstorm from a Sage Owl effect because of the influence this mechanic has upon an active hand. This effect alone can make brainstorm as good as Ancestral recall + 1/2 a Sage Owl or as bad as 1/3 an Ancestral Recall + 1/2 a Sage owl. That’s right, I said it: Brainstorm can be BETTER than Ancestral Recall. There is a two card quantity difference between the worst brainstorm and the best possible brainstorm, and most of the math behind understanding the value of a specific resolution of Brainstorm, as found between that spectrum, relies upon this mechanic.
Given the cantrip effect, the Hand+Library Manipulation effect is only a count of 2 cards. The value differences
To look at the Hand+Library Manipulation effect itself, we will neglect the cantrip and library manipulation components of Brainstorm for now.
Hand:
Card in Hand1 — Value in Hand=2, Value in Library=2 (in not particular order on library, just with X cards from the top)
Card in Hand2 — Value in Hand=3, Value in Library=0
Card in Hand3 — Value in Hand=1, Value in Library 1
Library
Card on Library1 — Value in Hand=3, Value in Library 1
Card on Library2 — Value in Hand=1, Value in Library 0
Current value of Hand+Library=7.
After a 2-card Hand+Library manipulation, where Hand1 is replaced with Library1 and Hand3 is replaced with Library2, the value of Hand+Library=10
This is rudimentary, but it shows the basic principle. It also shows how one can calculate the value of hiding valuable cards against discard effects.
Echoing truth against a High-tide/Reset deck is a useless card. Hand-Library manipulation increases your current hand value by putting echo onto your library. This, of course, is at the expense of future draw values. However, when you add shuffle effects, it turns dead cards into average card value. Essentially, Hand-library manipulation is a 3-fold utility:
1.) Gives immediate hand value increase equal to [Cards put into hand from library] - [Cards put on top of library].
2.) Combined with shuffle effect, increases library value to [Average library card value of X cards] - [current top X card library value]
3.) Hiding valuable cards that you don’t want discarded.
The permutation grid becomes more complicated than just this current turns active hand. Of course, we have to consider the effect upon future draws. This falls directly into the library manipulation arena.
Brainstorm does this effect like no other card for such a cheap cost. In so many cases, when your opening turn or 2nd turn really decides the match, brainstorm sculpts the early game. Some decks, for example control decks, have great need in sculpting their early game active hand (well beyond other archetypes' early game needs).
Card value–
I’ve talked alot about card value. But, I haven’t given any good definition for it. For now, I can't give you a really clean and easy to read definition. This is a much more complicated topic beyond the scope of this article (and I'm working on the article). Here is what I will say with regards to Brainstorm and card value:
Cantrips, Hand and Library manipulation are difficult to evaluate because it requires a system of identifying the exact of value of each card in a library in average circumstances/metagames and the mean variance of value between cards and functions in those circumstances/metagames.
Card values vary per metagame, but are static to any one specific metagame. Yawgmoth’s Will is inherently stronger in a format that is better at getting cards into the graveyard. Basic land is inherently stronger in a format that doesn’t have better alternatives. There are two types of metagames to consider:
Universal Metagame (format restricted metagame)
Specific Metagame (your circumstances)
If we are to understand the value of any card, including Brainstorm, to an exacting degree, it must be with the knowledge of what metagame we are speaking.
For the purposes of deckbuilding, card values are determined by their degree of influence on the offense/defense ratio of a deck. You are attempting to quantify how essential a card is to your win condition, or convertibly, your defense against an opponent's win condition (having a higher offense/defense ratio than your opponent).
A deck or card is meaningless without an opponent or metagame to interpret its value. Against a metagame/opponent with a 60-land deck, the best deck will be the one that has the highest and fastest average win condition. It isn’t just whether you won, it is the margin by which you win or lose that helps form the value of the cards in your deck against a 60-land deck metagame. The win-condition to be met, in this metagame, is simply reaching the stage where an opponent has 20 lifeloss, has been milled and can’t draw, or loses through a straight “win-ability” (Door to Nothingness, etc.). Whichever deck has the highest and fastest average chance of reaching that win-condition is the best deck in that metagame.
What if you went against a 60-FoW deck metagame? Perhaps the deck that was the best in a 60-land deck metagame would not be the best in this specific 60-FoW deck metagame. The combo deck that probably evolved in the 60-land deck metagame was not prepared to deal with permission. You might say it wanted “speed” at all costs. But, in reality, the deck is only as good as it matches against a specific metagame.
The value of a card has those two values, value in the universal context and the specific (and only one will matter when we speak of the value of a card). So, we should not be confused into thinking value isn’t calculable. Rather, we just need to see how to go about looking for and speaking about its value in the first place.
Brainstorm, likewise, has calculable values. We should work towards having mathematical reasons why we should or should not include Brainstorm in a deck, especially when it plays blue.
peace,
4eak
I separate brainstorm’s effect into 3 components:
1.) [Cantrip]
2.) [Library Manipulation]
3.) [Hand+Library Manipulation].
1.) [Cantrip]–Guaranteed +1 Card in your hand, but just as important, a guarenteed -1 card in your library. Brainstorm, like many cantrips, is 1 blue mana for 1 card, which is already a fair effect (not broken, but fair). This thins your deck like a street wraith or fetch land. A cantrip replaces lower quality cards in a deck, allowing you to see the highest quality cards of a specific function.
If you run 4 brainstorm in a 60 card deck, and you go through a quarter of your deck on average before the end of a game, then you are paying, on average, a single blue mana over the course of the game to have a 56 card deck. The scaling of the cost to cycle through more of your deck per average game is linear too.
Why would you want to turn 60 cards into 56? Some cards have a higher utility, value, or relevance to your deck or specific circumstance than others, and in reality, we only want to play those instead of lower quality cards if possible. Cantrips, like brainstorm, remove lower utility cards from the equation, allowing the remaining 56 cards to have a higher average utility value than the average 60 card deck. The question then becomes: was the average cost of using cantrips worth the card-quality gains?
I’ll use as straightfoward a case as I can think of to show you what a cantrip means to card quality. This case by no means showcases the brokenness that is Brainstorm that we might find in decks that abuse it best, but the case shows the principle behind cantripping.
Let’s say you were playing a deck that had 16 Volcanic Island, 40 Lightning Bolts (120 damage value), and 4 Shocks (8 damage value). Notice that shocks, on average, are 1/3rd less valuable than a bolt. We’ll say you see 11 cards per an average game, putting you at (11/60)*4 (total mana cost of cantrips in deck), or 0.733 of a U, on average, to go from a 60 card deck to a 56 card deck. What happens when we replace the lower quality cards of a deck, shock in this case, with a cantrip?
[Total value of win conditions]/[Total mana cost of win conditions]=[Average Win condition to mana ratio per card].
Shocks– 128/44=2.909
Cantrips– 120/(40+0.733)=2.946
The gain, in part, is one of mana efficiency. The otherside of the cantrip is how it affects your average “win-condition” value met per card.
[Total value of win conditions]/[Total cards in deck]=[Average Threat value per card] (Think of DPS for you MMORPGers)
Shocks– 128/60=2.133
Cantrips– 120/56=2.143
What if we made a deck with with 40 shocks and 4 ‘Flashback-less’ lava darts, how good is a cantrip then? Presume we see 13 cards per average game, or (13/60)*4=.866. Notice that Flashback-less lava darts are only 1/2 as effect as a shock, and contribute proportionately less to the total win condition value of the deck. The Shocks are more relevant to the bolt-deck than lava-darts are to the shock deck. This difference in proportion will illustrate the rising advantages on cantrips in decks and formats that have larger card quality disparities.
[Average Win condition to mana ratio per card]
Lava Darts– 84/44=1.909
Cantrips– 80/40.866=1.958
[Average Win condition value per card]
Lava Darts– 84/60=1.4
Cantrips– 80/56=1.429
The worse your cantrip-replaced cards proportionately compare to the average mana-efficiency and win condition values of the rest of your deck the better a cantrip becomes. Here are your comparisons:
Bolts’n'Cantrips/Bolts’n'Shocks Mana efficiency ratios– 2.946/2.909=1.013
Shocks’n'Cantrips/Shocks’n'Darts Mana efficiency ratios– 1.958/1.909=1.026
Bolts’n'Cantrips/Bolts’n'Shocks Win Condition density ratios–2.133/2.143=1.005
Shocks’n'Cantrips/Shocks’n'Darts Win Condition density ratios–1.429/1.4-1.021
As the win-condition value of the least valuable cards of a deck (those to be replaced with cantrips) proportionately decreases as compared to the more valuable cards in a deck, the proportionately better a cantrip becomes.
If all your spells have fairly equal win-condition value, then the effectiveness of a cantrip decreases. So, while the greater variation in the value or relevance of your cards the better a cantrip becomes, the other side of this equation is that perfectly balanced decks with card quality equivalence would not want to use cantrips. For example, if you ran 16 Volcanic Islands and 44 Bolts, would replacing 4 bolts with 4 cantrips be worth it? Let’s say you see 11 cards per game.
[Average Win condition to mana ratio per card]
Bolts–132/44=3
Bolts/Cantrips–120/40.733=2.946
[Average Win condition value per card]
Bolts–132/60=2.2
Bolts/Cantrips–120/56=2.14
Running straight bolts is simply better than having a cantrip. Why? There is too little variation in the mana-efficiency and win-condition values of the cards in a deck with nothing but bolts and land. In cases where all things in your deck are equal in value, then cantrips are not worth it. A perfectly balanced deck would not need cantrips. Building this “perfectly balanced” deck is more complicated than many would realize though. Remember the arguments against running more than 60 cards in a deck? Usually, because there is such extreme differences in the quality of cards in older formats, we seek the smallest decks possible to abuse the few cards that are just too darn good for their mana costs. Cantrips act as the glue between the broken cards of eternal formats in these cases. However, technically, there are cases where 65-card decks could be perfectly balanced, even better than 60 card decks. Perhaps you could make a 65-card deck that had a several functions, all maximal and equal quality cards for their slot and function, and any removal of a card would imbalance the deck’s card quality. Here, you would take a 65-card deck over a 60-card deck. But, how many decks do you know are this well made? The sort of perfection in balancing card value to make it such that a 65-card deck would be preferred to a 60 card deck is the same sort of calculation and balance that a deck would need in order not to consider cantrips. If there is any imbalance in the value of the cards in your deck, then cantrips are worth considering.
Since that perfection rarely exists, often due to format card pool constrainsts, we opt for cantrips. The proportionately less valuable a card is compared to the average card quality of a deck the more likely we should replace it with a cantrip.
Take a more extreme case, say 16 Volc-Islands, 40 4-damage for 1 mana burn cards (henceforth: Uber-Bolt), and 4 Flashback-less Lava darts. Say you’ll see 10 cards in an average game. (10/60)*4=.667 mana cost to goto 56 cards.
[Average Win condition to mana ratio per card]
Lava Darts– 164/44=3.727
Cantrips– 160/40.667=3.934
[Average Win condition value per card]
Lava Darts– 164/60=2.733
Cantrips– 160/56=2.857
Cantrips give proportionately larger gains when they replace cards of proportionately lower relevance. In this case, cantrips let us not run flashback-less lava darts and stick to straight uber-bolts, giving much higher mana efficiency and average card quality.
Cantrips aren’t the end-all-be-all solution though. Take a case where we ran 16 Volcanic Islands, 24 cantrips, and 20 Uber bolts vs. 16 land and 44 Uber Bolts. Say we’ll see 20 cards per game; (20/60)*24=8. Ouch, that is 8 mana, per game, spent on just lowering the count to 36. It will take too many turns to see the cards we need to see to be mana-efficient at all.
[Average Win condition to mana ratio per card]
Cantrips– 80/32=2.5
Uber Bolts– 176/44=4
[Average Win condition value per card]
Cantrips– 80/36=2.222
Uber Bolts– 160/2.667
This is an extreme example, but it shows that there is a specific number of cantrips we wish to run in any given deck. You can easily run too many or too few cantrips in a deck.
Burn, of course, can be a more straightfoward calculation than other decks. And, you’ll notice several burn decklists use bauble-cantrips to maximize and balance average card quality..for good reason. Other decks are certainly more complicated, but the principle still remains the same though:
The higher degree of disparity found between the relevance and value of the different cards in your deck, the more useful a cantrip becomes. Eventually, if you follow this path, you’ll see the extreme in silver-bullets and tool-box decks that rely upon card-quality, cantrips and tutors to consistently find the singleton card that may be the only relevant thing in your deck against an opponent.
This is why running Yawgmoth’s Will with cards that aren’t nearly as powerful would drive us to use cantrips: by running cantrips we will receive a higher average use and benefit of Yawgmoth’s Will over the course of many games. The benefit, often enough, is worth the cost of replacing weaker cards with cantrips.
Decks that have similar components are less likely to desire cantrips. An aggro deck, for example, may have few deviations from the mean value of cards in the deck. On the other hand, a combo deck may often find themself in situations where they have 2 of 3 combo pieces in hand, but need the last one. In this case, only the missing combo piece may be relevant to our situation, and cantrips increase the likelihood of finding the relevant cards.
The cantrip component of brainstorm fulfills a major glue-mechanic in which decks can more consistently run and play with higher quality cards with different functions and values in different circumstances.
2.) [Library Manipulation]–This is a more straight foward effect to consider. Think of this as Sage Owl. How many cards do you have left in your library? To what degree do the individual cards in a deck deviate from the mean win-condition value of cards in a deck? The higher this deviation, and the lower your library count, the more effective library manipulation becomes. Bare in mind, the mean win-condition value and deviations vary per metagame, per deck, per matchup, and per specific game circumstance. This makes it incredibly difficult to calculate, but it highlights the variance we see in even the mean values across a spectrum of conditions. Library manipulation, like a cantrip, helps isolate and condense the average variance from the mean value of win-condtions across the spectrum of play-conditions.
Sage Owling into a 40-card library with nothing but Lightning Bolts isn’t going to net you anything. However, Sage Owling into a 40-card library that has only three or four relevant card in the deck (perhaps you MUST Wrath of God next turn or you lose) increases your likelihood of seeing relevant cards sooner. Like the cantrip, library manipulation benefits the deck that has higher variance in card value from the average card value.
Let’s take a basic example:
If you had 36 cards left in your library, 3 WoG’s in the library, and Wrath was the only relevant card, what does a 4-card Library manipulation effect do for you?
Without the library manipulation you have a 1 in 12 chance to draw Wrath of God next turn.
With a 4-card library manipulation you have a 1 in 9 chance to draw Wrath of God next turn.
Library manipulation is still very good even beyond looking for 1-of a specific card in your deck. It lines up your deck plays too. It could be as simple as counting your land drops drops for the next several turns and making sure relevant spells are on top with land being placed exactly where you would need to draw it so you could make a land drop for the next several turns.
Library manipulation allows you to order cards in their relevance to your current game position. If you need a counterspell before you need a land, then go ahead and put the land under the counterspell. The land may be relevant, but maybe it is less relevant than the counterspell. Library manipulation increases the quality of your future draws. A basic permutation grid of a 4-card library manipulation ensues.
Actual Card1 (in Slot1) — Value at Slot1=1, V@S2=6, V@S3=8, V@S4=5
Actual Card2 (in Slot2) — Value at Slot1=3, V@S2=5, V@S3=4, V@S4=6
Actual Card3 (in Slot3) — Value at Slot1=0, V@S2=5, V@S3=6, V@S4=2
Actual Card4 (in Slot4) — Value at Slot1=4, V@S2=2, V@S3=3, V@S4=3
Card1 moves to Slot3, Card2 moves to Slot4, Card3 moves to Slot2, Card 4 moves to Slot1.
Originally, we have a top 4-card value of 15. After a 4-card library manipulation we have a top 4-card value of 23.
The permutation grid is actually much more complicated than I’ve provided. For example, what if Card 3 only has a value of 5 in Slot2 if and only if Card 4 is in Slot1? Multiply this type of value calculation, and you see that identifying the value specifics cards, even in a very specific circumstances, can be quit complicated. These are the sorts of mental calculations that we make on the fly. It seems obvious, but drawing out the reason why we do what we are doing is more complex than we initially thought.
Shuffle effects has a specific effect right here too. You know the value of the top X cards of your library. Is that value below the average value of X cards in your library? If it is below, then a shuffle effect increases the value by [Average Value of X cards]-[Value of Current X cards].
Decks with higher variations of value per slot make the most use of library manipulation. Again, perfectly rounded decks with zero variance from the mean value per card would not want library manipulation. It must be noted that this perfect balance might not be found at 60 cards in a specific format and metagame, and thus a perfect deck without cantrips might not be possible in many circumstances.
Brainstorm has a 2-card library manipulation value. However, it’s 3rd effect is the game-breaking ability that twists library manipulation into relevant and immediate card advantage and quality.
3.) [Hand+Library Manipulation] This is a very odd effect in magic. This is the effect that makes brainstorm more than just a mere cantrip and library manipulation. This ability might be seen as an extension of library manipulation, but we must distinguish this component of brainstorm from a Sage Owl effect because of the influence this mechanic has upon an active hand. This effect alone can make brainstorm as good as Ancestral recall + 1/2 a Sage Owl or as bad as 1/3 an Ancestral Recall + 1/2 a Sage owl. That’s right, I said it: Brainstorm can be BETTER than Ancestral Recall. There is a two card quantity difference between the worst brainstorm and the best possible brainstorm, and most of the math behind understanding the value of a specific resolution of Brainstorm, as found between that spectrum, relies upon this mechanic.
Given the cantrip effect, the Hand+Library Manipulation effect is only a count of 2 cards. The value differences
To look at the Hand+Library Manipulation effect itself, we will neglect the cantrip and library manipulation components of Brainstorm for now.
Hand:
Card in Hand1 — Value in Hand=2, Value in Library=2 (in not particular order on library, just with X cards from the top)
Card in Hand2 — Value in Hand=3, Value in Library=0
Card in Hand3 — Value in Hand=1, Value in Library 1
Library
Card on Library1 — Value in Hand=3, Value in Library 1
Card on Library2 — Value in Hand=1, Value in Library 0
Current value of Hand+Library=7.
After a 2-card Hand+Library manipulation, where Hand1 is replaced with Library1 and Hand3 is replaced with Library2, the value of Hand+Library=10
This is rudimentary, but it shows the basic principle. It also shows how one can calculate the value of hiding valuable cards against discard effects.
Echoing truth against a High-tide/Reset deck is a useless card. Hand-Library manipulation increases your current hand value by putting echo onto your library. This, of course, is at the expense of future draw values. However, when you add shuffle effects, it turns dead cards into average card value. Essentially, Hand-library manipulation is a 3-fold utility:
1.) Gives immediate hand value increase equal to [Cards put into hand from library] - [Cards put on top of library].
2.) Combined with shuffle effect, increases library value to [Average library card value of X cards] - [current top X card library value]
3.) Hiding valuable cards that you don’t want discarded.
The permutation grid becomes more complicated than just this current turns active hand. Of course, we have to consider the effect upon future draws. This falls directly into the library manipulation arena.
Brainstorm does this effect like no other card for such a cheap cost. In so many cases, when your opening turn or 2nd turn really decides the match, brainstorm sculpts the early game. Some decks, for example control decks, have great need in sculpting their early game active hand (well beyond other archetypes' early game needs).
Card value–
I’ve talked alot about card value. But, I haven’t given any good definition for it. For now, I can't give you a really clean and easy to read definition. This is a much more complicated topic beyond the scope of this article (and I'm working on the article). Here is what I will say with regards to Brainstorm and card value:
Cantrips, Hand and Library manipulation are difficult to evaluate because it requires a system of identifying the exact of value of each card in a library in average circumstances/metagames and the mean variance of value between cards and functions in those circumstances/metagames.
Card values vary per metagame, but are static to any one specific metagame. Yawgmoth’s Will is inherently stronger in a format that is better at getting cards into the graveyard. Basic land is inherently stronger in a format that doesn’t have better alternatives. There are two types of metagames to consider:
Universal Metagame (format restricted metagame)
Specific Metagame (your circumstances)
If we are to understand the value of any card, including Brainstorm, to an exacting degree, it must be with the knowledge of what metagame we are speaking.
For the purposes of deckbuilding, card values are determined by their degree of influence on the offense/defense ratio of a deck. You are attempting to quantify how essential a card is to your win condition, or convertibly, your defense against an opponent's win condition (having a higher offense/defense ratio than your opponent).
A deck or card is meaningless without an opponent or metagame to interpret its value. Against a metagame/opponent with a 60-land deck, the best deck will be the one that has the highest and fastest average win condition. It isn’t just whether you won, it is the margin by which you win or lose that helps form the value of the cards in your deck against a 60-land deck metagame. The win-condition to be met, in this metagame, is simply reaching the stage where an opponent has 20 lifeloss, has been milled and can’t draw, or loses through a straight “win-ability” (Door to Nothingness, etc.). Whichever deck has the highest and fastest average chance of reaching that win-condition is the best deck in that metagame.
What if you went against a 60-FoW deck metagame? Perhaps the deck that was the best in a 60-land deck metagame would not be the best in this specific 60-FoW deck metagame. The combo deck that probably evolved in the 60-land deck metagame was not prepared to deal with permission. You might say it wanted “speed” at all costs. But, in reality, the deck is only as good as it matches against a specific metagame.
The value of a card has those two values, value in the universal context and the specific (and only one will matter when we speak of the value of a card). So, we should not be confused into thinking value isn’t calculable. Rather, we just need to see how to go about looking for and speaking about its value in the first place.
Brainstorm, likewise, has calculable values. We should work towards having mathematical reasons why we should or should not include Brainstorm in a deck, especially when it plays blue.
peace,
4eak