Re: linear definition
A long time ago, I started writing an article about linear/non-linear and interactive/non-interactive decks. I was working with two definitions here.
Linearity: # of ways a deck can WIN. Interactivity: # of ways a deck can INTERACT with an opponent's non-life total resources.
Basically, you'd code every card in a deck to see how it contributed to linearity or interactivity, and then score decks based on those measures. Low score is more linear, less interactive. High score is less linear, more interactive. Note that interactivity doesn't include damaging players, but rather how your cards can interact with their other resources. This would create issues with Mill and Lantern, but works for other decks.
So let's preliminarily give this deck a linearity score of 10.. That's pretty darn linear. Yes, I realize there are different ways to count this and also combinations of ways to win (e.g. Grapeshot for 10, Electromancer swings for 6, Baral finishes for 4), but we're still effectively capturing that above by highlighting the three primary avenues.
How does this deck actually interact with an opponent? Here are the cards that are capable of interaction AND the different ways they can interact. I'm double counting copies of a card for every different interactive mode they have.
1. Electromancer can block (3 copies)
2. Baral can block (4)
3. Noxious Revival can screw with the top of an opponent's library (1)
4. Noxious Revival can screw with graveyard recursion (1)
5. Remand can counter an opponent's spells (3)
6. Remand can protect your spells (3)
7. Grapeshot can target creatures (3)
8. Grapeshot can target planeswalkers (3)
So if we add up all those different copies of cards capable of doing different things, we get an interactivity score of 21. So our final Storm score is 10 for linearity and 22 for interactivity.
Which would mean GDS is about twice as non-linear (half as linear) as Storm and 3 times as interactive as Storm. This fits the common understanding of Storm vs. GDS, which suggests there is something to the method.
The issue with this method is that decks like Burn and Affinity tend to also register as non-linear and highly-interactive because their cards technically have lots of modes. For instance, this list here (http://sales.starcitygames.com//deckdatabase/displaydeck.php?DeckID=117638) has a linearity score of 55 (even less linear than GDS because it has so many cards contributing to its win) and an interactivity score of 62, which makes it both less linear and more interactive than GDS. All of us know why that is going astray: even though Burn cards technically have lots of options, in practice they are all committed towards a single end, which makes it feel linear. But the method can't distinguish. Affinity (http://sales.starcitygames.com//deckdatabase/displaydeck.php?DeckID=117646) is also an oddball: linearity 45, interactivity 34.
So overall, this kind of method has promise but needs refinement/revision. I like it as a starting point to objectively rate deck linearity/interactivity based on their scores relative to other decks, even if it has some flaws. That's one of the reasons I never finished the article and am now submitting it here for some tinkering.
@ktkenshinx, That sounds like the most scientific approach that I've seen yet in these forums. It's very similar to how actual game theorists devise linear programming models, it seems, with expectiminimax trees and decisions. I would be very interested in reading your fleshed out explanation of this approach.
That's a very convenient editing of my original statement
When I say "lines of play", I'm talking winning strategies and tactics, or, the decision trees and their associated angles of attack to achieve a winning gamestate. We can take Storm as an example, since it can be both very linear and nonlinear, depending on how it is built. If the Storm player is relying wholly on Grapeshot, and has zero Empty the Warrens, nor any other card or set of cards that will be the final payoff of the mechanic by which the deck operates, then we can rather easily classify it as linear. We should probably note that, while playing and attacking with Baral and Goblin Electromancers is possible, it is not at all likely enough for consideration. However, if the Storm player has built their deck with both Grapeshot and Empty the Warrens, or some other additional payoff cards, then it has now opened up multiple lines of attack. Thus, if the Grapeshot is just not feasible (due to hexproof), then they still have another line that offers an alternate angle of attack, in using Empty the Warrens in an attempt to minimize the effectiveness of any protection the opponent might have (the number of blocking creatures is typically minimized to be meaningless using Empty the Warrens).
Why would you choose to edit that part out? Seems like it's very relevant to the conversation. If I recall correctly, I even used a case example, in which a recent tournament shows us a Humans player shutting down Storm entirely with a single Meddling Mage naming Grapeshot in game one, when the Storm player didn't have access to Empty the Warrens (because they didn't include on in the main). That is a great example of being hyper-linear, as the "alternate options" (attacking with Baral and Electromancers into the plethora of Humans the opponent has available) is simply not feasible enough to warrant consideration. Adding that single Empty the Warrens, however, would make it non-linear. Sure, it's still "more linear" than plenty of other decks, but strictly speaking, it is no longer non-linear. Blurring those lines is a great way to add confusion to the subject, and serves to support opinions not entirely based on fact.
I understand. The reason that I'm very strict in how I define "linear" and "non-linear" is because, as it relates to the study of game theory, adding another option for achieving a winning gamestate has opened up another decision tree. The term "line" itself infers a one-dimensional nature. Once we add that second option, that second decision tree available, then we are, strictly speaking, no longer one-dimensional. It may be fewer dimensions than other decks, but it is still no longer one-dimensional. Maybe it would benefit us to find some terminology that is more precise for the number of decision trees available without resorting to "rounding off" decision tree numbers to those which we just feel most comfortable using?
I was looking at my numbers above and thinking about GK's and thinkr's points. I think there are some hidden decision trees I may have missed in my initial analysis. One of them is the card type: for instance, instant-speed Bolt offers more flexibility than sorcery-speed Spike, and thus presents another decision. I also like thinkr's ideas of decision trees generally, and think it's a better definition for linearity than merely "how many ways can you win." Technically, a deck like Mill or Burn has tons of ways to win in its deck, but they all contribute to the same decision line. GDS, Jeskai, and other decks give more options and require more branching trees, and I think that's what most people mean when they talk about linearity.
So, new definition:
Linearity: # of decision trees in a deck
Basically, we'd look at a deck and see how many decisions any single card represents. This means timing (instant vs. sorcery), options when you play it, how you can use it, etc. We then multiply the number of decisions a card gives by the number of those cards in a deck, and then add up the scores for each grouping of cards. That would be a deck's linearity score. Let's try it out! High = lots of decision trees, low = less decision trees.
Less screwy than the last method but still some glitches. I think one of the issues here is that I am effectively trying to calculate card combo permutations by hand and I'm either over-counting or under-counting options. For instance, Snapcaster clearly creates more options and decisions with spells, but how is that expressed? Currently, I calculate the number of options a spell has (Opt has three: top, bottom, instant), multiply that by the number of Opts in the deck (2*3=6), and then multiply that by the number of Snapcasters (6*4=24), adding that to the total score. This gives GDS a ton of options, but the same method also gives Affinity a ton of options because of Ravager and Plating. So I might be weighting things incorrectly. I also might need to give Storm more Gifts piles options, which would naturally increase its score, and I might have missed some options entirely.
That said, this overall approach definitely makes more sense than the last method and feels like a better operationalization of linearity. It still needs refinement but this is probably the direction we should go in if we want to quantify linearity (at least, as long as we're comfortable with my linearity definition as "decision trees").
The other part of the discussion here is interactivity, and how that plays into the linearity of a given deck. While those two (linearity and interactivity) aren't the same, they do relate to each other. This may help clarify some things regarding ktk's current model; for example, Cranial Plating does present more options/decisions for Affinity, but those options end up with the same result, which in this case is increasing your creature's power and toughness. How does that weigh against something that offers options that all interact with your opponent, such as KCommand? Is there a way to put more weight on options that interact with your opponent? Is that even necessary/desirable? I don't know, it's 330am here and I'm spitballing, but it's something I thought should be discussed.
The other part of the discussion here is interactivity, and how that plays into the linearity of a given deck. While those two (linearity and interactivity) aren't the same, they do relate to each other. This may help clarify some things regarding ktk's current model; for example, Cranial Plating does present more options/decisions for Affinity, but those options end up with the same result, which in this case is increasing your creature's power and toughness. How does that weigh against something that offers options that all interact with your opponent, such as KCommand? Is there a way to put more weight on options that interact with your opponent? Is that even necessary/desirable? I don't know, it's 330am here and I'm spitballing, but it's something I thought should be discussed.
that's exactly what I thought aswell. Does interaction mean more decision trees?
I mean there is a reason why urx is harder to play than burn over a long event. Pros admit to this and thus choose the lighter decks for the sake of Thier mental endurance.
Interactivity offers more decision trees look at thoughtseize and a counterspell.
I can't find it right now but does anyone remember an article about the whole thought process of a turn one thoughtseize.
Proactivity and reactivity also affect decision trees. A thoughtseize has max 7 decisions a counterspell you can wait and pick.
Less decision trees-Linear and uninteractive (Storm)
Most decision tress-Non linear and interactive (First Uxx control cause it is reactive and then GBx cause it is proactive)
and there is an argument to be Made that the more decision trees the more skill required to play the deck.
Which is why people often call burn mindless and urx skillful.
Not that I'm saying that myself but it's just an observation.
And when you look at the top 10 decks in modern you do tend to see why people often like to use the word linear when it comes to modern. Even though there certainly are a certain amount of urx and bgx decks in there aswell.
Interactivity offers more decision trees look at thoughtseize and a counterspell.
I can't find it right now but does anyone remember an article about the whole thought process of a turn one thoughtseize.
Proactivity and reactivity also affect decision trees. A thoughtseize has max 7 decisions a counterspell you can wait and pick.
Less decision trees-Linear and uninteractive (Storm)
Most decision tress-Non linear and interactive (First Uxx control cause it is reactive and then GBx cause it is proactive)
I think this is one major reason that we should really be considering expectiminimax when discussing this. The term "interactive", as it's commonly used in these forums and in many MtG circles, is often defined as a self-centered argument: "They aren't interacting with me". The truth is, strictly speaking, it is rare, if ever possible, that any player actually interacts with the other. Strictly speaking, we only ever interact with the gamestate when use make game actions. Just as in chess, when a player makes a move, even when taking a piece, they aren't interacting with the opponent. They're interacting with the gamestate.
Storm does quite a bit of interacting with the gamestate. It just does it in a way that it doesn't typically interact with gamestate actions that the opponent has made (their creatures, their cards in hand, etc.). Otherwise, however, every spell a Storm pilot makes has interacted with the gamestate.
According to expectiminimax, the primary goal is to interact with the gamestate in such a way as to minimize the opponent's options available or significance of the opponent's options available to change the gamestate. Storm does exactly that, in that the deck operates in such a way as to bypass the significance of creatures or lands that an opponent has played, and, if possible, most non-creature spells.
Thus, as I mentioned, I think it's probably important that we make this distinction when we use the term "interaction" or "interactive". I mentioned before, and I think gkourou is very close to this point as well, in that the words "linear" and "non-interactive" are, for the most part, used as slander towards decks that people feel emotionally charged about rather than in their truest definitions. They are basically re-defined as seen fit to support emotional arguments against decks. While I understand, as some mentioned before, that if a poll was taken that many might agree to this layman definition of the terms, that in no way makes it accurate. It only makes it popular.
I actually got a beef with people calling linear decks bad for the format. The fact is that linear decks are where every person starts when building a magic deck. When people were first building Merfolk the obvious goal was to just fill the board with a glut of lords and steamroll, but as the deck evolved people put in new lines of play via cards with non-linear abilities, sacrificing power for card draw, counter magic, etc. Then we had Aether Vial join in to push the deck further along with the tribal block of Lorwyn / Shadowmoor. This is just going into merfolk and not even talking about the evolution of Jund and burn. Linear decks showing up that are not necessarily heavily established are a sign of growth, not necessarily a sign of a design flaw.
The T1 and T2 decks we see today should be considered the chronicles of modern magic history, to be frank.
Private Mod Note
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Rollback Post to RevisionRollBack
1. (Ravnica Allegiance): You can't keep a good esper control deck down... Or Wilderness Reclamation... or Gates...
2. (War of the Spark): Guys, I know what we need! We need a cycle of really idiotic flavor text victory cards! Jace's Triumph...
3. (War of the Spark): Lets make the format with control have even more control!
I actually got a beef with people calling linear decks bad for the format. The fact is that linear decks are where every person starts when building a magic deck. When people were first building Merfolk the obvious goal was to just fill the board with a glut of lords and steamroll, but as the deck evolved people put in new lines of play via cards with non-linear abilities, sacrificing power for card draw, counter magic, etc. Then we had Aether Vial join in to push the deck further along with the tribal block of Lorwyn / Shadowmoor. This is just going into merfolk and not even talking about the evolution of Jund and burn. Linear decks showing up that are not necessarily heavily established are a sign of growth, not necessarily a sign of a design flaw.
The T1 and T2 decks we see today should be considered the chronicles of modern magic history, to be frank.
Interactivity offers more decision trees look at thoughtseize and a counterspell.
I can't find it right now but does anyone remember an article about the whole thought process of a turn one thoughtseize.
Proactivity and reactivity also affect decision trees. A thoughtseize has max 7 decisions a counterspell you can wait and pick.
Less decision trees-Linear and uninteractive (Storm)
Most decision tress-Non linear and interactive (First Uxx control cause it is reactive and then GBx cause it is proactive)
I think this is one major reason that we should really be considering expectiminimax when discussing this. The term "interactive", as it's commonly used in these forums and in many MtG circles, is often defined as a self-centered argument: "They aren't interacting with me". The truth is, strictly speaking, it is rare, if ever possible, that any player actually interacts with the other. Strictly speaking, we only ever interact with the gamestate when use make game actions. Just as in chess, when a player makes a move, even when taking a piece, they aren't interacting with the opponent. They're interacting with the gamestate.
Storm does quite a bit of interacting with the gamestate. It just does it in a way that it doesn't typically interact with gamestate actions that the opponent has made (their creatures, their cards in hand, etc.). Otherwise, however, every spell a Storm pilot makes has interacted with the gamestate.
According to expectiminimax, the primary goal is to interact with the gamestate in such a way as to minimize the opponent's options available or significance of the opponent's options available to change the gamestate. Storm does exactly that, in that the deck operates in such a way as to bypass the significance of creatures or lands that an opponent has played, and, if possible, most non-creature spells.
Thus, as I mentioned, I think it's probably important that we make this distinction when we use the term "interaction" or "interactive". I mentioned before, and I think gkourou is very close to this point as well, in that the words "linear" and "non-interactive" are, for the most part, used as slander towards decks that people feel emotionally charged about rather than in their truest definitions. They are basically re-defined as seen fit to support emotional arguments against decks. While I understand, as some mentioned before, that if a poll was taken that many might agree to this layman definition of the terms, that in no way makes it accurate. It only makes it popular.
I would say chess is a game of attrition. One which a better board state is the goal. And outvaluing your opponent is the key.
Storm doesn't strike me as a very chess like deck, storm Feels more like solitaire even though it uses some but little interaction with the opponent.
Now jund for example actually cares about your board state and 1 for 1s or 2 for 1s you until they have control over it.
I actually got a beef with people calling linear decks bad for the format. The fact is that linear decks are where every person starts when building a magic deck. When people were first building Merfolk the obvious goal was to just fill the board with a glut of lords and steamroll, but as the deck evolved people put in new lines of play via cards with non-linear abilities, sacrificing power for card draw, counter magic, etc. Then we had Aether Vial join in to push the deck further along with the tribal block of Lorwyn / Shadowmoor. This is just going into merfolk and not even talking about the evolution of Jund and burn. Linear decks showing up that are not necessarily heavily established are a sign of growth, not necessarily a sign of a design flaw.
The T1 and T2 decks we see today should be considered the chronicles of modern magic history, to be frank.
They are bad, when its the best way to play.
exactly. I think people confuse: I hate a linear dominated game with;I hate linear dominant decks and don't think they should be allowed in modern. Although there are certainly some from the ladder.
For example, there are lines of play in chess that many people might consider "linear", or "non-interactive", in the layman terms as described here. I specifically teach my students to watch out for those openings, and how to combat it. Two examples are the scholar's mate and the queen's raid, and there are plenty of other examples to be seen.
In chess, however, minimax also works. It's not subject to expectiminimax because there is no chance involved, but the principle of minimizing the opponent's options, or the significance of their options, is a primary goal. There are a few ways to do this (setting up pins, forks, skewers, minimizing space available, getting a small material advantage and then playing attrition to the end, etc), but they all work for this goal.
You can see in my quote how Storm does this:
Storm does exactly that, in that the deck operates in such a way as to bypass the significance of creatures or lands that an opponent has played, and, if possible, most non-creature spells.
The opponent's creatures and/or permanents don't matter if the Storm player is able to bypass them by going straight to the face with a good Grapeshot. They similarly don't matter if the Storm player is able to reduce the significance of attackers or blockers the opponent has by amassing a large army of goblins. There are plenty of unique lines of play that accomplish one of these two goals, but those two "endpoints" of the decisions trees are what define how the deck is designed to work.
As for midrange decks like Jund, they similarly attempt to minimize the opponent's options, and/or the significance of their options. They use discard spells to remove options from the hand and removal spells to remove the opponent's options on the battlefield. They also use threats like Tarmogoyf, which are designed to minimize the opponent's battlefield options by out-sizing (or, outvalue-ing) them. An opponent's creature means very little if it cannot race a Tarmogoyf. Therefore, the significance of the opponent's creature on the battlefield is reduced.
In every case, we can see how expectiminimax is directly invoked to achieve a winning advantage. Every single competitive deck works best when it is allowed to solitaire, not just decks like Storm. The goal is to achieve a gamestate in which it might as well be solitaire. Even in chess, the primary principle that I teach students is to try to get one small pawn advantage, and then trade down until it becomes a king vs. king & pawn endgame. At that point, it's pretty much solitaire (if done correctly), and the win is inevitable. If done correctly, it shouldn't matter what the opponent does with their king, because they haven't allowed them to do anything significant. The only reason the opponent is even allowed a move at all is because it is required within the rules of the game.
While I understand that some might not find this "fun", that is the harsh reality of competitive games.
So how about that Bloodbraid Elf unban we should all agree on?
In all seriousness, Wizards may be more conservative than ever. Worlds was amazing to watch, but every single relevant team had a Temur/Sultai/4Color Energy Variant in full form forcing the sacrifices to be made on the other two decks. I forsee a Standard banning. I'm not sure if Wizards would compensate by giving Modern a few bones.
For example, there are lines of play in chess that many people might consider "linear", or "non-interactive", in the layman terms as described here. I specifically teach my students to watch out for those openings, and how to combat it. Two examples are the scholar's mate and the queen's raid, and there are plenty of other examples to be seen.
In chess, however, minimax also works. It's not subject to expectiminimax because there is no chance involved, but the principle of minimizing the opponent's options, or the significance of their options, is a primary goal. There are a few ways to do this (setting up pins, forks, skewers, minimizing space available, getting a small material advantage and then playing attrition to the end, etc), but they all work for this goal.
You can see in my quote how Storm does this:
Storm does exactly that, in that the deck operates in such a way as to bypass the significance of creatures or lands that an opponent has played, and, if possible, most non-creature spells.
The opponent's creatures and/or permanents don't matter if the Storm player is able to bypass them by going straight to the face with a good Grapeshot. They similarly don't matter if the Storm player is able to reduce the significance of attackers or blockers the opponent has by amassing a large army of goblins. There are plenty of unique lines of play that accomplish one of these two goals, but those two "endpoints" of the decisions trees are what define how the deck is designed to work.
As for midrange decks like Jund, they similarly attempt to minimize the opponent's options, and/or the significance of their options. They use discard spells to remove options from the hand and removal spells to remove the opponent's options on the battlefield. They also use threats like Tarmogoyf, which are designed to minimize the opponent's battlefield options by out-sizing (or, outvalue-ing) them. An opponent's creature means very little if it cannot race a Tarmogoyf. Therefore, the significance of the opponent's creature on the battlefield is reduced.
In every case, we can see how expectiminimax is directly invoked to achieve a winning advantage. Every single competitive deck works best when it is allowed to solitaire, not just decks like Storm. The goal is to achieve a gamestate in which it might as well be solitaire. Even in chess, the primary principle that I teach students is to try to get one small pawn advantage, and then trade down until it becomes a king vs. king & pawn endgame. At that point, it's pretty much solitaire (if done correctly), and the win is inevitable. If done correctly, it shouldn't matter what the opponent does with their king, because they haven't allowed them to do anything significant. The only reason the opponent is even allowed a move at all is because it is required within the rules of the game.
While I understand that some might not find this "fun", that is the harsh reality of competitive games.
Jund and storm are apples and oranges however. You could make them sound the same, but they attack from very different angles from each other.
Like ktk said there are varying degrees to interaction and linearity. Which mean decks don't operate in exact ways in modern and there are alot of them.
If you look at storm vs shadow you can see how one has more decision trees in ktks analysis.
In chess each player is playing the exact same pieces and must interact and play attrition if they wish to advance and win (especially so when the players are of a similiar elo) So using chess to compare to mtg mathematics isn't really a solid case.
I actually got a beef with people calling linear decks bad for the format. The fact is that linear decks are where every person starts when building a magic deck. When people were first building Merfolk the obvious goal was to just fill the board with a glut of lords and steamroll, but as the deck evolved people put in new lines of play via cards with non-linear abilities, sacrificing power for card draw, counter magic, etc. Then we had Aether Vial join in to push the deck further along with the tribal block of Lorwyn / Shadowmoor. This is just going into merfolk and not even talking about the evolution of Jund and burn. Linear decks showing up that are not necessarily heavily established are a sign of growth, not necessarily a sign of a design flaw.
The T1 and T2 decks we see today should be considered the chronicles of modern magic history, to be frank.
They are bad, when its the best way to play.
I agree with you, if the deck doesn't evolve into something more and just stagnates because the power of the linear play is just so good there isn't a reason to move things forward, then its a bad thing. Queue a giant rant on how parasitic mechanics like infect are a mistake and they should be designing things that work with existing mechanics. That or they just plug up their ears and make brick counters or have that entire deal with eldrazi happen again. Also now I'm imagining a future where they make TKS, and when I say that I'm talking Thought-Knot Sliver...
1. (Ravnica Allegiance): You can't keep a good esper control deck down... Or Wilderness Reclamation... or Gates...
2. (War of the Spark): Guys, I know what we need! We need a cycle of really idiotic flavor text victory cards! Jace's Triumph...
3. (War of the Spark): Lets make the format with control have even more control!
@hansolo, I think I see where you're coming from. I would agree that Jund and Storm work very differently, although I disagree that their angles of attack are different. It's the methods of which they reduce the opponent's options and significance of options that are different. They both attack face with either spells or creatures, however.
The key point that I was trying to get across is that they both are designed with expectiminimax at their core. They just use the theory for different decision trees. The end nodes (burn spell to face or creature damage, so long as the Storm player included Empty the Warrens) are pretty much the same, but the branches themselves look very different.
One of the reasons that I think it's important for us to consider the game in this sense is because it is possible for a player to make the opponent's deck non-interactive simply by the design of the player's deck.
For example, UW Control runs plenty of creature removal. I think we could agree that creature removal spells are interactive. However, what if we choose to play a deck that doesn't rely on creatures at all? Is it fair for us to say that UW Control isn't an interactive deck any more? By simply choosing to build our deck without using creatures we have reduced the UW Control deck's "interactive score". Another example is if we do use creatures, but we use cards like Cavern of Souls, which makes our creatures uncounterable. UW Control runs plenty of counters. Again, by the design of our deck, we've reduced the "interactive score" of the opponent's deck. Thus, when we talk about a deck being "non-interactive", this is a bit misleading, as we haven't considered the metagame and the ability of another deck to influence how interactive each other deck in that metagame can be.
I think, therefore, that it is important for us to recognize the distinction of a deck's ability to reduce the opponent's ability to interact in a significant manner, rather than how it "interacts". All of this, of course, has already been very well fleshed out and thoroughly described and proven. For us to adopt this method is just us recognizing that the common understanding of the game has yet to evolve to that level.
Other games have already adopted the understanding of expectiminimax, minimax, etc. That's the primary reason that chess is pretty much a solved game at this point, and why Alpha Go was able to beat humans. There are underlying principles within the game that the MtG community in general just haven't come to recognize as legitimate nor apply to the game, although it is painfully obvious that they do apply.
That is my goal here. I want us to evolve our understanding of the game, to catch up with what game theory has been saying for quite some time now. While we debate how we want to define as "linear" and "interactive", these definitions are already laid out and well known. While we, in this forum, argue what decks we like and what decks we don't like, there are principles of game theory that can be applied to figuring out how to define the strengths and weaknesses of those decks, to better understand how they work and how to play or combat them.
So while chess and MtG are different, the same principles of game theory are, without a doubt, applicable to both. We just have to reach that level where we understand and apply them. If our goal isn't to achieve a better understanding of the game, when what exactly is it?
@hansolo, I think I see where you're coming from. I would agree that Jund and Storm work very differently, although I disagree that their angles of attack are different. It's the methods of which they reduce the opponent's options and significance of options that are different. They both attack face with either spells or creatures, however.
The key point that I was trying to get across is that they both are designed with expectiminimax at their core. They just use the theory for different decision trees. The end nodes (burn spell to face or creature damage, so long as the Storm player included Empty the Warrens) are pretty much the same, but the branches themselves look very different.
One of the reasons that I think it's important for us to consider the game in this sense is because it is possible for a player to make the opponent's deck non-interactive simply by the design of the player's deck.
For example, UW Control runs plenty of creature removal. I think we could agree that creature removal spells are interactive. However, what if we choose to play a deck that doesn't rely on creatures at all? Is it fair for us to say that UW Control isn't an interactive deck any more? By simply choosing to build our deck without using creatures we have reduced the UW Control deck's "interactive score". Another example is if we do use creatures, but we use cards like Cavern of Souls, which makes our creatures uncounterable. UW Control runs plenty of counters. Again, by the design of our deck, we've reduced the "interactive score" of the opponent's deck. Thus, when we talk about a deck being "non-interactive", this is a bit misleading, as we haven't considered the metagame and the ability of another deck to influence how interactive each other deck in that metagame can be.
I think, therefore, that it is important for us to recognize the distinction of a deck's ability to reduce the opponent's ability to interact in a significant manner, rather than how it "interacts". All of this, of course, has already been very well fleshed out and thoroughly described and proven. For us to adopt this method is just us recognizing that the common understanding of the game has yet to evolve to that level.
Other games have already adopted the understanding of expectiminimax, minimax, etc. That's the primary reason that chess is pretty much a solved game at this point, and why Alpha Go was able to beat humans. There are underlying principles within the game that the MtG community in general just haven't come to recognize as legitimate nor apply to the game, although it is painfully obvious that they do apply.
That is my goal here. I want us to evolve our understanding of the game, to catch up with what game theory has been saying for quite some time now. While we debate how we want to define as "linear" and "interactive", these definitions are already laid out and well known. While we, in this forum, argue what decks we like and what decks we don't like, there are principles of game theory that can be applied to figuring out how to define the strengths and weaknesses of those decks, to better understand how they work and how to play or combat them.
So while chess and MtG are different, the same principles of game theory are, without a doubt, applicable to both. We just have to reach that level where we understand and apply them. If our goal isn't to achieve a better understanding of the game, when what exactly is it?
but that's what the term combo comes from. It's a method used to break the mechanics of the game. Sure they are both playing mtg which can be technically the same by your preferred theory. And every deck in modern is certainly trying to "win".
It still doesn't mean decks do not differ in linearity interactivity and overall "fun"ness to play against.
I think chess is much easier to solve with your preferred theory.
I feel there is a line that shouldn't be crossed when it comes to how efficient a deck is at preventing the opponents options in interacting.
That's why probe was banned for example. And that's why storm and dredge has been nerfed before.
Because even though they never broke any meta percentages they still where too efficient at creating a hard to interact with/unfun game by both Thier speed and resilience.
And if we used the expectiminimax theory on bans the format would have been dead long ago because that theory doesn't consider how fun the game is. It only shows that as long as people are trying to win and they also do not take up too much meta percentage they are fine.
Saying jund and storm attack from a similar angle because they deal damage to your face and try to prevent you from beating them is a bit disingenuous Imo.
but that's what the term combo comes from. It's a method used to break the mechanics of the game. Sure they are both playing mtg which can be technically the same by your preferred theory. And every deck in modern is certainly trying to "win".
It still doesn't mean decks do not differ in linearity interactivity and overall "fun"ness to play against.
I agree with all of this.
I think chess is much easier to solve with your preferred theory.
I also kind of agree with this. Expectiminimax is much more complicated that minimax, simply due to the necessity to calculate chance as well.
I feel there is a line that shouldn't be crossed when it comes to how efficient a deck is at preventing the opponents options in interacting.
That's why probe was banned for example. And that's why storm and dredge has been nerfed before.
Because even though they never broke any meta percentages they still where too efficient at creating a hard to interact with/unfun game by both Thier speed and resilience.
And I agree with this.
And if we used the expectiminimax theory on bans the format would have been dead long ago because that theory doesn't consider how fun the game is. It only shows that as long as people are trying to win and they also do not take up too much meta percentage they are fine.
I would disagree with this. This statement assumes that, for some reason, using expectiminimax couldn't identify the problem, with absolute certainty, and help find a solution. It assumes that we might apply expectiminimax and then do nothing with the deeper understanding. Applying it doesn't mean we just take no action whatsoever. Applying expectiminimax doesn't mean understanding the game on a deeper level and then just leaving it at that. It means approaching the game from a higher understanding, based on solid evidence, logic, and math, in order to grasp exactly why a deck is busted.
Saying jund and storm attack from a similar angle because they deal damage to your face and try to prevent you from beating them is a bit disingenuous Imo.
I understand if that's your opinion, but solid game theory shows, with certainty, without needing to rely on opinion, that the end nodes use the same function.
but that's what the term combo comes from. It's a method used to break the mechanics of the game. Sure they are both playing mtg which can be technically the same by your preferred theory. And every deck in modern is certainly trying to "win".
It still doesn't mean decks do not differ in linearity interactivity and overall "fun"ness to play against.
I agree with all of this.
I think chess is much easier to solve with your preferred theory.
I also kind of agree with this. Expectiminimax is much more complicated that minimax, simply due to the necessity to calculate chance as well.
I feel there is a line that shouldn't be crossed when it comes to how efficient a deck is at preventing the opponents options in interacting.
That's why probe was banned for example. And that's why storm and dredge has been nerfed before.
Because even though they never broke any meta percentages they still where too efficient at creating a hard to interact with/unfun game by both Thier speed and resilience.
And I agree with this.
And if we used the expectiminimax theory on bans the format would have been dead long ago because that theory doesn't consider how fun the game is. It only shows that as long as people are trying to win and they also do not take up too much meta percentage they are fine.
I would disagree with this. This statement assumes that, for some reason, using expectiminimax couldn't identify the problem, with absolute certainty, and help find a solution. It assumes that we might apply expectiminimax and then do nothing with the deeper understanding. Applying it doesn't mean we just take no action whatsoever. Applying expectiminimax doesn't mean understanding the game on a deeper level and then just leaving it at that. It means approaching the game from a higher understanding, based on solid evidence, logic, and math, in order to grasp exactly why a deck is busted.
Saying jund and storm attack from a similar angle because they deal damage to your face and try to prevent you from beating them is a bit disingenuous Imo.
I understand if that's your opinion, but solid game theory shows, with certainty, without needing to rely on opinion, that the end nodes use the same function.
unfortunately we don't have these methods at this point to apply your preferred theory with 100 percent proven accuracy with mtg, and have to use data from events as well as deductive reasoning. But that does not mean that those methods are bad.
And it certainly does not dismiss the fact that storm and jund are winning in different ways and different angles dispite in the end going for your hp and trying to both not allow interaction from your opponent.
I think process of identifying linear and interactive decks, assuming it is simply a scale from linear to non-linear could be solved much more simply by defining a linear deck as:
A linear deck's ideal lines of play (decision trees) are the same or exceedingly similar regardless whether an opponent exists or an opponent with no cards and 20 life existed.
An interactive deck can be defined as follows:
An interactive deck actively modifies the solution to the opponent's decision tree for finding the best path to victory.
There are a lot of cards that are "interactive" in the above sense like bloodmoon, ensnaring bridge, ghostly prison, chalice of the void etc etc that modify your opponent's decision trees, or changes what your opponent's path to victory are, but are not considered very fun to play against (but fun is 100% opinion).
Also I think any mention in the "number of decision trees" has nothing to do with linearity or whether a deck is interactive enough. That is simply the complexity of the deck. Something like storm has much higher complexity than most other decks (believe it or not) because there are so many forking options available. Making the absolute ideal decision at every fork is extremely difficult when you are constantly scrying, deciding, gifts packaging etc. When playing storm you have the options to play around almost every card that exists HOWEVER you would not say that storm is interactive even if there are huge decision trees because it does not actively modify its opponent's best path to victory.
@Monti, That sounds very intriguing. For your points:
A linear deck's ideal lines of play (decision trees) are the same or exceedingly similar regardless whether an opponent exists or an opponent with no cards and 20 life existed.
I agree that would make the "linear programming" of navigating decision trees, well, linear. The way that I see it in my head is that every deck has (nearly) infinite branches on the main tree, each according to the active changing of the gamestate from the opponent and how the deck resolves those changes. Each branch has an end node programmed into the deck, be it "direct damage", "infect damage", "normal creature damage", "mill", and so on. For the most part, many end nodes on each branch are the same, they just have different branches from the respective starting points to that end node. If all designed end nodes are the same, then that defines the deck as pure linear in form, concerning programmed win condition.
An interactive deck actively modifies the solution to the opponent's decision tree for finding the best path to victory. There are a lot of cards that are "interactive" in the above sense like bloodmoon, ensnaring bridge, ghostly prison, chalice of the void etc etc that modify your opponent's decision trees, or changes what your opponent's path to victory are, but are not considered very fun to play against (but fun is 100% opinion).
There seem to be two main ways for a deck to be "interactive". One is to "prune" the number of available branches on the opposing deck's decision tree, reducing the number of options available. Cards like Blood Moon and discard spells do exactly that. The other is to "wither" the branches available, or reduce the significance or meaningfulness of those branches. Cards like Ensnaring Bridge or Humility do this. Another way to prune branches is to not give the opposing deck the opportunity to choose a branch - By simply achieving a winning gamestate before the opposing deck has built up the required resources to choose that branch.
When I look to consider the effectiveness of a deck, and how it is designed or engineered, I look at four main characteristics:
How many branches in the decision tree are available, or built into, the deck? The more branches, the more likely that the deck can deal with an opposing deck pruning some of them.
How "healthy" are those branches? Or, how easy would it be for an opposing deck to wither those branches to be meaningless?
How effective is the deck at pruning branches on an opposing deck's decision tree?
How effective is the deck a withering an opposing deck's branches on their decision tree?
In my opinion, the more variety of decks that we have in a metagame that are equally balanced at doing those four things, the more "fun" the game can be, as each player can choose their favorite mechanic, design, or method to get to their end nodes.
EDIT: I should note that the number of branches available immediately drop from "near infinite" as soon as the choice is made for which deck the opponent is using. The branches involving the other cards available in the metagame are immediately pruned.
Storm rarely cares about the board state, it just wants to play 20 spells that your opponent can't really do much about and win, or vomit out 12 goblins at the end of their turn 2 or 3...
This Monday, I was on the draw for game 1, and kept a reasonable hand in the dark, I got to lay down 2 lands and on my opponents turn 3 he hit me for over 20 damage with the grapeshot+remand combo. Come on, man, you can write down all your eloquent scientific writing but that's what people mean by linear.
Maybe we should just have a scaling system. There's also been some really absurd arguments on reddit where people said that Affinity boards in spell pierce and thoughtseize postboard and is therefore an interactive deck
Storm rarely cares about the board state, it just wants to play 20 spells that your opponent can't really do much about and win, or vomit out 12 goblins at the end of their turn 2 or 3...
This Monday, I was on the draw for game 1, and kept a reasonable hand in the dark, I got to lay down 2 lands and on my opponents turn 3 he hit me for over 20 damage with the grapeshot+remand combo. Come on, man, you can write down all your eloquent scientific writing but that's what people mean by linear.
Maybe we should just have a scaling system. There's also been some really absurd arguments on reddit where people said that Affinity boards in spell pierce and thoughtseize postboard and is therefore an interactive deck
It's a stretch...
Yes, interacting with a variable of the gamestate is...interaction. I'm guessing you didn't read prior comments, but I suppose I'll explain again. The issue with how you are defining interaction is that it is a self-centered definition. You are defining interaction in such a way as to assume that you are entitled to having the opponent interact on an axis that you prefer. If you actually understand the basic game theory concept that we've been talking about, then you'll see that this is exactly what a competitive deck is designed not to do. If we interact on our opponents' terms, then we are allowing the branches of their decision trees to retain significance.
It doesn't matter what some majority, or even number, of MtG players "mean" by linear or interaction. Simply because a group of people believe a certain way does not make it true or accurate - that is called the bandwagon fallacy.
A long time ago, I started writing an article about linear/non-linear and interactive/non-interactive decks. I was working with two definitions here.
Linearity: # of ways a deck can WIN.
Interactivity: # of ways a deck can INTERACT with an opponent's non-life total resources.
Basically, you'd code every card in a deck to see how it contributed to linearity or interactivity, and then score decks based on those measures. Low score is more linear, less interactive. High score is less linear, more interactive. Note that interactivity doesn't include damaging players, but rather how your cards can interact with their other resources. This would create issues with Mill and Lantern, but works for other decks.
Here's an example using a classic Storm deck from the recent SCG Regionals event:
http://sales.starcitygames.com//deckdatabase/displaydeck.php?DeckID=117250
How can this deck win pre-SB? Let's count the ways the deck itself is trying to win.
1. Grapeshot burn (3 copies)
2. Baral beats(4)
3. Electromancer beats (3)
So let's preliminarily give this deck a linearity score of 10.. That's pretty darn linear. Yes, I realize there are different ways to count this and also combinations of ways to win (e.g. Grapeshot for 10, Electromancer swings for 6, Baral finishes for 4), but we're still effectively capturing that above by highlighting the three primary avenues.
How does this deck actually interact with an opponent? Here are the cards that are capable of interaction AND the different ways they can interact. I'm double counting copies of a card for every different interactive mode they have.
1. Electromancer can block (3 copies)
2. Baral can block (4)
3. Noxious Revival can screw with the top of an opponent's library (1)
4. Noxious Revival can screw with graveyard recursion (1)
5. Remand can counter an opponent's spells (3)
6. Remand can protect your spells (3)
7. Grapeshot can target creatures (3)
8. Grapeshot can target planeswalkers (3)
So if we add up all those different copies of cards capable of doing different things, we get an interactivity score of 21. So our final Storm score is 10 for linearity and 22 for interactivity.
Now let's look at a classic GDS deck:
http://sales.starcitygames.com//deckdatabase/displaydeck.php?DeckID=117255
How's our linearity?
1. Death's Shadow beats (4)
2. Gurmag Angler beats (2)
3. Snapcaster Mage beats (4)
4. Street Wraith beats (3)
5. Tasigur, the Golden Fang beats (2)
6. Burn spells (3)
7. Snapcaster burn recursion (3: only 3 spells can be flashed back)
That brings the deck's linearity to 21. What about interactivity?
1. DS block (4 copies)
2. GA block (2)
3. Snap block (4)
4. SW block (3)
5. Tasigur block (2)
6. Liliana discard mode (1)
7. Liliana sacrifice mode (1)
8. Liliana ultimate (1)
9. Fatal Push (3)
10. K-Command modes (all mode combos interact) x # of copies (12)
11. Bolt on creatures (1)
12. Bolt on planeswalkers (1)
13. Stubborn Denial (2)
14. Terminate (2)
15. Thought Scour an opponent to mess with scry/draws (4)
16. IoK (2)
17. TS (4)
That brings GDS to a whopping 59 for interactivity. Here's the final Storm and GDS comparison
Storm scores: 10 linearity, 21interactivity
GDS scores: 21 linearity, 59 interactivity
Which would mean GDS is about twice as non-linear (half as linear) as Storm and 3 times as interactive as Storm. This fits the common understanding of Storm vs. GDS, which suggests there is something to the method.
The issue with this method is that decks like Burn and Affinity tend to also register as non-linear and highly-interactive because their cards technically have lots of modes. For instance, this list here (http://sales.starcitygames.com//deckdatabase/displaydeck.php?DeckID=117638) has a linearity score of 55 (even less linear than GDS because it has so many cards contributing to its win) and an interactivity score of 62, which makes it both less linear and more interactive than GDS. All of us know why that is going astray: even though Burn cards technically have lots of options, in practice they are all committed towards a single end, which makes it feel linear. But the method can't distinguish. Affinity (http://sales.starcitygames.com//deckdatabase/displaydeck.php?DeckID=117646) is also an oddball: linearity 45, interactivity 34.
So overall, this kind of method has promise but needs refinement/revision. I like it as a starting point to objectively rate deck linearity/interactivity based on their scores relative to other decks, even if it has some flaws. That's one of the reasons I never finished the article and am now submitting it here for some tinkering.
Lantern Control
(with videos)
Uc Tron
Netdecking explained
Netdecking explained, Part 2
On speculators and counterfeits
On Interaction
Every single competitive deck in existence is designed to limit the opponent's ability to interact in a meaningful way.
Record number of exclamation points on SCG homepage: 71 (6 January, 2018)
"I don't want to believe, I want to know."
-Carl Sagan
Why would you choose to edit that part out? Seems like it's very relevant to the conversation. If I recall correctly, I even used a case example, in which a recent tournament shows us a Humans player shutting down Storm entirely with a single Meddling Mage naming Grapeshot in game one, when the Storm player didn't have access to Empty the Warrens (because they didn't include on in the main). That is a great example of being hyper-linear, as the "alternate options" (attacking with Baral and Electromancers into the plethora of Humans the opponent has available) is simply not feasible enough to warrant consideration. Adding that single Empty the Warrens, however, would make it non-linear. Sure, it's still "more linear" than plenty of other decks, but strictly speaking, it is no longer non-linear. Blurring those lines is a great way to add confusion to the subject, and serves to support opinions not entirely based on fact.
Lantern Control
(with videos)
Uc Tron
Netdecking explained
Netdecking explained, Part 2
On speculators and counterfeits
On Interaction
Every single competitive deck in existence is designed to limit the opponent's ability to interact in a meaningful way.
Record number of exclamation points on SCG homepage: 71 (6 January, 2018)
"I don't want to believe, I want to know."
-Carl Sagan
Lantern Control
(with videos)
Uc Tron
Netdecking explained
Netdecking explained, Part 2
On speculators and counterfeits
On Interaction
Every single competitive deck in existence is designed to limit the opponent's ability to interact in a meaningful way.
Record number of exclamation points on SCG homepage: 71 (6 January, 2018)
"I don't want to believe, I want to know."
-Carl Sagan
So, new definition:
Linearity: # of decision trees in a deck
Basically, we'd look at a deck and see how many decisions any single card represents. This means timing (instant vs. sorcery), options when you play it, how you can use it, etc. We then multiply the number of decisions a card gives by the number of those cards in a deck, and then add up the scores for each grouping of cards. That would be a deck's linearity score. Let's try it out! High = lots of decision trees, low = less decision trees.
Storm: 142
Burn: 194
Affinity: 500
GDS: 608
Electromancer attack: 3
Electromancer block: 3
Baral attack: 4
Baral block: 4
Ritual mana: 4
Ritual splice: 4
Manamorphose mana: 4
Manamorphose filter: 4
Manamorphose draw: 4
Revival recursion: 1
Revival GY screw: 1
Revival top of deck screw: 1
Revival instant: 1
Opt top: 2
Opt bottom: 2
Opt instant: 2
Pyretic mana: 4
Remand opponent: 3
Remand protection: 3
Remand Grapeshot: 3
Remand with Baral up: 3
Grapeshot removal: 3
Grapeshot planeswalker: 3
Grapeshot player: 3
SV bottom bottom: 4
SV bottom top: 4
SV top bottom: 4
Sleight choose A: 4
Sleight choose B: 4
Strand fetch island: 3
Strand fetch vents: 3
Strand instant: 3
Tarn fetch island: 4
Tarn fetch vents: 4
Tarn instant: 4
Gifts (4 common piles): 16
Gifts instant: 4
PiF cast: 2
PiF flashback: 2
Vents tapped: 4
Vents untapped: 4
Guide attack: 4
Guide block: 4
Guide reveal: 4
Swiftspear attack: 4
Swiftspear block: 4
Swiftspear prowess: 4
Ferocidon attack: 3
Ferocidon block: 3
Ferocidon damage management: 3
Eidolon attack: 4
Eidolon block: 4
Eidolon damage management: 4
Mesa for mountain: 4
Mesa for foundry: 4
Mesa for ground: 4
Mesa instant: 4
Foundry tapped: 2
Foundry untapped: 2
Ground tapped: 2
Ground untapped: 2
Foothills for mountain: 4
Foothills for foundry: 4
Foothills for ground: 4
Atarka's modes: 12
Atarka's instant: 12
Boros Charm modes: 12
Boros Charm instant: 12
Bolt creature: 4
Bolt planeswalker: 4
Bolt instant: 4
Blaze creature no landfall: 4
Blaze creature landfall: 4
Blaze planeswalker no landfall: 4
Blaze planeswalker landfall: 4
Blaze instant: 4
Skullcrack damage: 3
Skullcrack prevent life gain: 3
Skullcrack instant: 3
Lava Spike player: 4
Lave Spike planeswalker: 4
Rift Bolt creature: 4
Rift Bolt player: 4
Rift Bolt plansewalker: 4
Rift Bolt suspend: 4
Ravager attack: 4
Ravager block: 4
Ravager sacrifice: 4
Ravager modular: 4
Etherium attack: 4
Etherium block: 4
Etherium pump: 4
Memnite attack: 2
Memnite block: 2
Ornithopter attack: 4
Ornithopter block: 4
Signal attack: 4
Signal block: 4
Signal pump: 4
Overseer attack: 3
Overseer block: 3
Overseer pump: 3
Skirge attack: 3
Skirge pump: 3
Skirge phyrexian mana: 3
Ghirapur attack: 1
Ghirapur block: 1
Ghirapur sacrifice: 1
Blinkmoth attack: 3
Blinkmoth block: 3
Blinkmoth pump: 3
Blinkmoth activate: 3
Inkmoth attack: 4
Inkmoth block: 4
Inkmoth activate: 4
Spire lifeloss: 4
Plating equip: 4
Plating instant equip: 4
Drum activate: 4
Blast creature: 4
Blast player: 4
Blast planeswalker: 4
Blast metalcraft creature: 4
Blast metalcraft player: 4
Blast metalcraft planeswalker: 4
Blast instant: 4
Denial spike: 1
Denial negate: 1
Denial instant: 1
Thoughtcast 4U: 1
Thoughtcast 3U: 1
Thoughtcast 2U: 1
Thoughtcast U: 1
Ravager eats Ravager: 16
Ravager eats Master: 16
Ravager eats Memnite: 8
Ravager eats Thopter: 16
Ravager eats Pest: 16
Ravager eats Overseer: 12
Ravager eats Skirge: 12
Ravager eats Hope: 4
Ravager eats Citadel: 16
Ravager eats Blinkmoth: 12
Ravager eats Inkmoth: 16
Ravager eats Plating: 16
Ravager eats Drum: 16
Ravager eats Opal: 16
Ravager to Ravager: 16
Ravager to Master: 16
Ravager to Memnite: 8
Ravager to Thopter: 16
Ravager to Pest: 16
Ravager to Overseer: 12
Ravager to Skirge: 12
Ravager to Hope: 4
Ravager to Nexus: 12
Ravager to Inkmoth: 16
Plating to Ravager: 4
Plating to Master: 4
Plating to Memnite: 2
Plating to Thopter: 4
Plating to Pest: 4
Plating to Overseer: 3
Plating to Skirge: 3
Plating to Hope: 1
Plating to Blinkmoth: 3
Plating to Inkmoth: 4
DS attack: 4
DS block: 4
GA attack: 2
GA block: 2
GA delve: 2
SM attack: 4
SM block: 4
SM flashed: 4
SW attack: 3
SW block: 3
SW cycle: 3
SW cycle instant: 3
SW cycle life loss: 3
Tasigur attack: 2
Tasigur block: 2
Tasigur activated: 2
Tasigur activated instant: 2
Tasigur delve: 2
Liliana +1: 1
Liliana -2: 1
Liliana -6: 1
Mire for swamp: 4
Mire for crypt: 4
Mire for vents: 4
Mire for grave: 4
Crypt tapped: 2
Crypt untapped: 2
Delta for island: 4
Delta for swamp: 4
Delta for crypt: 4
Delta for vents: 4
Delta for grave: 4
Tarn for island: 3
Tarn for crypt: 3
Tarn for vents: 3
Tarn for grave: 3
Vents tapped: 2
Vents untapped: 2
Grave tapped: 1
Grave untapped: 1
Push <2: 3
Push <4: 3
Push instant: 3
Snapcaster modifier: 36
Kcommand modes: 12
Kcommand self: 3
Kcommand instant: 12
Snapcaster modifier: 108
Opt top: 4
Opt bottom: 4
Opt instant: 4
Snapcaster modifier: 64
Denial spike: 2
Denial negate: 2
Denial instant: 2
Snapcaster modifier: 24
Terminate: 2
Terminate instant: 2
Snapcaster modifier: 16
Scour self: 4
Scour opponent: 4
Scour instant: 4
Snapcaster modifier: 48
IoK opponent: 2
IoK self: 2
Snapcaster modifier: 16
SV bottom bottom: 2
SV bottom top: 2
SV top top: 2
Snapcaster modifier: 24
TS opponent: 4
TS self: 4
TS life-loss: 4
Snapcaster modifier: 48
Bolt opponent: 1
Bolt self: 1
Bolt creature: 1
Bolt planeswalker: 1
Snapcaster modifier: 16
Less screwy than the last method but still some glitches. I think one of the issues here is that I am effectively trying to calculate card combo permutations by hand and I'm either over-counting or under-counting options. For instance, Snapcaster clearly creates more options and decisions with spells, but how is that expressed? Currently, I calculate the number of options a spell has (Opt has three: top, bottom, instant), multiply that by the number of Opts in the deck (2*3=6), and then multiply that by the number of Snapcasters (6*4=24), adding that to the total score. This gives GDS a ton of options, but the same method also gives Affinity a ton of options because of Ravager and Plating. So I might be weighting things incorrectly. I also might need to give Storm more Gifts piles options, which would naturally increase its score, and I might have missed some options entirely.
That said, this overall approach definitely makes more sense than the last method and feels like a better operationalization of linearity. It still needs refinement but this is probably the direction we should go in if we want to quantify linearity (at least, as long as we're comfortable with my linearity definition as "decision trees").
I mean there is a reason why urx is harder to play than burn over a long event. Pros admit to this and thus choose the lighter decks for the sake of Thier mental endurance.
Which is why people often call burn mindless and urx skillful.
Not that I'm saying that myself but it's just an observation.
And when you look at the top 10 decks in modern you do tend to see why people often like to use the word linear when it comes to modern. Even though there certainly are a certain amount of urx and bgx decks in there aswell.
I think this is one major reason that we should really be considering expectiminimax when discussing this. The term "interactive", as it's commonly used in these forums and in many MtG circles, is often defined as a self-centered argument: "They aren't interacting with me". The truth is, strictly speaking, it is rare, if ever possible, that any player actually interacts with the other. Strictly speaking, we only ever interact with the gamestate when use make game actions. Just as in chess, when a player makes a move, even when taking a piece, they aren't interacting with the opponent. They're interacting with the gamestate.
Storm does quite a bit of interacting with the gamestate. It just does it in a way that it doesn't typically interact with gamestate actions that the opponent has made (their creatures, their cards in hand, etc.). Otherwise, however, every spell a Storm pilot makes has interacted with the gamestate.
According to expectiminimax, the primary goal is to interact with the gamestate in such a way as to minimize the opponent's options available or significance of the opponent's options available to change the gamestate. Storm does exactly that, in that the deck operates in such a way as to bypass the significance of creatures or lands that an opponent has played, and, if possible, most non-creature spells.
Thus, as I mentioned, I think it's probably important that we make this distinction when we use the term "interaction" or "interactive". I mentioned before, and I think gkourou is very close to this point as well, in that the words "linear" and "non-interactive" are, for the most part, used as slander towards decks that people feel emotionally charged about rather than in their truest definitions. They are basically re-defined as seen fit to support emotional arguments against decks. While I understand, as some mentioned before, that if a poll was taken that many might agree to this layman definition of the terms, that in no way makes it accurate. It only makes it popular.
Lantern Control
(with videos)
Uc Tron
Netdecking explained
Netdecking explained, Part 2
On speculators and counterfeits
On Interaction
Every single competitive deck in existence is designed to limit the opponent's ability to interact in a meaningful way.
Record number of exclamation points on SCG homepage: 71 (6 January, 2018)
"I don't want to believe, I want to know."
-Carl Sagan
The T1 and T2 decks we see today should be considered the chronicles of modern magic history, to be frank.
1. (Ravnica Allegiance): You can't keep a good esper control deck down... Or Wilderness Reclamation... or Gates...
2. (War of the Spark): Guys, I know what we need! We need a cycle of really idiotic flavor text victory cards! Jace's Triumph...
3. (War of the Spark): Lets make the format with control have even more control!
They are bad, when its the best way to play.
Spirits
Storm doesn't strike me as a very chess like deck, storm Feels more like solitaire even though it uses some but little interaction with the opponent.
Now jund for example actually cares about your board state and 1 for 1s or 2 for 1s you until they have control over it.
For example, there are lines of play in chess that many people might consider "linear", or "non-interactive", in the layman terms as described here. I specifically teach my students to watch out for those openings, and how to combat it. Two examples are the scholar's mate and the queen's raid, and there are plenty of other examples to be seen.
In chess, however, minimax also works. It's not subject to expectiminimax because there is no chance involved, but the principle of minimizing the opponent's options, or the significance of their options, is a primary goal. There are a few ways to do this (setting up pins, forks, skewers, minimizing space available, getting a small material advantage and then playing attrition to the end, etc), but they all work for this goal.
You can see in my quote how Storm does this:
The opponent's creatures and/or permanents don't matter if the Storm player is able to bypass them by going straight to the face with a good Grapeshot. They similarly don't matter if the Storm player is able to reduce the significance of attackers or blockers the opponent has by amassing a large army of goblins. There are plenty of unique lines of play that accomplish one of these two goals, but those two "endpoints" of the decisions trees are what define how the deck is designed to work.
As for midrange decks like Jund, they similarly attempt to minimize the opponent's options, and/or the significance of their options. They use discard spells to remove options from the hand and removal spells to remove the opponent's options on the battlefield. They also use threats like Tarmogoyf, which are designed to minimize the opponent's battlefield options by out-sizing (or, outvalue-ing) them. An opponent's creature means very little if it cannot race a Tarmogoyf. Therefore, the significance of the opponent's creature on the battlefield is reduced.
In every case, we can see how expectiminimax is directly invoked to achieve a winning advantage. Every single competitive deck works best when it is allowed to solitaire, not just decks like Storm. The goal is to achieve a gamestate in which it might as well be solitaire. Even in chess, the primary principle that I teach students is to try to get one small pawn advantage, and then trade down until it becomes a king vs. king & pawn endgame. At that point, it's pretty much solitaire (if done correctly), and the win is inevitable. If done correctly, it shouldn't matter what the opponent does with their king, because they haven't allowed them to do anything significant. The only reason the opponent is even allowed a move at all is because it is required within the rules of the game.
While I understand that some might not find this "fun", that is the harsh reality of competitive games.
Lantern Control
(with videos)
Uc Tron
Netdecking explained
Netdecking explained, Part 2
On speculators and counterfeits
On Interaction
Every single competitive deck in existence is designed to limit the opponent's ability to interact in a meaningful way.
Record number of exclamation points on SCG homepage: 71 (6 January, 2018)
"I don't want to believe, I want to know."
-Carl Sagan
In all seriousness, Wizards may be more conservative than ever. Worlds was amazing to watch, but every single relevant team had a Temur/Sultai/4Color Energy Variant in full form forcing the sacrifices to be made on the other two decks. I forsee a Standard banning. I'm not sure if Wizards would compensate by giving Modern a few bones.
Jund and storm are apples and oranges however. You could make them sound the same, but they attack from very different angles from each other.
Like ktk said there are varying degrees to interaction and linearity. Which mean decks don't operate in exact ways in modern and there are alot of them.
If you look at storm vs shadow you can see how one has more decision trees in ktks analysis.
In chess each player is playing the exact same pieces and must interact and play attrition if they wish to advance and win (especially so when the players are of a similiar elo) So using chess to compare to mtg mathematics isn't really a solid case.
Apples And oranges.
I agree with you, if the deck doesn't evolve into something more and just stagnates because the power of the linear play is just so good there isn't a reason to move things forward, then its a bad thing. Queue a giant rant on how parasitic mechanics like infect are a mistake and they should be designing things that work with existing mechanics. That or they just plug up their ears and make brick counters or have that entire deal with eldrazi happen again. Also now I'm imagining a future where they make TKS, and when I say that I'm talking Thought-Knot Sliver...
1. (Ravnica Allegiance): You can't keep a good esper control deck down... Or Wilderness Reclamation... or Gates...
2. (War of the Spark): Guys, I know what we need! We need a cycle of really idiotic flavor text victory cards! Jace's Triumph...
3. (War of the Spark): Lets make the format with control have even more control!
The key point that I was trying to get across is that they both are designed with expectiminimax at their core. They just use the theory for different decision trees. The end nodes (burn spell to face or creature damage, so long as the Storm player included Empty the Warrens) are pretty much the same, but the branches themselves look very different.
One of the reasons that I think it's important for us to consider the game in this sense is because it is possible for a player to make the opponent's deck non-interactive simply by the design of the player's deck.
For example, UW Control runs plenty of creature removal. I think we could agree that creature removal spells are interactive. However, what if we choose to play a deck that doesn't rely on creatures at all? Is it fair for us to say that UW Control isn't an interactive deck any more? By simply choosing to build our deck without using creatures we have reduced the UW Control deck's "interactive score". Another example is if we do use creatures, but we use cards like Cavern of Souls, which makes our creatures uncounterable. UW Control runs plenty of counters. Again, by the design of our deck, we've reduced the "interactive score" of the opponent's deck. Thus, when we talk about a deck being "non-interactive", this is a bit misleading, as we haven't considered the metagame and the ability of another deck to influence how interactive each other deck in that metagame can be.
I think, therefore, that it is important for us to recognize the distinction of a deck's ability to reduce the opponent's ability to interact in a significant manner, rather than how it "interacts". All of this, of course, has already been very well fleshed out and thoroughly described and proven. For us to adopt this method is just us recognizing that the common understanding of the game has yet to evolve to that level.
Other games have already adopted the understanding of expectiminimax, minimax, etc. That's the primary reason that chess is pretty much a solved game at this point, and why Alpha Go was able to beat humans. There are underlying principles within the game that the MtG community in general just haven't come to recognize as legitimate nor apply to the game, although it is painfully obvious that they do apply.
That is my goal here. I want us to evolve our understanding of the game, to catch up with what game theory has been saying for quite some time now. While we debate how we want to define as "linear" and "interactive", these definitions are already laid out and well known. While we, in this forum, argue what decks we like and what decks we don't like, there are principles of game theory that can be applied to figuring out how to define the strengths and weaknesses of those decks, to better understand how they work and how to play or combat them.
So while chess and MtG are different, the same principles of game theory are, without a doubt, applicable to both. We just have to reach that level where we understand and apply them. If our goal isn't to achieve a better understanding of the game, when what exactly is it?
Lantern Control
(with videos)
Uc Tron
Netdecking explained
Netdecking explained, Part 2
On speculators and counterfeits
On Interaction
Every single competitive deck in existence is designed to limit the opponent's ability to interact in a meaningful way.
Record number of exclamation points on SCG homepage: 71 (6 January, 2018)
"I don't want to believe, I want to know."
-Carl Sagan
It still doesn't mean decks do not differ in linearity interactivity and overall "fun"ness to play against.
I think chess is much easier to solve with your preferred theory.
I feel there is a line that shouldn't be crossed when it comes to how efficient a deck is at preventing the opponents options in interacting.
That's why probe was banned for example. And that's why storm and dredge has been nerfed before.
Because even though they never broke any meta percentages they still where too efficient at creating a hard to interact with/unfun game by both Thier speed and resilience.
And if we used the expectiminimax theory on bans the format would have been dead long ago because that theory doesn't consider how fun the game is. It only shows that as long as people are trying to win and they also do not take up too much meta percentage they are fine.
Saying jund and storm attack from a similar angle because they deal damage to your face and try to prevent you from beating them is a bit disingenuous Imo.
I agree with all of this.
I also kind of agree with this. Expectiminimax is much more complicated that minimax, simply due to the necessity to calculate chance as well.
And I agree with this.
I would disagree with this. This statement assumes that, for some reason, using expectiminimax couldn't identify the problem, with absolute certainty, and help find a solution. It assumes that we might apply expectiminimax and then do nothing with the deeper understanding. Applying it doesn't mean we just take no action whatsoever. Applying expectiminimax doesn't mean understanding the game on a deeper level and then just leaving it at that. It means approaching the game from a higher understanding, based on solid evidence, logic, and math, in order to grasp exactly why a deck is busted.
I understand if that's your opinion, but solid game theory shows, with certainty, without needing to rely on opinion, that the end nodes use the same function.
Lantern Control
(with videos)
Uc Tron
Netdecking explained
Netdecking explained, Part 2
On speculators and counterfeits
On Interaction
Every single competitive deck in existence is designed to limit the opponent's ability to interact in a meaningful way.
Record number of exclamation points on SCG homepage: 71 (6 January, 2018)
"I don't want to believe, I want to know."
-Carl Sagan
And it certainly does not dismiss the fact that storm and jund are winning in different ways and different angles dispite in the end going for your hp and trying to both not allow interaction from your opponent.
A linear deck's ideal lines of play (decision trees) are the same or exceedingly similar regardless whether an opponent exists or an opponent with no cards and 20 life existed.
An interactive deck can be defined as follows:
An interactive deck actively modifies the solution to the opponent's decision tree for finding the best path to victory.
There are a lot of cards that are "interactive" in the above sense like bloodmoon, ensnaring bridge, ghostly prison, chalice of the void etc etc that modify your opponent's decision trees, or changes what your opponent's path to victory are, but are not considered very fun to play against (but fun is 100% opinion).
Also I think any mention in the "number of decision trees" has nothing to do with linearity or whether a deck is interactive enough. That is simply the complexity of the deck. Something like storm has much higher complexity than most other decks (believe it or not) because there are so many forking options available. Making the absolute ideal decision at every fork is extremely difficult when you are constantly scrying, deciding, gifts packaging etc. When playing storm you have the options to play around almost every card that exists HOWEVER you would not say that storm is interactive even if there are huge decision trees because it does not actively modify its opponent's best path to victory.
I agree that would make the "linear programming" of navigating decision trees, well, linear. The way that I see it in my head is that every deck has (nearly) infinite branches on the main tree, each according to the active changing of the gamestate from the opponent and how the deck resolves those changes. Each branch has an end node programmed into the deck, be it "direct damage", "infect damage", "normal creature damage", "mill", and so on. For the most part, many end nodes on each branch are the same, they just have different branches from the respective starting points to that end node. If all designed end nodes are the same, then that defines the deck as pure linear in form, concerning programmed win condition.
There seem to be two main ways for a deck to be "interactive". One is to "prune" the number of available branches on the opposing deck's decision tree, reducing the number of options available. Cards like Blood Moon and discard spells do exactly that. The other is to "wither" the branches available, or reduce the significance or meaningfulness of those branches. Cards like Ensnaring Bridge or Humility do this. Another way to prune branches is to not give the opposing deck the opportunity to choose a branch - By simply achieving a winning gamestate before the opposing deck has built up the required resources to choose that branch.
When I look to consider the effectiveness of a deck, and how it is designed or engineered, I look at four main characteristics:
In my opinion, the more variety of decks that we have in a metagame that are equally balanced at doing those four things, the more "fun" the game can be, as each player can choose their favorite mechanic, design, or method to get to their end nodes.
EDIT: I should note that the number of branches available immediately drop from "near infinite" as soon as the choice is made for which deck the opponent is using. The branches involving the other cards available in the metagame are immediately pruned.
Lantern Control
(with videos)
Uc Tron
Netdecking explained
Netdecking explained, Part 2
On speculators and counterfeits
On Interaction
Every single competitive deck in existence is designed to limit the opponent's ability to interact in a meaningful way.
Record number of exclamation points on SCG homepage: 71 (6 January, 2018)
"I don't want to believe, I want to know."
-Carl Sagan
Storm rarely cares about the board state, it just wants to play 20 spells that your opponent can't really do much about and win, or vomit out 12 goblins at the end of their turn 2 or 3...
This Monday, I was on the draw for game 1, and kept a reasonable hand in the dark, I got to lay down 2 lands and on my opponents turn 3 he hit me for over 20 damage with the grapeshot+remand combo. Come on, man, you can write down all your eloquent scientific writing but that's what people mean by linear.
Maybe we should just have a scaling system. There's also been some really absurd arguments on reddit where people said that Affinity boards in spell pierce and thoughtseize postboard and is therefore an interactive deck
It's a stretch...
Yes, interacting with a variable of the gamestate is...interaction. I'm guessing you didn't read prior comments, but I suppose I'll explain again. The issue with how you are defining interaction is that it is a self-centered definition. You are defining interaction in such a way as to assume that you are entitled to having the opponent interact on an axis that you prefer. If you actually understand the basic game theory concept that we've been talking about, then you'll see that this is exactly what a competitive deck is designed not to do. If we interact on our opponents' terms, then we are allowing the branches of their decision trees to retain significance.
It doesn't matter what some majority, or even number, of MtG players "mean" by linear or interaction. Simply because a group of people believe a certain way does not make it true or accurate - that is called the bandwagon fallacy.
Lantern Control
(with videos)
Uc Tron
Netdecking explained
Netdecking explained, Part 2
On speculators and counterfeits
On Interaction
Every single competitive deck in existence is designed to limit the opponent's ability to interact in a meaningful way.
Record number of exclamation points on SCG homepage: 71 (6 January, 2018)
"I don't want to believe, I want to know."
-Carl Sagan