## Applying Controls Design Theory to Magic the Gathering

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I'm gonna go ahead a spitball here for a bit.

If you look at a game of Magic in terms of a complex multivariable, time-variant, and probabilistic system, hypothetically you could simulate this system via controls design.

Your "plant" in this system would be the design of the deck you are using. The input would be the various card choices that you make during the course of the game over a period of N turns, and the overall desired output would be some metric evaluating the successful completion of a game otherwise known as "winning the game". The metric could be the amount of potential damage per turn (or some other metric based on the style of deck being played: control, combo, aggro, etc.), with the desired output response having a certain percent overshoot of X% (here percent overshoot would be amount of overkill damage you could deal over a certain period of turns in order to make the resistance from your opponent negligible) and the settling time would be considered the number of turns it would take to cause your opponent's life total to go to zero. Steady-state error could be considered how close the deck's output is to the desired output. Obviously there is more complexity here, but you get the gist of it. The controls portion could be considered your sideboard options for the deck, or just changes to the deck that would take into account the resistance from your opponent, and the probabilistic error of the plant and improve the output to closely match the desired output that you want to achieve.

Most people would see the output has the number of turns it would take to kill the opposing player assuming no resistance or disturbance from that opponent, otherwise known as "goldfishing", but in standard play the number of "disturbances" can be narrowed down to a select amount of cards given the current format and meta (really this could be used for any format, it would just be harder for larger formats with more card choices). You could theoretically create a mathematical representation of the entire system taking into account these disturbances, account for error (probabilities that you don't draw into your core strategy, or draw into answers to opposing threats) and the overall frequency data (on average how long it takes for your deck to win in varying circumstances) and through this simulation you could find an optimal deck based on the given circumstances.

Some of the challenges with this are:

-Developing the system schematic that closely captures reality in an appropriate timeframe (as metas change dramatically from week to week and from set release to set release)
-Taking into account all variables and states of the system correctly
-Understanding that when you change your design from one iteration to the next, you might solve one issue the deck had, but you will invariably create new problems for the deck that it didn't have before you made a change to either the deck composition itself or the sideboard of the deck under scrutiny

I think that applying controls theory and systems design concepts to the design of MTG Decks would be an interesting application to pursue, but I'm not sure it would be entirely practical. A typical Magic player probably already does this to a certain extent just through experimentation and iterative design changes, and this would potentially get you similar results with more effort and time (depending on how well you understand the application).
I actually think I know of a post that covers this topic.

There was a dude who made a great equation possibly involving thermodynamics, but the locals of the place where he posted about it ran him over with critique, probably because he was too inteligent for the taste of the trolls resident at that place.

Would you like a public transcript of that post or would you like it on personal mail instead?

I actually think I know of a post that covers this topic.

There was a dude who made a great equation possibly involving thermodynamics, but the locals of the place where he posted about it ran him over with critique, probably because he was too inteligent for the taste of the trolls resident at that place.

Would you like a public transcript of that post or would you like it on personal mail instead?

Public is fine.

sage62
Joined Dec 1969
I think I've had enough of all this simulation and formula stuff from a certain forum dweller who apparently has a high horse he needs to get off of.
SAGE62:

It's actually a dwarven pony...

STOIC CHAMPOIN:

I'll see if i can find it.
sage62
Joined Dec 1969
I actually think I know of a post that covers this topic.

There was a dude who made a great equation possibly involving thermodynamics, but the locals of the place where he posted about it ran him over with critique, probably because he was too inteligent for the taste of the trolls resident at that place.

Would you like a public transcript of that post or would you like it on personal mail instead?

You are a paragon of humility.
Taldier
Joined Dec 1969
Im sure this wont stop these posts, but I just have to say: there are an almost innumerable number of relevant metaphors you can use to describe the process of deck building.  The problem occurs when you forget that your arbitrary metaphor is a metaphor.

Once you start actually attempting to extrapolate and apply results directly from the metaphor to the process that metaphor is describing, you are in trouble.  That way madness lies... and if you walk that way long enough you'll even run into wickeddarkman too...

Current decks
Comments or suggestions are always welcome

Modern
nothing at the moment

And what exactly would be the state space? I'm pretty sure the dimensionality of the state space, the control space (or both) makes th problem both analytically and computationally intractable. You are engaging in pure mental self-pleasure.
Glasir
Joined Dec 1969
You are engaging in pure mental self-pleasure.

"Go, then. There are other worlds than these." -- Stephen King, The Gunslinger

And what exactly would be the state space? I'm pretty sure the dimensionality of the state space, the control space (or both) makes th problem both analytically and computationally intractable. You are engaging in pure mental self-pleasure.

You are correct, without some sort of second order approximation (which is probably impossible in the case of the system I described), or shrinking your state space to a manageable level (both in size and dimensionality) through the elimination of a certain number of factors based on various acceptable assumptions (a challenge in and of itself), the analytical results found from this application wouldn't be very meaningful.

Yes, in making this post and brainstorming how this might work, it was in the pursuit of happiness (or as you would so eloquently put it, engaging in mental self-pleasure). My apologies if this was offensive to you.

Offended? Gosh no. Controls is fun. But I'd probably say that any game you could pose as a controls problem would probably be pretty boring.
Well in trying to control a system of this nature I'd be striving to predict states in a game where part of the fun is from the unpredictable courses that games can take.

Either way, I've already posited that actually playing the game, experimenting and iteratively changing your deck is most likely faster and more fun, but just attempting to conceptualize the game in terms of controls theory was a fun exercise (for me anyway).
Still looking!

But that shouldn't deter others from making an attempt at it before I find the other dudes post.
It's way back in time so I have to look through a lot of archive and the way I name my posts aren't always equally clear on topic!
STOICCHAMPION:

Found it.

Unfortunatelly it's from an insanely long and complicated discussion on tempo, where the guy put down an equation on tempo.

I have no idea if it's correct or even close to cover the aspects.
All I remembered from reading it was that it was consisting of TWO pages that I copyed and pasted into my fave forum and discussed it a lot, both before and after posting it.

His equation is here: (With his own words)
----------------------------------------------------------------------------------------------------------------------------
This whole experiment was to create categories for tempo has actually led to an experiment. Let me explain. The other night I was contemplating certain aspects of tempo. Questions such as “Can a player gain tempo but still lose the game?” “In a given state, can a player recognize tempo without seeing actual game play? The question that really inspired me was “What word would be the best to explain tempo: momentum, rate, or velocity?” I then reflected back on Noah Weil’s equation Time=Life + Mana. It suddenly dawned on me, tempo is physics, well, for magic at least.

Physics: the science that deals with matter, energy, motion and force.

Tempo: the theory of magic that deals with permanents, resources, and win conditions.

Common physics equations and terms.

D = distance V = velocity (distance/time) A = acceleration (distance/time squared)

T = time

Df = di + v0(t) + a1/2(t)2

Vf = vi + a(t)

Tempo equations and terms.

T = turns D = the win condition (Life, Poison counters, Milling) V = win condition/turns A = win condition/turns squared

I believe D is pretty self-explanatory. If you are playing a normal creature deck, your distance is 20 life. Add a player, 40 life. Another, 60 life. In a six player chaos game, the distance your deck must travel is 100 life. This relates equally well to poison counters. Milling an opponent will resemble the same aspect. Distance for one player, 60 cards. Another, 120 cards. In a six player chaos game, 300 cards. Before I move on, let me mention this.

Inertia is a non-quantifiable property of matter (not a unit of measurement) by which it remains at rest or in uniform motion in the same straight line unless acted upon by some external force

Magic exists in a state of Inertia. Once a player has gained velocity by playing a creature, that creature will maintain its velocity. Every turn (time), that creature such a grizzly bears will deal two damage per turn (miles per hour). As part of inertia, an external force (shock) must act upon it in order to decrease an opponent’s velocity. If the Grizzly Bears is not acted upon by an external force (shock), the Grizzly Bears will maintain its current velocity (two damage per turn).

In order to gain velocity as in a car, you must be able to accelerate to a certain speed. Acceleration occurs by using more energy or gas (mana) to gain a greater velocity. In magic, that involves by placing another Grizzly Bears onto the board. Once you have added another Grizzly Bears to the board, you will now be playing at a greater velocity (4 damage per turn).

Therefore, if an opponent is at 18 life and you have two Grizzly Bears in play, eliminating acceleration, how many turns will it take to travel 18 life. Here’s the answer.

Di = 18 life Df = 0 life V = -4 life per turn

Equation: (Note: velocity is constant)

Df = di + v(t) + a (t)2

Plugging in our numbers:

0 = 18 – 4 (t) + 0 (t)2

-18 = - 4(t)

t = 4.5 turns (In magic, we always round the numbers so its 5 turns)

This may seem like overkill. However, I think this brings to light a problem I think people have with tempo. When people are talking about tempo, they often argue about the speed of a deck. The problem, they are arguing about two different things. One person is talking about the potential of the deck to accelerate. The other is arguing about the velocity of the deck and so on and so on.

In reference to this: the Average Speed of the deck will consist of the average velocity of deck over turns or games. Thus, the average speed of a deck would be the ability of the deck to win or over multiple games. So, if somebody asked “What is the average speed of your deck?” You would say “it typically kills by turn 5.” Speed to me, in this sense, is the average velocity of your deck over time or games.

I’ll admit. These tempo equations work much better with creatures with haste (instantaneous velocity). It becomes more complicated for nonhastey creatures, but they still hold true. So, let’s add acceleration into the mix. Example, you are playing a deck with nothing but Raging Goblins. You play one Raging Goblin per turn (constant acceleration). If you opponent is at 20 life, how long will it take you to win the game.

Equation: (note: when you start the game, your initial velocity is zero)

Df = di + v0(t) + 1/2a (t)2

0 = 20 – 0 life (t) – 1/2 (t)2

-20 = – 1/2 (t)2

40 = t2

t = 6.3 turns (6 turns)

Kinda weird, but it works. If you don’t believe me, pull out a scratch sheet of paper or your cards. Ah, the power of math. This shouldn’t be a surprise since Magic, at its essence, is basically math.

Expanding on this, Tarmagoyf begins the game with a lower initial velocity of 0/ * +1. Later in the game, the Tarmagoyf attains a higher final velocity with the same amount of energy. (I knew somebody would ask, so I mentioned it.)

Energy into this system of magic is resources. As I stated in an earlier blog, resources is anything in magic. This relates well to a car traveling in this one-dimensional analogy. Cars that accelerate quickly, expend a great amount of energy. This holds true for decks as well. Aggro decks will expend a great amount of energy to increase a player’s velocity in early turns (time). Control decks must expend energy to slow an opponent’s velocity or hinder their acceleration.

As a FYI, Psychatog and Greater Gargadon are environmentally friendly by recycling your resources. Hmmm…maybe they should have been green cards.

One example that gets constantly thrown out there for tempo arguments is something like my Raging Goblins example. In the examples, it is stated that each player plays one Goblin on each other their turns for the rest of the game. Well, this is an example of constant acceleration pointing out that the player who went first will win. Sure, if I race a person in my car at the same constant acceleration as them, of course they are going to win. They got a head start. It’s about time (turns) not velocity or acceleration.

In physics, it is important to note that there a distinction between instantaneous velocity is made from velocity (average or constant). At any given time (turn) in a game, a judge approaching the table will see the instantaneous velocity of the game. Not the average or constant, but the instantaneous. Same holds true for acceleration. Acceleration happens whenever a creature (or permanents) is placed on the board. Every time you have played a creature, you have accelerated your game. At any time (a given turn) when you a play a spell, this is instantaneous acceleration.

How do I explain Auto-Win cards? Wormholes.

Combos? A perpetual motion machine that would violates the Second law of thermodynamics.

Darn it. Noah Weil was right. Magic is an equation.

Anyway, I’m done with this project. Don’t know what else I could really add. It would be a lot easier if Wizards of the Coast didn’t place Chaos theory into the game. Moving on to the characteristics of cards.

the basic version is that a player's "clock" is how fast they would be able to kill off the opposing player, based on what they have in play at the time. If your defenseless opponent has 18 life and you've got just a 3/3 creature in play, for now you've got a six-turn clock.

An excessively accessible article written for the most inexperienced Magic players managed to sum up your mishmash of unnecessarily formulas in just two sentences. A clock is how many turns it would take you to finish the opponent off if all things stay the same. A deck that tries above all else to speed up the clock and resists attempts to slow that clock is said to be focused on increasing tempo.

This is nothing new. It's the same thing that people have used to illustrate tempo for quite some time. And it's the reason that no one has taken heart to your concepts of seperate "Life Tempo" and "Mode Tempo"; they simply have no place in this model.
You are engaging in pure mental self-pleasure.

Everyone masters debate.
139359831 wrote:
Clever deduction Watson! Maybe you can explain why Supergirl is trying to kill me.
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