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I'm not saying that any future theories are required to have statistical or mathematical proof to help define them. There has been a lot of great articles lately like the discussions between Oscar and Geordie about card advantage or the niche article of"Harnessing the Power of Dirt." But I think the knowledge of the game will increase rapidly if we can apply a more rigorous mathematical aptitude to the study of it and I hope to show one approach to furthering the knowledge of Magic here.
Explanation of the Theory
The theory that I would like to present today is a tool to be used for deck building, deck playing and for general knowledge of the game. The theory is called Power Gap Theory. The theory states that the most important quality of a good deck is its ability to maintain a positive power gap on the board. By power, I mean the power of creatures on the board. This theory, however, does not hold true for all decks, as Magic is a unique game in that the creators constantly throw twists into the cards that allow alternate non-creature based ways of winning (Alternate Win Conditions, Combos, Burning Bridges, etc). However, the existence of these alternate decks does not make understanding the power gap any less important, as you must understand what decks you're facing when designing any new alternative deck.
One important note for explaining this theory is that cards are not defined within a vacuum. Though at one time I thought it might be possible to come up with a Grand Unifying Theory of Magic that could eloquently define each card independent of its environment, Oscar Tan in a series of emails, convinced me otherwise and has explained his rationale in some of his latest articles. However, I do think that when considering two opposing decks, that the value of each card can be defined. Oscar consistently challenged my ideas by asking if I could figure out the value of Ancestral Recall versus Dark Ritual. Within the grand architecture of the game of Magic that task is impossible. But by limiting the realm of analysis to two opposing decks then the value of those two cards can be defined.
Why Power Matters
The first thing to explain is why I use power as the determining factor. In any creature-based strategy, and by creature-based I mean any deck that has at least one creature in it, the goal is to do twenty damage. The primary focus of these decks is to deal this damage through attacking, which leaves power as the most important characteristic of the creature. The secondary characteristic that is important is the mana cost. Everything else is only a metagame choice. First strike, trample, flying, untargetability, toughness, regeneration, etc - these characteristics only mean something in the metagame and they are only there to gain card advantage (virtual or real).
I'm going to come back to why everything but power and casting cost is a metagame decision and only relates to card advantage, but first I need to further explain the theory so I can examine those details. Any good deck is designed to keep more of your power on the board in relation to your opponent. There are many ways to achieve this affect.
One method is via the weenie strategy. Create an initial short burst of power on the board using efficient power-to-mana creatures. Any deck that has a slight stall or doesn't have a good way of efficiently eliminating the weenies will fall. Another method is through the control strategy. It relies on eliminating all the power of the opponents creatures through removal and countermagic and then uses just a few of the most powerful creatures it can find (Psychatog, Morphling, Rainbow Efreet, Serra Angel, etc) to finish up. There are a myriad of variations on these themes, but the basic concept is the same - keep more power on the board than your opponent.
Essentially these decks are either creating more power on the board than the opponent can match, or limiting the opponent's power, which leaves the power you put on the board as the defacto power. But just describing power alone doesn't cut it, because each card costs mana. This is why the concept of the mana curve was developed.
Metagame Abilities
Let's talk about how all the abilities beyond power and casting cost are purely metagame decisions. I'll address toughness first, since I think this can be confusing. Toughness only matters if your deck or the metagame needs it and can only help with card advantage (which has been established as a way to help win the game) though card advantage is only there to help maintain the power gap. A high toughness will not help you win the game, since it does no damage. If a creature has a high toughness, then it might take more than one creature to kill it or a creature and a burn spell (card advantage). However, if you're playing against a Black deck filled with Terror effects then toughness was only a trade of cards.
Trample is usually only meaningful when combined with high toughness or regeneration. Flying only matters if you're planning on your opponent having ground-based blockers. So if you're building a deck that keeps your opponents creatures off the board through countermagic and removal, then flying is a meaningless ability (and therefore reduces the efficiency of your casting cost, since flying abilities cost more mana). Madness decks use Wonder to give their creatures an advantage (creating virtual card advantage) when other similar decks maintain a close power gap.
The Three Rates - Mana
There are three rates to consider when analyzing a deck. The first and most important is the mana rate. This is the rate that mana becomes available. This rate considers any resource that creates mana for use. There are two steps for checking the mana rate. The first is to look at the land drops, since this must follow a constant maximum of one per turn (Exploration notwithstanding). Luckily, this step is easy using the Excel program I posted to StarCityGames.com this summer.
For example, a twenty-four land deck will on average (50% probability) see this kind of mana available over the first ten turns:
Turn 1 2 3 4 5 6 7 8 9 10
Mana 1 2 3 4 4 5 5 6 6 6
The second step is to add in any other mana accelerants into your available mana. Using the excel program just use the"# of Particular Cards in the Deck" to calculate the mana boosts. If your deck has four Birds of Paradise then putting four into the # of cards slot tells you that, on average, you'll see a one mana boost by turn 4 and then again by turn 18. If the mana booster shows up on a turn before it could be used because of mana cost, just push it back until it can be used.
So the final mana availability with 24 lands and 4 Birds (they become active mana one turn later) looks like this:
Turn 1 2 3 4 5 6 7 8 9 10
Mana 1 2 3 4 5 6 6 7 7 7
The Three Rates - Power
Once you know how much mana is available at each turn, then that mana can be converted into a power per turn that you can put on the board. Now this power per turn is expressed in an average, which has its limitations, but it can be useful for comparison.
The way this is done is to first convert all creature cards into same casting cost groups. When you draw a one-mana creature, you're really drawing the average of all of the one-mana creatures you have. Let's say you have four creatures at 1/2 and four creatures at 2/1. When you're comparing the average power created from your one-mana card then get 1.5 power (remember we don't care about the toughness). So each casting cost group can be averaged out.
Now comes a more difficult part. We have three resources at work here: mana, cards and power. We know how much mana we have, and as we use that mana, we reduce the cards and increase the power on the board. We're also adding a card a turn, which is sometimes used to create mana (play a land). The way I expressed this relationship was to average out the casting cost and power. So at the time that two mana is available for all one-mana and two-mana spells, I averaged out all the one-mana and two-mana cards to generate an average CC/Power ratio. From the CC/Power ratio, an average card usage can be calculated so that cards can be subtracted from the system. So for example, if at the two mana slot (turn 2 for this theoretical deck) the Average Mana per Card is 1.5 and the Average Power per Card is 1.5 and we're spending two mana during this turn then:
2 Mana / (1.5 Mana per card) = 1.3 Cards used;
So we'll subtract 1.3 cards from the system. Now you might ask how can we subtract a third of a card. We'll we're looking at the average game and the statistical average will mean that sometimes you cast two one-mana creatures and sometimes you cast one two-mana creature and by taking all those events we get the average. And since we know the Average Power per Card is 1.5, then we can calculate that:
1.5 Power per card * 1.3 cards = 1.95 Power
There is one more hitch in these calculations, and that's the issue of probability of drawing these cards. The probability of drawing this average card must be accounted for when calculating the power on the board. I used the Hypgeomdist function to add the probability.
All this can be streamlined using an excel spreadsheet. I'm not going to go into details of the spreadsheet, but feel free to e-mail me and I'll send it to you so you can see the math. I also put in three different decks, and then played them out to verify the results (I played each deck twenty times to achieve a power-per-turn number). Each average of the twenty games matched the theoretical numbers within a statistically allowable margin (thus proving within this simplified model this way of looking at the game is true).
The Three Rates - Answers
Putting power on the board creates threats that must stopped with answers. Nearly all decks have answers in some form or another - removal (Terror), burn (Shock), countermagic (Mana Leak), etc. The rate of answers only matters on the type of threats it can deal with (and this is one reason why threats are better than answers, because sometimes you simply don't have the right answer).
When designing a deck that has lots of answers, you have to consider the decks that you'll be playing against to determine when and how many answers you need to have. The first thing is to figure out how fast the common decks out there can create power on the board. Then look at the turn by turn numbers to see if you're going to have 1) enough mana to cast your answers and 2) enough answers to deal with their threats before they can kill you. I don't have a good model on how to streamline this process, but some basic understanding how to use the Hypgeomdist function in excel can go a long way.
What about the other cards?
There are some cards that are not accounted for within the three rates. These are cards such as card drawers (Opportunity or Deep Analysis) and deck filters (Brainstorm or Impulse). Card drawers affect the game by adding more cards while spending mana. Deck filters only serve to dig through the deck to find better answers or threats (and this is why they are primarily used in decks that have a wealth of answers, because they always need to match the right answer with the right threat). In probability terms, deck filters reduce the Total Population in the Hypgeomdist function for a small investment of mana.
Also, some cards can affect other parts of the game. Burn doubles as both an answer, and serves to reduce the total damage needed to be done by creatures. Sligh decks during the Rath/Saga days didn't need to do twenty damage with their creatures, because Mogg Fanatics, Cursed Scrolls and burn would usually do the last five damage.
Cards need not be one type or the other. Some cards serve as both threats and answer (Meddling Mage, Mogg Fanatic, etc) or threat and mana (Llanowar Elves, Werebear, etc). The best cards serve in multiple fashions and are synergistic as well, but I don't think that's a revelation to anyone.
What use is this theory?
This theory is a model, and an incomplete model at that. Some of the areas that are lacking are within the use of the averages (power, mana and cards). When we use an average, we're assuming that the deck is consistent when some decks are bipolar. Sometimes they are explosive and sometimes they fall apart. A better way, though I'm unsure of how to do this, would be to turn that average at each turn to a distribution of the probabilities (which would look like a bell curve).
One of the other problems is defining synergy. Some cards get better when other cards are used, like Goblin Piledriver. These can be portrayed within the model. I approached this problem by using some general probabilities to get my answer. Another way would be to play the cards out to find out the usual number of other goblins that would be attacking with the Goblin Piledriver.
This brings up another issue - why use a model when you can play the cards out and figure out how the deck performs? For me, it's a time issue. Given a good team of intelligent players, they can break down many decks and their matchups and figure out what works best. However, for those that don't have a large resource of players to do all this you have to make your best theoretical guess at it. By applying some of the basics of this model, decks can be analyzed much faster. And it's a lot easier to figure out how changing a few cards can affect your deck using an excel spreadsheet, than it is to test the before and after by playing thirty to fifty games. This is definitely not a substitute for play testing and tuning decks, but it's a tool to help streamline your testing process (and hopefully provide more insight into the game).
Overall, there are still many subjective decisions within the game that require skill, but there are just as many decisions in playing and deckbuilding that are purely mathematical, and we should be able to define them. This is no different than professional poker that requires a thorough knowledge of the game theory to be successful, though the most successful know how to navigate that theory in real situations.
Even though I think that there are better ways to express these calculations, this should be a good starting point. Maybe someone with a real mathematics degree using a mathematical program (and not just an engineer using Excel), could better define these calculations. If you have any questions or comments feel free to email me.
Tom Carpenter
carpe@centurytel.net
