Why Statisical Analysis?

We use Statistical Analysis in almost every facet of society today. Engineering, Economics, Science, Medicine, Marketing, and Psychology are just a few fields which heavily utilize this branch of mathematics. In Statistical Analysis we seek to either optimize our gain, based on the probabilities presented, or we observe an optimal trend, and seek to find the probabilities behind it. Through my experiences in obtaining my B.A. in Biology, I was astonished when I learned that statistical models in Economics were virtually identical to statistical models in Ecology. By changing the meaning of certain variables , it is as if we could use money to explain population distribution and survivorship.

Game Theory, which takes root in Bayesian Logic, looks to optimize the probability of winning. The models of Game theory have been used in economics as well. Game theory uses descriptive, situational statistics, to build models , which describe the best decision in a given circumstance. In Magic the Gathering, one could apply Game Theory, by analyzing the current game environment, and modeling the best deck to play in the next sanctioned tournament. During sanctioned play, one could model the best play, at a critical moment in the match, again using descriptive probabilities, which will tell you what your opponent's deck is likely to have in it, or what your opponent is likely to play.

Inferential Statistics are probabilities built from inferences. Inferential models may be used to describe specific circumstances, but they can also describe trends in a broader scope. A skilled and ethical statistician will be able to define, and provide a clear definition of the assumptions in an inferential model. Assumptions are parameters designed to exclude variables from a model, which are believed to exhibit minimal or insignificant influence on the outcome. The groundwork of a descriptive model has no such assumptions, as the probabilities leading up to the model have already been observed. The only true assumption of a descriptive model is that the descriptive probabilities will not change, mid-circumstance. Assumptions can be seen as both the strength and weakness of inferential models.

I have chosen to pursuit inferential models, because first and foremost, I want to look at Magic the Gathering in broad scope. I hope to build models which will prove robust, as the game environment changes from expansion to expansion. Too often, I have seen veteran players leave the game, because of the inability to keep up with the latest trends in Magic. For some, the mechanics get too complex, for others the cards seem too over-powered. But with inferential models, we could gain a clearer understanding of the true power of cards, from past present and future expansions. In applying these models, the task of figuring out what to supplement your collection with, from the current set, would become much less daunting to the veteran player.

Why we keep decks trimmed.

Here is one of many blogs to follow, in which I will cover some basics in statistics and probability. I will post these in order to explain, piece by piece, the framework of my study, and eventual e-book. To most of us, it is common sense to keep our decks trimmed, because we know intuitively that doing so increases our chances of drawing into the cards we want to play. But without proof, our best guesses are nothing more than just that. Here is the proof, which can be found in the introductory chapter of most any Statistics text book:

IF: P(X) = 1/X

Prove: P(N) > P(N+1)

By substituting all instances of X for N when P(X) = 1/X, we get P(N) = 1/N.

By substituting all instances of X for (N+1) when P(X) =1/X, we get P(N+1) = 1/(N+1)

Let's arbitrarily call N 60, for a typical deck size of 60 cards.

We find that P(N) = P(60) = 1/60 = 0.01666,

and P(N+1) = P(60+1) = 1/(60+1) = 0.01639.

0.01666 > 0.01639, therefore

P(N) > P(N+1).


In otherwords, If you have one copy of a card in your deck, you will have a 1.666% chance of drawing into it from the top card of a 60 card deck, and a 1.639% chance of drawing into it from the top card of a 61 card deck. This might seem like an insignificant difference, but rest assured, as you add more and more cards to your deck, without thinning others out, the difference becomes significant. Later on I will cover finding Probability Without Replacement, which explains what happens to the probabilities of drawing into certain cards, as you deplete your library through the act of drawing. But for now chew on this; if you have 4 copies of a specific card in your deck, you have a 39.950% chance of drawing into at least one of those cards in your opening hand, from a 60 card deck. However if your deck is 70 cards, your chance of drawing into at least 1 of 4, in your opening hand, is reduced to 35.035%.

Introducing Myself

I have been collecting into, and playing Magic the Gathering for almost 15 years. I account myself one of the few casual-play veterans, who is pleased with the balanced direction the game has moved in. Over the years, I have watched a fluctuating, but growing population of players, gravitate towards MTG, and other trading card games. With MTG's overall popularity growth, I have seen this interest channeled into numerous web-pages, blogs, and forums, with increasing frequency. I have personally picked the brains of numerous posters, in constructing my own type two, and extended decks. The wealth of information I find, from players much younger than myself, is insightful and astonishing.

I am beginning a project, which I hope will be ongoing, where I will statistically analyze MTG at the most fundamental levels. Through the research I have already done, I have seen most serious analysis as outdated (in terms of modern game mechanics), or guarded (subscribe here, plug your numbers in here, get your answers here). Of coarse a major function of this initial blog is to share and network information on the subject of analyzing the game, which I may currently lack.

I feel strongly, that the research I will do, will be different than the quantitatively objective, or qualitatively subjective information already out there. Most analyses focus on overall game-play, and deck building strategies. I will refine my focus to individual cards, and individual mechanics, by revisiting the most fundamental levels of the game. I will construct probability models, which I hope will be transposable into future expansions, where I will be able to empirically measure the strength of future expansions.