The_Mathematics_Of_Magic_The_Gathering-Prywes.pdf

The Mathematics of Magic: The Gathering

A study in probability, statistics, strategy, and game theory

By Jon Prywes

May-June 1999

Here are some of Jon Prywes' Magic accomplishments:

He wrote a Magic magazine online called The Library of Leng, from 1995 to 1997;

He wrote three articles for Scrye Magazine in 1996 and 1997;

He started a Magic club at his high school in 1997, which ran through 1999;

He played in several semi-competitive tournaments including the 1999 Junior Super Series Eastern Divisional;

He wrote numerous articles for The Magic Dojo (featured on this page);

He wrote a paper about the mathematical components of Magic in 1999 (also featured on this page);

He has done hundreds of Magic eBay auctions;

He worked at a day camp teaching Magic strategy to kids in the summer of 1999

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Table of Contents

Introduction to Magic: The Gathering

A Mathematical Introduction....................................................................................................... 3

A Description of the Game ........................................................................................................... 4

A Sample Game............................................................................................................................. 6

Why Experience Counts................................................................................................................ 7

Two-Person Game Theory

What Does It Mean?...................................................................................................................... 9

The Basic Concepts ...................................................................................................................... 10

Two-Person Game Theory and Magic: The Gathering ............................................................... 12

Probability

Probability and Magic: The Gathering ....................................................................................... 15

Shuffling and Randomization .................................................................................................... 16

Drawing Cards From a Deck ........................................................................................................ 17

The Average Game ....................................................................................................................... 2

Chance Versus Skill ....................................................................................................................... 3

Deckbuilding

Deckbuilding and Magic: The Gathering ...................................................................................... 5

Deck Archetypes and Deck Strategy ............................................................................................. 6

Deck Comparison and Winning Ratios ........................................................................................ 7

Card Efficiency and Resource Management .............................................................................. 10

The Big Game

Probability, Statistics, Game Theory, and Magic ......................................................................... 11

Math and the Average Player ..................................................................................................... 12

Personal Applications ................................................................................................................. 14

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Introduction to Magic: The Gathering

A Mathematical Introduction

Magic: The Gathering is a game which some take seriously, and others lightly. Many understand it, even more do not.

Some play it nonstop and others never see the point. Personally, I think it is a great game. Only those who actually play

the game can truly understand how wonderful it is. Most people who write it off as pointless do not see the mechanics of

the game that make it simple, yet complex. In its five years of existence, the crowd that Magic has drawn can alone show

how great the game really is. On the weekend of May 1 and 2 I traveled with some friends to Pro Tour: New York in

Secaucus, New Jersey. Hundreds of people were gathered inside the Meadowlands Exposition Center just to play the

game they love. The Pro Tour is no day in the park; it is a great competition for the mind.

Zev Gurwitz, a junior at Scarsdale High School, along with myself, had both qualified for the Junior Super Series event.

Sixty-four players age seventeen and under qualify for the JSS at challenge tournaments held during the winter. We had

playtested for this tournament a fairly lengthy amount, yet I was still skeptical. As it turned out, I went 1-3 and dropped

out. Zev went 0-3 and dropped. A record of at least four wins was necessary to make the top eight, in order to play at

the JSS Championships where $250,000 in scholarship money is awarded. We tried, but what was it that kept us from

winning? What is it that keeps every Magic player from winning all the time? What keeps even the most skilled of players

at bay from winning every single game? It is a word called chance.

Chance is what makes Magic different from many other classic games. When one plays Chess, he gets to see all the

pieces. Both players have eight pawns and eight other pieces, either white or black. Every piece moved is public

information, and nothing is held secret. Chess is ruled by complete skill. Nothing random can possibly happen. Magic,

on the other hand, adds a new element. Not only are each person’s playing materials often different or varied (the

decks), but randomness exists in this game. One starts the game by randomizing his deck of sixty or more cards, then

drawing seven. He will definitely not draw the same seven cards in every opening hand! Variety is what makes this game

different than Chess. There are two elements of variety involved: Deck construction and random card draws.

Deck construction is part of what governs this element of variety. The card draws are simply a result of the deck

construction, as well as any cards played during the game which may alter the contents of what is remaining in the deck

(for example, removing cards from the deck). The Duelists’ Convocation International (DCI) is the authority governing

sanctioned tournaments. Although many players do not participate in tournaments, the DCI’s deckbuilding guidelines

provide for a fair game.

Some cards were printed before the research and development team realized they were overpowered. Hence, the cards are

banned from deck construction. Some cards are only restricted, meaning only one is allowed in a deck. Any card that is

not banned or restricted is allowed up to four copies per deck. A player has to think, “how many copies of a card do I

want in my deck? One, two, three, or four?” If he puts four copies in, he is likely to draw one very soon. If he only uses

one, his chances of drawing it are slim. Maybe having multiple copies is redundant and that is the reason for using fewer.

If he uses three copies of a card, then what will be his chances of drawing one by turn six? Probability pokes its nose up

into the game now.

In Chess, there is no probability, other than thinking about the number of possible moves. However, a player’s play style

is more likely to govern how he plays the game of Chess than probability will. For all intents and purposes, there is no

probability in Chess. Probability goes hand in hand with chance and random events. It is the chance that an event will

occur, or not occur. Will your adversary draw the card he needs to win the game this turn? Next turn? Within five turns?

Calculating the chances can help a player decide whether to play defensively and anticipate the other person drawing

their win card, or play aggressively and assume that the card will not be drawn by the other player.

Some of the other math involved in the game is less obvious. The “chance versus skill” question is a very hot topic

among high-level players. How much of playing a given deck is based on skill, and how much on chance? The obvious

solution would be to take a fairly inexperienced player and give him one deck, and give an experienced player the other.

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The experienced player will win with either deck if there is complete skill involved. If the deck is straightforward to the

point that there are no decisions to be made, and it beats the other deck almost automatically, then it is possible the

expert will lose games. In Magic, though, there are always at least some thoughtful decisions to make. The chance of

getting paired against a superior deck in a tournament does not mean an automatic loss. If your skill is lower to that of

the superior deck your chances may be slim, but if your skill is higher than your chances become much greater.

So then, what constitutes a superior deck? What deckbuilding techniques are necessary for one to make a deck that wins

more often than a deck which is not as good? How does one determine which card selections will guarantee a flawless

road to victory? Questions like these are what every player must try to answer when attempting to create the “perfect”

deck. Of course, I will aim to prove that the “perfect” deck does not exist. Laws of game theory are an invaluable aid in

proving this. While game theory is a very conceptual science, its laws very well do apply to games such as Magic. While

knowledge of these laws will not make one a Pro Tour Player, they can help one who is trying to understand the game

have an easier time making decisions. Game theory is simply a series of laws regulating how one goes about making

decisions in a situation. A “game” in game theory is not necessarily always a game as defined in common talk, but game

theory certainly has many game applications.

Through the laws of probability and game theory, along with statistical analysis and actual experiments, I will be working

on coming up with conclusions that can take these laws and correlate them to the game of Magic. First and foremost, I

will be explaining the basic concepts of game, in order for the reader who is unacquainted with Magic to familiarize

himself with these terms. In describing the game, I will attempt not to reiterate the entire rulebook, though I will

summarize the basics of how the game is played. Upon completion of these details, I will begin my outline of game

theory and begin on relating the math and game theory to its Magic applications. I will provide profiles of many of the

people who I consult with as part of my project. I will be discussing a survey I conducted in order to determine how

much math Magic players recognize as part of the game. This entire report will follow closely the outline given in my

project description.

A Description of the Game

The object of Magic is simple. You start with 20 life points, and when they are reduced to 0, the game is over and the

other person wins. You and another person both have a deck of sixty or more Magic cards, which are used in playing the

game. The cards can be from a variety of different sets, with (for the most part) no more than four of any one card. The

players alternate in taking turns. Each turn is a sequence of events that involve drawing a card, putting cards into play,

“attacking” the other player (using cards that represent creatures) and then discarding if necessary.

The basic resource in Magic is called mana. Mana comes from a Polynesian word meaning energy. Mana let you bring

certain cards into play, and use abilities on cards already in play. Some cards require more mana to use than others do.

There are five different colors of mana in the game, each representing a separate force. The forces represented are typical

of an adventure gaming genre: White mana represents the powers of good. Blue mana is for the powers of the mind.

Black mana is for the powers of evil. Red mana is for the powers of destruction and chaos. Green mana is for the powers

of nature and wildlife. Each color’s theme is represented in the cards of that color.

Thus a well-known adventure theme is presented in the cards, giving players something more than just a strategy game.

It is a game with a theme, with the strategy hiding in the background for the more advanced players to take notice of. In

addition, some cards do not have colors; these cards are either land (which produce mana), or artifacts (which are

colorless and can be played with any kind of mana). All cards in the game are referred to as “spells” for game purposes,

except for land. Lands are not spells and are not “cast”. They are simply placed into play. I will often use terms

interchangeably, however.

Another concept that the player must understand is the word “tap.” To tap a card is to rotate it sideways, indicating that

its powers have been used. This can be used to represent an attacking creature, a used artifact, or a land drawn for mana.

At the beginning of each turn you untap all your cards, so you effectively can use their powers once each turn (and/or

during your opponent’s turn).

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A spell’s casting cost is the amount of mana you need to use in order to play it. It is located in the top right corner of

the card. If a card’s casting cost is 2R, for instance, that means that you must spend one red mana and 2 of any color to

play it. Casting cost will be referred to a lot so this is very important to understand. The colored portion is specific, and

the numbered portion is generic, and can be paid using any color mana.

There are different types of cards in the game: Land, Artifacts, Creatures, Enchantments, Sorceries, and Instants. All cards

come in five different colors except for lands and artifacts. Lands are a special kind of card; you can play one each turn.

Lands can be tapped for mana, which is used to play any of the other kinds of cards. Artifacts are colorless, which means

you can use any kind of mana to play them. They may let you do anything from draw cards to affect cards in play.

Some artifacts are also creatures. Creatures can be used to do damage to your opponent (thus reducing his life total

from 20 to 0). However, if your opponent has creatures of his own out, he can use them to block yours. Enchantments

are cards played on an existing card, which modify what the card does, usually. An example would be an Enchant

Creature card, which would be played on a creature. It might make the creature weaker, or stronger. Some enchantments

are not played on other cards, and have a global effect on the game. Sorceries and instants cause a one-time effect on

the game. Sorceries can only be played during your turn; instants can be played anytime.

At the beginning of the game each player shuffles his deck. The players roll a die or flip a coin to determine who chooses

who goes first, then each draws seven cards. The player who chooses to go first does not draw a card on his first turn. The

sequence of a turn is as follows:

Untap phase: Untap all cards you control. This means to rotate them so they are all facing upward and not rotated

(tapped).

Upkeep phase: This is a maintenance phase. Some cards will make you do an effect during this phase, such as pay mana to

keep the card in play, for example.

Draw phase: Draw a card.

Main phase: You can do these things, in any order:

a. Put one land into play

b. Declare one attack

c. Play spells. You can play spells before or after your attack, as well as both. Note that all cards (except for instants) can

only be played on your turn during the main phase.

Discard phase: Discard down to seven cards.

Cleanup phase: Any effect that lasts until “end of turn” wears off now. Any damage on a creature, which does not

destroy it, wears off as well.

The attack works like this: You choose any untapped creatures you control that you have had in play at least one turn,

and tap them. Your opponent either blocks them or takes damage equal to their power. If he blocks, both creatures deal

damage to each other equal to their power. A creature has a pair of numbers in the bottom right corner. These are its

power and toughness. When a creature deals damage, it deals damage equal to its power. When it receives, the damage is

applied to its toughness. If it takes damage equal to or greater than its toughness it will be buried. That means it will be

played in a pile next to your draw pile called your discard pile. For example, a Giant Spider has a power and toughness of

2/4. It deals 2 damage to a creature blocking or blocked by it, and if unblocked during an attack, deals 2 to the player it

attacked. If it takes 4 damage during one turn, it will be placed in its controller’s discard pile.

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