Magic: The Solving

        “You are a Planeswalker.” Those are the first words a player of Magic: The Gathering hears when he sits down at his first game. Magic throws its players into a world of incredible creatures and sorcerers with unimaginable powers. Magic is a card game with strategic, combinatorial and chance components, and a winning strategy that is derived from all three.        

Magic: The Gathering was developed by Richard Garfield while he was a doctoral candidate at University of Pennsylvania in 1993. He worked with local volunteers to tweak and test the game. That same year, he game was introduced to Peter Adkison, the CEO of Wizards of the Coast (a gaming company), who loved the game and thought it would be a great seller to people at conventions who want something to play during down time (Adkison, 2009). The game grew in popularity after its release in the August of 1993, and continued to get a larger following until 1996 when Wizards announced its “Pro Tour” circuit, a series of competitions around the nation. The game has continued its climb of popularity, attracting a legion of loyal fans who make each new set released more successful than the last. Thousands of people compete in tournaments every weekend vying for prize pools of more than $30,000. The people who win at these tournaments devote a great deal of time to learning everything they can about the game. The most important thing a player can learn, though, is the rules.

        The rules of Magic are fairly straightforward.The object of the game is to attack with creatures or spells, and lower the opponent’s life total from 20 to 0.  Each player starts with a deck of 60 cards, chosen beforehand from cards the players own.. Players shuffle their decks and decide randomly who goes first. Each player then draws an initial hand of 7 cards that is kept hidden from his opponent. Players draw a card at the start of every turn, except the first. Players can play one land a turn, and can play as many spells or creatures as they are able to on any given turn. Lands and creatures stay in play from turn to turn, other spells such as instants or sorceries produce an effect and are then discarded. A land is a card that produces mana. Mana is the currency players use to pay for spells. In order to use a land the player taps by turning it 90 degrees to the side, indicating it has been used up for that turn and can’t be used again. Lands untap at the start of their controlling player’s turn, and can be used again.  Players have a variety of moves available to develop a winning strategy.

        The combinatorial component in Magic is the large number of choices available to players. Players can choose almost anything. They choose what deck to play, what spells to cast on any given turn, and what creatures to attack with. Turns in Magic are taken sequentially, and games have a large, but finite number of moves. While all information in Magic is initially imperfect, as the game progresses, and cards are played both players slowly gain more information about each other’s decks.

        The major strategic component in Magic is deck building. Decks fall into one of three categories: aggressive, controlling, or combo. Aggressive decks play as many creatures as possible, and try to win the game by lowering the opponent’s life total to 0 as quickly as possible. Controlling decks use cards that try and control the opponent by either countering his spells, or killing off all his creatures. Countering a spell means its effect does not happen, and the card is instead discarded. Combo decks try to win the game on one turn by playing a combination of cards that can kill the opponent in one fell swoop. There is a “rock, paper, scissors” aspect of Magic, in that control beats combo, combo beats aggressive, and aggressive beats control. All three decks win about 50% of the time against themselves (Eckstein).

Player, opponent

Combo

Aggressive

Control

Combo

0

+1

-1

Aggressive

-1

0

+1

Control

+1

-1

0

        In theory, there is no one deck that is completely dominant, as players should always be able to play a deck that counters it. In practice, however, certain decks have been so powerful for certain times that players had the choice of either using that deck, or losing to that deck. Currently the deck most popular is the Delver deck.By playing this deck, you give yourself a great chance of winning any sort of Magic tournament, assuming you play it correctly, and are lucky enough to draw the right cards.

        Is it better to be lucky or good? In Magic it’s often the former. Magic is a very probabilistic game. There is chance in almost every aspect of the game. Players shuffle their decks before the game starts, putting it in a random order. They draw cards off the of their decks at the beginning of your turn, and you have a chance of getting the card you need at any given time. Probability is also taken into account when playing spells. A Delver deck runs 8 counterspells, so the probability that you are going to play something against that deck, only to have the one card in your opponent’s hand counter it is about 8/60 (8 counterspells, 60 possible cards). Winning strategies for Magic are highly dependent on probability.

        Delver decks have taken the pro-tour by storm. At the most recent grand prix, 5 of the top 8 decks were delver decks, with the final match of the tournament actually being a Delver-Delver mirror match up (Van Lunen). In order to win with this deck players need two things: Luck, and Skill. Skill comes from playing the game over and over again, studying the other decks  that might be played against, and understand what the correct move is at any given time. Luck might seem like the more fickle of the two, but the reason delver decks work so well is that they manipulate their own luck by playing spells that let them draw more of their cards, or spells that let players rearrange the order of the cards on the top of their decks. A Delver deck typically consists of 4 copies of Delver of Secrets, the card that gives the deck it’s name. Delver of Secrets(above) is a creature that can deal one damage when it attacks, costs one mana to cast, and can be flipped by revealing an instant or sorcery, to turn into Insectile Aberration,  creature that deals three damage when it attack. When Delver of Secrets transforms into Insectile Aberration, it lowers the amount of turns its controller must attack with it from twenty to seven (1 damage a turn for 20 turns vs 3 damage a turn for 7 turns). Delver decks also play 4 copies of Ponder, a card that allows the player to look at the top 3 cards of his or her deck, rearrange them, or shuffle his or her deck, and then draw a card. Ponder allows players to effectively stack their decks for the next three draws, or to get a whole new set of cards on the top of their deck if they decide they don’t want the 3 that are on top. This deck also runs 20 lands, and 32 other spells that would flip delver. A Delver of Secrets played on the first turn, followed by it turning into Insectile Aberration the following turn, or using a Ponder to transform it on the third turn establishes a strong seven-turn clock that can easily win the game,. A winning hand contains of one Island, one Delver, and one Ponder and four other cards. The probability of an opening hand that contains at least one Delver, Island, Ponder, and four other cards follows a hypergeometric distribution. Hypergeometric distribuitons are a form of combinatorial probability that allows you to calculate probabilities of successive events without replacement of variables. The probability for the opening hand above is

P(hand)=(20C1)*(4C1)*(4C1)*(32C4)/(60C7)= 3%.

This probability is fairly low, but assuming a player got this hand he can play turn one Delver. At the start of his second turn, the player checks the top card of his library to see if it could flip delver. This probability is:

P(flip)= (cards that can flip delver)/(cards left)=31/53=58.3% .

 If the top card does not flip Delver, we use the Ponder to better our chances. Checking to see if at least one of the top 3 cards can flip Delver is the same as checking that none of the cards flip Delver, so P(Ponder flip)= 1- (31C0)*(21C3)/(52C3)=94.3%

The probability that we get at least one card that can flip Delver after Ponder is quite high, which is why the two cards see so much use together. Ponder’ing almost doubles the chance that you can flip a Delver, thereby increasing the likelihood that he will win.

        Magic: the Gathering is a fun and complicated game, which combinatorial, chance and strategic components. The winning strategy uses all three. By being able to see all available, moves, think strategically about the results of your moves, and by having good luck, or cards that let you manipulate your probability, you can win almost any game of Magic.