Two-Person Game Theory and Magic: The Gathering

Now that the ideas of perfect and imperfect information, zero-sum and non-zero-sum games, game trees and matrices are all clear, I can begin to correlate these ideas with their presence in the game of Magic: The Gathering. While ideas as these are an underlying basis of many games, understanding their applications is helpful in analyzing the roots of what make up the concepts behind the game. Combined with probability analysis this will prove most helpful to understanding the game.

Magic, as a whole, is a zero-sum game. At the end of a duel(1), there is both a winner and a loser, or a tie-game situation. Within the game lie non-zero-sum applications, however. For instance, both players start with 20 life points. The sum of the change in life points is not zero. As a matter of fact, it is far more likely to be negative, since most decks to not utilize cards that gain life. Usually, at the end of a game, one player's life will be zero and the other player's life will be a positive integer. This is not always true, because if both players reach zero simultaneously, the game is a draw. A game can also end because a player has to draw a card for whatever reason, and has no cards left. For the most part, though, reaching zero life is the end of a game, so that will be discussed for all practical reasons.

Magic is also a game of imperfect information, with a factor of chance within. You do not know your opponent's cards in hand unless a card which lets you see them is used by either player. The order of the remaining cards in your deck is also an unknown if you are not using cards that let you see this information. The exact contents of the opponent's deck are also an unknown that will not be available for you to see unless you have a card that lets you, such as Jester's Cap(2). Because you know the contents of your own deck, and do not get the same hand every game, there is an inherent factor of chance.

The decisions necessary in Magic can actually be drawn as game trees. Whenever you have to make a decision such as which card to play, whether to wait until later or play a card now, or whether to counter a spell or not, a game tree is involved whether you realize it or not. Allow the following hypothetical situation: Jim has out a Goblin Patrol(3), Mogg Fanatic(4), and Goblin Raider(5). He has three cards in hand: Shock(6), Mountain, and Mogg Fanatic. He has two Mountains in play, both untapped. His opponent, Jon, has out three Plains(7). Jon has no creatures out and five cards in hand. First Jim attacks Jon with the three creatures, dealing five points of damage, bringing Jon down to 12 life. The questions are, should Jim play the Mountain or hold onto it? Should Jim play the Mogg? Should Jim play the Shock? While I will not yet be analyzing the probabilities influencing the decision, I will analyze the several possible outcomes.

Jim knows that Jon is playing a solid white deck. That is, white is the only color of any of the colored cards in the deck. There are no cards of any other color besides colorless cards. That being said, there are many white cards that are capable of destroying what Jim has in play. Armageddon(8) and Wrath of God(9) are two of these cards. Also, Jon has Serra Angel(10) in his deck to block oncoming attackers with. Wrath of God will destroy all of Jim's creatures. With three lands out, Jon can possibly be holding a Wrath of God in his hand, along with the land he needs to cast it. Therefore, it may not be such a good idea for Jim to play any more creatures. Seeing that Jon is at 12 life, attacking twice more and casting the Shock will let Jim win. If he plays any more creatures, Jim will still have to attack twice to deal sufficient damage, yet will put Jon on the offensive.

However, if he waits too long, Jon may be able to take control of the board. If Jon can play a Serra Angel, he will be able to block one of Jim's attackers. Therefore having another creature to attack with might be a good idea. Mogg Fanatic can be sacrificed to deal one damage, so playing it is never too risky. Being able to attack with it, too, can be a benefit, nonetheless. If he plays the Mountain, he will not gain a sufficient advantage, and may leave himself susceptible to a mid-game Armageddon. If he plays the Shock now, he may give up the opportunity to use it on a creature Jon plays, or may just eliminate the surprise value. However, Jon may be pretty sure that Jim has one in his hand so that may not be an issue. These three decisions have many factors affecting them, as now demonstrated. Here is a general summary of Jim's options and Jon's resulting life total:

Therefore, if Jon plays a Wrath of God next turn he can have a shot at winning the game. If he does not, he will need one next turn, as having a blocker alone will not save him. Since Jim has a Shock in his hand, he can use it to do 2 damage to Jon. Combined with the two Mogg Fanatics, he can beat Jon simply by getting him to 4 life or less. However, if Jon plays the Wrath, Jim will have to sacrifice his Moggs immediately. Playing the Shock ends up not making a difference because to destroy the Serra Jim will need both the Shock and another creature or two. Therefore the efforts are better spent on reducing Jim's life total. Subsequently, if Jon is able to get out a Serra after casting a Wrath, Jim cannot do anything about it except hope to draw something good.

With the resources he currently has he cannot do anything. Playing the Mountain, however, does not seem worth it, since if Jon casts Armageddon, Jim will need to hope to draw another Mountain as well as the additional cards he already needs. Thinking ahead, Jim realizes the best plan is to play the Mogg Fanatic. Saving it will do no good since he already has enough attackers to beat Jon regardless of the Serra, provided he does not play a Wrath. By using this all-or-nothing strategy, Jim is assuming that Jon does not have a Wrath to cast next turn. If he does, the game will be grim for Jim. If he has one the turn after, that is where the only difference comes in.

By saving the Mogg, the advantage gained is that Jim will be able to play it after Jon casts the Wrath. However, since it is so late in the game, Jon will probably be ready to play his Serra the turn after, or something else that can control the Mogg. Hence it is better if he plays it and gets in an extra attack with the Mogg. Being able to put Jon down to 2 life, as opposed to 3 life, will help. If Jim draws another Shock he can win the game right then. Without the extra damage, he would need two direct damage(11) cards. Using an all-out aggressive strategy is therefore the best thing Jim can do in hopes of winning this game.

In order to demonstrate the concept of a game matrix in Magic: The Gathering, I will use a different example. Note that in the case of Magic, both players almost never make game decisions at the same time. The decision making process often alternates from one player to the other. However, the topic of deck selection has several game matrix applications. In Magic, any given kind of deck has both strengths and weaknesses. Deck A may win against deck B most of the time but lose to deck C just as often. Being as there are several types of strategies and sub-strategies within Magic, the choice of deck one uses may often be a great key in determining how he fares at a tournament. While play skill is important, deck match-ups and luck of the draw are factors of the game, also.

Some decks try to play slowly and take control of the game and then win, while others play aggressively and try to win immediately before the opponent can control the game. Jim has a fast red deck with fast creatures and direct damage, a fast green deck with many powerful low casting-cost creatures, and a slow blue control deck. Jon has a fast black creature deck, a white control deck, and a red control deck. For the sake of discussion, let it be given that Jim and Jon tested these decks against each other extensively and found how often their own decks won. Win percentages are from Jon's point of view.

Jon's Win % Jon Red Control Jon White Control Jon Fast Black
Jim Blue Control 70% 20% 80%
Jim Fast Red 50% 40% 20%
Jim Fast Green 30% 30% 50%

The game matrix given by Jim's decks versus Jon's decks can be solved in order for each player to determine which of his own decks is the best one to play against the other player. The simplest method would be to average the possible outcomes and find the result. By doing that, Jon's slow red deck wins 50% of the time, Jon's slow white deck wins 30% of the time, and Jon's fast black deck wins 50% of the time. Using 100 minus the number given since they are from Jon's point of view, Jim's slow blue deck wins 43% of the time, Jim's fast red deck wins 63% of the time, and Jim's fast green deck wins 63% of the time.

If each player uses one of his best overall decks, Jon will either be playing slow red or fast black against Jim's fast red or fast green deck. The simplified game matrix would be a 2x2 which looks like this:

Jon's Win % Jon Red Control Jon Fast Black
Jim Fast Red 50% 20%
Jim Fast Green 30% 50%

This being the case, Jon's slow red deck would win 40% of the time, and his fast black deck 35% of the time. Jim's fast red deck would win 65% of the time, and his fast green deck 60% of the time. That being the case, Jon would be more likely to use his red control deck and Jim more likely to use his fast red deck. Both players would then have a 50-50 chance of winning.

When going to a tournament, however, one must account for every possible deck match-up. That means that instead of just his own decks versus a friend's three decks, he must account for his own decks versus all the different types of decks he will expect to see. Additionally, he must take into account how many of those decks he expects to see. In Magic lingo, this is often referred to as the "metagame". This is the game within a game; determining what you think you are playing against. By determining your chance of being able to defeat a certain deck multiplied by the chance of facing it, and summing up all the products for "match-up probability times win probability" for each opposing deck, one can assess the overall win probability for any given deck. By doing this for each deck and finding which one has the highest probability of winning, a player guarantees himself the highest level of success.


Footnotes:

  1. A duel refers to a single game, while a match is the best two out of three games.
  2. Jester's Cap: Artifact, 4. Pay 2 mana, tap and sacrifice Jester's Cap to look through the opponent's deck and remove any three cards from the game.
  3. Goblin Patrol: Creature - Goblin, R. Echo. 2/1.
  4. Mogg Fanatic: Creature - Goblin, R. Sacrifice to deal 1 point of damage to a creature or player. 1/1
  5. Goblin Patrol: Creature - Goblin, 1R. Cannot be used to block. 2/2
  6. Shock: Instant, R. Target player or creature takes 2 damage. (Yes, this card is obviously a weakened Lightning Bolt, but when building decks out of the in-print pool of cards, only Shock can be used).
  7. Plains: Land. Tap for one white mana.
  8. Armageddon: Sorcery, 3W. Destroy all lands.
  9. Wrath of God: Sorcery, 2WW. Destroy all creatures. They cannot be regenerated (special ability that prevents them from being destroyed).
  10. Serra Angel: Creature, 3WW. Does not tap when attacking. 4/4, Flying.
  11. As opposed to creatures that have to go unblocked to do damage, cards which can do damage directly to a player immediately are often referred to as Direct Damage.

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