Probability: Chance Versus Skill

In Magic: The Gathering, there are two major factors which contribute to a victory. The deck and the player are these two factors. One player may play a deck to complete perfection, while another may lose horribly with it every game. Sometimes it may be the player's strategies that are flawed, and sometimes it may be the decks that he has to play against. Certain decks, as I will show, have inherent strengths against others, and that gives them a higher overall win percentage. Given that two players of equal skill are using the two given decks, the deck with an inherent strength against the other deck will win more often. Similarly, if two players of differing skill use the same pair of decks, the better player should usually win with either deck if the decks beat each other half of the time. However, there may be no "50% deck." Some may come close but there is no exact deck that fits this description. Instead, there are several different kinds of decks with varying win proportions.

In order to study this theory, I constructed four decks, and brought them over to my friend Jim's house. Jim is a Magic veteran from a couple years past when he used to collect. He no longer collects the cards, but he stays current with the game and plays online with Apprentice. By being able to represent cards in a computer program without actually having to obtain the cards, Apprentice is a wonderful tool. Of course, playing Magic in person is a completely different experience. Nonetheless, we played 12 sets of 4 games each. In each set, we each took a deck and played 4 games in order to determine a fairly rough estimate of the deck's win ratio, given the player. Note that a deck did not play against itself since there was only one copy of each deck. The results are represented in the following game matrix: (The scores are from Jon's perspective).

Chance and Skill Jim-Blue Jim-Green Jim-White Jim-Red
Jon-Blue N/A 0% 100% 25%
Jon-Green 100% N/A 75% 75%
Jon-White 75% 75% N/A 100%
Jon-Red 75% 0% 0% N/A

The following statistics can be obtained from this matrix:

Deck Overall Jon's Games Jim's Games
Blue 29% 42% 17%
Green 79% 83% 75%
White 63% 83% 42%
Red 29% 25% 33%

Jon won 58% of the games and Jim won 42% of the games. However, this is only an approximation since the card draws each deck gets vary a lot. A larger sample of games would be necessary in order to get a better approximation, but the number of games required would be cumbersome and uninteresting. This example is good enough to demonstrate the concept of chance versus skill. The green deck appears to be the overall best, with the white deck second overall best.

The idea of chance and skill interacting comes into play when deck match-ups are analyzed. For instance, when Jon played the green deck and Jim played the blue deck, the green deck won all four games. The same thing happened when Jim used the green deck. Therefore it appears that playing skill did not have an effect on this deck match-up. For the white deck against the green deck, however, this is not the case. It appears that the white deck required more skill to play correctly because Jon was able to beat the green deck three games with it, while Jim was only able to beat the green deck one game with it. Similar conjectures apply to white versus blue, and red versus green. Using a game matrix along with common sense ideas, one can determine where skill plays more of a role in the game than chance does. Even by just looking at the total wins, we see how Jon was able to win more with every deck except the red deck.

The larger the difference between Jim and Jon's wins is, the larger the amount of skill involved in playing the deck. Again, allow me to remind the reader that the amount of data used in this experiment only makes for a very crude estimation of chance versus skill. A bad hand will affect this experiment a lot worse than if there were, say, 100 games played in each trial. For instance, it does not make sense that Jim can play Jon's deck better than Jon can. Jon designed the red deck; so therefore, Jim should not be able to play it better in theory. Nonetheless, card draws can affect such an experiment as this in both ways. Still, it is possible that Jim may actually be the better player of this deck. While it is illogical, it is not to be ruled out as a possibility. The results are just not accurate enough to determine an actual result. However, they are good enough to demonstrate the main ideas.

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