# Magic Math – Match Win Percentages

When analyzing a certain matchup, we often say that one deck is a favorite to win the match if its match win probability is larger than 50%. Today, I’ll dive into the intricacies of this match win probability: I’ll show how important post-sideboard games are compared to pre-sideboard games, and I’ll analyze the effects of a large difference between playing first and drawing first.

## Post-sideboard games vs. pre-sideboard games

This formula tells us, for example, that when you have a probability of 40% to win game 1 and a probability of 60% to win a post-sideboard game, then you’ll have a probability of 55.2% to win the match. In other words, you’re a favorite to win!

But perhaps you can’t dedicate enough sideboard slots to the matchup to reach a 60% game win percentage post-sideboard. In that case, you may wonder which post-sideboard game win percentage you would need to guarantee at least a 50% match win probability. The following theorem provides the answer to this question as a function of your game-1 win probability, and the subsequent figure shows it graphically.

Several interesting things become apparent from this picture:

Even if your pre-board matchup is only 30%, you can still be a (very small) favorite to win the match if your post-board matchup is 60% or higher.
Conversely, if your pre-board matchup is 70%, you still need a better than 40% post-board matchup to be a favorite to win the match.
All in all, post-board games are much more important than pre-board games, so focus on the post-board games!

## The impact of a large difference between your GWPs on the play and draw

Consider a red aggro deck and a Knight of the White Orchid deck in Standard. Suppose that both decks expect to have a 60% overall game win probability against Deck X. (This number is the same on the play and on the draw.) Yet, there’s a difference: the red aggro deck is great on the play but weak on the draw, whereas the Knight of the White Orchid deck is always solid. To make things specific, let’s say that the red aggro deck is 80% to win a game on the play but 40% on the draw, whereas the Knight of the White Orchid deck is 60% to win both on the play and on the draw.

Which deck do you think will have the better overall match win percentage against Deck X?

A. The red aggro deck
B. The Knight of the White Orchid deck
C. It doesn’t matter—both have the same match win percentage

Ventured a guess? All right, now consider the same decks again, but in a different matchup against, say, Deck Y. Suppose that the red aggro deck is 60% to win a game on the play but only 20% on the draw, whereas the Knight of the White Orchid deck is 40% to win both on the play and on the draw.

I’ll ask the same question again: Which deck do you think will have the better overall match win percentage against Deck Y?

A. The red aggro deck
B. The Knight of the White Orchid deck
C. It doesn’t matter—both have the same match win percentage

What does your intuition tell you? Is there going to be a difference or not? Well, to find out the answer, let’s do the analysis.

This gives us a general formula where we can plug in the numbers and get an answer to our question, but it is looking complicated and doesn’t provide us with much insight yet. Instead, let’s try to gain some intuition by focusing on a specific class of settings (which, not completely coincidentally, encompasses the ones from my “puzzles”).

With this theorem, we can now easily answer my “puzzles”:

• The answer to the first question is A: The red aggro deck. To be precise, it will have a probability of 65.6% to win the match, whereas the Knight of the White Orchid deck will only have a probability of 64.8% to win the match. In other words, if you have a favorable matchup, then a big play/draw difference can yield a (very minor) additional benefit.
• The answer to the second question is B: The Knight of the White Orchid deck. Specifically, it will have a probability of 35.2% to win the match, whereas the red aggro deck will only have a probability of 34.4% to win the match. In other words, if you have a bad matchup, then a big play/draw difference can make things even worse (albeit, very slightly).

The differences between the match win percentages are arguably not large enough to take them into account when testing or choosing a deck for a tournament, but I still found it interesting to see the direction of the effect.