Game Theory for Duo Standard
A mathematically sound way to choose your deck in game 3.
(I’m just here for the spreadsheet)
The Problem
The Mythic Invitational will feature the biggest prize pool ever for a Magic tournament: $1 million total, including $250,000 for first place. The tournament will feature a new format called Duo Standard in which players bring not one, but two decks. The match is best of three. In the first game, each player’s deck is chosen randomly. In the second game, each player uses the deck they didn’t play in the first game. Finally, if the match is tied after two games, in the third and deciding game each player gets to choose their deck.
Choosing your deck for the third game seems like a random guessing game, but it turns out this strategic decision follows very specific rules worked out in the field of game theory, and applied extensively in poker. In this post I’m going to explain the rationale behind the theory and share a spreadsheet I made to get the optimal strategy for any match-up.
Imagine you’re playing for the $250,000 first prize in the finals. Let’s say you decided to bring Mono-blue Tempo and White Weenie, while your opponent is playing Sultai Midrange and Esper Control. You split the first two games. Now you have to play one game with everything on the line. How do you choose which deck to use?
Let’s say the match-up percentages look like this.
These match-up numbers are taken from this analysis. They may not, of course, be entirely accurate (more on this later), but for now let’s just say these are the numbers. Even with the match-up percentages set in stone, it’s not at all obvious what your strategy should be.
You could play Blue, hoping to capitalize on your favorable Sultai matchup; but if your opponent anticipates that, they could just choose Esper and get the upper hand. Aha! But if they’re going Esper, we’ll get the drop on them with White Weenie. But if we’re going White Weenie, they’ll switch back to Sultai…
The battle of wits has begun!
Clearly you can go round and round in circles with this kind of thinking. Is there a better way to approach this kind of decision?
A Solution from Game Theory
It turns out that the field of game theory has worked out a way of analyzing situations like this. Poker players are very familiar with this kind of analysis, but most Magic players aren’t (at least not yet!).
Returning to the final game of the Mythic Invitational, we’ve figured out that choosing either deck gives our opponent an obvious way to exploit us. At this point we might have the bright idea of flipping a coin to choose our deck. But it turns out then our opponent could always play Esper and earn themselves an overall winrate of 52%.
Mixing up our decks at random is a good idea, but if we don’t want to let our opponent exploit us by choosing one deck or the other, we need to mix at a frequency such that we don’t care which deck our opponent chooses. For these decks, that frequency turns out to be playing Blue 69% of the time and WW 31% of the time. This guarantees us an overall 51% win rate no matter which deck our opponent chooses. In fact, we don’t care which deck our opponent plays: we’re 51% either way.
You can figure out the exact frequencies by setting the expected value of each of our opponent’s deck choices to be equal, then solving for our frequency. Say p is the frequency we play Blue. Then our equation would be
p * 56 + (1-p) * 39 = p * 48 + (1-p) * 57
After doing a little algebra, it turns out p = .69. Of course, nobody wants to do all that math every time they need to make a deck choice. That’s why I made a handy spreadsheet that does it for you.
How to Use the Spreadsheet
First you’ll have to make your own copy of the spreadsheet. Then, edit the match-up numbers in the blue cells. You can also change the deck names to make it easier to keep track of things. Following poker naming conventions, “hero” is you and “villain” is your opponent.
The spreadsheet will automatically update to show the recommended strategies for both sides. It also shows your expected value: how often you should expect to win if both sides follow the optimal strategies. Finally, check (or un-check) the checkbox to get a recommendation drawn randomly from the suggested frequency.
What is the spreadsheet doing behind the scenes? First, it checks for “domination”: when one of your decks is better against both of your opponent’s decks. In that case, clearly you should always use that deck. You hardly need a spreadsheet to tell you that… but the spreadsheet will tell you that.
In the trickier case, where one of your decks is better against one of your opponent’s decks, but your other deck is better against their other deck, it does the math to figure out how often you should play each deck according to the method described in the previous section.
But hold on, we were talking about the Mythic Invitational finals. I’m guessing they won’t let you access an online spreadsheet during the match. So what’s the point?
Well, if Duo Standard eventually gets added to Arena, I think it’s reasonable for people to use a tool like this to help with deck selection in-game. But my main advice is to play around with the spreadsheet to develop your intuition about what the optimal strategies look like.
Nash Equilibrium
In the previous section we figured out our optimal strategy is to play WW 69% and Blue 31%. Doing the same analysis from our opponent’s point of view, we find that their optimal strategy is to play Sultai 35% and Esper 65%.
In game theory, this kind of strategy pair is called a Nash Equilibrium (yes, that Nash). It has the property that neither player can profit by unilaterally deviating from their strategy. In other words, even knowing your strategy, there’s no way I can change my strategy that will let me win any more often.
However, this also means your opponent can’t make a mistake (remember, we found our optimal strategy by ensuring both of our opponent’s deck choices had the exact same expected value). This raises the question of whether we should actually play the equilibrium strategy.
Well, it depends. Playing the equilibrium strategy sets a minimum value we can expect to receive regardless of what our opponent does. Thus it can be thought of as a defense against a clever opponent.
Even if you decide not to play the equilibrium strategy, knowing what it is can be very helpful for exploiting your opponent. For example, in the match-up we’ve been discussing, if we think our opponent will play Sultai more than 35%, we could exploit them by always playing Blue. It’s not obvious just from eyeballing the match-up numbers that 35% would be the cutoff point.
But what about the match-up numbers?
There’s just one problem: the strategies generated by this method are only as good as the match-up numbers. In Magic we often don’t have trustworthy match-up data. This is a real problem, but it’s still worth understanding the math of choosing your deck, for a few reasons.
Match-up percentages are just a way of expressing how decks perform against each other. If you have no idea about these percentages, you’re really saying that you don’t know which decks are good. That’s a problem, but it’s one you’ll have to try to solve anyway if you’re serious about winning a tournament.
More importantly, playing around with the calculator can yield valuable insights even without knowing the exact real match-up percentages. For example, what sorts of decks pair well together? Is it better to have extreme good and bad match-ups, or be closer to 50% across the board? Even with good match-up data, you can’t really answer questions like these without understanding the dynamics of choosing your deck in game three.
Conclusion
In this article I went over a mathematically sound way of choosing your deck for the third game of a Duo Standard match. I provided a spreadsheet to calculate the optimal strategies, based on match-up percentages. The best way to use this spreadsheet is to experiment with different match-ups to develop your intuition about the optimal strategies.
Historically, Magic players have made decisions mostly by feel. As the stakes of Magic tournaments increase, players will be incentivized to incorporate more data and math in their decision-making. While an understanding of game theory isn’t a substitute for having a strong understanding of match-ups or making good in-game plays, it’s an important piece of the puzzle for anyone who’s serious about winning $250,000 at the Mythic invitational.
