MTG Arena’s Opening Hand Algorithm and Smoothing: Some Real Outcomes

sphinx-of-foresight-art

Introduction

According to a research paper published by Cornell University, Magic is the most complex game ever known. In fact, the authors argue, it’s so complex that you could even use the game to emulate a Turing Machine. A Turing Machine is a mathematical model capable of simulating complex computer algorithms. This is a pretty big deal, or at least so my friend with a math PhD tells me. For us Magic players though, this complexity is very old news. We’re accustomed to expanding our understanding to include new rules, keeping in mind obscure old ones (quick, explain Banding in 30 seconds or less!), and endless errata. One recent example is Ajani’s Pridemate. Until 2019 Pridemate’s +1/+1 ability was optional, then the card changed and it became automatic. We adjust, we build, then we readjust when the next set drops (or sooner with Banned and Restricted announcements). Increasing complexity has been the story of the last 25+ years in a nutshell.

What players are not used to is having these processes happen behind the scenes. While MTGO has been digitally shuffling and coin flipping for almost two decades, Arena has introduced something new which seems to be creating some confusion. I’m speaking of course of Hand Smoothing (or just Smoothing) in best-of-one (BO1).

In this article I’ll provide some background about Smoothing and explain what it means as best I can with the information available from Wizards. Then I’ll demonstrate Smoothing’s effect with some data from Arena and give some examples of how to apply the information in practice.

BO1 matches are the most popular format on Arena. It’s a quick format that WotC has positioned prominently in the game GUI. As a result, understanding how Smoothing impacts gameplay is as important as understanding the cards, knowing when to mulligan, or adjusting to being on the play or draw.

To say Smoothing is misunderstood or controversial may be understating the case. It is never mentioned within the game, so many Arena players are completely unaware it exists. Some players are skeptical of its existence or effect. A small but occasionally vocal section of the population lament that it’s out to get them. The wise players not only know that Smoothing is real, they also build their BO1 decks in a fashion that exploits it. What makes the best Magic players the best is that they understand and exploit the rules better than their opponents.

What is Smoothing?

What is Smoothing though? In a typical game of Magic, you draw a random hand of seven cards from the top of your deck; this may be the oldest rule in Magic. In paper Magic we do this with physical decks but in MTGO, this is done digitally using a random number generator. Arena works the same way as MTGO, with a random number generator creating a random hand for you. However, starting in October of 2018 (not long after the launch of the open beta period), Smoothing was introduced to BO1 matches.

Smoothing was first announced on the official MTG Arena Forums with the gist of it being:

Best-of-One Starting Hand and Mulligan Rules

MTG Arena has a system for randomizing opening hands in single-game matches that is different from the traditional method. We knew early on that the ability to play a meaningful single game of Magic would be an important part of the MTG Arena play options for the time-constrained player, but in testing, losing a single-game match after a mulligan felt worse than normal without the fallback plan of winning the other two games.

To mitigate this loss of the “foul to give” built into best-of-three matches, we wanted to randomize starting hands in a way that would reduce the frequency of mulligans, but without incentivizing mana-base construction outside the strategic norms of the game. Here’s how it works:

  • The system draws an opening hand from each of two separately randomized copies of the decks, and leans towards giving the player the hand with the mix of spells and lands (without regard for color) closest to average for that deck.
  • You can then use standard Vancouver Mulligan rules to get a new hand with one fewer card. Mulliganed hands are created from a single randomized draw.

We feel this achieves the desired effect of reducing the frequency of mulligans without changing how players build their mana bases, but we will be looking at data and listening to feedback on this system, so let us know what you think. Remember that this won’t make all your starting hands perfect! You will probably have some stretches of hands where you think we must have turned this system off. The overall mulligan rate should be lower, though, and we hope that will create a better-feeling but strategically authentic experience for single-game matches.

This post resulted in a lot of guessing, questioning and hypothesizing. And a lot of math. Eventually, a follow-up was posted in May of 2019. This followup said that most of the hypothesizing was wrong; a few highlights include putting emphasis on WotC’s use of “leans toward”. What that distinction means is anyone’s guess, as Wizards seems set on not divulging any more than they already have.

State of the Beta for May 25th, 2018

Hi Folks,

Thanks for your patience with the delays this week. We’ve been working hard on the next major update, and we’re really looking forward to sharing it with you!

The Economy. Nick has covered where we’re at here.

June is Coming. Look out for more info next Wednesday on what is coming in our June update.

GP Vegas 2018. MTG Arena, Nate Price,and I will be at GP Vegas June 14-17th! If you’re going to be there, please swing by and we can chat about the game in person!

Matchmaking. There has been a bunch of discussion around matchmaking and how it works in various places on the forums. I’ve touched on this in the past, but I wanted to take a little time to solidify where we’ll be at for June.

  • Ranked Constructed – Will remain the same as straight MMR matchmaking.
  • Entry-Fee Events – Will switch to exclusively win-loss matchmaking for June. In July we will likely bring back the slight MMR skew for Quick events, depending on how the data shakes out.
  • Placement Matches – Everyone’s MMR and Ranking will be reset. Your first ten matches will determine your initial MMR/Ranking.
  • Ranks – Will be more fluid with this update but will also reflect your opponent’s and your actual MMR within a range.

Best-of-One Mulligan System. There have been a number of theories on how the system works, and we wanted to leave it alone to see how far perception drifted from reality. Ramora had a post recently that we wanted to add some additional information on to aid the discussion. Namely, the actual opening hand numbers we see with 17 lands in a 40 card deck. (This is over 100,000 opening hands)

P(0 Lands)P(1 Lands)P(2 Lands)P(3 Lands)P(4 Lands)P(5 Lands)P(6 Lands)P(7 Lands)
MTGA Theory0.00060.029360.272120.541640.144020.0120.000260
MTGA Actual0.000*0.0220.2560.4750.2270.0190.000*0.0000
Paper0.013150.092050.245460.322970.226080.083970.015270.00104

*: These will happen on occasion, but they’re rare.

A special guest appearance here from Senior Game Designer “Godot” to go into more detail:

Hey All,

I wanted to make a few of points on this topic.

First, there seem to be a lot of misconceptions about the system that I’d like to clear up. My original post on the current starting-hand system states:

The system draws an opening hand from each of two separately randomized copies of the decks, and leans towards giving the player the hand with the mix of spells and lands (without regard for color) closest to average for that deck.”

Many in the community have interpreted “leans towards giving” in that sentence as a strict “gives,” when that is not how the system works. We have not told the community exactly how the “leaning” works, but the deterministic charts and graphs in the OP are not an accurate reflection of the system.

The goal of the system is to reduce the number of mulligans and “non-games” in Bo1 matches without significantly altering optimal deck construction, such that there is no need to maintain a best-of-one (Bo1) and Bo3 version of the same deck. In particular, we want to reduce the number of new-to-Magic players that bounce off the game due to frustrating mulligans and mana issues early in their experience, especially those coming in from other games where there is no variance in the mana system.

Most suggestions I’ve read for this system tend to be for safeguards against bad openers that would alter optimal deck construction or favor certain deck types too much given our goals or for removing the protection entirely.

Many that want it removed entirely have expressed dislike for the idea of the system making a choice between two hands for them; that *they* should have that agency, or it shouldn’t happen at all. The point of the system and the “leaning” clause is to make the obvious choice when it presents itself (e.g., the system will never choose a zero- or seven-lander over a three- or four-lander), but to have the choice be increasingly fuzzier as it is increasingly less obvious. In the fuzzier cases, maybe you got the hand you would have chosen between the two, maybe you don’t, but both players are operating under that same situation, both players are getting a hand randomized from their deck, and both players are taking mulligans a lot less often in a match structure where it is extra painful to do so. I believe players who are unhappy with the idea of the system choosing from between two hands would be even unhappier with a significant increase in their mulligans and the resulting non-games.

We are continuing to look at data, and the system is likely to change again before the end of the Closed Beta, but if I had to choose between what we have now and removing it entirely, I would keep what we have now. Mulligans and non-games happen much less often, and optimal deck construction isn’t significantly different from what it would be without the system in place. It is certainly less significant than implied in the linked topic, and on a personal level, it doesn’t alter my deck construction at all.

If and when anything changes about the Bo1 starting hand system, we will of course let everyone know.

So we have no official explanation on the details of what this means. The basic understanding of what it is believed to mean is that if the average land count in an opening draw for your deck is 3 lands, and it pulls one hand with 4 lands and one with 3, the game will “lean towards” showing you, the player, only the hand with 3 lands. All of this happens in the background with no visibility to the player. That right there is where people start to get a little off the rails about Smoothing.

An invisible mulligan where we don’t get any input? Is this real? Is it out to get me? We can throw statistical calculations around all day (though I won’t in this case), but what’s the real impact?

The Experiment

opening-hand
Three lands are exactly what we wanted here…

I was pretty confident from my own experience playing paper Magic versus playing BO1 that Smoothing was real and what its effect probably was. A big indicator is that in paper Magic no-land hands happen pretty regularly. You draw, you grumble, then you mulligan and move on with your life. Since MTGO and Arena use random number generators to simulate that experience, no-land draws should be similarly common in Arena BO1, but in my time playing Arena I can only recall coming across such a hand once and it really stood out at the time. For a no-land hand to happen with Smoothing, both hands drawn by the algorithm would have to be no-land, a very unusual occurrence, so my anecdotal experience fits the understanding. But anecdotes aren’t evidence, they’re just stories about an experience.

To address these questions about Smoothing more concretely, I decided to collect some data. Others have run the numbers to give percentages and odds. With the veil WotC has pulled across Smoothing, I wanted to collect what the real outcomes look like. I took a 24 land deck and used it in Play, Standard Play and Ranked Play formats to see what my land counts were over a large number of hands. While we have no reason to believe that Smoothing is different for Ranked versus Play, they could be. As a control, I did the same thing with a paper deck for the sake of comparison.

One thing in the original post by Wizards about Smoothing that didn’t seem right is that it applies only to your opening hand draw and doesn’t apply to any Mulligans. So I mulliganed each Arena hand down to zero and tracked those numbers as well.

Finally, I decided to use a data set size of 104 hands. I settled on 104 because 7 mulligans in Arena gives you 8 hands to look at. Do that 13 times to get past 100 hands and you have a total of 104 hands. According to Hasbro’s recent Q4 earnings call, the average player spends 8 hours a week playing Arena. Based on my own experience, 6 games per hour is about average when playing a mix of deck styles. So 104 hands is about 17 hours of play, which is about 2 weeks of play for the average Arena player.

With the paper deck, I found that I averaged 2.78 lands per hand, which is about what you would expect from a 24 land deck. Then in Arena Bo3 104 hands averaged 2.70 lands, while Area Play B01 averaged 2.76 lands per hand. Finally, Ranked Play averaged 2.75 lands per hand. That’s a pretty consistent result between all four data sets considering the sample size. This was an expected outcome. Smoothing doesn’t change the average number of lands you would draw over a large sample size. The point of Smoothing is to reduce the individual hand extremes. Two hands with 3 lands a piece has the same average land count as one hand with 0 land and one hand with 6 land. The important difference in those two small groups is the difference in playability.

What will show the impact of Smoothing is how often we had any given number of lands per hand, from 0-7, and how many hands are keepable. While keepability for a given hand varies by deck type and other cards in hand, for a 24 land deck we’ll say that a keepable amount of lands per hand is in the 2-4 land range. How many of the individual hands were keepable based on number of lands then?

Paper Shuffling

With the paper deck, the distribution was about what you expect. The 6 and 7 land hands are a little bit high, but variance is part of data collection and they still accounted for less than 3% of draws. Based on our criteria, 26 of 104 hands were unkeepable.

# Lands# Starting Hands
71
62
56
419
334
225
112
05

MTG Arena Traditional Standard Play (BO3)

For Bo3 on Arena, the distribution broke out as you see below, with almost 20% of the draws having 0, 1, 5, 6, or 7 lands in the opening hand. That compares pretty close to what we got with the paper deck. That means we are considering a mulligan 20% of the time before we look at anything other than land. Even if you say that 5 lands is okay, 15% of hands are still not meeting the land draw criteria and only one third of hands are hitting the 3 land mark, which is arguably the ideal.

# Lands# Starting Hands
70
60
55
421
333
230
113
02

MTG Arena Play (BO1)

For Bo1 on Arena Play though, there was only one hand with a land count of 0, 1, 5, 6, or 7 lands. The land count was very consistent, with a whopping 60% of hands having an ‘ideal’ 3 lands. That’s almost double what we got in paper and Bo3.

# Lands# Starting Hands
70
60
50
410
360
233
11
00

MTG Arena Standard Ranked (BO1)

Arena Ranked Play was also incredibly consistent with only one hand having 0, 1, 5, 6, or 7 lands. Meanwhile 54 hands had 3 lands. The sample size is fairly small, but from these data sets it’s pretty clear that Smoothing has a real effect on opening hand consistency.

# Lands# Starting Hands
70
60
50
412
354
238
11
00

One important detail is that WotC’s second part of the Smoothing release was that it does not apply to Mulligans. This statement does not hold up under the collected data. The variation between the number of lands in the opening draw versus the Mulligans was minor. So not only is Smoothing definitely happening in Bo1, it happens every time you Mulligan.

What does this mean?

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One of the icons in Mono Red Aggro in 2019.

What does Smoothing mean for us as players? On the surface, Smoothing may seem like something designed to make things easier for weaker players. Smoothing does reduce the very un-fun aspect of getting lands with too many or too few lands to be playable. Exploiting Smoothing adds a layer of complexity to the game, and complexity always favors the good players over the weaker ones. Good players understand how this element of the game works. Once they understand it, they use it to their advantage in ways that less experienced players won’t understand.

When it comes time to build decks, whether you will play in Bo1 or Bo3 should be considered due to Smoothing. All deck types benefit to some degree with more consistent mana in the opening hand, from aggro to control. That said, some types will benefit a bit more.

In general, decks with three colors will benefit more than mono-colored decks. Compared to Bo3, they are more likely to have playable hands with a broad enough mana base to play all their spells. A strength of playing mono is that they have very stable mana bases, so the benefit there is less. Having a 60-70% chance of getting 3 or 4 lands in your opening hand and a less than 5% chance of having to mulligan due to 1, 5, 6, or 7 lands makes tri-color decks more consistent. It also makes them more comfortable mulliganing with a sub-optimal hand.

Among archetypes, Smoothing probably most benefits aggro decks. Many aggro decks run lighter on land. Smoothing facilitates that by giving players a better chance at a playable hand when running lighter on land. The consequence is being able to fit more threats into the deck mix.

Combo decks can also benefit a great deal from hand smoothing. Many combo decks, such as the current cat decks or even Reclamation decks, lean towards having at least one part of their main combos in their opening hand. Smoothing doesn’t effect that directly, but it does enable taking a mulligan with more confidence so the player can take another shot at finding that key card.

Ramp decks have a mixed bag with Smoothing. While Ramp likes having more consistent land in the opening hand, part of their strength is being more consistent due to their ramp effects. Ramp builders may want to build with fewer ramp effects when working in Bo1, and keep in mind that opponents won’t have the consistency problems they do in Bo3. Testing with Smoothing in mind is critical to find the right mix.

One specific deck that has benefitted a lot is Mono Red Aggro, also called RDW (red deck wins). RDW is one of the strongest decks in the meta in general right now, including strong deck selection at Worlds. RDW often runs light on lands, with only 20 lands in many best of 3 builds this is the deck’s biggest weakness. Having that weakness all but eliminated takes the deck up a notch. Personally, I took this a step further in my use of RDW earlier this season. I dropped down to 18 lands and replaced the two cut lands with low casting cost spells. While this would be foolish in Bo3, the deck had a 65% win rate in Bo1. The far more consistent mana base is huge in the high rate of RDW in Bo1.

Control decks probably get the short end of the stick on Smoothing. While all decks enjoy having more stable mana, classic Control may benefit less. Smoothing isn’t bad for Control, but it makes other archetypes consistent enough that Control has fewer openings in the Bo1 format.

It shouldn’t be a surprise that the decks mentioned are what we see a lot of in Bo1. Ramp, three color decks, and especially RDW make up a huge slice of the Bo1 metagame. Control is still present, but definitely seems less strong than in Bo3. How much of this is conscious and how much is incidental is difficult to say.

Conclusion

In conclusion, Smoothing is real in Best of One. Perhaps more important, Smoothing applies to mulligans, contrary to the statements by WotC. That doesn’t mean the improbable won’t happen. Crazy hands will always happen, but it will happen less often than in Bo3. Smoothing stabilizes your mana base and gives you a better chance at a playable mulligan if you don’t love your first draw. The question for each Bo1 player is: Now that you know Smoothing is real, how are you going to take advantage of it?

The impact of Smoothing on Best of One is not something that can be covered completely in one article. This is just one in a series of articles by MTGAZone covering the unique aspects of Bo1, from deck design and metagame analysis to Hand Smoothing and mulligans. As our understanding of the effects of Smoothing expands we’ll publish more pieces on tuning each archetype to best operate in Bo1. As always MTGAZone is committed to providing readers with the best information possible to master the format.

Terence

Korean gamer in Australia. Proud owner of the DotGG and its network of websites. I especially enjoy fantasy worlds, role-playing games, and collecting things! Favorite games include World of Warcraft, Magic: The Gathering, and League of Legends.

4 Responses

  1. Chelios says:

    It tempers with low mana cost spells too. MTG arena BO1 is dumbster fire i regret spending money on the game.

  2. Sky says:

    I think you guys need too look stuff over again. I’m running a tri-color deck right now and it has 26 lands total. I’m regularly going 5+ turns without drawing any mana. My friends experience the same issue when using a multi-color deck that isn’t pre-made. Maybe have the game order cards by spell type before it “randomizes” them? About 2/3 of my games have this issue regardless of mana curve and mana count, and that’s not enjoyable. Instead of MTG Arena, maybe you should call it MTG Mono… cuz single color decks are the only ones with decent randomization.

  3. Caue Dutra says:

    The article just ignore the most important complain that broken the game – the algorithm also messes up with your draws, not only the starting hand. I’m 100% sure that there is some kind of “handicap” the game will put to screw you if you gave too many wins, or to help you if you have too many loses. Becuase its almost inpossible to get 11 manas in a row in a cycling deck that started with 2 manas. And that shit happened twice, and both tines when I had 8+ winning streaks. For SURE the game puts a handicap, either by matchmaking agains bad match decks or by screwing you if you have too many wins. All this to make the win rate of all decks flutuates towards the 50%

  4. Sean says:

    It’s not just about the mana you draw in your opening hand either. What about the spells? When you draw a hand with your 3-4 lands, thats great. then you look at the other stuff and you notice all the casting costs of those are sometimes 5+. So on an opening hand, you literally have 0 plays besides lands. Your effectively top decking right off the bat. I’ve tested this in decks with a casting cost curve of 2.0 and have had it happen around 40% of the time within 200 games. That’s not a fun experience. In addition to that, you then have the the supposedly broken draw system as well. Games where you can go 5-10 turns with 0 lands, or nothing but lands. This wouldn’t be an issue since, well, that’s part of the game. The issue comes in when it happens consecutively in multiple games and your left with little or no chance of a win. I have gone back in some games, turn by turn, and thanks to top-decking, mana screw/flood there was 0% chance of a win. Wouldn’t have mattered the order the cards were played, what targets were chosen, attacks, etc. When players see stuff like that, they are going to postulate that the system is rigged somehow.
    In all my years of playing paper magic, since Ice Age block, I have never seen such terrible “randomness”