# What’s the Best MTG Arena Event for Expected Value, and Can You Go Infinite?

The esports announcement alluded to a lot of changes, including doubling the prize money for the tournament series formerly known as the Pro Tour. Since this will hopefully cover my flights, it has become more appealing to compete in them. So whereas I was originally planning to skip all the 2019 editions, I have now committed to attending them all. I follow the expected value, and the same type of thinking also applies to MTG Arena.

Once Ravnica Allegiance drops, I intend to play a lot of MTG Arena, if only to prepare for these paper tournaments. I have a bunch of grievances with the program that haven’t been adequately solved in over a year (such as the logic of the auto-tapper and the inability to set default stops—no, I never want to cast anything in my own beginning of combat step) but it also does a lot of things right, and I’ve been having fun playing games on MTG Arena, hitting daily quests, and growing my collection. MTG Arena also offers nice opportunities for mathematical analysis, so you can expect several Arena Math articles from me in 2019.

Today, I will analyze how the various types of events on MTG Arena stack up in terms of value. More specifically, I set out to determine the expected value, which describes a long-term average, of the various events’ prizes. The expected value (EV) is calculated by multiplying the prizes for each of the possible outcomes by the likelihood each outcome will occur, and then summing all of those values. Once I had those values, I also could determine whether it would be possible to “go infinite” in certain events, which is another way of saying that the expected prizes of some event are sufficient to pay for its entry fee.

I’m not the first person to analyze the value of MTG Arena events. On Reddit, for example, I found an EV calculator spreadsheet by CombatAnimal and an event reward analysis by AnnanFay. But I wanted to go a bit deeper on some topics and share my own take on the subject in article form. So let’s dive in.

# Probability of Reaching a Certain Record in “Stop at 3 Losses” Events

MTG Arena offers two types of events. In the first type of event, typically used for best-of-1 matches, you keep playing until you reach 3 losses or 7 wins, whichever comes first.

Suppose your game win rate is W. Then the probability of ending the event at N wins, for N smaller than 7, is given by:

$W^N (1-W)^2 \binom{N+2}{2} \cdot (1-W).$

This represents the binomial probability of getting N wins and 2 losses in N+2 games, multiplied by the probability of losing the final game. After all, every run with fewer than 7 wins must end in a loss.

The probability of stopping at 7 wins is given by:

$W^7 + 7 W^6 (1-W) \cdot W + 28 W^6 (1-W)^2 \cdot W.$

This represents the sum of three probabilities. The first is the probability of going 7-0. The second is the probability of going 6-1, multiplied by the probability of winning the next game. The third is the probability of going 6-2, multiplied by the probability of winning the next game.

With these formulas, we can obtain the following table.

 Win rate P(0 wins) P(1 win) P(2 win) P(3 win) P(4 win) P(5 win) P(6 win) P(7 win) 40% 21.6% 25.9% 20.7% 13.8% 8.3% 4.6% 2.5% 2.5% 45% 16.6% 22.5% 20.2% 15.2% 10.2% 6.4% 3.9% 5.0% 50% 12.5% 18.8% 18.8% 15.6% 11.7% 8.2% 5.5% 9.0% 55% 9.1% 15.0% 16.5% 15.2% 12.5% 9.6% 7.1% 15.0% 60% 6.4% 11.5% 13.8% 13.8% 12.4% 10.5% 8.4% 23.2% 65% 4.3% 8.4% 10.9% 11.8% 11.5% 10.4% 9.1% 33.7% 70% 2.7% 5.7% 7.9% 9.3% 9.7% 9.5% 8.9% 46.3% 75% 1.6% 3.5% 5.3% 6.6% 7.4% 7.8% 7.8% 60.1%

So if you are 50% to win an arbitrary game, then you are P(7 win) = 9.0% to take the grand prize for 7 wins. Note that P(7 win) is generally larger than P(6 win) because P(7 win) encompasses all 7-X records, whereas P(6 win) encompasses only the 6-3 record.

Another thing to note is that with a 50% win rate, you’re more likely to finish with 1 win or 2 wins than with 3 wins. This may be counterintuitive because a 3-3 record would be the most likely outcome in a 6-round Swiss event. But in the Arena events, you need to earn your three wins before your three losses. So you have to be 3-2 first before you can finish 3-3. If you start 2-3, you never get a chance to claw back to 3-3. This is why the average player is most likely to finish with 1 win or 2 wins in these “stop at 3 losses” events.

# Probability of Reaching a Certain Record in “Stop at 2 Losses” Events

In the second type of event offered on MTG Arena, which is typically used for best-of-3 matches, you keep playing until you reach 2 losses or 5 wins, whichever comes first.

If we let W denote your match win rate, then the probability of ending the event at N wins is W^N * (1-W)^2 * (N+1) for N smaller than 5 and W^5 + 5 * W^5 * (1-W) for N equal to 5.

 Win rate P(0 wins) P(1 win) P(2 win) P(3 win) P(4 win) P(5 win) 40% 36.0% 28.8% 17.3% 9.2% 4.6% 4.1% 45% 30.3% 27.2% 18.4% 11.0% 6.2% 6.9% 50% 25.0% 25.0% 18.8% 12.5% 7.8% 10.9% 55% 20.3% 22.3% 18.4% 13.5% 9.3% 16.4% 60% 16.0% 19.2% 17.3% 13.8% 10.4% 23.3% 65% 12.3% 15.9% 15.5% 13.5% 10.9% 31.9% 70% 9.0% 12.6% 13.2% 12.3% 10.8% 42.0% 75% 6.3% 9.4% 10.5% 10.5% 9.9% 53.4%

# Game Win Rate vs. Match Win Rate

Note that the two tables cannot be directly compared. Since most “stop at 3 losses” events are best-of-1, the win rate in the first table represents the game win rate. But since most “stop at 2 losses” events are best-of-3, the win rate in the second table represents the match win rate.

If all games were identical and independent, then a 60.0% match win rate would correspond to a 56.7% game win rate and a 75.0% match win rate would correspond to a 67.4% game win rate. These numbers are easy to derive by assuming that a game win probability G results in a match win probability of G^2 + 2 * G^2 * (1-G).

In reality, games aren’t identical and independent: There is skill in sideboarding, there are play/draw effects, and there’s the opening hand algorithm that grants better opening hands in best-of-1. Also, Swiss-style matchmaking can pull your results more toward the middle. Yet estimating the impact of all of these effects, some of which pull in different directions, is very difficult and can vary by format and by player. I purposefully disregard them in this article and will use the above-mentioned formula for the match win probability instead.

# What Match Win Rate Might the Top Pros Achieve?

Let’s consider best-of-3 matches that did not result in a draw and that were played at GPs and PTs, the two biggest types of competitive events. For these matches, the average competitor at the 2017 World Championship, based on the event’s media guide, had a lifetime match win rate of 62.8% (or more specifically, 64.6% at GPs and 59.3% at PTs). This is in line with the numbers gathered by the MTG Elo Project, where the average lifetime match win rate of the 30 highest-rated players at the time of writing is 62.1% (or more specifically, 64.0% at GPs and 57.7% at PTs).

So how does that translate to MTG Arena? Well, here some guesswork comes in. As my first guess, the average player at mythic or diamond level may be close to the skill level of an average Pro Tour competitor, and the average player at platinum level might correspond to an experienced Grand Prix competitor. Based on these estimates, the very best players in the game (say, the 32 players in the Magic Pro League) might be able to win 59% of their best-of-3 matches against a diamond ranked player and 64% of their best-of-3 matches against a platinum ranked player.

Yet there is only one type of event (best-of-1 Draft) that does matchmaking based on rank. For most events, players are paired based on win/loss record only, which mimics a Swiss-style event. Given this, I would estimate that a Magic Pro League member might be able to win 75%-80% of their best-of-3 matches against random opponents. This estimate is based on my own experience, but it matched the estimates of several actual Magic Pro League members when I asked them. I expect that a match win rate larger than 80% is out of reach even for the very best players in the game, at least on a long-term basis.

# What’s the Value of Packs and Cards?

A pack from the store is worth 1,000 gold or 200 gems. I use the same valuation for packs won as prizes in events. But a large amount of value from such a pack is associated with segments in the wild card track and the chance of opening a rare/mythic wild card (instead of the random rare/mythic) in that pack. Since Draft or sealed packs never contain wild cards and don’t advance the wild card track, their value is far less than 1,000 gold.

Although the “going infinite” analysis won’t require an estimate of the value of a Draft or sealed pack, such estimates are useful for comparing the overall worth of the various events. For my estimates, I use the following assumptions:

• A gem is worth 5 gold. This is based on the gem/gold ratio for buying packs in the store, and it’s close to the 5.26 gold/gem exchange rate seen in the entry fees of Constructed events.
• I value rares and mythics equally. If you build stock lists of Boros Weenie and Golgari Midrange, two of the most-played decks in Standard, then you need rares and mythics at a 1:4 ratio, which is exactly the ratio that the wild card track gives out. Since the majority of rare/mythic wild cards come from the track, the value of a rare wild card is at least very close to the value of a mythic wild card. I set them equal to each other for analytical tractability.
• I value a rare/mythic wild card at 6 times the value of a random rare/mythic, which could be owned or unowned. This is an impactful assumption, as the value of a wild card will differ from player to player. If you are just starting your collection and are interested in building various competitive decks, then about 33% of rares/mythics should be useful to you. But if you are looking to complete a particular deck or if your collection has grown large, then perhaps only 5% of rares/mythics will be relevant. My assumption strikes a middle ground, effectively saying that 17% of rares will be useful.
• Since I’m already swimming in commons and uncommons, I value common and uncommon wild cards at 0 gold, and I value random commons and uncommons according to the progress points they give for the vault. Since the vault contains 3 rare/mythic wild cards, these progress points imply that the commons and uncommons in an 8-card pack are worth 1.2% of the value of 3 rare/mythic wild cards and that the commons and uncommons in a 15-card pack are worth 2.2% of the value of 3 rare/mythic wild cards. This also seems like a reasonable valuation in case you didn’t already own playsets of every common and uncommon.
• Taking into account the published wild card track value per pack and the wild card upgrade chance per pack, a pack won in a Limited event grants 0.958 random rare/mythics and 0.208 rare/mythic wild cards in the long run. These numbers are based on the published 1:24 rate that represents the “average number of packs needed to open to acquire 1 wild card of the respective rarity.” Although statistics from MTGA Pro Tracker suggest that the actual drop rate is slightly more favorable, I didn’t feel confident using anything other than the officially publicized rates, even though their definition is muddled and ambiguous.

Taking the above assumptions for granted, we can equate the contents of a pack, including its corresponding vault and wild card progress and chances, to 1,000 gold. Subsequently, we solve for the value of a random rare/mythic. The result is that under the strong assumption that a rare or mythic wild card is worth 6 times the value of a random rare or mythic, a random rare or mythic is worth 412 gold and a rare or mythic wild card is worth 2,472 gold. Using these numbers, the value of a sealed pack is 577 gold.

A Draft pack may be worth slightly more or less, depending on your willingness to rare Draft, but I’ll use the same value for simplicity. Personally, I tend to rare Draft all Standard-playable rares instantly, but this also slightly hurts my chances in the Draft.

I stress again that this value of 577 gold is heavily dependent on the strong assumption that a rare/mythic wild card is worth 6 times the value of a random rare/mythic. The actual value should depend on your personal goals. If you value a wild card at 3 times the value of a random card (which could happen if you are just starting your collection and are interested in building various competitive decks), then a sealed pack would be worth 710 gold. If you value a wild card at 20 times the value of a random card (which could happen if you are looking to complete a particular deck or if your collection has grown large), then a sealed pack would be worth 398 gold, interestingly mostly captured in vault progress because wild cards are worth a fortune at that point.

The next type of reward whose value we need to estimate are the individual card rewards offered by Constructed events. They are either uncommon or rare Standard-legal cards. A rare card reward, as I already established under my assumptions, is worth 412 gold. An actual uncommon is worth 25 gold in vault progress, but the main value of an uncommon card reward lies in its upgrade chance: It may upgrade to a rare/mythic 15% of the time. Taking this into account, an uncommon card reward from an event is worth 83 gold on average.

# Best-of-1 Draft Events

Now let’s break down the events one-by-one, starting with best-of-1 Drafts. After translating the probability of getting a bonus pack into the corresponding expected value, we have the following payouts.

Focusing purely on gem prizes, we can combine this payout chart with the record distribution to find the expected gem payouts as a function of your game win rate.

Going infinite while doing best-of-1 Drafts is close to impossible, as you need a 74.7% game win rate to earn back the gems you spend in the long run. This game win rate corresponds to an 84.0% match win rate, which I believe is out of reach even for the very best players in the game. To make matters worse, a rank system was implemented for best-of-1 Drafts (though not for other Limited events) which attempts to pair, for example, diamond level Limited players with other diamond level Limited players. This means that the very best players will usually be paired against each other. Hence, I don’t believe anyone can chain infinite best-of-1 Drafts with just the gems earned in the event itself.

With a 50% game win rate, the expected prizes are 347 gems and 1.33 packs, plus the value of the cards you drafted. If we translate the prize packs into their corresponding gem value at the store exchange rate and use my valuation of the value of the three Draft packs, then the total value of all your spoils is 959 gems. That’s more than the 750 gem entry fee, but less than the equivalent value of the 5,000 gold entry fee (i.e., 1,000 gems). So you’re effectively making a profit if you’re paying with gems, but not if you’re paying with gold.

The gem/gold entry fee rate for best-of-1 Draft is the worst among all events. Given that 750 gems would be worth 3,750 gold in packs at the store, you might wonder why anyone would ever pay gold to enter this event. The reason is that it’s easy to get gold relative to gems. You can get up to 1,500 free gold every day just for completing quests, but the only way to get gems is by purchasing them in the store with real money or earning them as prizes in Draft events. And if you want to convert gold into gems, which is particularly interesting if you’re free-to-play, then the only way to do so is by entering these best-of-1 Drafts at a bad exchange rate.

Speaking of gems: If you would like to enter best-of-3 Drafts, then you need to pay with gems. There is no option to pay for entry with gold.

We can again find the expected gem value, but this time I am presenting it as a function of your match win rate. This means that you can’t directly compare it with the best-of-1 Draft chart, but the use of match win rate as a variable will be more intuitive to players who play lots of traditional drafts.

Going infinite while doing traditional Drafts is difficult, but it may be within reach for the top pros in the game. You need a 73.8% match win rate to earn back the gems you spend in the long run. In line with my discussion on what win rate the top pros might achieve, I believe that the very best drafters will be capable of this, but then we are talking good Pro Tour caliber players. Note that this is assuming that the average opponent is a random MTG Arena player, whereas traditional Drafts represent poor value for beginning or average players. If the average opponent is as good as an experienced Grand Prix competitor, then even members of the Magic Pro League may not be able to go infinite in traditional Drafts, but I don’t believe this is the case.

With a 50% match win rate, the expected prizes are 708 gems and 2.86 packs. If we translate the prize packs into their corresponding gem value at the store exchange rate and use my valuation of the value of the three Draft packs, then the total value of all your spoils is 1,626 gems. Subtracting the entry fee of 1,500 gems, you’re running a net profit of 126 gems. That’s smaller than the profit gained when paying gems to enter best-of-1 Drafts. This means that the average player is better off sticking to best-of-1 Drafts. If you’re curious about the net profit for your own win rate, then check out the chart at the end of this article.

# Sealed Events

While Sealed deck is a good way to build up your collection, no one is going infinite while playing Sealed. You need a 81.0% game win rate to earn back the gems you spend in the long run, which corresponds to an absurd 90.5% match win rate. No one is that good. The number is literally off the chart.

With a 50% game win rate, the expected prizes are 1,002 gems and 3 packs. If we translate the prize packs into their corresponding gem value at the store exchange rate and use my valuation of the value of the six sealed packs, then the total value of all your spoils is 2,295 gems. That’s a net profit of 295 gems, which is higher than either Draft event.

There are two downsides to playing Sealed on MTG Arena, however. First, the Sealed deck builder interface lacks the easy overview and user-friendly sorting tools that Magic Online offers. Second, Sealed is best-of-1, which means that you cannot choose to play or draw. For a format where drawing first has often been correct, it’s disappointing that you can’t make the choice.

# Best-of-1 Constructed Events

Going infinite while doing best-of-1 Standard is achievable for experienced players. You need a 56.7% game win rate, which corresponds to a 60.0% match win rate, to earn back your 500 gold in expectation. With this win rate, you’d be gaining 2.27 uncommon card rewards and 0.73 rare card rewards per event on average.

With a 50% match win rate, the expected prizes are 410 gold, 2.51 uncommon card rewards, and 0.49 rare card rewards. If you deduct the 500 gold entry fee and use my valuation for the card rewards, then you’re running a net profit of 320 gold (which is equivalent to 64 gems) per event.

Going infinite while doing traditional Standard is easier than in best-of-1 Standard: You need a 55.1% match win rate to earn back the 1,000 gold in expectation. With this win rate, you’d be gaining 1.64 uncommon card rewards and 1.36 rare card rewards per event on average.

With a 50% match win rate, the expected prizes are 863 gold, 1.83 uncommon card rewards, and 1.17 rare card rewards. If you deduct the 1,000 gold entry fee and use my valuation for the card rewards, then you’re running a net profit of 497 gold (which is equivalent to 100 gems) per event.

# Which Event is the Best Value?

Let’s put it all together, using the valuations of sealed packs and card rewards that I derived under my assumptions. This yields the following EV overview as a function of the game win rate, along with recommendations for various player types.

Note that the BO1 Draft occurs twice—once per entry option. Whereas the gold/gem ratio for most event’s entry fee is close to the 5:1 ratio seen for purchasing packs in the store, this is not the case for BO1 Draft. To clarify this difference, I included the event twice. The gem entry line is equal to the gold entry line, lifted by 250 gems.

If you increase the value of rare/mythic wild cards relative to random rares/mythics, then the numbers shift around, but the overall preferences and curve shapes stay roughly the same, with one main exception: The value of Sealed drops. Once you value a rare/mythic wild card at 15 random rares/mythics or more, then the 50% win-rate player is better off playing BO1 Draft instead of Sealed. At that point, the difference in expected rewards between Limited events and Constructed events becomes smaller as well.

Another thing worth noting is that the above chart doesn’t take into account the time required to finish events, which is not completely fair—for a 50% win rate player, a BO1 event lasts 5.7 games and a BO3 event 8.4 games on average. If you would study the gem equivalent value per game, then the overall picture would remain roughly the same, although small differences would emerge. Most notably, the Traditional Standard curve would intersect the BO1 Standard curve at a higher win rate. This is why I wrote “BO1 Standard is also fine” at the top of the chart for average players. Likewise, the Sealed curve would intersect the Traditional Draft curve at a higher win rate.

And in the end, an essential factor is simply which format you enjoy the most. Netting an expectation of 200 extra gems (which are worth about one dollar) per hour is nice and all, but playing the formats you like the best is more important.

As a player with a high lifetime win rate myself, I plan to build up my Ravnica Allegiance collection by chaining a bunch of Traditional Drafts. I might be able to go infinite, but it won’t be easy. Then, once I’ve built several Standard decks using the Ravnica Allegiance guilds, I intend to enter Traditional Constructed events. Fortunately, these are exactly the two formats of Mythic Championship Cleveland. The rank rewards currently give me no incentive to make a push for mythic on the ladder, but it should still be fun to get there, and there are plans to expand these rank rewards in the future.

Speaking of the future, this entire analysis is predicated on the current state of the economy, which will likely change. The biggest problem at the moment is that you can receive cards that you already have a complete playset of. Although fifth commons and uncommons contribute surprisingly significantly to vault progress, fifth rares and mythics have very low value. As Game Director Chris Clay wrote in a post on the Beta forums: “We are very aware that it is The Issue™ with MTG Arena right now.” Their leading contender is a smarter system that provides duplicate protection.

Depending on the new system, the EV of the various MTG Arena events could change. But since the new system is scheduled for “sometime in Q1 of 2019,” I would be surprised if this happens before the release of Ravnica Allegiance. So I hope that my analysis will remain useful for at least a while. In any case, I’m already looking forward to evaluating the new system, and I’m hoping that MTG Arena will reach new heights in 2019.

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