On December 13, a new ranking system was added to MTG Arena, and Preseason 1 ran until the end of January. For Preseason 2, which spans the month of February, several progression changes were made. At the moment, the ranking system works as follows:
- There are six ranks: Bronze, Silver, Gold, Platinum, Diamond, and Mythic. Each non-Mythic rank has four tiers, with tier 1 being the best and tier 4 being the worst within that rank. For Constructed, each tier consists of 6 steps. For Limited, Bronze consists of 4 steps and all other ranks consist of 5 steps.
- Ranks operate separately for Constructed and Limited. To rank up in Constructed, you’ll want to play in the “Ranked” best-of-one games or in the “Ranked” best-of-three games. Both contribute towards the same rank. To rank up in Limited, you’ll want to enter the best-of-one Draft events.
- Everyone initially starts at Bronze tier 4. You gain steps with wins: in best-of-one, it’s 2 steps gained per win in Bronze and Silver and 1 step gained per win in Gold, Platinum, and Diamond. You lose steps with a defeat: in best-of-one, it’s 0 steps lost per win in Bronze and 1 step lost per win in all other ranks.
- In best-of-three, these gains and losses are doubled, and only the match result counts. So if you win a match 2-1 in Gold, you move up 2 steps. Personally, I like the choice to count the match result only. The alternative of counting game results, under which you would move 1 step with 2-1 and 2 steps with 2-0, would unduly punish decks that are poor against the field in game 1 but that have excellent sideboards. This alternative system might therefore yield a different metagame than in a tabletop best-of-three event. Since the current implementation only considers the match result, it avoids this issue.
- Once you get enough steps, you’ll move into the next tier at that rank, or if you’re at tier 1, into the next rank.
- Once you ascend into a new rank, you’re safe for the rest of the season. No matter how many times you lose, you can’t fall from a rank (e.g., from Gold 4 to Silver 1).
- Although losses can push you down a tier within a rank (e.g., from Gold 3 to Gold 4), there is some protection when you move into a tier. Although to my knowledge the exact workings have never been officially announced, I discovered empirically that in best-of-one you cannot fall a tier within a rank within 3 games of reaching a new tier. So if you advance from Gold 4, step 5 to Gold 3, step 0 and then go loss-loss-loss or win-loss-loss, you’ll still be Gold 3, step 0 because of that 3-game protection. But if you lose again in your next game from that spot, you will drop to Gold 4, step 5. If you win immediately after that drop, then you’re back to Gold 3, step 0, with a 3-game protection.
- In best-of-three, I discovered empirically that tier protection applies to only one match.
A natural question is how many games you should expect to play to advance to the next rank, or all the way to Mythic. This article is dedicated to that question.
How to Calculate the Expected Number of Games Required to Hit Mythic?
Let’s start by considering an illustrative example of a fictive ranking system with one tier and 3 steps. If your win rate is 60% and you change one step with any win or defeat, then you can represent it as the following Markov chain:
For any state X, let E[X] describe the expected number of games required to reach the state “MYTHIC” when we’re currently in state X.
For the state “START,” 40% of the time we play one game and are then back in the same state, and 60% of the game we play one game and advance to state “1 win.” Hence, E[START] = 40% * (1 + E[START]) + 60% * (1 + E[1 win]). We don’t know yet what the values of E[START] or E[1 win] are, but the equation is valid.
For the state “1 win,” a defeat pushes us back to state “START,” whereas a win advances us further. Hence, E[1 win] = 40% * (1 + E[START]) + 60% * (1 + E[2 wins]). Finally, E[2 win] = 40% * (1 + E[1 win]) + 60% * 1.
This represents a system of equations. With three unknowns and three equations, we can use standard linear algebra techniques to solve it. The problem is small enough to do it by hand, which yields the following results:
- E[1 win]=6.30.
- E[2 win]=3.52.
For the MTG Arena ranking system, we can apply the same approach. The only difference is that the system of equations is larger and that the tier protection system requires a bit of extra bookkeeping. My source code, written in Python, is available here.
A Brief Review of Related Works
For Preseason 1, several analyses of the MTG Arena rank system appeared. Both Reddit user Ramora_ and Lucas Buccafusca determined the expected number of games to rank up as a function of your win rate, along with several other insights. Whereas Ramora_’s method was equivalent to mine, Buccafusca’s was based on Monte Carlo simulation. Monte Carlo simulation yields an accurate, fast approximation for high win rates, and it’s easy to implement. But it’s inaccurate and slow for low win rates where ranking up would be a rare event.
In Hearthstone, there is an online tool at Primedope that uses simulation to estimate the number of games to reach the legendary rank. An article by Jan van der Vegt shows how to calculate these numbers exactly with a Markov chain model and the linear algebra method that I explained to him back then.
The main contribution of the present article is to provide updated numbers for Preseason 2. Due to changes to the progression system and the introduction of best-of-three for the Constructed rank, the results of Ramora_ and Buccafusca no longer apply. Moreover, I introduced tier protection in my analysis to ensure that the model more accurately captures the real MTGA rank system. The impact of this is surprisingly large.
Expected Number of Games to Reach the Next Rank in Limited
In Limited, the number of steps per tier is the same for Gold, Platinum, and Diamond. Therefore, the expected number of games to reach the next rank as a function of your game win rate is the same for these three ranks.
|Game win rate||Games from Bronze to Silver||Games from Silver to Gold||Games from Gold to Platinum|
Let’s consider an above average player who has 70% to win a game in Bronze, 64% in Silver, 60% in Gold, 56% in Platinum, and 52% in Diamond. According to the table, in expectation this player would need 11.4 games to go from Bronze 4 to Silver 4, then 20.3 games from Silver 4 to Gold 4, then 73.4 games from Gold 4 to Platinum 4, then 99.7 games from Platinum 4 to Diamond 4, and then 153.0 games from Diamond 4 to Mythic. Since you can never fall in rank, this makes for a grand total of 358 games to go from Bronze to Mythic.
At 8 games per hour, which approximates the pace that I reach, 358 games requires 45 hours of play. Drafts take a bit of time as well, but it seems fair to say that an experienced drafter will usually reach Mythic if they play for two hours per day on average. (It’s not guaranteed since these are expected values and the variance is large, but it’s a decent indication for month-long seasons.)
Players with very low win rates, say 40%, may still rank up if they hit a long enough winning streak. Reaching Gold in particular is feasible, as you’re gaining two steps per win in Bronze and Silver. Going from Bronze to Gold would take a 40% win rate player 78.9 games in expectation (i.e., about 10 hours). But such a player would need 16,911 games in expectation to go from Bronze to Mythic. Since this requires 71 hours of play per day, that’s clearly not feasible.
The impact of tier protection is substantial. My understanding of tier protection, as incorporated in my calculations, is that you cannot fall a tier within a rank (e.g., from Gold 3 to Gold 4) within 3 games of reaching a new tier. If tier protection was not in place, then a player with a 60% win rate would require 90 games in expectation to go from Gold to Platinum (compared to 73.4 with tier protection) and a player with a 50% win rate would require 420 games in expectation to go from Gold to Platinum (compared to 205.0 with tier protection). Especially for lower win rates, this little bit of protection along the way represents a huge difference.
Expected Number of Games to Reach the Next Rank in Constructed
For Constructed, you can rank up either in best-of-one or in best-of-three. Both contribute to your overall Constructed Rank. For best-of-one, I used the exact approach to determine the expected number of games to rank up. For best-of-three, I could use the same approach to determine the expected number of matches to rank up, but I wanted to figure out the number of games to enable comparison with best-of-one.
For game win rates other than 50%, the expected number of games per match is different for matches won and matches lost, which adds an extra layer of complexity to the problem. It’s more tractable to account for this in a simulation. The resulting numbers for 45% win rates in best-of-three may be slightly inaccurate, and it was too time-consuming to simulate them for even lower win rates. But the expected numbers for larger win rates in best-of-three are unlikely to be off by more than one game.
|Game win rate||Games from Bronze to Silver in best-of-one||Games from Bronze to Silver in best-of-three|
Moving from Bronze to Silver is easy. After all, you only need three wins to advance to the next tier in best-of-one, and you don’t lose any steps with a loss. The difference between best-of-one and best-of-three is small, although you’re slightly better off with best-of-one if your win rate is low.
|Game win rate||Games from Silver to Gold in best-of-one||Games from Silver to Gold in best-of-three|
Moving from Silver to Gold is also relatively easy. Basically, everyone who plays regularly can achieve Gold every season. Even if you have a 45% win rate, which is rather low, you can make it from Bronze to Gold in 79.1 games in expectation (i.e., in about 10 hours).
You can earn packs and gold at the end of each season based on which rank you achieve. The reward for Bronze is 1 pack (although you do need to play at least one game to get it) and the reward for Gold is 2 packs, 1000 gold. If your win rate is lower than 52%, then your expected winnings per game (under my valuation for ICRs) are larger if you play ladder until you rank up to Gold than if you merely grind the best-of-1 Constructed Events. For win rates of 52% or more, you’re better off playing Constructed Events if you just want to build up your collection.
|Game win rate||Games from Gold to Platinum in best-of-one||Games from Gold to Platinum in best-of-three|
The rest of the climb is more tedious, and the rewards for reaching Platinum or higher ranks are laughable. In Constructed, the expected number of games to reach the next rank as a function of your game win rate is the same for Gold, Platinum, and Diamond. In any case, you’re better off playing best-of-three than best-of-one.
Consider again our above average player who has 70% to win a game in Bronze, 64% in Silver, 60% in Gold, 56% in Platinum, and 52% in Diamond. According to the tables, in expectation this player would need 17.1 games to go from Bronze 4 to Silver 4 and 24.3 games from Silver 4 to Gold 4 while playing best-of-one. Then, switching to best-of-three, they require 78.1 games from Gold 4 to Platinum 4, then 107.7 games from Platinum 4 to Diamond 4, and then 173.8 games from Diamond 4 to Mythic. Since you can never fall in rank, this makes for a grand total of 401 games to go from Bronze to Mythic. At 8 games per hour, this requires 50 hours of play. (As a remark, the expected number of games would be 521 if there was no tier protection.)
We can compare this to a seasoned pro who might have 80% to win a game in Bronze, 75% in Silver, 70% in Gold, 66% in Platinum, and 62% in Diamond. This player would require 203 games in expectation to go from Bronze to Mythic, or about 26 hours of play. Dedicated players can therefore spend most of their time battling it out in Mythic rank.
Limitations of the Study
Quantitative models never capture all aspects of reality:
- I compared best-of-one and best-of-three under the assumption that your game win rate is constant in all modes. So for a game win probability P, your match win probability is P^2 + 2(1-P)P^2. Yet if the best players move to best-of-three, then best-of-one may be easier. On the other hand, there is skill in sideboarding that the formula doesn’t account for.
- I compared best-of-one and best-of-three under the assumption that average game length is the same in both modes. If best-of-one is filled with fast Mono-Red decks, then it may be faster to rank up in that mode. Likewise, if you play fast Mono-Red decks yourself, you may be able to jam more games per hour.
- Given the complexities of tier protection and best-of-three, there is always a chance that an error slipped into my calculations. If you spot one, please let me know. But I have confidence that my implementation is correct because I separately coded an exact calculation and a simulation, and the results matched closely. For win rates between 50% and 60%, they differed by no more than one game.
- The numbers I provided are expected values. Please keep in mind that the actual number of games that anyone needs can vary wildly. There is a lot of variance.
- I don’t know how long it will take to reach the Top 8 of the Mythic rank. For reference, according to Chris Clay, the top player in the Mythic Rankings has the highest MMR in the game, and they’re using a Glicko system for MMR.
In expectation, an above average player (with a 56% win rate in Platinum, and 52% in Diamond) requires approximately 45 hours of play to go from Bronze to Mythic in Limited and approximately 50 hours of play to go from Bronze to Mythic in Constructed, at a rate of 8 games per hour. These numbers take into account the huge impact of tier protection, which can substantially reduce the expected number of games required to reach the next rank.
Before the unveiling of the Preseason 2 changes, there were fears that ranking up in best-of-three would take considerably more time than ranking up in best-of-one. But in the current implementation, my results show that you get to Mythic faster if you jam best-of-three matches from Gold onward.