How Long is the Road to Masters? - Spacenaut - December 3, 2018

 
Screen Shot 2018-12-01 at 1.43.38 AM.png

A few years ago, I played Hearthstone as a mostly free-to-play player and was really frustrated how I could never climb very high on the ladder.  I eventually quit HS and came to play Eternal.  Among other nice things about Eternal, I found I rose up through the ladder much more easily and eventually hit Masters without having to shell out a ton of money for a Tier 1 deck. How different are the ladder experiences of the two games?  I wrote a simulation program to find out for Eternal and used an off-the-shelf simulator to simulate HS. First, a quick overview of the different ladder systems. If you are familiar with both, scroll down until you see a graph. 

Eternal - there are 4 tiers ["bronze", "silver", "gold", "diamond"] with 3 separate sub-tiers (3, 2, 1, usually expressed in Roman Numerals, III, II, I). New players start out at Bronze III. After earning 100 points in a sub-tier, players are promoted to the next one higher.  For example, getting 100 points in Silver I puts a player in Gold III and getting 100 points in Diamond I gets a player into Masters. 

On a game win in the 3 lowest tiers, players earn 20 points on a win (maybe a few more based on MMR, but my simulations assume always 20) and lose 10 points on a loss.  Once in diamond, a win gains 15 points and a loss loses 15 points.

Once in a tier, players cannot go down a tier (e.g. at Silver III, 0 points a player can lose 1000 times in a row and not go to Bronze I), a so-called "hard floor".  The sub-tiers have "soft floors", meaning players can lose a few games at Silver II before dropping to Silver III.  There was a change a few months ago that guaranteed that number to be at least 3 losses - I have modeled it as each loss after 3 at 0 points is a 50/50 to go down a tier.

Hearthstone - there are 25 ranks (tiers). New players start at rank 25 and progress towards 1 (there is now an option to start at rank 50 to get some bonus rewards, but it is skippable). After earning 5 stars in a rank, players advance to the next rank. Winning at rank 1, 5 stars will put a player into Legend. 

Players gain 1 star on a win and lose 1 star on a loss. For tiers 25-6, there is a win streak mechanic that grants 2 stars for the 3rd and subsequent win in a row.

At ranks 25-21, players cannot lose stars. Losing a star when at 0 stars for a rank will result in going down to the next lower rank at 4 stars.  The exceptions to this are ranks 20, 15, 10, and 5, which have hard floors; this means players can lose any number of games at rank 5, 0 stars without going down to rank 6. 

Simulating that ranking behavior while varying a deck's winning percentage resulted in the following graph:

graph 1.png

I computed the range of win percentages from 48-65%. I more or less made up these numbers as reasonable win percentages a deck could likely have. A deck is unlikely to have more than a 65% match-up against a field of decks when averaged out over 100+ games. In Magic the Gathering, a somewhat similar game, they report, "historically the best deck in a late-season format settles to around 52–53% against the field."  

Here's the data in table form, with a few rows elided due to space concerns:

Screen Shot 2018-12-03 at 1.41.35 PM.png

This data backs up my experience, climbing to Masters for the first time was at least twice as fast, possibly more assuming a deck with a <55% win rate against the field.

graph 2.png

What about on returning Masters/Legend players?  Masters players in Eternal go to Gold III, 0 points at the start of a new season and Legend players in HS go to rank 4, 0 stars, so there is less of a climb.  See the above chart for this scenario.

Screen Shot 2018-12-03 at 1.41.54 PM.png

Eternal is faster up until about the 57% win percentage, which might be attainable by a very skilled player with a very familiar deck that attacks the meta in just the right way.  With a deck that has a 50/50 chance to win, Eternal is still twice as fast to climb. 

Why the difference?  The "soft floors" make a large part of the difference.  Let's compare the Diamond tier of Eternal and Rank 5 through Legend of HS.  For HS, you simply need to get 25 more wins than losses, which will take a while.  Look up binomial distribution for a way to exactly calculate the odds of getting 38+ wins out of 50 games played; it's low. In contrast, the soft floors in Eternal turn this into 3 sets of "7 more wins than losses", which are more likely to happen, even with less performant decks.  When sitting on 0 points, a loss usually doesn't set you back, and so you get to "save" your progress - requiring fewer wins overall to rank up.

To test this explanation, I removed the soft floors from Eternal and the numbers became more similar.  Eternal still had an edge, especially at lower deck win percentages, but that's due to the second cause - Eternal's ladder is shorter. A 100% win rate in Eternal would take 66 games to go from the bottom to the top, where as in HS, it takes 75 wins (including the discount from win streaks).  The size of the ladder from the last "hard floor" is smaller too - Diamond III to Masters takes 21 straight wins, Rank 5 to Legend takes 25 straight wins. The two effects of the soft floor and the shorter ladder combine for a quicker rank-up time of Eternal over HS.

Final thoughts:

It's nice to see some numbers that back up how my ranked experience has been on these two games.  I'd be curious to see if they match up with others experiences. Notice a bug in my code? Are my assumptions misaligned? I look forward to any discussions on the Reddit Thread. 

If anyone has data on MTG: Arena's rank up/down system, I'd be curious to try to simulate that as well - to me, a non MTGA player, the ranked system there seems a bit opaque.