The Elder Scrolls

A statistical simulation about reachable rank depending on win rate and time needed to reach legend.

TheElderScrolls10 - A statistical simulation about reachable rank depending on win rate and time needed to reach legend.

In Elder Scrolls: Legends the ranking system does not allow deranking. It is therefore easy to see that with a win rate of 50% everybody should be able to reach Legend rank at some point, because you would statistically earn more points than lose (because if you lose in snake rank you are not losing a star).

Obviously the conclusion to draw is, that with infinite time even a person with a tiny win rate would eventually reach legend rank. Even though mathematically correct, unfortunately in a season there is not infinite time at our disposal. I was then interested in finding out, what is the minimal win rate necessary to actually reach legend rank in a real-world setting with time constraints and then further how many days of playing would be needed at a certain win rate to actually get to legend rank. I will start first with the results and then for those interested a little extra in the end with details about the simulation parameters.

First the results of the achievable rank depending on the winrate (see my definition of achievable in the detailled section):

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Achievable rank depending on winrate

As can be seen, a very low win rate of e.g. 20% would only achieve statistically rank 9 at most, far from legend rank. At about 30% win rate the player starts being able to reach rank 5 and then finally with a winrate of 37% legend rank (rank 0) is achievable during a season.

Here an other example with a higher bonus round probability, showing some more steps but similar outcome:

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Achievable rank with a 20% bonus round probability

The next interesting question is then, how many matches would a player with a certain win rate need to play in order to reach legend rank.

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Number of matches to be played to reach legend rank dependent on win rate

In order to judge these results, we need to convert this into "how long would it take for somebody with a certain winrate to actually reach legend rank". I chose four different player types who play 10, 5, 2 or half an hour a day and playing on average 5 matches per hour.

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y-axis shows days to be played

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https://i.redd.it/d6f3b4021zi21.png

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https://i.redd.it/xqxhlr451zi21.png

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https://i.redd.it/rr26vos51zi21.png

It surprised me, that if you start from rank 12 even players playing a lot but having a win rate worse than 50% need half the month to actually reach legend rank. Not to mention that the casual player, even with an incredible 65% win rate would not be able to reach legend rank in the first month.

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Obviously, with the mechanic of milestone ranks, a low win rate player would start out in rank 9 or 5 in the next month and continue their journey from there to grind themselves further up. Hence for the total amount of days displayed in those graphs are actually a lower bound (no deranking due to season end) and the casual player playing 30 mins a day at a win rate of 50% might need much more than 200 days to actually grind their way up to legend rank.

Now some added bonus for those interested in the details of the simulation. The simulation count was always 10000 tries on reaching a rank. Hence 10000 players playing with a given win rate and then looking where they land. The amount of games that are played are variable (with the maximum game count exception as will be explained in the next paragraph) while the target rank is fixed. The bonus round probability was set to 12% (I couldn't find any statistic and just guessed from my own experience). There are two snake ranks and there is no star loss in the higher ranks 11 and 12. Bonus round probability is 0 when rank 5 or lower is reached.

First the definition of which rank I deemed theoretically achievable depending on win rate. For this I first defined a maximum of matches possible in a month by guessing that somebody playing 30 days for 24 hours each day and being able to play a whooping 6 matches per hour would reach this semi-theoretical maximum. This leads to a superhuman maximum match count per month at 4320 matches.

I then set up a simulation to play to a target rank, starting with the highest rank 11 going all the way to legend. If the simulation reached a rank after x matches, this was saved and a second simulation was run reaching the target rank after y matches etc. until reaching 10000. Then taking the median average of the number of matches necessary to reach a rank et voilà you know how many matches need to be played at a certain winrate to reach a target rank.

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Now at last the maximum match count comes into play. As said in the introduction, if you had infinite time at hand, everybody would reach legend at some point. Setting the maximum matches played to 4320 lead to a conditional loop, that excluded this datapoint from the calculation and further set a counter to tell how often the rank had not been reached. If this happened more than every third time of the 10000 runs, I deemed the rank as not achievable.

Of course there are many tweaks and optimizations one could come up with, like a variable win rate depending on rank (high win rate in easy ranks, lower win rate in harder ranks) or in general a gaussian win rate distribution to make the simulation more real. However, I hope this answers the general question of whether a monkey playing Elder Scrolls Legends would eventually reach Legend rank and if yes, how long they would need ;).

I hope you enjoyed this little journey. Cheers.

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