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Comprehensive KoT Guide (De Rege Latronum)

guide probability odds chance spinner throne upgrades data pdf

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#1 Aluce

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Posted 20 March 2016 - 07:45 PM

Over the course of the last few months, I have compiled and expanded on a comprehensive and (more or less) complete document about King of Thieves, discussing about all things related to costs and probability as well as strategy and statistical tests:

De Rege Latronum

 

If you are on mobile and the file doesn't open in your browser, please try and save the file (hold the link) first.

 

The file is a work in progress. More details and tables will be added, by my own judgment. Requests are much appreciated!

 

Currently the file contains information on the following, in order of appearance:

  • Throne crafting costs and stats
  • Throne crafting strategy
  • Corrected gem pick chances
  • A complete list of unique gems (credits to Chaos; see also this thread)
  • All castle and trap upgrade costs, including Merchant correction
  • Included a complete table about trap upgrades beyond level 30
  • Included complete tables about Tree upgrades
  • A table showing which trap levels get certain thrones to one star after 7 deaths, 6 deaths (to correct for a Revitalizing Heartbeat trigger), as well as one for 5 deaths and for 4 deaths
  • Both the above tables now show trap levels beyond level 30
  • A summary showing which trap levels are most important to keep in mind (including >30)
  • Some information about the Maze and its spinner
  • A section listing information about seasons and loyalty statuses
  • A section about guild management and delta manipulation tips
  • A list of general tips and tricks concerning upgrades and trap placements
  • A table showing probabilities of Revitalizing Heartbeat kicking in 12, 3 or 4 times with varying chances per trigger
  • Respin strategies (for gems) showing a priori gem steal chances and orb efficiency
  • Tables showing the probability that your ritual will be interrupted (at least once)
  • Tables showing the expected amount of times your ritual will be interrupted
  • Descriptions and examples expanding on how the above tables should be interpreted
  • Probability of getting Foreseer 1 and 5 respectively after n total rituals given certain respin strategies for ingredients
  • Tables showing critical areas for statistical tests concerning the spinner's legitimacy (added columns for 15% and 30% spins to allow retrieve spinner testing)
  • Orb-sustainable strategies to craft Foreseer 1 or Foreseer 5 based on some assumptions on minimal orb income and average ritual completion time
  • Terminology (closely related to the dictionary thread)

 

Please, if you have any ideas for additions or corrections to this document, let me know in this thread (or by PM). I want this document to be as complete as possible, so that anyone who has any questions about probability, statistics and other information can simply open up this document and find their answer!

It took me a lot of time to compile this document, so I hope it will be considered useful.  ^_^

 

P.S.: The entries in the table of contents are clickable for easy navigation.


Edited by Aluce, 04 March 2017 - 07:40 PM.
Update

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#2 Skydancer

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Posted 20 March 2016 - 07:58 PM

Coming from another math-addict like you, this is beautiful.

You may want to add trap trap levels needed to get down to 1star after 7 deaths.

Other than that, I can't see anything wrong about calculations, the " foreseer after n rituals" part is awesome

 

/E reading everything now, where did you get the info that " best gem pick " is triggered before the 1/3 given by the random pick chance?

I've always been wondering about this one but I never had the patience to get the necessary amount of tests.


Edited by Skydancer, 20 March 2016 - 08:13 PM.

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#3 Sotherogue

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Posted 20 March 2016 - 08:14 PM

God damn brilliant. Pin it now, please.


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#4 Serboy

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Posted 20 March 2016 - 08:26 PM

That's awesome man. Brilliant guide from a briliant mind. Will be beneficial for both beginners and pro players. (Btw congratz, this will be pinned for sure)



#5 Aluce

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Posted 20 March 2016 - 08:29 PM

 

/E reading everything now, where did you get the info that " best gem pick " is triggered before the 1/3 given by the random pick chance?

I've always been wondering about this one but I never had the patience to get the necessary amount of tests.

That's my own finding. It is the only logical explanation given the fact that it's happened plenty of times to me that the unique highest tier gem was selected, but without the "best" label appearing. There's been some debate about this, however. Unfortunately it is difficult to run a proper test, since a lot of the time you cannot tell which gem actually has the highest value; it's only possible if there is one gem with the highest tier. So I just went with the (only) logical option. I remember Dan commenting on this trait before, but he had no conclusive answer.

There is also a lot of fuzziness about how exactly gem defence and gem steal bonuses work on eachother. I suspect that, say, a 40% steal chance boost means that you have 1.4 times the default chance (5%, 25% or 50%), and that there is a similar reduction for gem defence. However, this is hard to test, as a raider cannot see the exact gem defence level of the raidee, and there might be a max gem steal chance (similarly to how there is a minimum of 5%).

 

Thanks for the positive feedback guys. Makes me happy.  :lol:


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#6 Skydancer

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Posted 20 March 2016 - 08:42 PM

 

 

That's my own finding. It is the only logical explanation given the fact that it's happened plenty of times to me that the unique highest tier gem was selected, but without the "best" label appearing. There's been some debate about this, however. Unfortunately it is difficult to run a proper test, since a lot of the time you cannot tell which gem actually has the highest value; it's only possible if there is one gem with the highest tier. So I just went with the (only) logical option. I remember Dan commenting on this trait before, but he had no conclusive answer.

There is also a lot of fuzziness about how exactly gem defence and gem steal bonuses work on eachother. I suspect that, say, a 40% steal chance boost means that you have 1.4 times the default chance (5%, 25% or 50%), and that there is a similar reduction for gem defence. However, this is hard to test, as a raider cannot see the exact gem defence level of the raidee, and there might be a max gem steal chance (similarly to how there is a minimum of 5%).

 

Thanks for the positive feedback guys. Makes me happy.  :lol:

 

I see, maybe i'll try to run some tests in the next 3 days about the " best gem pick" thingy, i'll get demoted probably so I'll have 3 free days :lol:

 

About the gem defense / gem steal, I believe it's just additive, and then affected by the multiplicator given by the star rating(1,0.5,0.1)

So, foreseer 5 vs 0 gem defense is theoretically 120%, capped at 80%, like the minimum is 5%.

No way to proof this, just my own consideration as I observed quite a bit the behavior of foreseers from 1 to 5 vs maximum gem defense.


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#7 Oliver666666

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Posted 20 March 2016 - 09:17 PM

Now im impressed. i wasnt expecting so well written formulas or such a well formatted guide.

I study maths at uni, and really, im impressed. This has A LOT of work in it.


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#8 Aluce

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Posted 20 March 2016 - 09:40 PM

Now im impressed. i wasnt expecting so well written formulas or such a well formatted guide.

I study maths at uni, and really, im impressed. This has A LOT of work in it.

Thank you! I study maths at uni as well. Which probably explains my love for and usage of LaTeX. ;)


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#9 Synchro246

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Posted 21 March 2016 - 02:50 AM

I already gave you props and I shall again
I have used similar techniques to tackle medical decision making and I do think this is well crafted

#10 Dub

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Posted 21 March 2016 - 03:17 AM

Nice job Aluce!

#11 DanFariasM

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Posted 21 March 2016 - 03:42 AM

Won't open for me :( (iOS) I'll try on pc soon, I'm dying to look at it since I kinda know what to expect haha

#12 Bryce

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Posted 21 March 2016 - 04:22 AM

The only input I think could possibly make it better is maybe to add a conclusion to some of the sections, such as the probability of respins. For example, where would you draw the line on an unreasonable amount of respins according strictly to the percentages instead of gem size? Kind of like what scenario is the best bang for your buck.

 

EDIT: punctuation


Edited by Bryce, 21 March 2016 - 04:22 AM.


#13 Aluce

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Posted 21 March 2016 - 10:05 AM

The only input I think could possibly make it better is maybe to add a conclusion to some of the sections, such as the probability of respins. For example, where would you draw the line on an unreasonable amount of respins according strictly to the percentages instead of gem size? Kind of like what scenario is the best bang for your buck.

It's very difficult to define properly. First of all, the way you spend your orbs depends on how many orbs you earn on average in a round; if you can win Totemius often or manage to demote and promote consistently, you will have more orbs to your disposal and it'd be reasonable to use more orbs than if you can't. 

Furthermore, the most orb-efficient strategy is to obviously not use any orbs at all (but that'd destroy its purpose); the second most orb-effective is to always respin only once for whatever is important enough to spend 5 (but no more) orbs on. There's a correlation between orb usage and delta results, and it's really up to the player to decide which side on the spectum (low or high orb/delta) he plays on.

 

However, I can calculate how many orbs high end players will have at their disposal on average by regarding the ways to obtain orbs (leagues, guild wars and locks) and depending on how successful players are at obtaining them, I can show which strategies are orb-sustainable such that it has the highest expected rewards. But to do this properly, I'll need to do various assumptions (I don't know the probability of finding an orb/pot under a lock, and pots may cost 7 orbs, and whether pots are used in a raid depends on the base), so it cannot be truly definite.


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#14 Skydancer

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Posted 21 March 2016 - 10:20 AM

The only input I think could possibly make it better is maybe to add a conclusion to some of the sections, such as the probability of respins. For example, where would you draw the line on an unreasonable amount of respins according strictly to the percentages instead of gem size? Kind of like what scenario is the best bang for your buck.

 

EDIT: punctuation

 

I've done some thoughts on respins in the past, honestly it depends on how much you value orbs.

As a freebie, I see orbs as a way to progress into the game ( aka orbing big gems when in a streak of bad bases, or storing them, basically trading orbs vs inventory space).

If I have, let's say, 3 sh1t bases in a row, a semiperfect would look like 2 orbed rituals in my book, something like (( 200+200+200)+200+200), since I still fuse deep spikeds even in bad bases.

This leads to 216orbs/1m gems, so 1orb = ~ 5k.

In this way, I respin only if the expected value of the gem does not exceed MY OWN orb value.

Example: I'm raiding a triple deep gold, and so they are 99% triple semi. The 999k is worth 216 orbs so, based on the stars, I would respin, on 3 stars, aka 50%( to simplify calculations) : 5+10+20+40+80=155 orbs. That's 3/4 the value of the gem, and i have a 1-(1/2)^6 chance not to get it, so my gain is still positive. If I went for the 160 respin, I would face an expected loss

 

/E I don't like to relate orbs at disposal with the ones you gain from leagues, since totemius is an absolute gamble( either you find a semi and you get 200 orbs, or you don't). I remember finding SIX babygoldens during a totemius and I was still behind the guy who found a semi 2hours after the league start and went zombiemode upgrading traps for three days, so I don't really like to manage my orb supply around totemius. It's like.. if I win " oh, nice 200orbs". If i lose " oh, w/e who cares".

 

You can still play around weekly challenges, as this is much more consistent than leagues


Edited by Skydancer, 21 March 2016 - 10:30 AM.

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#15 Bryce

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Posted 21 March 2016 - 01:10 PM

I've done some thoughts on respins in the past, honestly it depends on how much you value orbs.

As a freebie, I see orbs as a way to progress into the game ( aka orbing big gems when in a streak of bad bases, or storing them, basically trading orbs vs inventory space).

If I have, let's say, 3 sh1t bases in a row, a semiperfect would look like 2 orbed rituals in my book, something like (( 200+200+200)+200+200), since I still fuse deep spikeds even in bad bases.

This leads to 216orbs/1m gems, so 1orb = ~ 5k.

In this way, I respin only if the expected value of the gem does not exceed MY OWN orb value.

Example: I'm raiding a triple deep gold, and so they are 99% triple semi. The 999k is worth 216 orbs so, based on the stars, I would respin, on 3 stars, aka 50%( to simplify calculations) : 5+10+20+40+80=155 orbs. That's 3/4 the value of the gem, and i have a 1-(1/2)^6 chance not to get it, so my gain is still positive. If I went for the 160 respin, I would face an expected loss

 

/E I don't like to relate orbs at disposal with the ones you gain from leagues, since totemius is an absolute gamble( either you find a semi and you get 200 orbs, or you don't). I remember finding SIX babygoldens during a totemius and I was still behind the guy who found a semi 2hours after the league start and went zombiemode upgrading traps for three days, so I don't really like to manage my orb supply around totemius. It's like.. if I win " oh, nice 200orbs". If i lose " oh, w/e who cares".

 

You can still play around weekly challenges, as this is much more consistent than leagues

Very thought out. Thanks for your insight.



#16 Rachid

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Posted 21 March 2016 - 02:52 PM

Won't open for me :( (iOS) I'll try on pc soon, I'm dying to look at it since I kinda know what to expect haha


No problem opening it on my iPad.
MzDGziC.jpg

#17 Aluce

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Posted 21 March 2016 - 02:54 PM

About the gem defense / gem steal, I believe it's just additive, and then affected by the multiplicator given by the star rating(1,0.5,0.1)

So, foreseer 5 vs 0 gem defense is theoretically 120%, capped at 80%, like the minimum is 5%.

No way to proof this, just my own consideration as I observed quite a bit the behavior of foreseers from 1 to 5 vs maximum gem defense.

I now have data to support this. On my alt account, I have been raided at three stars by a Brave Heart while my gem defense was at 27%. The wheel showed a green part just very slightly smaller than a quarter. So the gem defense chance works in an additive way. I'm quite sure that the gem steal bonus for thrones is additive as well. Thanks! I will start incorporating probabilities about thrones vs. gem defense soon.


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#18 Rachid

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Posted 21 March 2016 - 02:55 PM

Very good job. Really clear and extensive, think this will be useful for a lot of people (especially the new players).
MzDGziC.jpg

#19 Buguinho

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Posted 23 March 2016 - 05:38 PM

I get a blank page on Android :(

Anyway, congratulations on the project ^^

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#20 Astaramis

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Posted 23 March 2016 - 06:43 PM

This downloaded automatically in my phone lol o.o but I can see it clearly (android user here)
I can't understand the math part sorry xD though I get the basics, pretty useful ö
Thanks for take the time to do this! Gonna show it to some friends (⌒▽⌒)

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