Kingdom Tribute--the math


#1

Okay, so I have been leveling my kingdoms up to get that all-elusive tribute, but it doesn’t seem to be coming in as quickly as I thought it should. So I did the math to see if my expectations were unreasonable. They were. The game has it right, and tribute is calculated the way they say it should be. So how often should you expect to see tribute?

The following list gives the chances of receiving tribute from at least one Kingdom. It is assumed that you have all 19 Kingdoms, and that every kingdom is the same level. Obviously, if one of both of those assumptions does not apply to you, you would use a different formula.

19 level 1 kingdoms: 17.4%
Level 2: 31.9%
Level 3: 43.9%
Level 4: 54.0%
Level 5: 62.3%
Level 6: 69.1%
Level 7: 74.8%
Level 8: 79.5%
Level 9: 83.3%
Level 10: 86.5%

Taking into account the cost of leveling, my take is that the best bang for your buck is to get all kingdoms to level 5 as quickly as possible. That leaves you sitting at a solid 5/8s winning percentage, which would drive any Vegas casino nuts. Beyond that, the payoffs are not as good for what you have to put in.


#2

This is actually very good information, thanks! And 86% is actually exactly how it feels.


#3

I started on the same thought path, but here is where I took it next.

Your home kingdom doubles the tribute.
One question to ask yourself is “what do you want out of your tribute”.
For me, the answer was gold. I wanted to get as much gold as possible, so I could spend that on:

  • levelling more kingdoms
  • continuing to battle

A friend of mine wanted glory because glory buys rewards and maps, maps make gems and gold. His build pattern differs to mine for that reason.

There are six kingdoms I put higher importance on.
My home kingdom is Khaziel, because it gives 250 gold. When it hits tribute (as it just did right now when I went to check my info) it gives 500 gold. Good times.
The five others are:
Stormheim (200 gold, 2 glory)
Broken Spire (200 gold, 4 souls)
Adana (175 gold, 2 glory, 2 souls)
Maugrim Woods (150 gold, 2 glory, 4 souls)
Zaejin (150 gold, 3 glory, 2 souls)

My formula was to level all kingdoms evenly, except level those five +1 on all other kingdoms, except level Khaziel +1 on top of that.

So when my others got to 5, those 5 were 6 and Khaziel was 7.

Levelling up this way skewed my tributes to be high in gold.

However, after this point, the gold cost to continue in such a fashion became astronomical.
I realised that I had better impact levelling Khaziel to 10 from this point, not for the extra 3% tribute (I was already hitting roughly 7 in 10 times, and half of those were doubles or better) but for the +1 health for troops.
Winning battles is still the best way to earn gold, and having more health always helps that.

Once I had done so, I calculated my next best play to upgrade Stormheim to level 10, which I did.
Now I’m working on Broken Spire.

Since this patch, I have earned about 250 gems - enough to make the purchase of my dragon armour.
I’m still on 67.9% to get a successful tribute (of some kind. This includes doubles, triples, and even that six-tribute I had last week)

However, once I started mathing this, questions arose that affected my calculations.

Can a tribute occur more than once from the same city in the same day?
If so, the math remains the same. If not, the math gets more complicated.

Also, if this is the right place, I wish there was a feature to scroll between kingdom screens, from one kingdom to the next. So much easier than closing the window, finding the next city, clicking it, clicking kingdoms, closing it, finding the next kingdom, etc.


#4

Because I can’t help myself, I had to explore the probabilities further. My results were surprising. More numbers follow.

Again, assumption is that you have 19 kingdoms, all leveled the same. The following list shows the chances of getting EXACTLY 1 tribute.

19 level 1 kingdoms: 15.9%
Level 2: 26.4%
Level 3: 32.9%
Level 4: 36.4%
Level 5: 37.7%
Level 6: 37.4%
Level 7: 36.0%
Level 8: 33.9%
Level 9: 31.3%
Level 10: 28.5%

And this is the probability of getting 2 OR MORE tributes.
Level 1: 1.5%
Level 2: 5.5%
Level 3: 11.0%
Level 4: 17.5%
Level 5: 24.5%
Level 6: 31.7%
Level 7: 38.8%
Level 8: 45.6%
Level 9: 52.0%
Level 10: 58.0%

This is the probability of getting 3 OR MORE tributes.
Level 1: 0.1%
Level 2: 0.6%
Level 3: 1.8%
Level 4: 3.8%
Level 5: 6.7%
Level 6: 10.2%
Level 7: 14.4%
Level 8: 19.1%
Level 9: 24.1%
Level 10: 29.5%

The chance of 4 OR MORE tributes: From here on, I will streamline numbers below 1%.
Level 1 to level 3: < 0.3%
Level 4: 0.6%
Level 5: 1.3%
Level 6: 2.4%
Level 7: 4.0%
Level 8: 6.0%
Level 9: 8.5%
Level 10: 11.5%

Now the big one: The chances for 5 OR MORE tributes.
19 level 1 to level 6 kingdoms: < 0.5%
Level 7: 0.9%
Level 8: 1.5%
Level 9: 2.3%
Level 10: 3.5%

Just for kicks, I’ll show the numbers for 6 OR MORE tributes as well, so those of you have had such luck can feel special.
Level 1 to level 5 kingdoms: < 0.05%
Level 6: 0.1%
Level 7: 0.1%
Level 8: 0.3%
Level 9: 0.5%
Level 10: 0.9%

Seven or more tributes occurs 0.1% of the time with level 9 kingdoms and 0.2% of the time with level 10. The probability of any other event not mentioned is less than 0.05%.

I still think you need to get all your Kingdoms to level 5 as quickly as possible, but level 10 looks pretty sweet.


#5

Nice work. Seems I got extremely lucky to get a 6 tribute :smiley:


#6

Interesting debate so far - and well done with all the maths (not math, sorry, I am English).

No one so far has noted that getting to level 10 also brings a global stat gain to all your troops - which over-riding all the tribute stuff is a darned good reason to get to level 10 everywhere!


#7

Following patch 1.08, I’ve done a binomial distribution check on how the new equates to the old…

Prior to 1.08, if you had all 20 Kingdoms at Level 10:

Chance of 0 Tribute: 12.2 %
Chance of 1 Tribute: 27 %
Chance of 2 Tributes: 28.5 % (+ 2 gems)
Chance of 3 Tributes: 19 % (+ 10 gems)
Chance of 4 Tributes: 9 % (+ 15 gems)
Chance of 5 Tributes: 3 % (+ 20 gems)
Chance of 6 Tributes: 1 % (+ 20 gems)
Chance of 7 Tributes: 0.2 % (+ 20 gems)

The end result was that, on average, you could expect 4.66 gems on average per cash collect.

Since patch 1.08, if you had all 20 Kingdoms at Level 10 and below Kingdom Power Level 3:

Chance of 0 Tribute: 12.2 %
Chance of 1 Tribute: 27 %
Chance of 2 Tributes: 28.5 % (+ 1 gems)
Chance of 3 Tributes: 19 % (+ 2 gems)
Chance of 4 Tributes: 9 % (+ 3 gems)
Chance of 5 Tributes: 3 % (+ 4 gems)
Chance of 6 Tributes: 1 % (+ 5 gems)
Chance of 7 Tributes: 0.2 % (+ 6 gems)

The end result was that on average you would expect 1.12 gems on average per collect (that’s about a quarter the amount beforehand).

Then I was wondering if it would re-balance when uncapped if I got all 20 Kingdoms to Power Level 3 (double tribute chance)…

Chance of 0 Tribute: 1.2 %
Chance of 1 Tribute: 5.7 %
Chance of 2 Tributes: 14 % (+ 1 gems)
Chance of 3 Tributes: 21 % (+ 2 gems)
Chance of 4 Tributes: 22 % (+ 3 gems)
Chance of 5 Tributes: 17 % (+ 4 gems)
Chance of 6 Tributes: 11 % (+ 5 gems)
Chance of 7 Tributes: 5.5 % (+ 6 gems)
Chance of 8 Tributes: 2.2 % (+ 7 gems)
Chance of 9 Tributes: 0.7% (+ 8 gems)

You would expect 2.99 gems per cash collect. So even after that miraculous effort of leveling troops and adding traits…one can expect to earn 36% less gems via the Tribute route than at the end of 1.07.


Help with math as it pertains to Cursed and Bats
Tributes provide less gems
#8

What is the probability of getting all 19 tributes at once? :grin:

Factoring in the gold cost for each level, I would say the sweet spot for minimum level is 6. It costs 8k each to get kingdoms from lvl5-lvl6. Whereas, it costs 15k each to get lvl6-lvl7.


#9

To answer on a previous post, yes 1 kingdom can give tribute multiple times a day.


#10

haha…a little under 1 in 1 trillion (with all kingdoms at power level 3)

If you get 14, then you’re 1 in a million :dizzy_face:


#11

This is the highest tribute i ever have (18 keys and 9 gems). I have 19 3-stars+ kingdoms.


#12

Very nice :slight_smile:
In past 24 hours I’ve got one with 7 tributes, though i only have 1 kingdom with 3 stars. .


#13

Anyone is better than me :smile:


#14

3 month necro is pretty good but nowhere close to the top :stuck_out_tongue:


#15

Hey
I know this thread is old but I wanted to do the math by myself and failed :confused:
For 0 Tributes I understand what to do:
0,9^20 = 12,2% like your post also shows

And now I struggle how to do for chances of 1,2,3… Tributes.
My approach is something like this:
Wouldn´t it be a tree diagramm with 2^20 possible outcomes ? And the Calculation above is only right because for zero Tributes there is only 1 outcome out of 2^20 ?
So for 1 Tribute I assume something like:
0,9^19 * 0,1 * (outcomes out of 2^20 that belongs to 1 Tribute)
But I don´t know what that number would be.

Since I have the solution out of your post I can calculate the number backwards
1 Tribute: 0,9^19 * 0,1 * 20 = 27 % -> 20: number of Kingdoms ??
2 Tributes: 0,9^18 * 0,1^2 * 190 = 28,5% -> 190: No clue where that comes from

But my approach could also be completly wrong :smile:

Could you (or someone else) please explain how the Math is done ?
I would be very grateful :pray:


#16

I’ll do 5 tributes from 20 kingdoms as an example, and hopefully you can extrapolate from there.

First you calculate the chances for each to give the expectation. 90% chance at no tribute, 10% at tribute so you have

0.9^15 for the 15 that didn’t get
0.1^5 for the 5 that did

Next, you need to know how many different ways you can chose 5 kingdoms. It could be kingdoms 1 2 3 4 and 5, it could be 1 2 3 4 and 6, etc…

So if you had 20, how many ways could you choose the first to get tribute? 20 choices.
The next would be 19, then 18, so the total is 2019181716

But, that is too many since it would count 1 2 3 4 5 and 5 4 3 2 1 as different choices, even though it is the same set. So you need to divide by how many ways you can order those 5 choices. That works out to be 54321

Putting it all together you get:
0.9^15 * 0.1^5 * 2019181716 / 54321 = 3.19%

In formula version, for x tributes:
(0.9)^(20-x) * (0.1)^x * 20!/(20-x)! / (x!)


#17

Thank you!!

I did not think about dividing by the ways you can order :sweat_smile:

Now it makes sense :slight_smile:


#18

Yeah, that is where the 190 came from: 2019 / 21 = 190


#19

At some point I made a little spreadsheet to quench my curiosity - you can check there the chances for a given number of tributes based on number of kingdoms with up to 2 different tribute probability (10 and 20% usually for 3* kingdoms and less, all level 10). I added some other GoW data there later, but you can make a copy of the sheet, enter your values (top row only in Tribute chances tab) and get the results if you want.

EDIT: for some reason the link doesn’t seem to work, so I’m repeating it below with an added space, so that the forum treats it as a text you can easily copy:

https:// docs.google.com/spreadsheets/d/17W_62X371JoaKrIPWHQW3wJbmBwePbQRRji_sLcW-bg/edit?pref=2&pli=1#gid=0


#20

Yeah - drawing a tree can work but is very very long once you start trying to capture every branch when there are now 24 kingdoms.

Easiest way is to use the binomial distribution formula in excel.