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[Guide] Self-sustaining condition for unbirthday party

DeletedUser27525

Introduction
The unbirthday party quest (UB) is a recurring quest that is available in all eras. The player is asked to pay a certain amount of coins and supplies for a random reward. In this guide we ask whether it is possible to achieve self-sustaining UB. We define self-sustaining as having no net loss in coins and supplies.

Notation and Definition
The condition for UB to be self-sustaining is simply
P(large coins) x large coins + P(small coin) x small coin >= C(coins)
P(large supplies) x large supplies + P(small supplies) x small supplies >= C(supplies)​
where P() denotes the probability and C() denotes the cost of UB.

To simplify the notation, we define:
R1 = small coins/supplies reward
R2 = large coins/supplies reward
P1 = P(small coins) = P(small supplies) = 1/7
P2 = P(large coins) = P(large supplies) = 1/14​
so that the self-sustaining condition is rewritten as
P1R1 + P2R2 >= C​
for both coins and supplies.

UB with CF boost
Using the data for "random rewards" provided on wikia, it is easy to see that UB cannot be self-sustaining with its base rewards. However, Chateau Frontenac provides a boost factor of
unbirthday_eq2.png​
for level L > 10. Rearranging terms in the self-sustaining condition, we obtain the critical level of CF for self-sustaining UB:
unbirthday_eq3.png​
for both coins and supplies. We denote the corresponding critical level as Lc and Ls, respectively, and effectively L = max (Lc, Ls).

Results
1. UB without other recurring quests
In this section we first consider the case of UB alone without other recurring quests. This shall give us an upper bound for self-sustaining UB.

The results for all eras are plotted below and tabulated in the Appendix (values for BA are theoretical since one cannot build CF in BA). We see that the critical level for self-sustaining UB is the lowest in IA, where L = 73. Generally speaking, the critical level increases (but not monotonically) through the eras from IA to CA and remains fairly stable till AF, beyond which the critical level becomes dauntingly high (385 for OF and 583 for VF).

unbirthday.png

2. UB with 2nd recurring quest
It is possible to use the coins/supplies reward from UB to fulfill the "collect x coins/supplies" recurring quest, which may in turn give you coins and supplies. A complete solution to this problem is too difficult and therefore we make a simple approximation to obtain an analytical solution.

Let us recall the self-sustaining condition without a 2nd quest:
B(P1R1 + P2R2) >= C​
The rewarded coins/supplies can be used to fulfill a 2nd quest that requires you to collect Q coins/supplies. (The values for Q are obtained from "recurring quests" on wikia.) We make an approximation that the 2nd reward can be immediately retrieved as a fraction of the 2nd quest. The self-sustaining condition then becomes
unbirthday_eq5.png​
We make the variable substitution X = P1R1 + P2R2 to simplify the algebra:
BX ( 1 + B X / Q ) >= C
X^2 B^2 + Q X B - Q C >= 0.​
Solving for the positive root of the quadratic equation, we obtain:
unbirthday_eq7.png​

Results are plotted in the following figure (as crosses) together with the upper bound obtained in the previous section (as dots). They are also tabulated as L2c, L2s, L2 = max(L2c, L2s) in the Appendix. Note that:
  • Starred numbers are theoretical since the 2nd recurring quest is not possible in those ages.
  • Only two eras require a CF level below 100: IA (73) and LMA (96).
  • Overall, we see a reduction of about 20% in the critical CF level to reach self-sustaining UB.
The fractional-reward approximation in the above analysis is a lower bound estimate for self-sustaining UB with a 2nd quest, since in practice the player can only do worse. Hence, the actual L2 should be somewhere between L and L2, but probably closer to L2.

unbirthday2.png

3. UB with 3rd recurring quest
Similar to the previous section, we make some simplifying assumptions:
  • 1 quest for coins and 1 quest for supplies. (The cases of 2 coins quests or 2 supplies quests are ignored.)
  • Fractional rewards (as explained above).
This model allows us to store both coins and supplies from UB. The self-sustaining condition now contains an additional cross term that involves both coins and supplies, and we use the respective subscripts c and s to distinguish them. For example, the condition for coins is:
unbirthday_eq5.png​
and the condition for supplies can be similarly derived. The two conditions are rewritten as
unbirthday_eq5.png​
where X_{c,s} are P1R1 + P2R2 for coins and supplies, respectively. The solution is
unbirthday_eq6.png​

Results are plotted below (as open squares) and tabulated in the Appendix. The critical CF level is reduced by about 30-40% compared to UB alone. This reduction is most significant in VF, where the CF level is brought down from 583 to 323.
unbirthday3.png

Final Remarks
To assess the self-sustaining condition for UB, I have provided an exact upper bound on the CF level (without 2nd recurring quest) and an approximate lower bound with 2nd and 3rd recurring quest. For most players with 2 recurring quests, L2 should serve as a good estimate.

"Self-sustaining" ignores city production by definition. In practice, city production will enable players to achieve a pseudo-self-sustaining state more easily.

Happy Unbirthday! :)

Appendix
AgeLcLsLL2cL2sL2% change between L2 and LL3cL3sL3% change between L3 and L
BA*296240296226182226-23.6186157186-37.2
IA72737349*49*49*-32.9*37*37*37*-49.3*
EMA11611611684*84*84*-27.6*67*67*67*-42.2*
HMA147147147109*109*109*-25.9*90*90*90*-38.8*
LMA100128128749696-25.0617777-39.8
CA210215215168169169-21.4142145145-32.6
IndA255194255205156205-19.6172137172-32.5
PE240179240198147198-17.5168131168-30.0
ME257189257211157211-17.9182139182-29.2
PME240167240195137195-18.8165121165-31.2
CE209174209165141165-21.1141121141-32.5
TE177167177140135140-20.9121115121-31.6
FE206240240157185185-22.9134152152-36.7
AF198222222154170170-23.4130143143-35.6
OF349385385245296296-23.1215232232-39.7
VF535583583400369400-31.4303323323-44.6
(* = theoretical value)
 
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DeletedUser31440

You'll need to factor in the 2nd RQ slot for LMA and above then the third from whenever it comes in to get more accurate values.
 

DeletedUser29726

Ozyman Tremble Weaklings said:
You'll need to factor in the 2nd RQ slot for LMA and above then the third from whenever it comes in to get more accurate values.
Click to expand...

The 3rd comes when you're "done" the story. There is (or at least was) a way to get pauses from it in FE story which you could be doing as early as CE BUT I'm not sure the exact trigger that causes the pause and lets you get 3 slots. It happened to me on my main in FE but I don't remember my choices at all (at the time I didn't even realise how you took the continent map changed the story in places). A friend who tried to get it to happen never got the break though.

Usually the 3rd slot only comes if you're able to complete the very last bit of story in the game (which i believe there's a hard block in OF now, so you'd have to be at least OF)

As for the impact of the 2nd quest slot, it's actually really hard to account for if you're trying to 'optimise' return to get closer to infinite quests. i.e. if you currently have a collect supplies with 1 small reward progress and get rewarded a coin reward (small or large) do you switch? (answer depends on age AND level of chateau). In some ages the impact is not slight at all as a large coin/supply reward might be another quest all to themself.
 
Last edited by a moderator:

DeletedUser

xivarmy said:
The 3rd comes when you're "done" the story. There is (or at least was) a way to get pauses from it in FE story which you could be doing as early as CE BUT I'm not sure the exact trigger that causes the pause and lets you get 3 slots. It happened to me on my main in FE but I don't remember my choices at all (at the time I didn't even realise how you took the continent map changed the story in places). A friend who tried to get it to happen never got the break though.

Usually the 3rd slot only comes if you're able to complete the very last bit of story in the game (which i believe there's a hard block in OF now, so you'd have to be at least OF)

As for the impact of the 2nd quest slot, it's actually really hard to account for if you're trying to 'optimise' return to get closer to infinite quests. i.e. if you currently have a collect supplies with 1 small reward progress and get rewarded a coin reward (small or large) do you switch? (answer depends on age AND level of chateau). In some ages the impact is not slight at all as a large coin/supply reward might be another quest all to themself.
Click to expand...
Don't ask me how, but I have 3 RQ slots in CE right now on Vingrid.
 

DeletedUser29726

A rough spreadsheet i made for it in PE :
upload_2018-10-15_23-4-1.png

Key points :

1) The assumption of getting a 'double quest' from an unbirthday is 1/7 - this is actually low because at these chateau levels a large coin or supply package is a double quest - and the chance of that is 1/7. The occasional consecutive string of small coin or supply packages would actually make it a bit higher than 1/7.

2) The Bonus Quests column is how many extra quests you get just from the coin/supply income of doing 50 base quests (for example from FP donation or from your own coin/supply collections). At the point where you're doing 359 quests a day at level 117 that might already be enough.

3) It did take it down from 180 to 250 to a mere 148 to go infinite - this is not slight.
 

DeletedUser27525

xivarmy said:
A rough spreadsheet i made for it in PE :
View attachment 10993

Key points :

1) The assumption of getting a 'double quest' from an unbirthday is 1/7 - this is actually low because at these chateau levels a large coin or supply package is a double quest - and the chance of that is 1/7. The occasional consecutive string of small coin or supply packages would actually make it a bit higher than 1/7.

2) The Bonus Quests column is how many extra quests you get just from the coin/supply income of doing 50 base quests (for example from FP donation or from your own coin/supply collections). At the point where you're doing 359 quests a day at level 117 that might already be enough.

3) It did take it down from 180 to 250 to a mere 148 to go infinite - this is not slight.
Click to expand...
Thank you for the feedback! I'll take a closer look at the effect of the 2nd slot and update the guide.
 

DeletedUser27525

xivarmy said:
3) It did take it down from 180 to 250 to a mere 148 to go infinite - this is not slight.
Click to expand...
The numbers I have for PE are:
Lc 240
Ls 179
L2c 198
L2s 147
So it seems we're in the same ballpark. :)
 

DeletedUser36572

Laozi said:
Introduction
The unbirthday party quest (UB) is a recurring quest that is available in all eras. The player is asked to pay a certain amount of coins and supplies for a random reward. In this guide we ask whether it is possible to achieve self-sustaining UB. We define self-sustaining as having no net loss in coins and supplies.

Notation and Definition
The condition for UB to be self-sustaining is simply
P(large coins) x large coins + P(small coin) x small coin >= C(coins)
P(large supplies) x large supplies + P(small supplies) x small supplies >= C(supplies)​
where P() denotes the probability and C() denotes the cost of UB.

To simplify the notation, we define:
R1 = small coins/supplies
R2 = large coins/supplies
P1 = P(small coins) = P(small supplies) = 1/7
P2 = P(large coins) = P(large supplies) = 1/14​
so that the self-sustaining condition is rewritten as
P1R1 + P2R2 >= C​
for both coins and supplies.

UB with CF boost
Using the data for "random rewards" provided on wikia, it is easy to see that UB cannot be self-sustaining with its base rewards. However, Chateau Frontenac provides a boost factor of
B = (L-10) x 0.05 + 2.5​
for level L > 10. Rearranging terms in the self-sustaining condition, we obtain the critical level of CF for self-sustaining UB:
L >= [ C / ( P1R1 + P2R2) - 2.5 ] x 20 + 10​
for both coins and supplies. We denote the corresponding critical level as Lc and Ls, respectively, and effectively L = max (Lc, Ls).

Results
1. UB without other recurring quests
In this section we first consider the case of UB alone without other recurring quests. This shall give us an upper bound for self-sustaining UB.

The results for all eras are plotted below and tabulated in the Appendix. We see that the critical level for self-sustaining UB is the lowest in IA, where L = 73. Generally speaking, the critical level increases (but not monotonically) through the eras from IA to CA and remains fairly stable till AF, beyond which the critical level becomes dauntingly high (385 for OF and 583 for VF).

View attachment 11002

2. UB with 2nd recurring quest
It is possible to use the coins/supplies reward from UB to fulfill the "collect x coins/supplies" recurring quest, which may in turn give you coins and supplies. A complete solution to this problem is too difficult and therefore we make a simple approximation to obtain an analytical solution.

Let us recall the self-sustaining condition without a 2nd quest:
B(P1R1 + P2R2) >= C​
The rewarded coins/supplies can be used to fulfill a 2nd quest that requires you to collect Q coins/supplies. (The values for Q are obtained from "recurring quests" on wikia.) We make an approximation that the 2nd reward can be immediately retrieved as a fraction of the 2nd quest, i.e., on top of the 1st reward from UB we get another factor of
B(P1R1 + P2R2) / Q.​
The self-sustaining condition then becomes
B(P1R1 + P2R2) [1+B(P1R1 + P2R2) / Q] >= C.​
We make the variable substitution X = B(P1R1 + P2R2) to simplify the algebra:
X ( 1 + X/Q ) >= C
X^2 + QX - QC >= 0.​
The solution to the quadratic equation is
X = [ -Q +/- sqrt(Q^2 +4QC) ] / 2​
where obviously we only take the positive root in this context:
X = [ sqrt(Q^2 +4QC) - Q ] / 2.​
Substituting back for the boost B and level L, we obtain:
B = [ sqrt(Q^2 +4QC) - Q ] / 2(P1R1 + P2R2)
L = { [ sqrt(Q^2 +4QC) - Q ] / 2(P1R1 + P2R2) - 2.5 } x 20 + 10.​

In the following figure, results are plotted (as crosses) together with the upper bound obtained in the previous section (as dots) and tabulated as L2c, L2s, L2 = max(L2c, L2s) in the Appendix. Overall, we see a reduction of about 20% in the critical CF level to reach self-sustaining UB.

Notice that IA only needs a CF level of 49 (assuming it is possible to get the 2nd quest!). There are two other eras that needs a CF level of below 100 (again, if a 2nd quest is possible): 84 for EMA and 96 for LMA.

The fractional-reward approximation in the above analysis is a lower bound estimate for self-sustaining UB with a 2nd quest, since in practice the player can only do worse. Hence, the actual L2 should be somewhere between L and L2, but probably closer to L2.

View attachment 11003

3. UB with 3rd recurring quest
Unfortunately, this is beyond the scope of this guide. We can certainly expect L3 to drop even further. Nonetheless, L2 should serve as a good estimate.

Appendix
AgeLcLsLL2cL2sL2% change
BA296240296226182226-23.6
IA727373494949-32.9
EMA116116116848484-27.6
HMA147147147109109109-25.9
LMA100128128749696-25.0
CA210215215168169169-21.4
IndA255194255205156205-19.6
PE240179240198147198-17.5
ME257189257211157211-17.9
PME240167240195137195-18.8
CE209174209165141165-21.1
TE177167177140135140-20.9
FE206240240206185206-14.2
AF198222222154170170-23.4
OF349385385245296296-23.1
VF535583583400369400-31.4
Click to expand...

Eeek ... :confused:
Happy Unbirthday; drinks are on the house.
 

DeletedUser31440

This is a link to a CF calculator I made awhile ago, it has an option to calculate with a 2nd RQ giver. May be useful for you in fine tuning your findings:

https://docs.google.com/spreadsheets/d/1CSHpQepk-w32xInH6Id2_I85gXgf9lvZ1NzHltWZSWc/edit?usp=sharing

For 2nd RQ giver this is my reccomendation for using it:

Choose a quest that would not otherwise be used during collections, I chose collect supplies as it doesn't interfere with doing the production quest, spend fp's quest or collect coins quest so it can always be active, at least with how I go about questing.

Determine the number of RQ's that you will do on a daily basis (collect coins, spend fp's, produce x of x) do not include UBQ's. - This is the field E5 - Other Daily RQ's

Determing the amount of supplies and coins that you plan on spending on UBQ's each day - These are fields B4 & B5 -- Coins/Supplies to use on UBQ

If in an age that has a 2nd RQ giver, choose Yes in field H7 and enter the number of supplies needed to complete the collect supplies quest in H8.

If you have production buildings that you use for the produce x of x, figure out the number of supplies produced daily from those and enter into H9.

How the calculator works:

Modify A2, B2, D2-I2 for the age you are in and trying to calculate (https://forum.us.forgeofempires.com...rontenac-a-hopefully-healthy-obsession.20436/)

Enter in the number of supplies and coins that you will spend on UBQ's each day in B4 & B5, enter number of non-UBQ RQ's done each day in E5, and fill out the 2nd RQ giver space if using a 2nd RQ giver.

Calculator takes the number of supplies and coins to use on UBQ's to determine the most limiting factor. (Divides number of supplies/coins to be used by the amount needed for a UBQ and returns the integer value so it always rounds down) this will populate in D5 to give your daily number of UBQ's.

Calculator adds up the number of daily UBQ's and other RQ's. Takes the total number done and then determines the amount of supplies and coins produced from doing those then determines the number of extra UBQ's that can be done with those. This populates in B7 & B8 - 1st return level.

Calculator takes the extra UBQ's from 1st return level and then determines amount of supplies and coins that will be produced and number of UBQ's.

Calculator will do 10 return levels to ensure that it catches them all.

All the extra UBQ's from all return levels are summed up in field F5, Bonus UBQ's.
 

DeletedUser29726

There is a collect coins quest in FE - not sure why the number for it is missing from wiki. Will track it down and add it there.

Edit: Value is 800k coins - issue on wiki is that it was listed as a second supplies quest
 

DeletedUser27525

xivarmy said:
There is a collect coins quest in FE - not sure why the number for it is missing from wiki. Will track it down and add it there.

Edit: Value is 800k coins - issue on wiki is that it was listed as a second supplies quest
Click to expand...
Appreciate it! Fixing it right away
 

DeletedUser27525

Ozyman Tremble Weaklings said:
This is a link to a CF calculator I made awhile ago, it has an option to calculate with a 2nd RQ giver. May be useful for you in fine tuning your findings:

https://docs.google.com/spreadsheets/d/1CSHpQepk-w32xInH6Id2_I85gXgf9lvZ1NzHltWZSWc/edit?usp=sharing

For 2nd RQ giver this is my reccomendation for using it:

Choose a quest that would not otherwise be used during collections, I chose collect supplies as it doesn't interfere with doing the production quest, spend fp's quest or collect coins quest so it can always be active, at least with how I go about questing.

Determine the number of RQ's that you will do on a daily basis (collect coins, spend fp's, produce x of x) do not include UBQ's. - This is the field E5 - Other Daily RQ's

Determing the amount of supplies and coins that you plan on spending on UBQ's each day - These are fields B4 & B5 -- Coins/Supplies to use on UBQ

If in an age that has a 2nd RQ giver, choose Yes in field H7 and enter the number of supplies needed to complete the collect supplies quest in H8.

If you have production buildings that you use for the produce x of x, figure out the number of supplies produced daily from those and enter into H9.

How the calculator works:

Modify A2, B2, D2-I2 for the age you are in and trying to calculate (https://forum.us.forgeofempires.com...rontenac-a-hopefully-healthy-obsession.20436/)

Enter in the number of supplies and coins that you will spend on UBQ's each day in B4 & B5, enter number of non-UBQ RQ's done each day in E5, and fill out the 2nd RQ giver space if using a 2nd RQ giver.

Calculator takes the number of supplies and coins to use on UBQ's to determine the most limiting factor. (Divides number of supplies/coins to be used by the amount needed for a UBQ and returns the integer value so it always rounds down) this will populate in D5 to give your daily number of UBQ's.

Calculator adds up the number of daily UBQ's and other RQ's. Takes the total number done and then determines the amount of supplies and coins produced from doing those then determines the number of extra UBQ's that can be done with those. This populates in B7 & B8 - 1st return level.

Calculator takes the extra UBQ's from 1st return level and then determines amount of supplies and coins that will be produced and number of UBQ's.

Calculator will do 10 return levels to ensure that it catches them all.

All the extra UBQ's from all return levels are summed up in field F5, Bonus UBQ's.
Click to expand...
Thank you for sharing this tool! I have now added the analysis for 3 recurring quests.
 

DeletedUser35867

I'd like to note that you also get FP5 packs from rewards, which can then be used to complete "Spend X Forge Points" quests. This should reduce the minimum level to go infinite by a small, but nonzero, amount.
 

DeletedUser32824

tl;dr (I'm about to eat dinner). In LMA am I self sustaining with a 100 or 128? A guy in my arc 80 group has his at like 119 in LMA! I'm leveling a Dynamic Tower and Voyager to give me a higher daily supply production to offset less levels in my CF (Gives more FP to increase supplies than CF to self-sustaining in the short term)

I'll read this in more detail later. Looks yummy though!
 

DeletedUser27525

amiar said:
tl;dr (I'm about to eat dinner). In LMA am I self sustaining with a 100 or 128? A guy in my arc 80 group has his at like 119 in LMA! I'm leveling a Dynamic Tower and Voyager to give me a higher daily supply production to offset less levels in my CF (Gives more FP to increase supplies than CF to self-sustaining in the short term)

I'll read this in more detail later. Looks yummy though!
Click to expand...
Thanks for your interest! For LMA:
Lv 128 for UB alone.
Lv 95 for 2 recurring quests.
Lv 77 for 3 recurring quests.
 

DeletedUser35867

NINJAxPENGUIN92 said:
I'd like to note that you also get FP5 packs from rewards, which can then be used to complete "Spend X Forge Points" quests. This should reduce the minimum level to go infinite by a small, but nonzero, amount.
Click to expand...

Would you be willing to apply this to your calculations?
 
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