Analytical results of DRN checks or Why fate weaving needs some love

[greengrocer]

The effect of cursed luck

Because of the truncation of negative results from DRN - drn, the effect of cursed luck is +3.53 att and +3.53 str of the attacker.

The chance of n=2 critical hit goes from 1/36 to 38%, a 35% increase.

The math to get 3.53 and 38% will be explained in the later section.

Let's say the defender has protection of 24, which is a reasonablely high number before alt9 Army of Gold is online. It is effectively +2.1 str to attacker

In the 50% active situation, cursed luck gives you +3.53 att and +5.63 str.

The final effect is generally < +2 att and < +3 str.

Fate weaving is at 16 vs mr casting cursed luck. As a S7 and mandatory Misfortune 1 bless, it really is too weak from a combat point of view.

I recently played a disciple game as MA Marignon pretender with 2 astral bonus bless point. The other 2 guys are Ashdod and Phaeacia. Counter Will of the Fates is a good bonus for the giants. It is the major reason I picked fate weaving knowing its subpar combat effect.

Through the game, using Mari Chalice, fate weaving feels pretty good for a S5 bless. In my opinion, it is OK for S6 and Misfortune 1. I just can not take it as at the current S7 price point.

You might want to argue the Arcane Nexus point. But a S6 pretender with boosters could also do the same. So why would I pick fate weaving?

As cursed luck works in a roll_dice(100) > 50 fashion, it is pretty easy to make fine tweak to the chance of active.

I am proposing to make cursed luck has a 60% of active. Or at least 57%, so the effect of -drn roughly reach its face value of Exp(drn)=4.

My 2 cents ends here. All the following are just math now. I used Sagemath 9.0. You can go to cocalc.com to try out the formulas.

Probability Generating Functions (PGF) of DRN checks

This all starts from a post about DRN on physicsforums [www.physicsforums.com] . gill1109 suggested Probability Generating Function (PGF), which is usually used for summation of discrete random variables, n>=0.

In Dominions we care about drn-drn (GE strength check), DRN-DRN (regular combat), DRN-(DRN-drn) (fate weaving/cursed luck combat), and the negative results of subtraction are always truncated. For this purpose, I extend PGF to n <0 and always separate n=0, n>0 and n<0 series into PGF= GF_neg + p0 + GF_pos

The details are in my wiki [dom5.imd3.com]. Here I list some results.

PGF of drn

pgf_drn(z)=1/6*(z+z^2+z^3+z^4+z^5)/(1-z**5/6)

This is only PGF that needs a infinite summation. After this, all are just add, multiply, fraction and expansion of rational functions.

Expectation of drn is its PGF's 1st derivative value at z=1

Exp(drn) = pgf_drn'(1)=4

Variance of drn is
Var(drn)= pgf_drn''(1) + pgf_drn'(1) - [Exp(drn) ]^2 = 8

These can be easily verified by a infinite summation.


PGF of DRN - DRN

p(0)=29/343

gf_pos(z) = - 1/343*(29*z^10 + 28*z^9 + 16*z^8 - 16*z^7 - 77*z^6 - 524*z^5 - 658*z^4 - 796*z^3 - 919*z^2 - 1008*z)/(z^10 - 12*z^5 + 36)

This is a symmetric distribution, so the negative side GF_neg can be get by just substitute z by 1/z in GF_pos
gf_neg = gf_pos.substitute (z=1/z)

The chance of hit table in rational number is here [dom5.imd3.com]. It match the numbers in Illwinter manual.

PGF of DRN-drn and truncate(DRN-drn)

PGF of DRN-drn

p(0)= 3/49

gf_pos(z) = -1/49*(3*z^10 - 6*z^9 - 23*z^8 - 48*z^7 - 81*z^6 - 158*z^5 - 174*z^4 - 177*z^3 - 167*z^2 - 144*z)/(z^10 - 12*z^5 + 36)

gf_neg(z) = 1/49*(13*z^4 + 9*z^3 + 6*z^2 + 4*z + 3)/(6*z^5 - 1)

Before truncation, Exp(DRN-drn)=4. But after you truncate all the negative term, PGF is turned to

p(0)= 10/49
gf_pos(z)=-1/49*(3*z^10 - 6*z^9 - 23*z^8 - 48*z^7 - 81*z^6 - 158*z^5 - 174*z^4 - 177*z^3 - 167*z^2 - 144*z)/(z^10 - 12*z^5 + 36)

The new expectation is Exp(trunc(DRN-drn)) = 219/49 ~= 4.47, around 12% increase.

The effect of a drn drops from 4 to (8-4.47)=3.53

PGF of DRN - truncate(DRN-drn)

p(0)=134/2401

gf_pos(z) = 1/2401*(356*z^10 + 1063*z^9 + 1997*z^8 + 3170*z^7 + 4577*z^6 + 6824*z^5 + 6992*z^4 + 6848*z^3 + 6355*z^2 + 5508*z)/(z^10 - 12*z^5 + 36)

gf_neg(z) = 1/2401*(4068*z^9 + 3323*z^8 + 2647*z^7 + 2073*z^6 + 1609*z^5 - 118*z^4 - 163*z^3 - 167*z^2 - 153*z - 134)/(36*z^10 - 12*z^5 + 1)

The change of hit chance is here [dom5.imd3.com] .

The change of damage is here [dom5.imd3.com].

Both results match an effect 3.53.

[oldrocker99]

Great googlymoogly! What a labor of love, and mathematically accurate.

[Alias]

That aligns with my evaluation of Fateweaving worth for regular rolls. Mind you, though, Fateweaving can be used to boost otherwise hard-to-boost rolls like Auras and Trample.

[Zonk]

Through the game, using Mari Chalice, fate weaving feels pretty good for a S5 bless. In my opinion, it is OK for S6 and Misfortune 1. I just can not take it as at the current S7 price point.

Are you considering that "Cursed Luck" also negates Luck on the unit? That must be worth something because mass Luck (for example Will of the Fates) is nice.

EDIT: Maybe it should negate Twist Fate too.

[Silence]

Does cursed luck affect MR rolls too ?!

[Zonk]

Also, what are you using to know the % of critical hits? The manual, or this Dom4 code http://dominionsmods.com/index.php?showtopic=2332 which MIGHT still be true except for -50% protection being replaced by -25%?

[Mr. Mogger]

So what you're saying is that +att +str(maybe bloodsurge) +fateweaving MA Marignon FAB pretender (maybe dormant) is a thing?

Lets see: MA Marignon has fire damage (somewhat easy to counter without magic), some astral (MR can counter). With your math and +str its like 20 damage flags and about 24 damage knights. That's pretty good, pretty much a 1 shot on most indie units.

I haven't really used cursed luck but does it last after the battle?

[greengrocer]

I totally agree. That's why I like fate weaving. I was the one who said 'we should do fate weaving' in that disciple game and used Exp(drn)=4 to persuade the others. Hoof of cavalry does not get str buff, while fate weaving even buff your non-sacred. Counter mass luck is good for Ashdod giants, etc.

The game finished before we even ran into a mass luck. The other 2 guys were nor very impressed with the melee combat bonus of Fate weaving.

I started out to run all these calculation to prove Fate Weaving is good for melee combat.

I was quite surprised by the outcome.

Originally posted by Alias:

That aligns with my evaluation of Fateweaving worth for
regular rolls. Mind you, though, Fateweaving can be used to boost otherwise hard-to-boost rolls like Auras and Trample.

[greengrocer]

We are referring to the same source then. I believe the reverse compiled code in your link is actually from Dom 3 code diving.

I did some code diving myself into Dom 5.35 and saw similar code. At the time I did not understand the 'immobilized' tag. It was check some attribute bit.

Someone in our group discussion did some tests and confirmed the test results match the formulas.

Here I copy the effect of cursed luck / fate weaving on all cases.

< = 2, 671/1764 vs regular DRN roll 1/36

< = 3, 212/441 vs regular DRN roll 3/36

< = 4, 73/126 vs regular DRN roll 6/36

Originally posted by Zonk:

Also, what are you using to know the % of critical hits? The manual, or this Dom4 code http://dominionsmods.com/index.php?showtopic=2332 which MIGHT still be true except for -50% protection being replaced by -25%?

[greengrocer]

Counter mass luck is how I talked Ashdod and Phaeacia into fate weaving. I have no doubt it will be a killer feature in the late game, especially for giants. Sadly our game finished before we even ran into one mass luck.

Another problem of fate weaving is that it competes against magic weapon. Ashdod and Phaeacia do not care as their sacred are born with magic weapon. It is not the case for most other nations.

Even MA Marignon needs magic weapon. In 1 battle against vampires, vampires' attack are nearly all repelled, so fate weaving is not triggered. Yes, repel is the very first check in the melee combat code and if repel happens, no other effect will trigger.

My knights of chalice can not cut through 25 prot, so both sides just standing there hit air. It is kinda hilarious.

Originally posted by Zonk:

Through the game, using Mari Chalice, fate weaving feels pretty good for a S5 bless. In my opinion, it is OK for S6 and Misfortune 1. I just can not take it as at the current S7 price point.

Are you considering that "Cursed Luck" also negates Luck on the unit? That must be worth something because mass Luck (for example Will of the Fates) is nice.

EDIT: Maybe it should negate Twist Fate too.

[greengrocer]

It was a disciple game with Ashdod and Phaeacia. I actually had F6S7N7B4 awake Baphomet after MA Marignon F2S2 bonus.

My Mari mostly provided the bless, update allies capitals and forged some lanterns. The other 2 guys did most of the heavy lifting.

I probably will not use such a bless in single player. After all, fate weaving needs the enemy to take a swing at your sacred first, to run the 16 vs MR check. After you win the 16 vs MR check, it is 50% chance if it is working.

Flags could easily die after the first enemy swing. I mostly mix them with other non-sacred like swordsman, to make sure somebody will survive to take advantage of the effect.

No, it does not last after battle. I am not sure how long cursed luck lasts. I do remember seeing it fade away after maybe 4-5 turns in combat.

Originally posted by ItWasn'tMe:

So what you're saying is that +att +str(maybe bloodsurge) +fateweaving MA Marignon FAB pretender (maybe dormant) is a thing?

Lets see: MA Marignon has fire damage (somewhat easy to counter without magic), some astral (MR can counter). With your math and +str its like 20 damage flags and about 24 damage knights. That's pretty good, pretty much a 1 shot on most indie units.

I haven't really used cursed luck but does it last after the battle?