OW describes the "Zeroing Effect" for Begginers

[OmikronWarrior]

Hinjo: We are willing to reduce your [Belkar's] prison sentence by 5 years if you contribute to the defense of Azure City by helping defeat the invaders.

Belkar: Well I'm no legal expert, but I'm pretty sure I haven't actually been convicted of anything.

Roy: Wait! He pleads guilty.

Belkar: What? No I don't.

Roy: Yes, you do.

Belkar: No, I don't.

Roy: Yes, you do! Look, I'm the one who talked Hinjo into reducing the charges to manslaughter... Now the minimum penalty for manslaughter in Azure City is four years in prison. Now do you see where I'm going with this?

Belkaer: … No.

Roy: *sigh* You plead guilty, get sentenced to four years, then defend the city... and your sentence gets reduced by five years.

Belkar: How do the hell do I server -1 years in prison?

Roy: You don't, you idiot, you get set free!

Belkar: Oh, I get it. Ok, then, I plead guilt.

Hinjo: Very well... for the crime of voluntary manslaughter, I sentence you to spend a term in prison equal to six years.

Roy: Ah, crap.

-Rich Burlew, Order of the Sick #420

Roy probably should not have discussed cheating the legal system in front of Hinjo, but it is Belkar we are interested in. He isn't the sharpest tool in the shed when it comes to adding and subtracting. Unfortunately, a few to many of my fellow Pretenders do not understand how negative numbers work in the Dominions 3 world. For example, let us look at the spell Relief. On paper, it seems underwhelming. Though it affects every friendly unit on the battlefield, it only restores a scant 1 fatigue per round. Yet, experienced players swear by this spell and its ability to save armies in the long battles. Why the discrepancy? It is due to something I call the “Zeroing Effect.”

Simply put, the spell doesn't restore 1 fatigue, but like most Dominions mechanics restores 1+DRN-DRN. This is crucial. A novice in statistics might say, “so what? This may introduce some variance, but don't the two random rolls cancel each other out in the long run and thus the spell still averages 1?” No! If the negative random number is greater, does the unit loose fatigue? It does not. Say the spell does -9 fatigue reduction. This becomes zero. Meanwhile, the equally likely 10 fatigue reduction actually does 10 fatigue reduction and thus the two do not “cancel” each other out to leave the average at 1. The actual average will be higher.

What does this mean for dominions? It means spell that have a low effect on paper actually have a much greater effect in practice. Freezing Mists, Bone Grinding, etc. become much more worthwhile to consider casting. Though the math is very hard to calculate, I tentatively will say Relief averages at least 5 points of fatigue reduction per round. Thats almost enough to ignore the fatigue reduction from Heat from Hell, provided the battle does not last too long.

Obviously, the further the base number is from zero the less influence the “zeroing effect” has. Therefore, the ideal spell to take advantage of this phenomenon has: a) an unassumingly low “effect.” b) affects multiple targets, the more, the better. c) has no other check such as protection (meaning Armor Negating) or Magic Resistance. Now, with this mind, hit the grimoire and start rethinking magic and randomness in the Dominions 3.

[Psycho]

Freezing Mists is evil. I had my huge sacred yavana army reduced to nothing in only a few castings by Ctis's W9 A-something pretender when storming the capital.

[WraithLord]

Psycho, can you please give more details re. this battle and freezing mists effect. like:
What was the temperature?
Was it under quagmire?
Other globals?
How much damage did the mists do, max, avg (your impression of course)?

[Psycho]

Both me and Ctis had heat dominion. No quagmire. No globals. He had about 5 sauromancers in his capital, some low end summons and his lady of rivers. I can't say how much one freezing mists did on average. They stay for three turns and with the W9 of the pretender, they had a huge AoE. My army was blocked on the entrance and got covered by several clouds. It was over very fast for me. In my case it didn't matter, but it's worth knowing that this is AN damage. And the funniest thing is that Ctis player didn't even script it. He just had that much evoc researched.

[WraithLord]

Thanks for the details.
I wonder whether or not CR 50% would have made a significant difference.

[Fantomen]

It doesn´t, I´ve played a lot with freezing mists as LA Atlantis. It is very effective against anything without 100% CR.

[Baalz]

Yeah, I mentioned that in my LA Atlantis guide. That spell is so good as a combo move its worth passing out some gems to cast if you need to, but you really need to add in winter ward if you're using it with your troops because it tears up even cold resistant heavy infantry very quickly. Works great with skellispam.

[Agema]

If anyone wants to check this sort of phenomenon mathematically, I've attached an MS Excel file I made months ago. I don't know programming, so I constructed it longhand with Excel, and it's accurate up to DRN rolls about 40-50, which means it's over 99.99% accurate. The stuff you want is on the first page, the later pages are my long-winded working, which I've tripled checked and should be right.

* * *

First table lists the chance of any one particular roll happening, the cumulative probability of getting that number or less, and the probability of not getting that number or higher.

Second table lists the DRN vs. DRN (Att v Def) as you'll find early in the manual. Note my figures differ from the manual; I'm pretty sure my working is accurate and that I've understood the concepts, so I'm not sure where the disparity arises. Whatever, the differences are small.

Third table lists the average damage per hit from a hit (Str-Prot). You could tell from this table than a 0AN damage hit would do an average of 2.4 damage per hit, and the Relief will gain on average 2.9 per unit per round.

Fourth table lists how mages you're likely to lose with astral duels depending on the opposition.

[Pehmyt]

Originally Posted by Agema

If anyone wants to check this sort of phenomenon mathematically, I've attached an MS Excel file I made months ago. I don't know programming, so I constructed it longhand with Excel, and it's accurate up to DRN rolls about 40-50, which means it's over 99.99% accurate.

I think I should comment that as far as I can understand your spreadsheet, the results are ~10% off as soon as the open-ended rolls enter the picture. Did you take into account that 6 equals "5 +extra roll"? Not that it matters for the present discussion.

For example, the probability of getting 7 should be that of throwing one of (2;5), (3;4), (4;3), (5;2), (6,1;1), (1;6,1) (with ";" separating the two dice), that is 4/36 + 2/216 = 13/108 = 0.1204.

The results corresponding your column A can be computed exactly by writing the value of DRN roll as X=5k+r+1, with r between 0 and 4, and using

P(DRN = 5k+r+1) =  ((6-r)5k +r )/6^(k+2)

The chances of DRN being at least as large as a given number X=5k+r+1 can also be computed exactly, although this is not directly useful. It could be practical for generating a table similar to that in manual without losing too much precision. For 6-sided dice it reads

P(DRN >= 5k+r+1) = ( (48 - 13r + r^2)5k + 72 + r - r^2 )/( 2 * 6^(k+2) )

[Agema]

Originally Posted by Pehmyt

I think I should comment that as far as I can understand your spreadsheet, the results are ~10% off as soon as the open-ended rolls enter the picture. Did you take into account that 6 equals "5 +extra roll"? Not that it matters for the present discussion.

For example, the probability of getting 7 should be that of throwing one of (2;5), (3;4), (4;3), (5;2), (6,1;1), (1;6,1) (with ";" separating the two dice), that is 4/36 + 2/216 = 13/108 = 0.1204.

The results corresponding your column A can be computed exactly by writing the value of DRN roll as X=5k+r+1, with r between 0 and 4, and using

P(DRN = 5k+r+1) =  ((6-r)5k +r )/6^(k+2)

The chances of DRN being at least as large as a given number X=5k+r+1 can also be computed exactly, although this is not directly useful. It could be practical for generating a table similar to that in manual without losing too much precision. For 6-sided dice it reads

P(DRN >= 5k+r+1) = ( (48 - 13r + r^2)5k + 72 + r - r^2 )/( 2 * 6^(k+2) )

Yes, I did account for 6s being re-rolled, however there was an error in my working, I misplaced some columns in my working as if an extra roll was calculated as 6+d6, not 6-1+d6, which was where the 10% arrived. I've revised my tables accordingly, and I'll check them agains the book and repost them.

I had a look at those equations, and although my maths is pretty rusty these days I don't think they work for 2d6 which are the basis of DRNs in Dom 3. I'm not entirely clear what k and r represent - k = number of 6s rolled and [i]r[/r] = d6-1 excluding 6? It would work with something more like X=5k+r+r'+2.