Dice vs Armor table

[fas723]

After some requests I made a Die vs. Armor table where you can see how much expected damage you will deal depending on your amount of dice vs. the amount of armor.

Tables in here always look strange. I hope you can read it. I have put it in here for you who don't want to download it. If you like to print the table I made an ok looking tab in Excel suitable for this which you can bring to your games. In there where you also find the calculation. You find it here:

https://drive.google.com/folderview?id=0Bz0fnLKKUKlxaUtBU1dhcDNCTkE&usp=sharing

It is quite hard to write large tables right into the forum, so please let me know if you find any errors. The printable version should be fine though, so it is safe.

The table show the expected value in a normal distribution based on the amount of dice vs. your opposing armor. The two columns next to each "my" are one standard deviation, "sigma", up and down from the expected value. When you read the table "my" will represent the most likely result and within +/- 1 "sigma" you have a 68.2% hit rate.

Note that resiliant can be interpreted as infinit armor. In this table you can use armor levle 9 to simulate this as long as the amount of dice do not exceeds 7 4. EDIT: It should be 4 dice not 7. Table updated with Resilient and Incorporeal.

Enjoy!

1                       2                       3               Armor       μ-σ       μ       μ+σ       μ-σ       μ       μ+σ       μ-σ       μ       μ+σ       0       0,18       1,00       1,82       0,85       2,00       3,15       1,59       3,00       4,41       1       -0,08       0,67       1,41       0,38       1,44       2,51       0,97       2,30       3,62       2       -0,26       0,50       1,26       0,06       1,11       2,16       0,54       1,81       3,09       3       -0,26       0,50       1,26       -0,04       1,03       2,09       0,32       1,61       2,90       4       -0,26       0,50       1,26       -0,08       1,00       2,08       0,21       1,52       2,83       5       -0,26       0,50       1,26       -0,08       1,00       2,08       0,19       1,50       2,82       6       -0,26       0,50       1,26       -0,08       1,00       2,08       0,18       1,50       2,82       7       -0,26       0,50       1,26       -0,08       1,00       2,08       0,18       1,50       2,82       Resil       -0,26       0,50       1,26       -0,08       1,00       2,08       0,18       1,50       2,82       Incorp       -0,14       0,33       0,80       0,00       0,67       1,33       0,18       1,00       1,82                                                                                                           4                       5                       6               Armor       μ-σ       μ       μ+σ       μ-σ       μ       μ+σ       μ-σ       μ       μ+σ       0       2,37       4,00       5,63       3,17       5,00       6,83       4,00       6,00       8,00       1       1,64       3,20       4,76       2,37       4,13       5,90       3,13       5,09       7,04       2       1,10       2,59       4,08       1,74       3,43       5,12       2,43       4,31       6,19       3       0,79       2,26       3,73       1,32       2,97       4,62       1,92       3,74       5,56       4       0,60       2,09       3,58       1,05       2,70       4,35       1,57       3,37       5,16       5       0,51       2,03       3,54       0,90       2,57       4,25       1,35       3,16       4,97       6       0,48       2,00       3,53       0,83       2,52       4,21       1,22       3,06       4,90       7       0,47       2,00       3,53       0,80       2,51       4,21       1,16       3,02       4,88       Resil       0,47       2,00       3,53       0,79       2,50       4,21       1,13       3,00       4,87       Incorp       0,39       1,33       2,28       0,61       1,67       2,72       0,85       2,00       3,15                                                                                                           7                       8                       9               Armor       μ-σ       μ       μ+σ       μ-σ       μ       μ+σ       μ-σ       μ       μ+σ       0       4,84       7,00       9,16       5,69       8,00       10,31       6,55       9,00       11,45       1       3,93       6,06       8,18       4,76       7,04       9,32       5,60       8,03       10,46       2       3,16       5,22       7,28       3,93       6,16       8,38       4,73       7,11       9,49       3       2,57       4,56       6,55       3,27       5,42       7,57       4,00       6,31       8,62       4       2,14       4,09       6,03       2,76       4,85       6,95       3,42       5,66       7,91       5       1,85       3,79       5,74       2,40       4,47       6,54       2,99       5,19       7,39       6       1,66       3,63       5,59       2,15       4,23       6,32       2,68       4,87       7,07       7       1,56       3,55       5,54       1,99       4,10       6,21       2,47       4,69       6,91       Resil       1,48       3,50       5,52       1,84       4,00       6,16       2,21       4,50       6,79       Incorp       1,09       2,33       3,58       1,33       2,67       4,00       1,59       3,00       4,41

[webcatcher]

Very helpful, thanks.

[Aylin]

Awesome! 

[jacksmack]

Its clear you got math skills i do not posses.

But I have a hard time accepting that rolling 7 dice vs 9 armor = 3.5 average damage.
Average roll per dice = 0.5 normal and 0.5 crit.
So rolling 7 dice = average 3.5 normal and 3.5 crit.
Just taking the 3.5 crit and it matches your number from the table.
However... sometimes the normal damage will spike up and actually exceed the 9 armor. Shouldnt that add something to the 3.5 average?
Is it because its so rare that it has been rounded down to 3.5? - my guestimate would be that 7 dice do closer to 4 damage on average than 3.5

Especially the 3 dice table interests me because of the hydra.
The 1.61 average vs 3 armor .  i thought it would be higher - the normal damage only provides 0.11 on average.

I going to make a postulate and there is a big chance im wrong here:
Your table somehow takes into account the chance of armor being applied - 3 dice has almost 30% (29.7% IIRC) chance of not triggering armor due to only crits and nulls being rolled - but somehow doesnt take spikes of normal damage exceeding the armor into account.

Like i said - my foundation of math to base all this on is both weak and flawed so dont take it amiss.

[svvcDark]

Hooray for statistics!

Thanks for this. It's great.

[IndyPendant]

Actually jacksmack, his math is good here.  I think the problem here is twofold: 1) fas is tabling probabilities, while you're describing statistically improbable anecdotes; and 2) fas has decided to round to 2 decimal places for his table, which (as you speculated) tends to indicate no difference where, statistically speaking, there are indeed very small differences.  However, those differences are negligible:

The chance of spiking normal damage does add something to the average; a very small something.  In your example (7 dice vs 9 armour), the average damage actually works out to 3.505 (rounded).  If crits were treated like normal damage for the purposes of armour, the average would be a flat 3.500; the chance of spike-normal damage adds that 0.005 variance.  Which, for the purposes of players in the game, can safely and accurately be rounded to 3.50.

[fas723]

Good that you liked it!

Anyone who printed it out? Did it work  out ok?

I will correct my statement regarding Resilient, and update the OP. I wasn't thinking clear when I mentioned 7 dice to be the limit. It is rather 4 dice. So, the table is ok for Resilient with 9 armor and up to 4 dice.

Question:
How often have you seen 8 or 9 armor? Is it better I change this to Resilient and Incorporeal?  I don't want to expand the table because it won't fit one paper then (A4 size). If 7 armor could be removed the curve at the bottom could be enlarged as well. Comments?

[fas723]

Its clear you got math skills i do not posses.

But I have a hard time accepting that rolling 7 dice vs 9 armor = 3.5 average damage.
Average roll per dice = 0.5 normal and 0.5 crit.
So rolling 7 dice = average 3.5 normal and 3.5 crit.
Just taking the 3.5 crit and it matches your number from the table.
However... sometimes the normal damage will spike up and actually exceed the 9 armor. Shouldnt that add something to the 3.5 average?
Is it because its so rare that it has been rounded down to 3.5? - my guestimate would be that 7 dice do closer to 4 damage on average than 3.5

Especially the 3 dice table interests me because of the hydra.
The 1.61 average vs 3 armor .  i thought it would be higher - the normal damage only provides 0.11 on average.

I going to make a postulate and there is a big chance im wrong here:
Your table somehow takes into account the chance of armor being applied - 3 dice has almost 30% (29.7% IIRC) chance of not triggering armor due to only crits and nulls being rolled - but somehow doesnt take spikes of normal damage exceeding the armor into account.

Like i said - my foundation of math to base all this on is both weak and flawed so dont take it amiss.

Jacksmack,
I think Indy explained it well. I have rounded the figures with 2 decimals. If you like to see the "real" value, go into the calc tab in the Excel and scroll down to the copied table. Either you change the shown decimals or you just click the cell you like to see and it will show the result with all (almost) decimals. 7 dice vs 9 armor have true expected value of 3,5049189814925.

As Indy said the spikes are there. In the 7 dice vs 9 armor case you have to roll at least 8 point of normal damage with your 7 dice (not including all your perfect rolls). This will occur in just a fraction (0,04%) of the times you roll something else. That is why it give such small impact.

Regarding the 3 vs 3 Hydra case it is the same thinking. With 3 armor you have to roll at least 4 normal damage to make an impact. This will occur 3% of the times you make that roll (if i'm not mistaken...).

Use the standard deviations to get a better feel for your probabilities. 

[Shad0w]

I will be moving this

[ringkichard]

So after some procrastination, I got around to looking at this, and I like it a lot.
I'd been working on something similar, but I'd been hoping to avoid using a complete brute force solution and instead try to automate a permutations / combinations solution. The fact that you beat me to it shows how well that went for me

If you CC license your code, I wouldn't mind stealing your VBA (now that I know that LibreOffice for Mac supports Basic macros) and extracting a bit more data. Not that I'm anything but a dabbling amateur, but standard deviation never really feels like a good substitute for graphing the whole distribution, and it gives me a good excuse to make pretty pictures.

-- EDIT

I can't get the macro to run on my system, even after changing its name. This is likely my fault, as I don't really know what I'm doing, but it could also be version incompatibility. It was nice to see how you did it, though! If anyone wants to teach me a complete permutations / combinations solution to mage wars dice math, I'd be interested to learn!

[fas723]

So after some procrastination, I got around to looking at this, and I like it a lot.
I'd been working on something similar, but I'd been hoping to avoid using a complete brute force solution and instead try to automate a permutations / combinations solution. The fact that you beat me to it shows how well that went for me

If you CC license your code, I wouldn't mind stealing your VBA (now that I know that LibreOffice for Mac supports Basic macros) and extracting a bit more data. Not that I'm anything but a dabbling amateur, but standard deviation never really feels like a good substitute for graphing the whole distribution, and it gives me a good excuse to make pretty pictures.

-- EDIT

I can't get the macro to run on my system, even after changing its name. This is likely my fault, as I don't really know what I'm doing, but it could also be version incompatibility. It was nice to see how you did it, though! If anyone wants to teach me a complete permutations / combinations solution to mage wars dice math, I'd be interested to learn!

Glad you liked it. Sure go ahead and use the code, it is no rocket sience. As you might have seen already I just made a loop that rolled every signel possilbe out put, stored each result and went to "my" and "sigma" from there.

Humm...so you didn't get it to work even when you changed the function name as described?
Go in to the VBA editor. At the top for each function you find this code: "Function expected_value1(dice, armor)" & "Function sigma1(my, dice, armor)". Change both of them to "Function expected_value(dice, armor)" & "Function sigma(my, dice, armor)", without the "1". After this you have to go to one cell and hit "enter". I did this so to prevent the loop to start each time someone would open the file. It took my computer quite some time run through it all. At the end there is 6^9 (10077696 pcs) possible combination for each roll with 9 dice, and in here this is done 27 times (and then we have all other # dice 27 times as well). 

[fas723]

Table updated with Resilient and Incorporeal instead of Armor 8 and 9.

[DeckBuilder]

@Fas723

Can I challenge you to use your great maths skill to work out something please?

I roll X attack dice against Y armour (after piercing)
I always re-roll all X dice (once only) if I roll less than average net damage - but I must take the re-roll

How much extra damage is this ability to re-roll worth? (As X vs. Y tables above)
Could you then compare this to just rolling X+1 and X+2 dice once at the start?

If this is really complicated (or too challenging for you - play on the ego!), then no problem.
I've had a bet our Forum Statistician could solve this but it's not an issue if it's too hard for you.

[Savvy players will know the game development reason why I'm asking this huge favour...]

[fas723]

@Fas723

Can I challenge you to use your great maths skill to work out something please?

I roll X attack dice against Y armour (after piercing)
I always re-roll all X dice (once only) if I roll less than average net damage - but I must take the re-roll

How much extra damage is this ability to re-roll worth? (As X vs. Y tables above)
Could you then compare this to just rolling X+1 and X+2 dice once at the start?

If this is really complicated (or too challenging for you - play on the ego!), then no problem.
I've had a bet our Forum Statistician could solve this but it's not an issue if it's too hard for you.

[Savvy players will know the game development reason why I'm asking this huge favour...]

Sure I can help you the best I can (when I have time...). 
I don't really understand your example though. You are saying that you can re-roll your dice? How?
Ok, let's say you can, then you want to re-roll every time you are below average [my]. Is that correct? And your question is how much you would gain from doing that?

Well, if I get your question correct, I would say that in average you will get the difference between your first result and your next expected value [my]. Once you re-roll you will reset your chances of getting something.
If you compare this diff with the difference you get between X dice and X+1 dice you can then determine your best option.

To calculate this you must predict or choose a value for your first roll (which you later on re-roll). Or I can look for the breakeven point when you should have chosen to go for +1 depending on your first roll (I'm not sure why you want to know this since in a game you can never know which way you should go before you have made the first roll, and then it is to late.)

Did I understand your question correct?

[DeckBuilder]

Hi fas723

Thanks for getting back on this, it's much appreciated. I might as well be blunt.

There is a promo card called Akiro's Favour and its persistent enchantment benefit is this:
Once per round when attacking. this creature may either reroll ALL its attack dice using the new roll or reroll a d12
In my "Promo Cards Feedback" thread, I slated it (and Ballista) as overpowered as printed.

Now evaluating the re-roll of d12 (for effects or for Daze), even a dumbo like me can do that
For example: if I am Dazed, before I had 50% miss, now I have a 25% miss if I use it that way
Re-rolling a Daze miss takes priority of its use but 50% of the time, I would not need to use it that way
(Daze is situational, the effect die is worse, Arc Lightning's Stun 9+ becomes Stun 5/9 with reroll option)

What is beyond my rusty maths (maybe not SAS/SPSS/Excel) is to evaluate the value of the optional reroll all keep 2nd roll effect

My maths intuition tells me that it's worth between +1 (Hawkeye) and +2 (Bear Strength) dice
I also know that it will reduce the variability making the attack more certain to deal a threshold damage
Note the versatility of this spell - it's just awesome (especially for mages with all those control effects)

Because it's a promo card (and I have already slated its cost), I'm not divulging any design secrets here
I came to you because I worry AW may be making a big mistake here (they appreciate it's undercosted)
And I care deeply about the game and don't want a ubiquitous card so would like it costed appropriately

So can you please use your maths skills to evaluate the benefit of this card?
You would be doing the game an invaluable service if you could put your mind to this very soon?

(I'm a firm believer in tapping into a fan base and crowd sourcing, it's a free asset, everyone feels good, win-win)

I may get in trouble for being so frank...