Dice vs Armor table

[Wildhorn]

Without being a math guru, but on average, if you reroll 50% of time your dice and then you 50% chance to get average or more damage, it means an increase of about 25% damage. So take the chart up here and increase numbers by 25%.

[jacksmack]

Without being a math guru, but on average, if you reroll 50% of time your dice and then you 50% chance to get average or more damage, it means an increase of about 25% damage. So take the chart up here and increase numbers by 25%.

Its not 50% chance to roll average or more.

With 1 and 2 dice it is 33,3% to roll average, 33,3% to roll above and 33,3% to roll below.
Which means its 66,6 to roll average or above.

I believe the chance of rolling excately average drops from 3 dice and up. Thus the chance of rolling "Average or above" drops as well.

[webcatcher]

I'll let someone else do the numbers on this one but an attack rolling a low number of dice definitely benefits more on average from this ability than one rolling a high number because the standard deviation goes down as you add more dice.

[Wildhorn]

Without being a math guru, but on average, if you reroll 50% of time your dice and then you 50% chance to get average or more damage, it means an increase of about 25% damage. So take the chart up here and increase numbers by 25%.

Its not 50% chance to roll average or more.

With 1 and 2 dice it is 33,3% to roll average, 33,3% to roll above and 33,3% to roll below.
Which means its 66,6 to roll average or above.

I believe the chance of rolling excately average drops from 3 dice and up. Thus the chance of rolling "Average or above" drops as well.

You are right.

[ringkichard]

In the abstract, damage re-rolls aren't that good. For example, with 1 die and no armor, a reroll has 1/3 chance of contributing 1 damage, on average.   Expected value of 1/3 damage per roll.

With two dice and no armor, a reroll has a total of 2/9+2/9 = 4/9 expected damage improvement per attack.

With 3 dice and no armor, a reroll has 3/27 + 6/27 + 6/27 = 5/9 expected damage improvement per attack.

With 4 dice and no armor, a reroll has 4/81 + 12/81 + 20/81 + 12/81 = 48/81 expected damage improvement per attack.

And so on. The extra damage is, overall, quite low when considered strategically. It's tactically very useful, however, potentially halving the chance of failure on an important roll.

--
Edit, I should clarify that's fractional damage added to the mean damage per attack. Compare that to Bear Strength's 2 damage.

[Kharhaz]

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

You can talk about promo cards

[fas723]

@DeckBuilder
Hummm Akiro's Favour, why didn't I think of that one? I have it here somewhere myself.
Ok, I see what you are saying. It can be done. Give me some time and I'll see what results I can bring.

Basically what I think I will do is to run each roll twice. For the first roll I just eliminate the lower half of the results, and merge it together with the second roll. In theory it will give twice as many "good" results as "bad".

Without being a math guru, but on average, if you reroll 50% of time your dice and then you 50% chance to get average or more damage, it means an increase of about 25% damage. So take the chart up here and increase numbers by 25%.

Its not 50% chance to roll average or more.

I think you are partly right both of you. In a perfect normal distribution half (almost) of the samples falls below average and half (almost) falls above. In the three dice example it is not a continues distribution, rather a step vise one, and in these cases the exact average [my] can contain a substantial portion of the solution space.  The more dice there is, the closer to a smooth distribution it will be.
The 25% increase is however a quite rough approx. This depends on number of dice and armor (It will maybe occur but in one or two cases perhaps).

[Aylin]

New averages with reroll: (all values are approximate, with ~ +/- 0.05 error)

                1 armour   2 armour   3 armour   4 armour   5 armour   6 armour
1 die         1.33           0.99           0.82           0.82           0.82           0.82
2 dice        2.46           1.91          1.54            1.46          1.46            1.42
3 dice        3.55           2.82          2.32            2.15          2.06            2.02
4 dice        4.60           3.85          3.20            2.90          2.66            2.64
5 dice       5.76            4.90          4.12            3.62          3.41            3.27
6 dice        6.8             5.89          5.05            4.48          4.10            3.89


Based on this in the 1-6 dice range Akiro's Favor adds between 1/3 and 1 die worth of damage, with it being worth "more dice" against higher armoured targets or if you're already rolling a lot of dice.

Adding in the rest of the armour and dice values will be fairly simple if more is requested.

In each scenario, the dice were rolled, converted to the no damage/normal damage/critical damage we have here in Mage Wars, then checked against the average (rerolling the dice iff we got below average damage). After 5000 trials of each, the damages were averaged.

Do note that the method I used didn't allow for standard deviations due to memory concerns, so those will not be appearing.

Code

There is a practical issue as well to determining Akiro's Favor's "worth"; with so many combinations of dice and effective armour, people are not likely to reroll only if they get below average (they may reroll higher than average results or keep below average results, for example). That is outside the scope of what I'm able to analyze with this program, but it is something to consider.

EDIT: Also important to note is that the reroll would probably go to the effect die first if needed, since that has a more tangible result in most cases.

I suppose on a personal note I should be a little offended, since evidently I wasn't considered capable of solving the problem.

[fas723]

@All - Edit
New updated tables 2014-02-24

@Aylin
Ahh didn't see you had done this over nigh. I made a chart as well and processed it all night. My figures are almost the same (which is good), but there are some differences due to that you have used 5000 samples for each while I have used all possible combinations.

@All

Akiro's Favour - table                                                                                                          1                       2                       3               Armor       μ-σ       μ       μ+σ       μ-σ       μ       μ+σ       μ-σ       μ       μ+σ       0       0,31       1,33       2,35       1,13       2,67       4,21       1,88       4,89       7,90       1       0,00       1,00       2,00       0,59       2,21       3,83       1,27       3,60       5,94       2       -0,24       0,83       1,91       0,20       1,82       3,44       0,80       2,57       4,34       3       -0,24       0,83       1,91       0,08       1,71       3,35       0,56       2,39       4,23       4       -0,24       0,83       1,91       0,03       1,67       3,31       0,43       2,31       4,18       5       -0,24       0,83       1,91       0,03       1,67       3,31       0,40       2,28       4,17       6       -0,24       0,83       1,91       0,03       1,67       3,31       0,39       2,28       4,17       7       -0,24       0,83       1,91       0,03       1,67       3,31       0,39       2,28       4,17       resil       -0,24       0,83       1,91       0,03       1,67       3,31       0,39       2,28       4,17       incop       -0,12       0,56       1,23       0,09       0,96       1,84       0,29       1,74       3,20                                                                                                           4                       5                       6               Armor       μ-σ       μ       μ+σ       μ-σ       μ       μ+σ       μ-σ       μ       μ+σ       0       2,90       5,53       8,16       3,55       8,02       12,50       4,40       9,58       14,76       1       2,01       5,05       8,09       2,78       6,55       10,31       3,58       8,06       12,54       2       1,47       3,88       6,29       2,18       5,23       8,28       2,91       6,67       10,42       3       1,09       3,60       6,12       1,76       4,19       6,63       2,44       5,47       8,50       4       0,86       3,44       6,02       1,44       4,01       6,58       2,01       5,21       8,41       5       0,76       3,37       5,98       1,27       3,92       6,57       1,74       5,04       8,35       6       0,72       3,34       5,96       1,19       3,88       6,57       1,60       4,95       8,30       7       0,72       3,33       5,95       1,16       3,86       6,57       1,53       4,90       8,27       resil       0,76       2,79       4,82       1,15       3,86       6,57       1,50       4,88       8,25       incop       0,57       2,12       3,68       0,86       2,43       4,01       1,13       2,70       4,27                                                                                                           7                       8                       9               Armor       μ-σ       μ       μ+σ       μ-σ       μ       μ+σ       μ-σ       μ       μ+σ       0       5,25       11,13       17,00       6,11       12,67       19,24       6,97       14,22       21,46       1       4,41       9,59       14,77       5,24       11,12       17,00       6,09       12,66       19,23       2       3,68       8,13       12,59       4,47       9,63       14,79       5,28       11,14       17,00       3       3,14       6,84       10,54       3,86       8,25       12,64       4,61       9,71       14,81       4       2,58       6,53       10,48       3,40       7,02       10,65       4,09       8,39       12,69       5       2,38       5,54       8,71       2,95       6,81       10,68       3,53       8,14       12,76       6       2,17       5,44       8,71       2,64       6,66       10,68       3,33       7,08       10,84       7       2,05       5,39       8,72       2,46       6,56       10,66       3,08       6,99       10,89       resil       1,96       5,34       8,72       2,30       6,46       10,62       2,80       6,87       10,94       incop       1,38       3,66       5,95       1,70       3,92       6,13       2,01       4,13       6,26                                                                                                                                                                                                                                                                         Akiro's Favour gain - table (Akiro's Favour table vs Standard table)                                                                                                  1       2       3       Armor       μ       μ       μ       0       0,33       0,67       1,89       1       0,33       0,76       1,31       2       0,33       0,71       0,76       3       0,33       0,69       0,78       4       0,33       0,67       0,78       5       0,33       0,67       0,78       6       0,33       0,67       0,78       7       0,33       0,67       0,78       resil       0,33       0,67       0,78       incop       0,22       0,30       0,74                                                                                                   4       5       6       Armor       μ       μ       μ       0       1,53       3,02       3,58       1       1,85       2,41       2,97       2       1,29       1,80       2,36       3       1,34       1,22       1,73       4       1,35       1,31       1,84       5       1,34       1,35       1,88       6       1,34       1,36       1,89       7       1,33       1,36       1,88       resil       0,79       1,36       1,88       incop       0,79       0,77       0,70                                                                                                   7       8       9       Armor       μ       μ       μ       0       4,13       4,67       5,22       1       3,53       4,08       4,63       2       2,91       3,47       4,03       3       2,28       2,83       3,40       4       2,45       2,17       2,73       5       1,75       2,34       2,95       6       1,81       2,43       2,21       7       1,84       2,46       2,30       resil       1,84       2,46       2,37       incop       1,33       1,25       1,13                                                                                                                                                                              As you said Aylin, Akiro's favour doesn't give much statistically. I would say that it is most suitable when rolling effect dice. Then you only have one die which gives the highes effect of Akiro (apply even for attack dice, see table). If you want to see the code just go to the link in the first post and download the Excel. I have updated the file in there. @Deckbuilder Happy?

[ringkichard]

Based on this in the 1-6 dice range Akiro's Favor adds between 1/3 and 1 die worth of damage, with it being worth "more dice" against higher armoured targets or if you're already rolling a lot of dice.

Thanks for doing this. It's good to have my intuition confirmed that AF is good against Resiliant.

There is a practical issue as well to determining Akiro's Favor's "worth"; with so many combinations of dice and effective armour, people are not likely to reroll only if they get below average (they may reroll higher than average results or keep below average results, for example). That is outside the scope of what I'm able to analyze with this program, but it is something to consider.

That's a good point. Against resiliant it's mercifully easy to calculate expected damage, but e.g. a 6 die attack vs 3 arm isn't so simple. My intuition says the breakpoint is 4 (resisting the urge to cheat now and look it up), but I'm not sure I'd reroll a 3 unless it would mean the death of the target.

Which its the real interesting use of AF, I think. If I have to kill a target with an attack, it gives a strong % boost to my chance of success.

(unreached musings ahead) AF could find good use on something like Royal Archer against a swarm of foxes, significantly increasing the chances of a one hit kill (~.4 -> ~.6 ; approximately 50% improvement), which I think is on par with Hawkeye.

[DeckBuilder]

Yes, very much, thank you both for your hard work.
 
There seem to be a few anomalies in both results
With Aylin (e.g. 1 die vs. 2+ should be constant, 2 die vs. 4+ should be constant, I attributed it to her Monte Carlo sampling technique.
But it seems you have used an "every permutation" mathematical model? (Wow!)

I am a very simple person so I will just look at 1 die vs. 1 armour
Half the time (0, 0, 1), I will score 0 net damage so I will re-roll.
This re-roll gives me a 1/3 chance to score 1 damage (2, 1*) and 1/6 chance to score 2 damage (2*)
Which equates to +0.67 damage in 50% situations which equates to +0.33 damage.
Yet when I look at the delta chart of 1 die vs. 1 armour, I see 0 (yet +0.33 in 1 vs. 0).
I'm sure there must be something I've not considered here but I don't know what it is...

Other anomalies was me expecting the deltas to erode away with extra armour but it doesn't always erode?
E.g. in 7-9 dice attacks, the deltas above 2 armour feels random (for a method that doesn't use sampling).

I suspect some of these issues may due to difficulty transposing tables into posts, errors creep in easy.
None of this takes away from the hard work that you (and the strangely quiet Aylin) have done.
And I am very grateful for this feedback (and so amazingly quickly too). Many thanks.

At first glance, it seemed like my maths intuition was wrong about its benefit.
But if I identify how much +1 die gains in original table, deltas are comparable and in some cases superior.

I think what can't be captured is probability of hitting a target damage level (anything excess is irrelevant).
Why do you say the deltas on standard deviation is "not possible" to demonstrate lower variability / higher consistency?

In some ways, this may demonstrate luck plays less of a role in the attack dice than any I ever anticipated.
Which is heartening (and explains my initial phobia of d12 effects).

Yes, Akiro's Favour is most definitely geared to the effect die.
But it's also a good insurance against outlier poor attack rolls.
And grants more certainty achieving a target kill damage level.

Hmmm, I was surprised by these results but these gains have to be compared against +1 die and +2 dice.

Very thought-provoking. Thank you, both of you!

[Aylin]

There seem to be a few anomalies in both results
With Aylin (e.g. 1 die vs. 2+ should be constant, 2 die vs. 4+ should be constant, I attributed it to her Monte Carlo sampling technique.

Yes; different runs of the program also give slightly different results due to this.

I am a very simple person so I will just look at 1 die vs. 1 armour
Half the time (0, 0, 1), I will score 0 net damage so I will re-roll.
This re-roll gives me a 1/3 chance to score 1 damage (2, 1*) and 1/6 chance to score 2 damage (2*)
Which equates to +0.67 damage in 50% situations which equates to +0.33 damage.
Yet when I look at the delta chart of 1 die vs. 1 armour, I see 0 (yet +0.33 in 1 vs. 0).
I'm sure there must be something I've not considered here but I don't know what it is...

No, you are correct here. Explanation below.

I suspect some of these issues may due to difficulty transposing tables into posts, errors creep in easy.
None of this takes away from the hard work that you (and the strangely quiet Aylin) have done.
And I am very grateful for this feedback (and so amazingly quickly too). Many thanks.

In all honesty, I haven't played Mage Wars since early-January. The group I play with has been focused on playtesting some games made my a local designer.

Very thought-provoking. Thank you, both of you!

You're welcome.

@Fas723

I believe there is a mistake in your table. For 1 die against 2+ armour/resilient, it should be:
1/6 + 2/6 + 2/3(1/6+2/6) = 1/2 + 2/3*1/2 = 3/6 + 2/6 = 5/6 = ~0.833333
whereas you have 0.50. Perhaps you should double-check your other numbers as well, to ensure they are free from error.

Also, could you post your method? Your 5 and 6 die examples are much greater than mine, and I'm unsure if it's due to an error in my program or not. In either case, I'm interested in seeing how you did it.

[fas723]

There is something fishy with my table. My results are equal to Aylins results but at a different row...and they don't seems to be completely correct. I'm not sure where these error originates from, if it is from the input or calculation. Don't use my table until I have reviewed it. I'll be back with an update. Sorry...

[fas723]

@Aylin
Again you posted just before me. 
The code is within the Excel, maybe you want it in another format?
I saw that the file I have shared also were not updated, so there must have been something strange going on before I left home today. This is what happens when you rush things. 

[Aylin]

@Aylin
Again you posted just before me. 
The code is within the Excel, maybe you want it in another format?
I saw that the file I have shared also were not updated, so there must have been something strange going on before I left home today. This is what happens when you rush things. 

I don't see it in the excel file. It might be because I'm using Libre Office (my OS is Mint, not Windows). When I open it all I see are a bunch of '#'s.

If you could put it into another format that would be great.