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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.
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
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...]
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%.
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.
I may get in trouble for being so frank...
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.
Akiro's Favour - table
Akiro's Favour gain - table (Akiro's Favour table vs Standard 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? :) |
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.
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.
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.
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...
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.
Very thought-provoking. Thank you, both of you!
@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. :P
@Aylin
Were you able to reach the code now?
You can see from this analysis that the expected damage for the 4-dice piercing attack is 4 whereas the expected damage for the 6-dice attack is only 3.74.
Any different from this?
http://forum.arcanewonders.com/index.php?topic=13562.0 (http://forum.arcanewonders.com/index.php?topic=13562.0)
You are going to make Gozer madAny different from this?
http://forum.arcanewonders.com/index.php?topic=13562.0 (http://forum.arcanewonders.com/index.php?topic=13562.0)
I am merging the threads 8)
You are going to make Gozer madAny different from this?
http://forum.arcanewonders.com/index.php?topic=13562.0 (http://forum.arcanewonders.com/index.php?topic=13562.0)
I am merging the threads 8)
A graph that shows when piercing is better than +2 dice for many combinations of dice and armor would be very helpful to this discussion. Tables of numbers are great but it can be hard to see patterns in them.
Base Attack | No Armour | 1 Armour | 2 Armour | 3 Armour | 4 Armour | 5 Armour |
1 | 2 | 1.296 | 0.815 | 0.611 | 0.856 | 1.005 |
2 | 2 | 1.198 | 0.593 | 0.259 | 0.642 | 0.914 |
3 | 2 | 1.132 | 0.428 | -0.029 | 0.404 | 0.76 |
4 | 2 | 1.088 | 0.307 | -0.259 | 0.168 | 0.569 |
5 | 2 | 1.059 | 0.219 | -0.44 | -0.046 | 0.364 |
6 | 2 | 1.039 | 0.156 | -0.581 | -0.234 | 0.162 |
7 | 2 | 1.026 | 0.111 | -0.688 | -0.394 | -0.028 |
8 | 2 | 1.017 | 0.078 | -0.769 | -0.525 | -0.199 |
9 | 2 | 1.012 | 0.055 | -0.83 | -0.632 | -0.349 |
10 | 2 | 1.008 | 0.039 | -0.876 | -0.717 | -0.476 |