Strife:Calculating Expected Statistics

'''This section is completely optional and for fun for those that enjoy maths. The following is the tools I made and use to balance the game, explained with examples so you can calculate your theoretical combat effectiveness.'''

DPA, DPTS, and DPT
To see your combat effectiveness in terms of damage, players can calculate several things with their means of damaging, i.e., weapons, spells, etc… To find the average of these, then you add the maximum and minimum magnitudes and then half it. To find the maximum of these then just use the maximum magnitudes. To find the minimum of these then just use the minimum magnitudes.
 * DPA: Damage Per Action. This is the total damage one action will cause. May be multiplied by the number of targets it will affect.
 * DPTS: Damage Per Turn Second. This is the total damage an action will cause per second caused. It is a great measure of comparing different damaging methods that take different time lengths out of your turn to use the action. To calculate this it is DPA/(Time taken to do damaging action)
 * DPT: Damage Per Turn. This is the total damage you do in a whole turn. It is all of the DPA’s you want to be in an action added together to total up to on or under your turn’s length.

'Note that for calculating the effectiveness of damaging actions, average DPTS is best'' as it theoretically can compare all actions individually with any other action. Criticals are not considered as are negligible in comparisons if you discount it everywhere, and overly complex.''' Average DPTS with criticals considered in the ranges ARE fun to look at, though!

Lets to a couple of examples in various sections below. First, one item, a glass longsword using it's Thrust attack: Its average DPA is $$(4+30)/2=17$$, with its Maximum DPA being 30 and minimum DPA being 4. This can be multiplied by Strength Level or any other modifier to calculate outgoing damage if needed. This makes its average DPTS $$17/2=8.5$$, maximum DPTS $$30/2=15$$ and minimum DPTS $$4/2=2$$, as it’s If we just do a turn of two hits (totalling 4 seconds) this is an average DPT of $$8.5 \times 4=17 \times 2=34$$, maximum DPT of $$15 \times 4=30 \times 2=60$$ and minimum DPT of $$2 \times 4=4 \times 2=8$$.

As a final note, to take short blade balancing into account for DPTS, we can instead use $$DPTS=(3 \times DPA)/4$$, as is the average of 3 actions over 4 seconds. To say larger short blades take 1 second to hit when under 75 Agility is a bit of an under-exaggeration!

Advantage or Disadvantage damage rolls
Sometimes, like with Hand-to-hand, you may have advantage or disadvantage damage. For Maximum and Minimum values this is trivial; just the Maximum and Minimum dice values, but for Average this is different:

For an Advantage dice: $$Average\ DPA=\sum_{i=1}^{dice}\frac{2i(i-1)}{(dice)^2}$$

For a Disadvantage dice: $$Average\ DPA=dice-\sum_{i=1}^{dice}\frac{2i(i-1)}{(dice)^2}$$

Where "dice" is the size of the advantage or disadvantage dice. The Average DPA can then be converted into Average DPTS by dividing by the time it takes to perform the action, i.e., if the time taken to perform the action is $$T$$, then:

For an Advantage dice: $$Average\ DPTS=\sum_{i=1}^{dice}\frac{2i(i-1)}{T(dice)^2}$$

For a Disadvantage dice: $$Average\ DPTS=\frac{dice}{T}-\sum_{i=1}^{dice}\frac{2i(i-1)}{T(dice)^2}=\frac{1}{T}\left(dice-\sum_{i=1}^{dice}\frac{2i(i-1)}{(dice)^2}\right)$$

The Average DPA can also be used in finding the Average DPT, and may even be roll weighted and compared with other forms of damage.

For an example of finding the Average DPTS of a 67 strength level heavy gauntlet hand-to-hand punch, we want an advantage roll at $$dice=67$$ and $$T=1$$, hence:

$$Average\ DPTS=\sum_{i=1}^{dice}\frac{2i(i-1)}{T(dice)^2}=\sum_{i=1}^{67}\frac{2i(i-1)}{1(67)^2}=\sum_{i=1}^{67}\frac{2i(i-1)}{4489}=\frac{1}{4489}\sum_{i=1}^{67} {(2i^2-2i)}\approx 45$$

Hence in this instance, this player has an Average DPTS of 45 when using their hand-to-hand punches. A single advantage or disadvantage may also be approximated by simply multiplying the average of that dice roll by 4/3 or 2/3 respectively.

Comparisons using Average DPTS (example)
We can compare two high level items with enchantments at 100 Strength Level using average DPTS: Mehrunes’ Razor and the Ice Blade of the Monarch. Therefore, at any strength level with equal long blade and short blade levels, the Mehrunes’ Razor is marginally better than the Ice Blade of the Monarch, assuming no resistances or weaknesses on the wearer (other than from the weapon).
 * Mehrunes’ Razor: $$Physical\ average\ DPTS=[(12+17)/2]/1=14.5$$, It’s weakness to poison is 5%, so the enchantment (poison) damage is multiplied by 1.05, so $$Enchantment\ average\ DPTS=([(6+20)/2]/1)\times1.05=13.65$$, hence $$average\ DPTS=14.5+13.65=28.15$$
 * Ice Blade of the Monarch: $$Physical\ average\ DPTS=[(5+50)/2]/2=13.75$$, $$Enchantment\ average\ DPTS=[(10+40)/2]/2=12.5$$, $$average\ DPTS=12.5+13.75=26.25$$

Roll Weighted Average DPTS (advanced)
Another way to compare is roll weighted average DPTS, taking the rolls you need to make into account. Roll weighted average DPTS is short-term inaccurate because you either succeed a roll in the moment or don’t but is a great way of comparing different damaging methods over a long period of use. Note: this discounts luck, as is negligible in comparisons if you discount it everywhere, and overly complex to calculate.

For example: Fire Storm. Fire Storm does a d10 of Fire damage for 10 second in 10ft on a target. Assuming this is on one target, it would do $$([(1+10)/2]/2)\times10=27.5$$ average DPTS, where the “×10” is the 10 seconds it lasts for. If the caster had 40 Willpower, using light armour, and 50 Destruction level, then they’d need to succeed an advantage willpower at level 40 $$1-(0.6\times0.6)=0.64=64\%$$ chance of success) and a 50 Destruction level standard roll (50% chance of success) giving the full roll combination $$0.5\times0.64=0.32=32\%$$ chance of success, as you need to succeed both. This means the roll weighted average DPTS is $$0.32\times27.5=8.8$$

Comparisons using Roll Weighted Average DPTS (example)
As an example, I will bring back Mehrunes’ Razor average DPTS = 28.15 and Ice Blade of the Monarch average DPTS = 26.25



We can take these comparisons a step further: Say if $$M\times28.15=N\times26.25$$ then $$N/M\approx1.07$$, meaning if we do some analysis (image), we can see that if N is the long blade skill level and M is the short blade skill level, then over the long term, using the Ice Blade of the Monarch is actually better than using the Mehrunes’ Razor if the users long blade skill level is roughly 2-6 levels (higher gap at higher skill levels) above the users short blade skill level. If the player had an exact value for these levels, then we could refine the range down to a single digit, but as I used randomly chosen levels as an example, this is a range.

Comparisons using Roll Weighted Average DPTS when taking FP/MP into account
If resource consumption (FP or MP) is drastically different on both methods of damaging calculations may need to be different. Instead, you may want to calculate how many (with combat restoration) of those actions you can do in a row, calculate its average damage over that length of time, and then roll weight that figure.

For example, if we have 51 Maximum MP, light armour, 40 Willpower, 70 Mysticism, and 50 Destruction then we can compare Absorb Health and Fire Storm. Absorb Health has an $$average\ DPTS=[(5+52)/2]/2=14.25$$ and the user has an $$0.64\times0.7=0.448=44.8\%$$ chance to cast Absorb Health meaning the roll weighted average DPTS is $$0.448\times14.25=6.384$$, making Fire Storm’s roll weighted average DPTS (at 8.8) better in this instance. But, as we have 51MP, we can cast Absorb Health 5 times and Fire Storm only twice (both with combat regeneration). This means $$Absorb\ Health\ RW\ average\ DPTS\times5=31.92$$ and $$Fire\ Storm\ RW\ average\ DPTS\times2=17.6$$, making Absorb Health lower damage, but (damage wise) more consistent and better for long term use, not even considering its ‘restorative’ effects.