## Edit: Debunking that SH is less useful with higher EV

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### Edit: Debunking that SH is less useful with higher EV

Spoilered this because otherwise people think I am mathematically modelling the entrety of crawl for some reason
Spoiler: show
This is going to be more of the basic maths to modelling survivability, which will apply real life situations and most games. The maths is very simple and basic. I wouldn't be going into actual crawl game mechanics.

In crawl terms the assumption will be plain melee 1 vs 1. The character is brainlessly meleeing the monster and vice versa. This is to simplify everything. Yes I know range attacks exists and all sorts of spells and abilities that ignore certain defences and game play styles and so forth. Nevetheless, I hope everyone can gain, or affirm understanding of the principles.

---

Terms defined. Layman style.

P = Probability

1 =100% chance
0 = 0% chance
0.5 = 50% chance
0.1 = 10% chance

x is defined as to multiply

P = 1 equates to certain to occur
P = 0 equates to no chance to occur

---

Survivability of an encounter = (attack) x (defence)

This stands the real life(tm) model that a 100% chance to kill is normally a 100% chance to "win".
Also that a 100% defence is normally a 100% chance to "win".
Edge cases occur when either number is 0% in which case both nominally survives or both dies.

This has parallels in crawl, where paradoxially sounding, it is normally recommended to invest skill experience in damage (attack) ability in the early game as it gives greater gains than investing skill experience in defence.

---

In crawl there are four main methods of defence, that is to say to withstand an attack. This is HP, AC, EV, SH.

What I am interested is is debunking that SH is less useful innately if EV is high.
In probability theory, EV = chance to fail to evade. EV = (1 - chance to evade)
In probability theory, SH = chance to fail to shield. SH = (1 - chance to shield).
Assume that EV and SH are equal to each other and operates in the exact same way.
Assume that EV and SH operates by having a chance to prevent each attack.

---

Imagine that the "power level" of defence is the effective HP that is created to withstand an attack.
Personally I don't like using % as it adds a layer of confusion that tends to confuse people.

Imagine HP = 100. If EV would prevent half of attacks, then survivability is doubled, time to be killed is doubled.
That is to say that effective HP is 200% An alternative is to say that that effective HP is doubled to 200 HP.

---

Actually I think a more useful descriptor of "power level" of defence would be "Time taken before dying". The greater the time taken before dying, the more powerful defences are.

So T = Time taken before dying.

Standard T = 1
EV =0.5, then T = 2
I hope we can all agree on this.

If EV prevents a third of all attacks then time taken before dying is 3 times longer than before.
If EV would prevent a quarter of all attacks, then the time taken before dying would be 4 times longer than before.
If EV = 0.25, then T = 4.
If EV = 0.2 (a fifth), then T = 5
If EV = (a sixth) then T = 6
If EV = (a seventh) then T = 7
If EV = (an eigth) then T = 8
If EV = (A ninth) then T =9
If EV = 0.1, then T = 10
If EV = 0.01 then T = 100
If EV = 0.001 then T = 1000
If EV = 0, then T = infinity

Meaning that if you prevent every single attack from hitting your character then you will never die. This is not debateable.

---

The simple thing to plug into your calculator or pen and paper or mental arithmetic is T = (1/EV)

Likewise if EV = 1, then T = (1/SH)

Due to the assumption that EV and SH operates by preventing an attack, the method to find the new time taken before dying is :

T = (1/EV) x (1/SH)

---

Example character A: Chance to evade is 30%. EV = chance to fail to evade. EV = (1 - chance to evade). Chance to evade is 30%, which is P= 0.3. EV = (1 - 0.30), therefore EV = 0.7

Time taken before dying = (1/EV) = (1/0.7) = 1.43

Example character B; Chance to evade is 70%. EV = chance to fail to evade. EV = (1 - chance to evade). Chance to evade is 70%, which is P= 0.7. EV = (1 - 0.70), therefore EV = 0.3

T = (1/EV) = (1/0.3) = 3.33

---

Hopefully as noticed, character B, with a 70% chance to evade has greater time taken before dying that character A. What we want to look at is the impact of the chance to shield an attack. Lets take that probability to shield an attack is 20% chance.

For both @A and @B, the probability to shield an attack is 20%. SH = chance to fail to shield. EV = (1 - chance to shield). P= 0.2. EV = (1 - 0.20), therefore EV = 0.8

T = (1/EV) = (1/0.8) = 1.25

As can be seen, rather unsuprisingly, a 20% chance to shield is less useful than either a 30% chance to evade or a 70% chance to evade. However what we are interested in is the impact of SH. It's not neccesary in actuality, but for the sake of completeness.

---

For @A, T = (1/EV) x (1/SH)

T = (1/0.7) x (1/0.8) = 1.78

Impact of shield = (time taken before dying with shield / time taken before dying without shield) = 1.78 / 1.43 = 1.25

(Actual numbers used, not rounding error used)

For @B, T = (1/EV) x (1/SH)

T = (1/0.3) x (1/0.8) = 4.17

Impact of shield = (time taken before dying with shield / time taken before dying without shield) = 4.17 / 3.33 = 1.25

The 25% increase in effectiveness for both characters A and B is not suprising because the maths used is very simple, but the explanation and refutation behind it less so.

Btw, when dodge probability reaches 100%, time to die is infinity, and so a shield will increase the time dying to infinity x 1.25, which is still infinity, but a different type of infinity. Infinity is funny like that.

---

There's several other ways of explaining this like the "effective HP" method.

If anyone wants further explanation at any point, point out any mistakes seen, please do so, and I'll explain/rectify it.
Last edited by Plantissue on Tuesday, 31st July 2018, 14:55, edited 4 times in total.

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### Re: The maths of Survivability

You are correct within your model, and take the correct approach ("effective HP"). I didn't check all the details, but the basic point is fine.

However, the main thing to keep in mind is that "all models are wrong, but some are useful". Your model isn't too useful. Crawl isn't anywhere close to you and a pure melee monster fighting 1v1.

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### Re: The maths of Survivability

You can click the "value of ac/ev/sh" link in my sig for a more detailed look at it. You start with T=(1/EV_Pass) * (1/SH_Pass) * (1/AC_Pass), where EV_Pass is the percent of damage tested against evasion that gets past evasion, SH_Pass is the percent of damage tested against SH that gets past SH, and AC_Pass is the percent of damage tested against AC that gets past AC.

Now take the logarithm base 1.1 of both sides:
log T = log(1/EV_Pass) + log(1/SH_Pass) + log(1/AC_Pass)
This allows us to look at our total defenses as a sum of the three independent types of defenses. I call log T the "total defensive value" of your character. log(1/EV_Pass), log(1/SH_Pass), and log(1/AC_Pass) are the defensive values due to EV, SH, and AC.

This lets us ask questions like "if we increase SH from 15 to 16, how much does total defensive value increase by?" It turns out that the total defensive value always increases by the same amount when you increase SH from 15 to 16, regardless of the values of AC and EV. This lets us define the "marginal defensive value" of SH. Anyway, you can click the link to see some more discussion and pretty graphs on the subject.
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Ziggurat Zagger

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### Re: The maths of Survivability

AC 19, no shield vs stone giant, EV 7, 12, 17, 22, 27, 32:
Code:
`           AvHitDam | MaxDam | Accuracy | AvDam | AvTime | AvSpeed | AvEffDamDefending:     12,8 |     36 |      82% |  10,6 |   100  |  1,00 |     10,6Defending:     13,1 |     36 |      70% |   9,3 |   100  |  1,00 |      9,3Defending:     12,8 |     36 |      61% |   7,9 |   100  |  1,00 |      7,9Defending:     12,7 |     36 |      49% |   6,3 |   100  |  1,00 |      6,3Defending:     13,1 |     36 |      38% |   5,1 |   100  |  1,00 |      5,1Defending:     13,2 |     36 |      30% |   4,0 |   100  |  1,00 |      4,0`

AvEffDam is decreased by 1.3, 1,4, 1.6, 1.2, 1.1 when EV is increased by 5

Code:
`           AvHitDam | MaxDam | Accuracy | AvDam | AvTime | AvSpeed | AvEffDamDefending:     12,4 |     36 |      55% |   6,8 |   100  |  1,00 |      6,8Defending:     12,7 |     36 |      47% |   6,1 |   100  |  1,00 |      6,1Defending:     12,7 |     36 |      39% |   5,0 |   100  |  1,00 |      5,0Defending:     12,7 |     36 |      33% |   4,3 |   100  |  1,00 |      4,3Defending:     12,3 |     36 |      25% |   3,1 |   100  |  1,00 |      3,1Defending:     12,9 |     36 |      20% |   2,6 |   100  |  1,00 |      2,6`

AvEffDam is decreased by 0.7, 1.1, 0.7, 0.8, 0.5 when EV is increased by 5 so apparently extra EV is less useful when you have a shield, no surprise here.

How much does SH help?
10.6 is decreased to 6.8, that's 3.8 or 35%
9.3 is decreased to 6.1, that's 3.2 or 34%
7.9 is decreased to 5.0, that's 2.9 or 36%
6.3 is decreased to 4.3, that's 2.0 or 31%
5.1 is decreased to 3.1, that's 2.0 or 39%
4.0 is decreased to 2.6, that's 1.4 or 35%

So shield saves you from less damage the higher EV you have, obviously and no surprise again.

Edit. I forgot the most important thing i.e. accuracy.
82% is decreased to 55%, that's absolute 27% or relative 32%.
70% is decreased to 47%, that's absolute 23% or relative 32%.
61% is decreased to 39%, that's absolute 22% or relative 36%.
49% is decreased to 33%, that's absolute 16% or relative 32%.
38% is decreased to 25%, that's absolute 13% or relative 34%.
30% is decreased to 20%, that's absolute 10% or relative 50%.

So yeah, I have just proved that SH becomes less useful the higher EV you have because what matters is absolute accuracy decrease and 27% is much better than 10%.
Also note the test was done vs Stone Giant which has a single attack, that's ideal situation for SH.
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Plantissue

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### Re: The maths of Survivability

But relative is what matters, not absolute. Relative improvement for the shield remains constant.

You are able to stand there tanking hits for (your HP)/AvEffDam turns. In every case from your fsim, you can tank hits for about 50% more turns with the shield than without it. To put it another way, the shield lets you kill 50% more stone giants (if they fight you one at a time). It's always 50%, in each case that you tested.
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Ziggurat Zagger

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### Re: The maths of Survivability

Berder wrote:But relative is what matters, not absolute. Relative improvement for the shield remains constant.

No, absolute accuracy is more important. For instance, you are not fighting, just quaffing heal wounds and waiting for teleport. High EV character has 20% chance to be hit instead of 30% and low EV character has 55% chance to be hit instead of 82%. Of course the shield helps more to the latter.
Another example is that you can basically think "I have 10 characters at 1 HP remaining, how many characters will survive next turn". Shield for high EV will save you 1 character (7 dead instead of 8) and for low EV it will save you 2.7 characters (5-6 dead instead of 8), and then next step is to realize that it does not matter how many HP or what AC you have, shield saves the same HP to all characters when we compare it to evasion.
Last edited by VeryAngryFelid on Sunday, 29th July 2018, 09:09, edited 1 time in total.
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Ziggurat Zagger

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### Re: The maths of Survivability

Berder wrote:You are able to stand there tanking hits for (your HP)/AvEffDam turns. In every case from your fsim, you can tank hits for about 50% more turns with the shield than without it. To put it another way, the shield lets you kill 50% more stone giants (if they fight you one at a time). It's always 50%, in each case that you tested.

I think chance to be hit which has linear dependency with number of HP lost is a better parameter than "number of dead giants". One character lost 55 HP instead of 82 HP, another character lost 20 HP instead of 30 HP. For whcih character shield was more useful?
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### Re: The maths of Survivability

That's an interesting point about the 1 HP scenario. Still, you would always rather have the shield than not have it. The real question is whether you could better spend the skill XP on something else. For example, could you prevent yourself from getting to 1 HP more effectively by training the shield, or by training an attack skill? See below.

I think chance to be hit which has linear dependency with number of HP lost is a better parameter than "number of dead giants". One character lost 55 HP instead of 82 HP, another character lost 20 HP instead of 30 HP. For which character shield was more useful?

If they're only fighting a single giant and can then rest, it doesn't matter, assuming they had more than 82 HP they survived no matter what they did. A more reasonable circumstance is the game throws N giants at you (or N threats equivalent to giants) and the question is whether your HP would be positive or negative at the end of it.

Consider it this way. There is a tradeoff between training the shield or training more attack skills. You have a choice between getting the shield, which results in 50% more survival time (because of 33% less incoming damage), or putting the same amount of XP into an attack skill, which results in, let's say, +40% damage. And you don't have enough XP to get both. You will *always* kill more giants (on average over many trials) if you choose the shield, no matter what your current EV is, because 50% is more than 40%.

This can be demonstrated like this: the amount of giants killed before training either skill is roughly (Player HP) * (AvEffDam from player)/((Giant HP) * (AvEffDam from giant)) = BaseGiantsKilled. If you train the shield skill, the amount of giants killed is (Player HP) * (AvEffDam from player)/((Giant HP) * (AvEffDam from giant * 0.66)) = BaseGiantsKilled * 1.5. If you instead train the attack skill, the amount of giants killed is (Player HP) * (AvEffDam from player * 1.4)/((Giant HP) * (AvEffDam from giant)) = BaseGiantsKilled * 1.4.

BaseGiantsKilled*1.4 is always less than BaseGiantsKilled*1.5, so you see that it's always the correct decision to train the shield instead of the attack skill, regardless of how much EV you currently have. At least, it is the correct decision under the assumption that training the attack skill instead of the shield skill increases your damage by 40%.

Oh, a little explanation for the formula for BaseGiantsKilled:
BaseGiantsKilled = (turns spent alive whacking giants) * (AvEffDam from player) / (giant HP)
and
(turns spent alive whacking giants) = (Player HP) / (AvEffDam from giant)
therefore
BaseGiantsKilled = (Player HP) * (AvEffDam from player)/((Giant HP) * (AvEffDam from giant))
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### Re: The maths of Survivability

Notice that I am not arguing whether shield is useful or not. I am arguing about "is shield less useful to character with high EV than it is to character with low EV". Sorry, you write quite complicated formulae and I am lazy to keep arguing.
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### Re: The maths of Survivability

Well, to summarize the point I was making, the real question is whether a shield is a better XP investment than spending the same XP on something else. And it turns out that if you're fighting enemies like stone giants, the right decision doesn't depend on your EV.
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VeryAngryFelid

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### Re: The maths of Survivability

Do you guys suppose that the correct amount of skill assignment (shield/armour/dodge/fighting) from a fixed exp pool, in terms of maximizing # turns survived vs, say, a stone giant, can be easily determined, like solving a linear optimization problem, or is it better to monte-carlo it, like vaf is doing?
For a given starting xp level and shield/armour gear set-up, ofc.
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### Re: The maths of Survivability

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### Re: The maths of Survivability

Do you guys suppose that the correct amount of skill assignment (shield/armour/dodge/fighting) from a fixed exp pool, in terms of maximizing # turns survived vs, say, a stone giant, can be easily determined, like solving a linear optimization problem, or is it better to monte-carlo it, like vaf is doing?
For a given starting xp level and shield/armour gear set-up, ofc.

Yes, it could be done. It has not yet been done. The link "value of ac/ev/sh" in my sig is a big piece of the puzzle. The missing pieces that would have to be added to that and bundled together in one piece of software or formula are:
• How much XP does it cost to reach a given skill level from a given starting skill level (simple)
• How much ac/ev/sh/hp does a character have, for a given skill level, stats, and equipment set
From this it would be a simple matter to calculate the total Defensive Value of your build, for different levels of skilling on a limited XP budget, and optimize them.

For it to be more useful you would also want to take into account the damage the character is dealing. Part of the benefit of fighting is that it increases your damage as well as your HP. That would be a larger project, even just if you are only looking at melee damage.
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Spider Stomper

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### Re: The maths of Survivability

bel wrote:You are correct within your model, and take the correct approach ("effective HP"). I didn't check all the details, but the basic point is fine.

However, the main thing to keep in mind is that "all models are wrong, but some are useful". Your model isn't too useful. Crawl isn't anywhere close to you and a pure melee monster fighting 1v1.

The point is as I have written to show that the usefulness of SH doesn't change depending on EV value. A lower or higher EV does not make SH anymore or any less useful. The model is just there to argue that point. I explained it in simple words, then veryangryfelid and damerell basicall threw maths into it. Only that their maths is utter and complete nonsense as they basically threw around numbers and calculations that made no sense. i've explained my reasoning from basic points and is able to do so, becuase it make sense. Neither of them can do so.

---

Berder wrote:You can click the "value of ac/ev/sh" link in my sig for a more detailed look at it. You start with T=(1/EV_Pass) * (1/SH_Pass) * (1/AC_Pass), where EV_Pass is the percent of damage tested against evasion that gets past evasion, SH_Pass is the percent of damage tested against SH that gets past SH, and AC_Pass is the percent of damage tested against AC that gets past AC.

Now take the logarithm base 1.1 of both sides:
log T = log(1/EV_Pass) + log(1/SH_Pass) + log(1/AC_Pass)
This allows us to look at our total defenses as a sum of the three independent types of defenses. I call log T the "total defensive value" of your character. log(1/EV_Pass), log(1/SH_Pass), and log(1/AC_Pass) are the defensive values due to EV, SH, and AC.

This lets us ask questions like "if we increase SH from 15 to 16, how much does total defensive value increase by?" It turns out that the total defensive value always increases by the same amount when you increase SH from 15 to 16, regardless of the values of AC and EV. This lets us define the "marginal defensive value" of SH. Anyway, you can click the link to see some more discussion and pretty graphs on the subject.
Thank you for confirming that the defensive value of SH doesn't change on the values of AC or EV. I was only concerned with SH and EV. but it is good for someone to affirm the same with AC as well.

---

VeryAngryFelid wrote:
Spoiler: show
AC 19, no shield vs stone giant, EV 7, 12, 17, 22, 27, 32:
Code:
`           AvHitDam | MaxDam | Accuracy | AvDam | AvTime | AvSpeed | AvEffDamDefending:     12,8 |     36 |      82% |  10,6 |   100  |  1,00 |     10,6Defending:     13,1 |     36 |      70% |   9,3 |   100  |  1,00 |      9,3Defending:     12,8 |     36 |      61% |   7,9 |   100  |  1,00 |      7,9Defending:     12,7 |     36 |      49% |   6,3 |   100  |  1,00 |      6,3Defending:     13,1 |     36 |      38% |   5,1 |   100  |  1,00 |      5,1Defending:     13,2 |     36 |      30% |   4,0 |   100  |  1,00 |      4,0`

AvEffDam is decreased by 1.3, 1,4, 1.6, 1.2, 1.1 when EV is increased by 5

Code:
`           AvHitDam | MaxDam | Accuracy | AvDam | AvTime | AvSpeed | AvEffDamDefending:     12,4 |     36 |      55% |   6,8 |   100  |  1,00 |      6,8Defending:     12,7 |     36 |      47% |   6,1 |   100  |  1,00 |      6,1Defending:     12,7 |     36 |      39% |   5,0 |   100  |  1,00 |      5,0Defending:     12,7 |     36 |      33% |   4,3 |   100  |  1,00 |      4,3Defending:     12,3 |     36 |      25% |   3,1 |   100  |  1,00 |      3,1Defending:     12,9 |     36 |      20% |   2,6 |   100  |  1,00 |      2,6`

AvEffDam is decreased by 0.7, 1.1, 0.7, 0.8, 0.5 when EV is increased by 5 so apparently extra EV is less useful when you have a shield, no surprise here.

How much does SH help?
10.6 is decreased to 6.8, that's 3.8 or 35%
9.3 is decreased to 6.1, that's 3.2 or 34%
7.9 is decreased to 5.0, that's 2.9 or 36%
6.3 is decreased to 4.3, that's 2.0 or 31%
5.1 is decreased to 3.1, that's 2.0 or 39%
4.0 is decreased to 2.6, that's 1.4 or 35%

So shield saves you from less damage the higher EV you have, obviously and no surprise again.

Edit. I forgot the most important thing i.e. accuracy.
82% is decreased to 55%, that's absolute 27% or relative 32%.
70% is decreased to 47%, that's absolute 23% or relative 32%.
61% is decreased to 39%, that's absolute 22% or relative 36%.
49% is decreased to 33%, that's absolute 16% or relative 32%.
38% is decreased to 25%, that's absolute 13% or relative 34%.
30% is decreased to 20%, that's absolute 10% or relative 50%.

So yeah, I have just proved that SH becomes less useful the higher EV you have because what matters is absolute accuracy decrease and 27% is much better than 10%.
Also note the test was done vs Stone Giant which has a single attack, that's ideal situation for SH.

Actually since you have very nicely fsim it, you have confirmed that SH is just as useful no matter the EV.

Code:
`           EV | AvEffDam | AvEffDam with Shield | Extra time taken before dying 7  |     10.6 |                  6.8 |   1.56 |12 |     9.3 |                   6.1 |   1.52 |   17 |     7.9 |                   5.0 |   1.58 |   22 |     6.3 |                   4.3 |   1.47 |   27 |     5.1 |                   3.1 |   1.64 |   32 |     4.0 |                   2.6 |   1.54 |   `

As can be seen, there is no correlation in the effectiveness of shields as it changes from 7 EV to 32 EV.

To take your 1 HP example, i.e. instant death from a hit.
No matter the EV, you will always have an approx 55% extra chance to the probability to live. That is what your info says.

So VAF and damerall, can we finally lay this to rest? To say that shield is less useful to character with high EV than it is to character with low EV is not true and is just as absurd as to say that EV is less useful to a character with high HP.

Of course this doesn't answer questions of skill investment at any stage of the game, but that's not what I am debunking.

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### Re: The maths of Survivability

Plantissue wrote:So VAF and damerall, can we finally lay this to rest? To say that shield is less useful to character with high EV than it is to character with low EV is not true and is just as absurd as to say that EV is less useful to a character with high HP.

I can just agree to disagree. I believe I have proved my point, you believe you have proved your point. Because math alone is nothing, its results depend on its interpretation and that's where we fundamentally disagree: you believe we should maximize number of dead monsters, I believe we should minimize number of lost HP.
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### Re: The maths of Survivability

Plantissue wrote:EV is less useful to a character with high HP

It is kind of joke but it is actually true because having higher HP means higher regeneration so the character has easier time getting hit often but for less damage. It is so painful to wait hundreds of turns to restore lost HP as XL 1 Fe/Sp.
Difference between Sp with high AC/low EV and with low AC/high EV is more pronounced than difference between Mi with high AC/low EV and with low AC/high EV.
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### Re: The maths of Survivability Edit (Debunking the assertati

You can agree to disagree, but you will have to say why. You did write that what you care about was whether the usefulness of SH doesn't change depending on EV value.
Spoiler: show
VeryAngryFelid wrote:I am arguing about "is shield less useful to character with high EV than it is to character with low EV".

In the other thread, I explained very simply in words why the value of EV doesn't change the effectiveness of SH, but you decided to bring maths into this. Except that your maths doesn't make any sense at all which is why you cannot explain the method behind your reasoning. But I thank you for going ahead and fsim to bring real numbers which is much appreciated as it brings real numbers to confirm that the usefulness of SH doesn't change depending on EV value.

Your fsim has confirmed that the usefulness of SH doesn't change on the value of EV.
berder has, unknown to me, independently confirmed that in another thread.

VeryAngryFelid wrote:
Plantissue wrote:EV is less useful to a character with high HP

It is kind of joke but it is actually true because having higher HP means higher regeneration so the character has easier time getting hit often but for less damage. It is so painful to wait hundreds of turns to restore lost HP as XL 1 Fe/Sp.
Difference between Sp with high AC/low EV and with low AC/high EV is more pronounced than difference between Mi with high AC/low EV and with low AC/high EV.

If you are hit half as much as previously no matter the HP you only have to wait half the time to restore HP anyways.
Last edited by Plantissue on Monday, 30th July 2018, 14:37, edited 1 time in total.

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### Re: The maths of Survivability Edit (Debunking the assertati

Ok, last attempt.
Let's compare 3 theoretical characters:
1) EV 0. You get hit every time, shield is a very useful thing.
2) EV 10000000. You are never hit, shield is useless.
3) EV 30. You are hit often/sometimes depending on monster accuracy, shield is useful but not that useful as for first character.
Thus I believe the formula for "shield usefulness" is a declining function of EV, it decreases from "very useful" to "useless" when EV grows.
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### Re: The maths of Survivability Edit (Debunking the assertati

In scenario 1) and scenario 3) of EV 0 and EV 30, the shield is just as useful. I don't get why you don't understand this. You can fsim it again if you like.

In scenario 2) at EV 10 000 000, will you never be hit? You'll have to fsim it for me.

Assuming that you will never be hit at EV 10 000 000, then HP, AC, resistances are also equally pointless as SH. Would you also too be arguing that HP, AC and resistances are all less useful the value of EV as they would all be useless if you never get hit? (Assuming HP is bigger than 0 of course.)

But I'll leave you with this quote from the opening post as I thought you would go with that line of reasoning:
Spoiler: show
Btw, when dodge probability reaches 100%, time to die is infinity, and so a shield will increase the time dying to infinity x 1.25, which is still infinity, but a different type of infinity. Infinity is funny like that.
Last edited by Plantissue on Monday, 30th July 2018, 14:51, edited 1 time in total.

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### Re: The maths of Survivability Edit (Debunking the assertati

By the way there was an error in my fsim.
It should be
82% is decreased to 55%, that's absolute 27% or relative 32%.
70% is decreased to 47%, that's absolute 23% or relative 32%.
61% is decreased to 39%, that's absolute 22% or relative 36%.
49% is decreased to 33%, that's absolute 16% or relative 32%.
38% is decreased to 25%, that's absolute 13% or relative 34%.
30% is decreased to 20%, that's absolute 10% or relative 33%.
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### Re: Edit: (Debunking that SH is less useful with higher EV

Btw, just so you know, Extra time taken before dying, is basically the same metric as Effective HP, is basically the same metric as minimising the HP lost before dying. The sole metric i am using is "defensive power", not "offensive power".

Anyways, in terms of absolute extra turns/time, waiting for a tele to kick in while trapped scenario, adding SH to a higher EV, actually make SH more valuable the higher the EV is, as you now got more turns to survive before the tele activates, just like if you got more AC or more HP, you got more time/chance to live the higher your EV is.

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### Re: Edit: (Debunking that SH is less useful with higher EV

I think I know what's our problem.
For high EV characters chance to be hit is really low: 30% with EV 32. You basically can kill the giant while being hit once or zero so shield seems to do nothing.
For low EV character your chance to be hit is really high: 82% with EV 7. You basically cannot melee the giant, it will kill you in 2 attacks. And then shield shines, because instantly you have just 55% chance to be hit so you can melee the giant (4 attacks are expected) provided you can kill it in 3-4 attacks too if lucky.
But of course you can claim that there is no difference: first character can kill 3 giants without shield and 4 giants with shield, second character can kill 0.3 giant without shield or 0.45 giant with shield. 3/4 = 0.3/0.45
Shield saves 40 HP per giant's attack for first character and 4 HP for second character, again no difference
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### Re: Edit: Debunking that SH is less useful with higher EV

Another test. You are at full HP and want to melee the giant. What chance do you have to be hit 3 times in a row?
Ev7, no shield: 0.82^3=55%, you cannot melee it.
With shield: 0.55^3=16%, you can melee it. Your chance to die is decreased by 39 absolute percents. Really useful shild, going from likely to die 55% to comfortable 16%.
Ev 32, no shield: 0.3^3=2.7%, you easily can melee it.
With shield:0.2^3=0.8%, still easy. Your chance to die decreased by 1.9 absolute percents. Almost useless shield, went from 2.7% to 0.8%
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### Re: Edit: Debunking that SH is less useful with higher EV

There are many ways to measure improvement. Here are some:
1. Change in damage received per turn, measured in points of HP
2. Change in to-hit chance per turn, measured in points of to-hit probability
3. Change in turns to live, measured in number of turns
4. Current damage received per turn, divided by the previous damage received
5. Current to-hit chance, divided by previous to-hit chance
6. Current turns-to-live, divided by previous turns-to-live
7. Log_1.1(Current turns-to-live / previous turns-to-live)
According to measures 1 and 2, SH gives a smaller benefit if your EV is higher. According to measures 4, 5, 6, and 7 SH gives the same benefit if your EV is higher. According to measure 3, the benefit of SH actually increases if your EV is higher.

Now let's cut through the fog. The important thing is what *character build decisions* you should make. Try this: set up a wizmode character with a ring that grants +20 EV. Explain why you think training shields would be less effective against a stone giant than putting the same skill XP into, say, armor skill, and now remove the EV ring and explain why you would make the opposite decision in this case.

You can't set up such a character. Go ahead and try. The decision of where to spend the XP will be the same regardless of your EV. If you should train shields with the ring, you should train shields without it. If you should instead train armor skill with the ring, you should train armor skill without it.

You can use http://crawl.chaosforge.org/Skill#Experience_Required as a guide for your XP budget. Easiest if you set your race to human so all aptitudes are 0.

Now, there is one training decision that a +20 EV ring affects: it affects whether you should train *Dodging*. And not in a linear manner. The details are there: viewtopic.php?t=14989 . So, don't try to compare training dodging to training shields with and without the ring, because you actually might find a difference; instead compare training armor to training shields with and without the ring.
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### Re: Edit: Debunking that SH is less useful with higher EV

You have already provided the relevant formula before. If you already have high EV, then it is more efficient to train fighting/armour even without taking defense-ignoring attacks into account. Especially with high level skills becoming more expensive. So a buckler becomes attractive for casters in robe who have trained dodging/fighting to 10+.
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### Re: Edit: Debunking that SH is less useful with higher EV

Berder wrote:There are many ways to measure improvement. Here are some:
1. Change in damage received per turn, measured in points of HP
2. Change in to-hit chance per turn, measured in points of to-hit probability
3. Change in turns to live, measured in number of turns
4. Current damage received per turn, divided by the previous damage received
5. Current to-hit chance, divided by previous to-hit chance
6. Current turns-to-live, divided by previous turns-to-live
7. Log_1.1(Current turns-to-live / previous turns-to-live)
According to measures 1 and 2, SH gives a smaller benefit if your EV is higher. According to measures 4, 5, 6, and 7 SH gives the same benefit if your EV is higher. According to measure 3, the benefit of SH actually increases if your EV is higher.

Math is useless without interpretation. Let's check metrics where shield gives different benefit.

Where high EV characters benefit less:
1. shows how you survive being adjacent to 8 giants. Really useful metrics.
2. shows how you survive a single giant when you have 1 HP. Extremely useful metrics.
x. metrics from my previous post "getting 3 hits in a row" shows your chances to die from 100 HP when adjacent to 3 stone giants. Really useful metrics.

Where high EV characters benefit more:
3. shows that with 100 HP the shield improves your time from 10 turns to 16 turns and from 25 turns to 40 turns. The second increase is greater in numbers but is still less useful. If you don't see why, increase the EV (50 turns instead of 80 has even greater increase and is even less useful) or just let your current HP go below 45 HP (that's max damage of stone giant) and suddenly see how you instantly switch to metrics 2 (can I die this turn?). Almost useless metrics.
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### Re: Edit: Debunking that SH is less useful with higher EV

VeryAngryFelid wrote:You have already provided the relevant formula before. If you already have high EV, then it is more efficient to train fighting/armour even without taking defense-ignoring attacks into account. Especially with high level skills becoming more expensive. So a buckler becomes attractive for casters in robe who have trained dodging/fighting to 10+.

Go ahead and actually try to set up the situation in fsim like I said. You will find that, whether you train shields or armor, you get a percentage reduction in your AvEffDam defending, and that this is the same percentage regardless of whether you have the EV+20 ring or not. This makes the decision independent of the EV+20 ring. You just choose whichever one gives the greater percentage reduction, and this will in both cases lead to the greater absolute reduction in AvEffDam, compared to training the other skill.
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### Re: Edit: Debunking that SH is less useful with higher EV

Berder wrote:Go ahead and actually try to set up the situation in fsim like I said. You will find that, whether you train shields or armor, you get a percentage reduction in your AvEffDam defending, and that this is the same percentage regardless of whether you have the EV+20 ring or not. This makes the decision independent of the EV+20 ring. You just choose whichever one gives the greater percentage reduction, and this will in both cases lead to the greater absolute reduction in AvEffDam, compared to training the other skill.

How do you interpret percentage reduction? As number of turns in percents? It is useless, see my previous post.
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### Re: Edit: Debunking that SH is less useful with higher EV

percentage reduction is (AvEffDam_old - AvEffDam_new) / AvEffDam_old. For example, if 5 AvEffDam is reduced to 3 AvEffDam, this is a 40% reduction.

And the point is that if you choose the skill option that gives the greater *percentage* reduction in AvEffDam, this is also the option that gives the greater *absolute* reduction in AvEffDam.

To illustrate, say AvEffDam_old = 5, and you have two skilling options: one that gives AvEffDam_new = 3, and one that gives AvEffDam_new = 4.
Under option 1 (train shield), you have 40% reduction in AvEffDam, and the absolute reduction in AvEffDam is 5-3 = 2.
Under option 2 (train armor), you have 20% reduction in AvEffDam, and the absolute reduction in AvEffDam is 5-4 = 1.
You see that the option with the greater percentage reduction in AvEffDam is also the option with the greater absolute reduction. That would be option 1, which you should choose.

Now, go back to before you chose either option and imagine your EV has been increased by a lot, so that only half the attacks get past your EV as before. Now your AvEffDam_old is 2.5 instead of 5. The situation now looks like:
Option 1 (shield skill), AvEffDam_new = 1.5. This is still a 40% reduction in AvEffDam. The absolute reduction now is 2.5-1.5 = 1.
Option 2 (armor skill), AvEffDam_new = 2.0. This is still a 20% reduction in AvEffDam. The absolute reduction now is 2.5-2.0 = 0.5.
You see, again, that the greater percentage reduction in AvEffDam is also the option with the greater absolute reduction. That's still option 1. The absolute reduction decreased because your EV is higher this time. However, the percentage reduction stayed the same, and the correct choice (option 1) also stayed the same.

I hope you can see that no matter what you increase the EV by, the situation will look the same. If you increase EV so much that only 1/10 of the attacks get past your EV as in the first situation, AvEffDam_old is now 0.5 instead of 5, and the options will look like this:
Option 1 (shield skill). AvEffDam_new = 0.3. 40% reduction, absolute reduction of 0.2.
Option 2 (armor skill). AvEffDam_new = 0.4. 20% reduction, absolute reduction of 0.1.
You see again that option 1 is the correct choice. It will always be the correct choice no matter what your EV is.
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### Re: Edit: Debunking that SH is less useful with higher EV

Then I don't see how percentage reduction makes shield give equal benefit to high ev and low ev chracters. Because this means that high ev characters get less damage reduction (their damage without shield is lower) and hence shield is less useful to them.
Notice how I described situations which use measures from your list. Absolute damage reduction can be seen in real game, percentage reduction is just math fantasy to "prove" a claim. It is not that different from ln(Damage reduction) or sin(Damage reduction) which don't apply to anything shield-related either.

PS. I am not arguing about armour skill or training in general, of course I agree with that analysis. So far we don't see any real game measure where shield would be better for high EV characters than for low EV characters while we see several measures where the opposite is true. So my claim is proved, case closed
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### Re: Edit: Debunking that SH is less useful with higher EV

There are a couple metrics where the benefit of SH is larger, the higher your EV. For example, the absolute increase in the number of stone giants you can kill in a row.
If you can kill 10 stone giants in a row due to your high EV, and your shield grants +50% additional time to live, then you get to kill 5 additional stone giants. Whereas if your EV is low and you can only kill 5 stone giants in a row, the same shield will only let you kill 2.5 additional stone giants.

Here's another question: would you say AC is less useful for high EV characters than for low EV characters? You will note that, if due to high EV your AvEffDam defending is already only 1.0, then the most additional AC can do for you is reduce absolute AvEffDam defending by 1.0 at the absolute limit. Whereas if with low EV your AvEffDam defending is 10, then AC can reduce your AvEffDam defending by up to 10 points.
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### Re: Edit: Debunking that SH is less useful with higher EV

Sorry, that metrics seems useless to me. First you need to fight the monsters one on one and second still shield is less useful to high ev characrer here because it already can kill more monsters than low ev character can. Just increase ev to ridicioulus value to see what I mean, is it important that you can kill 1000 giants instead of 800? The difference 200 is very large and yet it is completely irrelevant and shield is useless.
Last edited by VeryAngryFelid on Tuesday, 31st July 2018, 05:15, edited 1 time in total.
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### Re: Edit: Debunking that SH is less useful with higher EV

I am not sure about AC. What I am sure is that every character needs offense also. So getting high EV from loot can result in me switching to train magic, melee, ranged, fighting or armour . I don't have strict rules or feelings here.

Edit. Of course in case of avveffdam 1 ac does not become useless as it decreases damage spikes also. Even if avveffdan is decreased from 1.0 tp 0.5 it can mean that max damage is reduced from 50 to 25 which is a big deal. Shield does not become absolutely useless either. But basically yes, AC becomes less useful the higher EV you have.
Similarly both AC and EV become less useful the higher SH you have.
It is easy to understand/explain: if you block 90% attacks, your ac/ev do not matter as much as when you block only 10% attacks.
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### Re: Edit: Debunking that SH is less useful with higher EV

I thought more about that "number of monsters killed" measure.

Code:
`          AvHitDam | MaxDam | Accuracy | AvDam | AvTime | AvSpeed | AvEffDamDefending:     12,8 |     36 |      82% |  10,6 |   100  |  1,00 |     10,6Defending:     13,1 |     36 |      70% |   9,3 |   100  |  1,00 |      9,3Defending:     12,8 |     36 |      61% |   7,9 |   100  |  1,00 |      7,9Defending:     12,7 |     36 |      49% |   6,3 |   100  |  1,00 |      6,3Defending:     13,1 |     36 |      38% |   5,1 |   100  |  1,00 |      5,1Defending:     13,2 |     36 |      30% |   4,0 |   100  |  1,00 |      4,0`

Code:
`           AvHitDam | MaxDam | Accuracy | AvDam | AvTime | AvSpeed | AvEffDamDefending:     12,4 |     36 |      55% |   6,8 |   100  |  1,00 |      6,8Defending:     12,7 |     36 |      47% |   6,1 |   100  |  1,00 |      6,1Defending:     12,7 |     36 |      39% |   5,0 |   100  |  1,00 |      5,0Defending:     12,7 |     36 |      33% |   4,3 |   100  |  1,00 |      4,3Defending:     12,3 |     36 |      25% |   3,1 |   100  |  1,00 |      3,1Defending:     12,9 |     36 |      20% |   2,6 |   100  |  1,00 |      2,6`

Let's assume our character deals to the monster "double average received" damage i.e. something like
Code:
`Attacking:     25,4 |     36 |      49% |   12,6 |   100  |  1,00 |      12,6Defending:     12,7 |     36 |      49% |   6,3 |   100  |  1,00 |      6,3`

It is reasonable so you can melee the giant and lose about 50% HP before killing it when you have EV 17.
How does shield change it?
Now you lose 4.3 i.e. you are expected to lose about 33% HP before killing it. Useful, but does not change much.

What about lower EV character without shield?
You lose 10.6, deal the same 12.6 meaning it is a very close fight, the monster should be avoided.
Let's add shield. You lose 6.8, that's about 50% HP lost before you kill the monster, meaning the shield is really useful.

What about higher EV character without shield?
You lose 4.0 i.e. you are expected to lose about 33% HP.
With shield you lose 2.6 so you are expected to lose about 20% HP. Even less important than for EV 17 character.

And now some interpretation staff. What is the meaning of "number of turns" measure when you beat each other with avveffdam? Does it mean the monster starts missing when you are expected to live longer? Or do you start missing? No. Some of you will die and who dies is determined by "damage taken/damage received", number of turns is irrelevant here. True, you can live longer if you are unlucky with your attacks but it is not determined by number of turns, it is determined by "damage taken per turn". Does the number of turns is reponsible for "now I can melee an ogre instead of goblin"? No, again it is determined by "damage taken/damage received". So I don't see how measure "how long a fantasy fight can last" is important or even relevant. If anything, I started to hate my current X-crawl character who uses quickblade in plate armour and melees iron golems in Depths, the fights lasted so long that I suspended the game. Longer fights result in loss of piety and food
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### Re: Edit: Debunking that SH is less useful with higher EV

What determines whether you can kill a monster or not in a simple slugging fight is the following formula:
CombatPower = (AvEffDam attacking) * (your HP) / ((AvEffDam defending) * (enemy HP))
If this value is >1, you will on average win, and the higher the value is the more likely you are to win. If it is equal to 1, it's a coin toss who wins. If it is <1, you will on average lose. If the value is N, you will on average be able to kill N of these monsters one at a time before dying. I hope this explains why CombatPower is supreme. You should generally choose attack and defense skills so that CombatPower is maximized.

Now, (AvEffDam defending) = (1 - block chance) * (1 - dodge chance) * (damage per hit after armor)
therefore the formula is equivalent to:
CombatPower = (AvEffDam attacking) * (your HP) / ((1 - block chance) * (1 - dodge chance) * (defending damage per hit after armor) * (enemy HP))

A 50% increase in (AvEffDam attacking) has the same effect on CombatPower as a 50% increase in 1/(1-block chance), which has the same effect on CombatPower as a 50% increase in 1/(1-dodge chance), which has the same effect on combat power as a 50% increase in 1/(defending damage per hit after armor), which has the same effect on combat power as a 50% increase in (your HP). These factors are all independent, they are just multiplied together.

In your last post you wondered about the meaning of turns to live. Here: CombatPower can also be written as (AvEffDam attacking) * (turns to live) / (enemy HP). This is the significance of (turns to live): it is the product of several of the multiplicative terms in the CombatPower formula. Increasing (turns to live) by 50% has the same effect on CombatPower as increasing (AvEffDam attacking) by 50%. So, that is the importance of turns to live. A percentage increase in turns to live is just as valuable as the same percentage increase in your attack damage.

The last time I brought up these formulas you complained you were too lazy to figure them out, but this is really what you need to understand.
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### Re: Edit: Debunking that SH is less useful with higher EV

I see now, thank you. Sorry, I was really tired when I said "lazy". Now it is morning and I feel better

A percentage increase in turns to live is just as valuable as the same percentage increase in your attack damage.

No. That fsim had max damage from stone giant 36 and average received damage for character with EV 32 equal to 4.
Now compare 3 characters:
1) 4 incoming damage and 20 outcoming damage.
2) 2 incoming damage and 10 outcoming damage
3) 1 incoming damage and 5 outcoming damage.
Are the characters really equal to each other? I would use the first one anytime. Because max incoming damage is usually much higher than average damage (36 for the fsim) so the faster I kill the giant the better, no matter how tiny damage I am supposed to get, because bad luck happens and I can be hit twice in a row. There is no significant difference between 4 incoming damage and 1 incoming damage, because 36 max damage is what matters and both 1 and 4 are too small to matter. A shield for high EV character actually does even less, it brings incoming damage from 4 to 3.

Also I still didn't change my mind on
3. shows that with 100 HP the shield improves your time from 10 turns to 16 turns and from 25 turns to 40 turns. The second increase is greater in numbers but is still less useful. If you don't see why, increase the EV (50 turns instead of 80 has even greater increase and is even less useful) or just let your current HP go below 45 HP (that's max damage of stone giant) and suddenly see how you instantly switch to metrics 2 (can I die this turn?). Almost useless metrics.

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Slime Squisher

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### Re: Edit: Debunking that SH is less useful with higher EV

Berder wrote:A percentage increase in turns to live is just as valuable as the same percentage increase in your attack damage.

No, it's not.

Lack of attention resulting from induced boredom of repeat low-impact keypresses is an important factor in DCSS survivability. Humans are piloting these characters.

Spider Stomper

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### Re: Edit: Debunking that SH is less useful with higher EV

At this point I am just wondering what is the point you are trying to make theangryfelid. No matter what metric you are using, as long as it makes sense, as you have gone from minimize number of lost HP, 1 Hp left scenario, fsim, turns to live, the result is still the same. The result of your varying metrics have not changed. The effectiveness of a shield is completely independent from EV.

Your 3 character example above is irrelevant. Whether or not you would prefer to recieve higher damage in exchange for a higher damage output is irrelevant but it's an interesting point to make, as the strategical decision making will turn from maximising defence to maximising offence, but still irrelevant. You can make a whole new thread about it if you like on the topic. The effectiveness of a shield is still completely independent from EV.

@Implojin, assuming that you don't get bored tapping a key, (and from what I can see of the current generation of mobile games some people don't), it's a perfectly fine metric of a power of a character modelling a simple melee combat, for the purposes of working out whether SH is less useful or not with higher EV.

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### Re: Edit: Debunking that SH is less useful with higher EV

Plantissue wrote:At this point I am just wondering what is the point you are trying to make theangryfelid... The effectiveness of a shield is completely independent from EV.

I have proved that effectiveness of a defense depends on effectiveness of another defense.
With high HP and good regen you don't need AC/EV/SH that much (compare naked Tr to naked Fe).
With high AC you don't need HP (Gr vs Hu)/ EV (Na in CPA vs Fe) / SH (any race) that much
With high EV you don't need HP (Sp vs Ce), AC (Fe), SH (any race) that much.
With high SH you don't need AC (OpFi vs HuGl)/EV( TrFi vs HuGl)/HP (KoFi vs HuGl) that much.

This is very simple to explain. SH vs EV is the most obvious case here, you can check EV before SH and nothing will change, it will even make SH more useful than it really is. Then you realize that shield is checked ONLY if EV check fails and that means that SH is checked in 20% cases for one character and in 80% cases for another character. Of course it means SH cannot be equally useful to the characters.
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### Re: Edit: Debunking that SH is less useful with higher EV

The effectiveness of a defence depends on the effectiveness of another defence because, well it does. If HP was doubled, but EV effectiveness was halved, then broadly speaking, your defences are the same. If HP was halfed but in relative terms the EV effectiveness is doubled, then broadly speaking your defences are the same. No one should be disputing this. But if at any point in the game, if you are seeking to improve your defences, the effectiveness of SH is independent from EV, no matter what metric or maths you use.

If, as your argument goes, that defence values of both EV and SH is good enough that you would rather put training to improve your offensive power, then the question of improving EV or SH wouldn't come up, as you aren't seeking to improve either.

Berder's "combat power" formula could help you decide whether to improve offensive capability over defensive capabilities or not, if at any point you are undecided as to which to go for, rather than trying to get a feel for it intuitively through thousands of games.

But your original argument was that if your EV is high, you are better off training EV than SH , whilst if EV was low, you should consider training SH. The only reason we are even introducing maths was because in the other thread when I tried explaining with words, you introduced maths into it and made a total cock-up of it, and then argued it wasn't a total cock-up.

---

Hence why I decided to write an ironclad mathematical modelling. Which was a mistake because some people still think I am trying to mathematically model the entirety of crawl.

By the way if anyone is interested if someone wants a formula for defences it is quite simple. It goes something like this.

Total Defence = (effectiveness of defence A x effectiveness of defence B x effectiveness of defence C x effectiveness of defence D x effectiveness of defence E) and so on and so forth.

In other words, you multiple the effectiveness of each together. For instance, if you want to work out whether or not you should put on that amulet of vitality or that rN+ ring or whatever to improve your flat defence against a single torment attack, knowing that formula, you can just just work out effectiveness of each and plug it in. it's a bit more complicated for multiple torments, but not that much more if you are mathematically literate.
Last edited by Plantissue on Tuesday, 31st July 2018, 15:40, edited 1 time in total.

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### Re: Edit: Debunking that SH is less useful with higher EV

This is just math fantasy. (Yes, I am tired again). I provided at least 3 situations where sh gives much higher chance to save life to low ev character comparing to high ev character and now I see again those "but you can fight 500 giants instead of just 250, it's the same as you can fight 1.5 giants instead of 0.75".
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### Re: Edit: Debunking that SH is less useful with higher EV

But your original argument was that if your EV is high, you are better off training EV than SH

Lol, what? My initial and current and future claim is that SH is less useful to high EV character than it is to low EV characters.
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### Re: Edit: Debunking that SH is less useful with higher EV

It's not a maths fantasy. I didn't even use any real maths in my last post, just words! If you can't do maths, you should have never introduced maths into it in the first place. The maths says that SH is independent of EV. The game fsim says the same thing. Describing it in words says the same thing. Just because yours and damerell's intuition says something different, it doesn't mean it is true.

And please, for the last time, stop comparing absolute values again. I thought we gone through this already and you agreed not to do so. Imagine your absolute +50% dodge rate Imagine going for 0% to a 50% dodge rate. You have now doubled the chance to save the life right? Sounds resonable right? A totally reasonable metric isn't it? That's just your intuition fooling you, because when you go from 50% dodge rate to 100% you don't double the chance to save the life, your character now cannot be hit and you now made your character invincible. How many times can this be explained?

And if you are wondering about real crawl situations, Titans are the #1 killer in crypt, if I remember correctly in another thread, so maybe the hypothetical situations where you get surrounded by 5 angry titans aren't as rare as you think.

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### Re: Edit: Debunking that SH is less useful with higher EV

Ok, here is my math fantasy.
Both you and Berder use quite complicated formulae in the form of A*a * B*b * (C*c) etc. where A, B, C are some constants for specific character and a is defence multiplier from SH value, b is one from AC value, c is one from EV value.
Then you claim that a in that formula benefits different characters in exactly the same way. To put it simple 500*a=2*a. How do you prove it? Very simple, you introduce a new measure which is ratio of those values i.e.(500×a)/500 and (2×a)/2 and eventually a=a. Of course it is, but still 500×a is way greater than 2×a. If we have a monster which requires 300 to beat, the first character does not even need a shield where the second character would have problems eveb with +8 large shield of protection.
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### Re: Edit: Debunking that SH is less useful with higher EV

Sorry, I leave the thread. It is no longer fun
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### Re: Edit: Debunking that SH is less useful with higher EV

VeryAngryFelid wrote:But your original argument was that if your EV is high, you are better off training EV than SH

Lol, what? My initial and current and future claim is that SH is less useful to high EV character than it is to low EV characters.

Really?...

viewtopic.php?f=5&t=25583 Friday, 27th July 2018, 14:26

VeryAngryFelid wrote:
Plantissue wrote:Why would higher EV result in SH become less useful?

I am lazy to run fsim so here is an easy example.
Let's compare 2 characters. One has 30% chance to dodge attacks, another has 70% chance to dodge attacks.
Now let's add a shield which blocks 20% attacks.
First character blocks 20% attacks and dodges 0.8*0.3=24% attacks so it blocks/dodges 44% attacks which is roughly 46% (44/30*100%-100%) improvement.
Second character blocks 20% attacks and dodges 0.8*0.7=56% attacks so it blocks/dodges 76% attacks which is roughly 9% (76/70*100%-100%) improvement.

---

You introduced maths to "prove" that higher EV results in SH becoming less useful, is just totally bogus.
It's simply chucking numbers around, resembling what the calculations should look like, to "prove" your argument.

So I did an ironclad maths to show that is not the case.
Then you did run an fsim. Showing that is not the case.
Then you changed metrics a number of times. But no matter the metric, properly done, that is not the case.

The maths used is not complicated. It is no more complicated than the maths you used. The only complication is fsim, which you generously provided yourself. The only difference is that I explained the reasoning behind the maths with words. If you think the maths is fantastical, you only have to point out where it doesn't make sense. The maths isn't even needed at all, except to debunk the totally bogus maths you and damerall used to support your position. BODMAS is not complicated. When do they teach bodmas in school anyways? There's not even any powers or roots involved. to be honest the brackets are also superfluous. Anyone after primary schooling in UK can do this maths.
Last edited by Plantissue on Tuesday, 31st July 2018, 16:28, edited 1 time in total.

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### Re: Edit: Debunking that SH is less useful with higher EV

Also I still didn't change my mind on

3. shows that with 100 HP the shield improves your time from 10 turns to 16 turns and from 25 turns to 40 turns. The second increase is greater in numbers but is still less useful. If you don't see why, increase the EV (50 turns instead of 80 has even greater increase and is even less useful) or just let your current HP go below 45 HP (that's max damage of stone giant) and suddenly see how you instantly switch to metrics 2 (can I die this turn?). Almost useless metrics.

CombatPower = (AvEffDam attacking) * (your HP) / ((1 - block chance) * (1 - dodge chance) * (defending damage per hit after armor) * (enemy HP))
Perhaps what you really mean to say is that when CombatPower is high (not just when EV is high) then a 50% increase in CombatPower is less useful no matter where the source: whether you get it from training your attack damage, your HP, your block chance, your dodge chance, or your armor. They are all less useful if CombatPower is already high, because when CombatPower is low you go from killing 1 giant to 1.5 giant which is really significant to your game-winning chances, and when CombatPower is already super high you go from killing 10 to killing 15 which is perhaps an unnecessarily large amount of power since you were going to win anyway.

On the other hand, which skill you train: attack, HP, shields, dodge, armor, will be independent of your current CombatPower. You get less advantage from training shields if CombatPower is already high - from a certain way of looking at it - but you also get proportionally less advantage from training *everything else*. The "reducing" benefit from shields is not different from the reducing benefit from attack, HP, dodge, and armor. The decision you make of what to train is independent of your current CombatPower (but does depend on the skill cost of increasing the different skills, among other things).

Now compare 3 characters:
1) 4 incoming damage and 20 outcoming damage.
2) 2 incoming damage and 10 outcoming damage
3) 1 incoming damage and 5 outcoming damage.
Are the characters really equal to each other? I would use the first one anytime. Because max incoming damage is usually much higher than average damage (36 for the fsim) so the faster I kill the giant the better, no matter how tiny damage I am supposed to get, because bad luck happens and I can be hit twice in a row. There is no significant difference between 4 incoming damage and 1 incoming damage, because 36 max damage is what matters and both 1 and 4 are too small to matter.

The fact you have only 1 incoming damage in the third case, makes getting hit for 36 max damage much less likely. 1/4 as likely each turn as in case number one.

Actually, the third character is the one with the lowest chance of losing the game, because while all three will kill the giant with the same expected HP loss, if the the third character starts to lose it can break off the fight and run away much more safely. Also, the third character can benefit more from regeneration than the first one. It might be more tedious to play, but your intuition would mislead you if you think it's weaker. Your intuition would mislead you because a decent weapon is one of the fastest ways to increase your CombatPower, and should be the first choice to train when you find one. A character is not generally faced with a choice between dealing 10 damage and receiving 2, or dealing 5 damage and receiving 1. Instead characters who have just found a decent weapon tend to be faced with a choice between dealing 10 damage and receiving 2, or dealing 5 damage and receiving 1.8, and they should generally choose the high damage option just because it has a higher CombatPower for the skill cost.

Then you realize that shield is checked ONLY if EV check fails

It doesn't matter, but in fact shield is checked first, then EV is checked only if shield fails.

or just let your current HP go below 45 HP (that's max damage of stone giant) and suddenly see how you instantly switch to metrics 2 (can I die this turn?).

If you break off a fight as soon as your HP goes below 45, you might use an alternative CombatPower stat that depends on your current HP minus 45. All the calculations are the same, except that you multiply by (HP-45) instead of by HP. Against a stone giant, which is ranged and is in a place where other ranged things will shoot you, it's actually better to break off a fight as soon as your HP goes below 50%, and again you can use an alternative CombatPower stat to handle this.
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Ziggurat Zagger

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### Re: Edit: Debunking that SH is less useful with higher EV

This looks like a conflict over some silly mis-understandings over statements and terms

VAF appears to be trying to say "If you have enough defense, more defense isn't as helpful", that he labeled "enough defense" as "EV" and "more defense" as "SH" is a mistake he doesn't seem to recognize.

Berder et.al. are trying to say "It doesn't matter whether 'more defense' is EV, SH, AC(well, really 'damage reduction' here, which AC is a close, if slightly inaccurate, proxy for), HP or even attack power, mathematically, if the proportions are the same, the result is the same" hence relabeling this thing 'combat power' which makes sense since it's not actually defense specifically at all..

So you guys are comparing apples and oranges, but VAF's underlying point (which he stated in an incorrect manner) is correct, in that if you have enough combat power, more combat power is *in an absolute sense* less useful, in that you need "enough combat power to kill the things you are going to face before you are able to rest", once you have that, more "combat power" than that doesn't do anything.

Of course how many "things you are going to face before you can rest" is random, and unknown, and is at best an estimate, so more is always better, but the returns diminish and costs increase, so it's going to be easier to increase any one of those variables by a a fixed percent than any of the other ones if I say "I want to increase my combat effectiveness by 30%" there's definitely an answer to which of the values it's easiest to do that with.

It's arguable that if you could quantify the ability to control the "number of things that you should kill before you can rest" it could be added to the "combat power" formula, and that if you could add points to that mythical stat, it would have equal value with the rest (in that if you can halve the number of monsters you have to kill.

VAF seems to be trying to point out (poorly) that at a certain point, *avoiding more attacks isn't going to change the end result by a significant margin* because you already avoid enough attacks to make a reasonable combat where avoiding attacks makes a difference a success (in the "you will get to the point where the monsters are dead and you can rest" point)

Some side points that might effect someone's decision beyond a simple combat power calculation:

In addition to increasing your overall combat effectiveness, HP also reduces the likelihood of dying to the worst case scenario
AC reduces the chances of getting to the worst case (and some types of damage reduction reduce the likelihood of dying to the worst case as well)
More attack, EV and SH don't do this.
EV and SH also have a subset of attacks that ignore them.
AC does too, but it's a pretty small subset, and there's nothing I can think of that ignores AC, but not EV or SH
Nothing ignores having more HP.

So in my analysis, *given similar amounts of increase in combat power*, HP > AC > Offense > EV > SH, however it's also frequently easiest to increase offense (you can usually see a larger increase in combat power more quickly either by changing spells, equipment, or training up to min delay) and it's very very possible that it'll be easier to increase say, SH by a larger value than AC, offense or EV.

Of course some times you have a really really high "ability to limit the number of creatures you have to fight before you rest" stat (for example spriggans can mostly walk away from everything, and rarely have to fight more than one thing before resting) which may drastically reduce the need for more combat power.
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### Re: Edit: Debunking that SH is less useful with higher EV

Plantissue wrote:Really?

Oh, it's fun again so I am back for one post
Really.

me wrote:True, I benefit from EV equipment no matter if I have shield or not, but it helps more if I have higher EV. Also the higher EV I have the less useful SH becomes.

You are welcome to point at word "training" in this post.
Or in the post which you referenced:
me wrote:I am lazy to run fsim so here is an easy example.
Let's compare 2 characters. One has 30% chance to dodge attacks, another has 70% chance to dodge attacks.
Now let's add a shield which blocks 20% attacks.
First character blocks 20% attacks and dodges 0.8*0.3=24% attacks so it blocks/dodges 44% attacks which is roughly 46% (44/30*100%-100%) improvement.
Second character blocks 20% attacks and dodges 0.8*0.7=56% attacks so it blocks/dodges 76% attacks which is roughly 9% (76/70*100%-100%) improvement.

Sorry, I cannot stop myself from personal attacks so I'd better leave now.
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### Re: Edit: Debunking that SH is less useful with higher EV

Berder

Sorry, I have to leave the thread so I can just repeat my post which was written after the post you are replying to. Thank you for discussion! (no sarcasm)
SH vs EV is the most obvious case here, you can check EV before SH and nothing will change, it will even make SH more useful than it really is. Then you realize that shield is checked ONLY if EV check fails and that means that SH is checked in 20% cases for one character and in 80% cases for another character. Of course it means SH cannot be equally useful to the characters.

Edit. Siegurt, mostly right but then EV vs SH is really unique (see this post).

Siegurt wrote:AC does too, but it's a pretty small subset, and there's nothing I can think of that ignores AC, but not EV or SH

Some hydra simulacra. AC does not do much when you get 150+ damage which ignores it. Acid Blob is fun too.
Last edited by VeryAngryFelid on Tuesday, 31st July 2018, 17:04, edited 2 times in total.
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