Effectiveness of one more point of ac/ev/sh (pictures!)


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Post Friday, 30th January 2015, 06:48

Effectiveness of one more point of ac/ev/sh (pictures!)

The following graphs show the effectiveness of getting one more point of ac/ev//sh against five different plain melee enemies. Each curve shows you how much defensive value there is in getting one more point of the attribute for that curve. With these graphs, you can make statements like "The first twelve points of AC with GDR 0.34 are much more valuable against a yak than subsequent points of AC," or "the value of a point of EV against a death yak peaks between EV 20 and EV 30." Eyeballing the heights of the curves and the area under the curves in these diagrams can tell you just about everything about how effective different kinds of defenses are against the listed monsters.

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All the curves for a given monster are on the same scale. That means, for example, that if you look up your current AC, EV, and SH on the chart, and the point on the curve for SH is higher than the points for AC and EV, then one more point of SH will help you more against that monster than one more point of AC or EV would. It doesn't mean that SH is currently helping you more than AC or EV - just that the incremental advantage of one more point of SH would help more.

Why is there that weird spikiness on the left side of the AC curves for GDR 0.34? That's because the GDR amount is capped at half your AC. So on even values of AC, that cap increases by 1, which makes a big difference and accounts for the spike. On odd values of AC the cap doesn't change so there is a smaller improvement. Your GDR amount is also capped at 0.34 * maximum damage, so once half your AC is at least that amount, the maximum GDR amount stops increasing, and the spikiness stops happening. The curve then converges with the AC curve for GDR 0. It can be said that the benefit of having a high GDR is all at low values of AC; once your AC is above the threshold, another point of AC helps you only about as much as it would with GDR 0.

For the spiky part of the AC curve, you can imagine the "real" marginal improvement as a curve that lies midway between the "high" spikes and the "low" spikes. It averages out to that.

The total defensive value for a character consists of the sum of the area under the curves for your AC, EV, and SH. The higher that amount is, the better your defenses are against that monster.


How did I make these curves? To get a little technical, first I calculated the damage reduction factor for AC, EV, and SH against the different monsters. The total damage reduction factor is obtained by multiplying together those three values. Then I took the logarithm with base 1.1 of these damage reduction factors for AC, EV, and SH, which gives me "defensive value." The total defensive value (the logarithm of the total damage reduction factor) is obtained by adding together your three defensive values for AC, EV, and SH. Now the curves shown are the differences between successive defensive values (a discrete derivative of the defensive value curve). For instance, the point for EV = 20 is obtained by subtracting the defensive value for EV 20 from the defensive value for EV 21.
Last edited by Berder on Friday, 30th January 2015, 12:07, edited 3 times in total.
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Post Friday, 30th January 2015, 08:40

Re: Effectiveness of one more point of ac/ev/sh (pictures!)

is this real life
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Post Friday, 30th January 2015, 08:44

Re: Effectiveness of one more point of ac/ev/sh (pictures!)

kroki wrote:is this real life

No, it's Crawl.
...{HEMoDEHuDDAs}{HaBeKoAK}CeVM{MfWnMiAK}TeAMDrIE{FoVMVSFi}{MuVMGhGlVpMo}HaWrSpWz
{OgGlTrMo}{CeWnMfBeMiSk}DrEE{GrFiFoGl}DgEnFeNe{OpGlHuSu}DDArHaCKSpAEGrTmDgFEDsCjGhMo
HuVM{HaAMBaEn}{HuMoHOWn}DsWzDDHu{DgWnGnBe}FeIE{MiEnMfCj}SpNeBaEEGrFE{HaAKTrCK}DsFESpHu

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Post Saturday, 31st January 2015, 12:58

Re: Effectiveness of one more point of ac/ev/sh (pictures!)

For completeness, and in case the earlier graphs were confusing, here are the graphs of the actual defensive values (not the marginal defensive values). Your total defensive value can be obtained by taking the points on the curves for your AC, EV, and SH, and just adding them together. As an example, in the yak graph, the SH curve (pink) crosses the AC at GDR 0.34 curve (green), at 30. That means against a yak, 30 SH is equally useful as 30 AC; a character with 0 AC 0 EV 30 SH would take an equal amount of damage against a yak as a character with 30 AC 0 EV 0 SH with GDR 0.34, assuming he's only getting hit once per turn.
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Explanation: these are the logarithms (base 1.1) of the damage reduction factors mentioned in the original post. The graphs shown in the original post are the "derivatives" of these graphs, showing how the slope of these graphs change.

I think you can tell more from the graphs in the original post, though; the marginal value of another point of AC is more useful to know than the actual value of all your AC. And you can kind of estimate these graphs from those in the original post just by looking at the area under the curves.
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Post Saturday, 31st January 2015, 18:16

Re: Effectiveness of one more point of ac/ev/sh (pictures!)

While this is not exactly CYC material, it's not exactly dungeon crawling advice either. You make no specific and useful claims as to what a player should be doing to win, nor do you pose any meaningful question. I don't care about "spikes in my marginal defensive value versus a yak". If I see (or know from experience) that a Yak has too much damage output for my character to melee, I find other ways to kill or greatly weaken it at range, or I find a way to avoid it. I never need to consult a curve in making this decision. As for planning my character I tend to know whether I'm going for more AC/GDR or more EV earlier based on the background, species, and also what I've found in the dungeon.

If you're just trying to show off your ability to make graphs and play with numbers, this belongs in CYC. If you don't want this moved to CYC, try to make some dungeon crawling (i.e. gameplay) advice out of it that is both useful and non-obvious.

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Post Saturday, 31st January 2015, 21:48

Re: Effectiveness of one more point of ac/ev/sh (pictures!)

gammafunk wrote:I don't care about "spikes in my marginal defensive value versus a yak".

While I would agree it might have been more useful to smooth that region of the plot, picking this as your flagship reason to denigrate this posting really makes it look like your posting is pure anti-intellectualism rather than any attempt at fair and reasonable criticism of this posting. The rest of the paragraph only supports this interpretation.

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Post Saturday, 31st January 2015, 22:10

Re: Effectiveness of one more point of ac/ev/sh (pictures!)

gammafunk wrote:While this is not exactly CYC material, it's not exactly dungeon crawling advice either. You make no specific and useful claims as to what a player should be doing to win, nor do you pose any meaningful question. I don't care about "spikes in my marginal defensive value versus a yak". If I see (or know from experience) that a Yak has too much damage output for my character to melee, I find other ways to kill or greatly weaken it at range, or I find a way to avoid it. I never need to consult a curve in making this decision. As for planning my character I tend to know whether I'm going for more AC/GDR or more EV earlier based on the background, species, and also what I've found in the dungeon.

If you're just trying to show off your ability to make graphs and play with numbers, this belongs in CYC. If you don't want this moved to CYC, try to make some dungeon crawling (i.e. gameplay) advice out of it that is both useful and non-obvious.


There's a ton of information in these plots. Your objection reads like "I'm sorry, you didn't state your answer in the form of a question." And what difference does it make how you plan your character? Lots of RPG fans are interested in this kind of min-max-y information. What is the point of pointing out a particular question that these plots bear on when they obviously bear on a lot of different questions?

More generally, what's up with the hostility to Berder's number-heavy approach?
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Post Saturday, 31st January 2015, 22:17

Re: Effectiveness of one more point of ac/ev/sh (pictures!)

mps wrote:More generally, what's up with the hostility to Berder's number-heavy approach?
Mainly that he spams it everywhere and that it's only useful if you're fighting 1 yak and all of your keyboard's keys are broken except for tab

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Post Saturday, 31st January 2015, 22:25

Re: Effectiveness of one more point of ac/ev/sh (pictures!)

duvessa wrote:
mps wrote:More generally, what's up with the hostility to Berder's number-heavy approach?
Mainly that he spams it everywhere and that it's only useful if you're fighting 1 yak and all of your keyboard's keys are broken except for tab


Not so sure. Among other things, it gives an account of where to concentrate gains to get the most of your resources at different points of the game. In normal gameplay, this probably doesn't make a huge difference, but there are presumably challenge-type situations where you operate with very tight resources and having this information makes a difference.
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Post Saturday, 31st January 2015, 23:14

Re: Effectiveness of one more point of ac/ev/sh (pictures!)

The charts can be used to make equipment decisions. Say you must choose between a ring giving +3 AC and a ring giving +3 EV - you can use the marginal defensive value charts to see at a glance which is better for you; just look at the points for your current AC and EV curves, and see which is higher. Or say you want to upgrade to heavier armor or start using a shield. For example, suppose the armor gives you +5 AC (plus more GDR) and -3 EV. Look at the marginal defensive value curves for your AC and EV, and if the point for your current AC is at least 3/5 as high as the point for your EV, then the heavier armor is better defensively. Also consider your attack speed - every 10% decrease in attack speed or damage costs you about 1.1 point of defensive value on these charts. Every 10% increase in attack speed or damage gives you exactly 1 point.

The charts are also good for answering long term strategic questions like, "Is it worth it to use a shield on my character?" Maybe you already have a shield and are wondering if it's really doing anything for you. You can see at a glance how much defensive value the shield is giving you against some typical enemies, assuming you have about 20 SH with a +5 shield. You can compare that to the loss in damage output of using a shield instead of a 2h weapon (if any - 10% damage gives you 1 point) to estimate if it's worth it. Of course, the shield is typically not worth it if you aren't able to hold a choke point, unless there is no loss in damage output.

Note that for early game levels of SH and EV against a yak (taken to represent a difficult pre-lair monster), the marginal defensive value (MDV) of either is roughly 0.5 points. So suppose you're a fighter. What's your starting shield doing for you? Your starting shield for most races gives you about +9 SH and -3 EV, which is a gain of about 6/2 = 3 points of DV. If it isn't slowing your attack speed by 30%, it's worth wearing in the early game. This is just an estimate, of course, since you won't be fighting actual yaks for a while.
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Post Saturday, 31st January 2015, 23:30

Re: Effectiveness of one more point of ac/ev/sh (pictures!)

First example with the rings is mostly correct. There are times I would prefer an increase in AC over EV even if the "damage reduction factor" of one increase was slightly less, and vice versa, depending on the character. But for the most part, the graphs can potentially be helpful here, in much the same way that fsim can be helpful when making simple comparisons between damage output of certain weapons (though in both cases there are things the comparison will not adequately capture).

For the other two, the increase from SH would have to give much better defenses than the AC even to be considered, seeing as how SH is the least dependable form of defense (a fact that is not apparent if one bases his or her judgments on the assumption that combat in Crawl exists in a vacuum except for you and a melee-only enemy hitting each other). Ditto for the third example. Killing an enemy more quickly is valuable because it gives less turns for the enemies that are actually the most dangerous in the game (for the most part, things that are not melee-only) to do their dangerous not-melee stuff, like summon more bad guys or hit you with hexes, etc.

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Post Saturday, 31st January 2015, 23:46

Re: Effectiveness of one more point of ac/ev/sh (pictures!)

and into wrote:First example with the rings is mostly correct. There are times I would prefer an increase in AC over EV even if the "damage reduction factor" of one increase was slightly less, and vice versa, depending on the character. But for the most part, the graphs can potentially be helpful here, in much the same way that fsim can be helpful when making simple comparisons between damage output of certain weapons (though in both cases there are things the comparison will not adequately capture).

For the other two, the increase from SH would have to give much better defenses than the AC even to be considered, seeing as how SH is the least dependable form of defense (a fact that is not apparent if one bases his or her judgments on the assumption that combat in Crawl exists in a vacuum except for you and a melee-only enemy hitting each other). Ditto for the third example. Killing an enemy more quickly is valuable because it gives less turns for the enemies that are actually the most dangerous in the game (for the most part, things that are not melee-only) to do their dangerous not-melee stuff, like summon more bad guys or hit you with hexes, etc.

Absolutely; these charts don't give you the full picture, they just give you a rough estimate. The charts let you see how much the shield helps in regular combat, but then you do have to consider summons and nasty hexes and all that stuff, and make a judgment call.
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Post Sunday, 1st February 2015, 00:07

Re: Effectiveness of one more point of ac/ev/sh (pictures!)

Berder wrote:The charts are also good for answering long term strategic questions like, "Is it worth it to use a shield on my character?" Maybe you already have a shield and are wondering if it's really doing anything for you.

I was rather surprised at how much more effective SH is against orb guardians than other defenses; do you know why it works out that way, and what sorts of enemies would have similar plots?

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Post Sunday, 1st February 2015, 00:18

Re: Effectiveness of one more point of ac/ev/sh (pictures!)

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Post Sunday, 1st February 2015, 00:30

Re: Effectiveness of one more point of ac/ev/sh (pictures!)

Hurkyl wrote:
Berder wrote:The charts are also good for answering long term strategic questions like, "Is it worth it to use a shield on my character?" Maybe you already have a shield and are wondering if it's really doing anything for you.

I was rather surprised at how much more effective SH is against orb guardians than other defenses; do you know why it works out that way, and what sorts of enemies would have similar plots?

It's because orb guardians do a lot of damage with each hit, making armor less effective, and because they have the fighter flag which increases their to-hit value and makes evasion less effective. They only have 15 hit dice, similar to a death yak, and they don't do multiple attacks, which means shields are nearly as effective against them as they are against death yaks.
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Post Sunday, 1st February 2015, 00:35

Re: Effectiveness of one more point of ac/ev/sh (pictures!)

Oh, right, fighter flag exists and doesn't work on SH. Then it makes sense.

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Post Sunday, 1st February 2015, 01:38

Re: Effectiveness of one more point of ac/ev/sh (pictures!)

Orb guardians doing a lot of damage each hit doesn't mean armour is less effective. In fact GDR is more relevant for large-damage hits since it's reducing more damage.

Deciding between +3 EV vs +3 AC rings or optimizing my D:1 Fighter gameplay in case of a Yak guarding all three downstairs to D:2 doesn't sound very important in terms of decision making. I tend to make my shield decisions in terms of questions like: am I playing a character that will do relatively little melee for which defense is more important than having a 2h weapon, or am I playing a species with a large EV penalty that can otherwise deal with the loss of 2h melee damage, or does a decent shield exist in the game yet? I've used the "maximize the sum of AC and EV" rule to good effect for armour decisions involving more marginal differences between AC and EV, but thankfully crawl doesn't care too much if you don't find a local maximum of a hidden function.

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Post Sunday, 1st February 2015, 02:24

Re: Effectiveness of one more point of ac/ev/sh (pictures!)

The greatest value is that it's interesting to see how these things actually work. You can see that AC derives the most benefit per point at low amounts of AC, and EV is not so good at low amounts but at high amounts outperforms AC against many enemies. And you can see that AC and SH actually are comparable to EV against many enemies.

There's a factor I haven't mentioned, which is that a high AC character is able to keep fighting safely to a lower amount of HP than a high EV character can do. If your safety rule is to abandon the fight or take other emergency measures as soon as there is a 1% chance of dying in the next two turns, the amount of HP at which you have to abandon it is greater for high EV than it is for high AC, due to the higher damage variance you get with an EV defense. This is going to depend on your max HP as well as your defenses. So it would boost the effectiveness of AC somewhat.
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Post Sunday, 1st February 2015, 02:35

Re: Effectiveness of one more point of ac/ev/sh (pictures!)

gammafunk wrote:

[...] "maximize sum of AC and EV" [...] but thankfully crawl doesn't care too much if you don't find a local maximum of a hidden function.


Got plots? I realize this information is available in the tables from the other thread, but it'd be nice to see it graphically.

Also, re: Berder's latest post, it might be interesting to include a measure of damage variance in these analyses, e.g. standard deviation, damage quartiles, etc. I suspect that's information people are less familiar with.
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Post Sunday, 1st February 2015, 04:03

Re: Effectiveness of one more point of ac/ev/sh (pictures!)

mps wrote:Got plots? I realize this information is available in the tables from the other thread, but it'd be nice to see it graphically.

Plots of what?

Also, re: Berder's latest post, it might be interesting to include a measure of damage variance in these analyses, e.g. standard deviation, damage quartiles, etc. I suspect that's information people are less familiar with.

I think the number that would be most useful for that would be the 99th percentile of damage dealt in a hit. Or in two hits.
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Post Sunday, 1st February 2015, 04:21

Re: Effectiveness of one more point of ac/ev/sh (pictures!)

Berder wrote:
mps wrote:Got plots? I realize this information is available in the tables from the other thread, but it'd be nice to see it graphically.

Plots of what?


Oh, sorry, I deleted part of what I meant to post. Got plots of average damage reduction for where AC + EV is held constant and one, say AC, varies?


Also, re: Berder's latest post, it might be interesting to include a measure of damage variance in these analyses, e.g. standard deviation, damage quartiles, etc. I suspect that's information people are less familiar with.

I think the number that would be most useful for that would be the 99th percentile of damage dealt in a hit. Or in two hits.


Agreed, although quartiles would give a finer view of what a typical encounter looks like.
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Post Sunday, 1st February 2015, 04:46

Re: Effectiveness of one more point of ac/ev/sh (pictures!)

mps wrote:
Berder wrote:
mps wrote:Got plots? I realize this information is available in the tables from the other thread, but it'd be nice to see it graphically.

Plots of what?


Oh, sorry, I deleted part of what I meant to post. Got plots of average damage reduction for where AC + EV is held constant and one, say AC, varies?

They all vary independently of each other; how useful AC is (in terms of its damage reduction factor) depends on how much AC you have, but not on how much EV or SH you have. (Not ---quiiite-- correct in the case of SH if enemies are attacking multiple times in a turn with unequal attacks, but very nearly). So you don't need separate plots where some variables are held constant. You can think of each of the curves for DV in this thread as the total DV if the other two attributes are held constant at 0. Complicated slightly by the fact that the damage reduction factor of EV 0 is greater than 0.

If you're talking about that AC/EV table I posted a while ago, most of that table was unnecessary; the first row and first column tell you everything you need to know, and the rest of the table is just from multiplying together the AC and EV damage reduction factors.

This thread uses DV instead of damage reduction factor because DV is additive, whereas damage reduction factor multiplies.
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Post Sunday, 1st February 2015, 05:54

Re: Effectiveness of one more point of ac/ev/sh (pictures!)

I understand all of that, but you don't make graphs just to display data -- you can do that better with a table. The point is to expose detail to visual intuition.

Upthread gammafunk suggests there's little or no point in looking at these values in much greater detail than the usual AC+EV heuristic. Your plots show there's enough nonlinearity that there could be more to it, but without visualizing it, it's a little hard to see what's there that can be used without having a graph or table in front of you.
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Post Sunday, 1st February 2015, 09:58

Re: Effectiveness of one more point of ac/ev/sh (pictures!)

Well, as you requested:
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ac+ev=25 is typical for a character around xl10
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ac+ev=50 is typical for a character around xl20.

As you can see, the shape of this kind of chart substantially depends on the total. The ac+ev=50 chart has a bump on the right because ev starts being more effective around 20-30 against many monsters. The ac+ev=25 chart has a dip in the middle because ev is less effective than ac at low levels, and then it goes up again at the right because that's approaching the bump from the ac+ev=50 chart.

I think you can tell a lot more about this from the marginal defensive value charts in the OP. From those, it's easy to see that if you have 40 ac and 25 ev, then another point of ev is roughly twice as good as a point of ac against all but the orb guardian. It's hard to see things like that from the charts in this post.
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Post Sunday, 1st February 2015, 23:28

Re: Effectiveness of one more point of ac/ev/sh (pictures!)

gammafunk wrote:Orb guardians doing a lot of damage each hit doesn't mean armour is less effective. In fact GDR is more relevant for large-damage hits since it's reducing more damage.

You're looking at the wrong number; the amount of damage reduced is not relevant on its own. Taking one less damage per hit means a lot when you're being hit for five damage. Taking two less damage means very little when you're being hit for fifty damage. The relative amount of damage reduced is a much better proxy for what actually matters.

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Post Monday, 2nd February 2015, 01:01

Re: Effectiveness of one more point of ac/ev/sh (pictures!)

I'm not following your method for making the original graphs, specifically "defensive value". I'll provide an example of my thinking:

Assume the character is fighting a yak and initially has 0 AC, 0 EV and 0 SH. Now assume the character has 1 AC, 0 EV and 0 SH. The marginal increase in defense (damage reduction) should be just under 0.5 because it's equally likely to roll either a 0 or 1 for AC damage reduction, plus I believe there's a small chance to miss regardless of EV so that must be taken into account, plus there is a chance the yak could roll 0 for damage (the miss chance and 0 damage are why it's just under 0.5 instead of 0.5).

EV and SH work differently by either totally negating the attack or doing nothing, so their marignal increase in defense will depend both on the incremental probability increase of negating the attack as well as the average power of the attack (negating a yak attack versus an orb guardian attack which has higher average damage). Further complications come into play versus creatures with multiple attacks or when considering AC/EV/SH together or considering GDR.

For the initial graphs you come up with "defensive value" by multiplying three "defensive factors" together and taking a logarithm. How did you calculate the defensive factors, and why multiply them then take a logarithm base 1.1?

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Post Monday, 2nd February 2015, 01:14

Re: Effectiveness of one more point of ac/ev/sh (pictures!)

monsters cant roll 0 for damage

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Post Monday, 2nd February 2015, 02:05

Re: Effectiveness of one more point of ac/ev/sh (pictures!)

SaidTheAlligatorKingToHisSon wrote:I'm not following your method for making the original graphs, specifically "defensive value". I'll provide an example of my thinking:

Assume the character is fighting a yak and initially has 0 AC, 0 EV and 0 SH. Now assume the character has 1 AC, 0 EV and 0 SH. The marginal increase in defense (damage reduction) should be just under 0.5 because it's equally likely to roll either a 0 or 1 for AC damage reduction, plus I believe there's a small chance to miss regardless of EV so that must be taken into account, plus there is a chance the yak could roll 0 for damage (the miss chance and 0 damage are why it's just under 0.5 instead of 0.5).

EV and SH work differently by either totally negating the attack or doing nothing, so their marignal increase in defense will depend both on the incremental probability increase of negating the attack as well as the average power of the attack (negating a yak attack versus an orb guardian attack which has higher average damage). Further complications come into play versus creatures with multiple attacks or when considering AC/EV/SH together or considering GDR.

For the initial graphs you come up with "defensive value" by multiplying three "defensive factors" together and taking a logarithm. How did you calculate the defensive factors, and why multiply them then take a logarithm base 1.1?

You're focusing on defenses subtracting from the incoming damage, but that's the wrong way to go about it. It makes sense for AC but as you say it gets complicated for EV and SH. Also, it can't be easily compared across different monsters that do different amounts of damage or that have multiple attacks. It's better to start by looking at how much your defenses divide the incoming damage. You might think that AC isn't dividing, it's subtracting, but if the incoming damage is 18 and the result after AC is 6, then that can be considered as dividing the incoming damage by 3.

Okay, first, the "damage reduction factor" is the amount by which incoming damage is divided. If an enemy would deal 18 damage on average to someone with no defenses, and with your defenses they only deal an average damage of 2 damage, then your "damage reduction factor" is 18 / 2 = 9. Incoming damage from that enemy is reduced by a factor of 9 on average.

A player has a damage reduction factor from AC, EV, and SH. Perhaps their AC reduces that damage (if it hits) from an average of 18 to an average of 6, which corresponds to a damage reduction factor from AC of 18 / 6 = 3. Let's denote this by DRF_AC = 3. Let's say their EV lets them dodge half of incoming attacks that aren't blocked. That means DRF_EV = 2. And say their SH lets them block 1/3 of all incoming attacks, letting 2/3 through, so DRF_SH = 1 / (2/3) = 1.5. That means the total factor by which incoming damage is reduced is 3 * 2 * 1.5 = 9. Let's denote this by DRF_tot = 9. If the enemy would deal 18 damage on average if the player had no defenses, then the enemy is only dealing the player 2 damage on average with his current AC, EV, and SH.

However, since the damage reduction factors multiply together, it's harder to visualize how much your DRF_AC, DRF_EV, and DRF_SH affect your DRF_tot. It's hard for humans to look at three numbers and get a sense of their product. Therefore I take the logarithm of each damage reduction factor. Note

DRF_tot = DRF_AC * DRF_EV * DRF_SH
log (DRF_tot) = log(DRF_AC * DRF_EV * DRF_SH)
log (DRF_tot) = log(DRF_AC) + log(DRF_EV) + log(DRF_SH)

As you see, to find log(DRF_tot), you just have to add three things together. I'll denote log(DRF_*) as DV_*, which stands for Defensive Value. So we have
DV_tot = DV_AC + DV_EV + DV_SH
So to find your total defensive value, you only have to add together the defensive values from your AC, EV, and SH. That's nice and is why I prefer DV to DRF in these charts.

Now the charts in the original post are about the marginal defensive value (MDV), which is how much defensive value you get from one more point of AC, EV, or SH. Getting one more point of AC increases your DV_AC by an amount equal to MDV_AC, which increases your DV_tot by that same amount. Note that to maximize your defenses, you want to maximize DRF_tot, which is the same as maximizing DV_tot. So you want to make defensive choices that add as much as possible to your DV.


I hope that answers your questions.

(Why am I taking the logarithms to base 1.1? Well, that makes the marginal defensive values fall generally between 0 and 1, which is nice because if I used a larger logarithm base then the marginal defensive values would only be small fractions of 1. Also it means that a 10% increase in attack speed is equivalent to 1 point of DV).
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Post Monday, 2nd February 2015, 02:18

Re: Effectiveness of one more point of ac/ev/sh (pictures!)

gammafunk wrote:While this is not exactly CYC material, it's not exactly dungeon crawling advice either. You make no specific and useful claims as to what a player should be doing to win, nor do you pose any meaningful question.
I agree that there is no proper place on the forum for a posting like this, but I don't think it's non-content in any way. (Nobody's forced to read this and it is about Crawl, after all... unlike some other threads.)

While it's clear that these numbers bear no immediate meaning on most gameplay situations (too many other factors), I believe this kind of number-crunching is actually useful. Galehar was quite fond of it, and I think it can positively inform design decisions.

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Post Monday, 2nd February 2015, 02:36

Re: Effectiveness of one more point of ac/ev/sh (pictures!)

Oh - both of those tradeoff charts were for GDR 0. With GDR 0.34, the first few levels of ac are more valuable, as you'd expect. Here are the charts for GDR 0.34.

Image
Image
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Post Monday, 2nd February 2015, 20:02

Re: Effectiveness of one more point of ac/ev/sh (pictures!)

I follow your reasoning now, though would you mind running a sample calculation? Total defensive value (not mariginal) for 10 AC (0 GDR) vs. a yak? I'm curious as to how you're determining average damage reduction for AC, as my understanding of it would require a fairly laborious calculation; enumerate all possible outcomes, then determine expected value of attack.

duvessa wrote:monsters cant roll 0 for damage

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Post Tuesday, 3rd February 2015, 06:14

Re: Effectiveness of one more point of ac/ev/sh (pictures!)

SaidTheAlligatorKingToHisSon wrote:I follow your reasoning now, though would you mind running a sample calculation? Total defensive value (not mariginal) for 10 AC (0 GDR) vs. a yak? I'm curious as to how you're determining average damage reduction for AC, as my understanding of it would require a fairly laborious calculation; enumerate all possible outcomes, then determine expected value of attack.

That is exactly how I calculate it. For each combination of dice rolls for damage and AC, I sum up the damage that would be dealt without AC, and the damage that is prevented, and in the end divide the damage-without-ac total by the damage-with-ac total. I use Python to do it.

The others are done similarly, though SH is tricky with multiple attacks. Here are the relevant sections of my code (the rest is just printing and plotting).

  Code:
# calculates the average factor by which incoming dmg is reduced by AC
def ACReduction(ac, dam, gdr = 0):
     totreduction = 0
     gdr_amt = int(gdr * 100 * dam) / 100
     if gdr_amt > ac / 2:
          gdr_amt = ac / 2
     totdam = 0
     for roll1 in xrange(1, dam + 1): # damage roll
          for roll2 in xrange(ac + 1): # ac roll
               if roll2 < gdr_amt:
                    roll2 = gdr_amt
               if roll2 > roll1:
                    roll2 = roll1
               totreduction += roll2
               totdam += roll1
     if totreduction == totdam:
          return float("inf")  # 100% damage reduction, infinite reduction factor
     return float(totdam) / (totdam - totreduction)

def ACReductionMultipleAttacks(ac, dam, gdr = 0):
     acreduc = 0
     if ac == 0:
          return 1.0
     elif type(dam) == list:
          treduc = 0
          for d in dam:
               treduc += d - (float(d) / ACReduction(ac, d, gdr))
               acreduc = sum(dam) / (sum(dam) - treduc)
     else:
          acreduc = ACReduction(ac, dam, gdr)
     return acreduc


def calcToHit(hitdice, fighter):
     tohit = 18
     if fighter:
          tohit += int(25 * hitdice / 10)
     else:
          tohit += int(15 * hitdice / 10)
     return tohit

# calculates the average factor by which incoming dmg is reduced by EV
# provide _either_ hitdice and fighter, _or_ tohit, but not both
def EVReduction(ev, hitdice = 0, fighter = False, tohit = None):
     if tohit == None:
          tohit = calcToHit(hitdice, fighter)
     totreduction = 0
     totdam = 0
     if ev == 0:
          return 1 / 0.975
     for roll1 in xrange(tohit + 1):
          for roll2 in xrange(2 * ev):
               for roll3 in xrange(2 * ev + 1):
                    if roll1 < (roll2 + roll3) / 2:
                         totreduction += 1 # 1 damage dodged
                    totdam += 1
     # normalize to 1.0 totdam and acct for the fact the preceding calculation only applied to 95% of cases
     totreduction = 0.95 * (float(totreduction) / totdam)
     # now account for 2.5% automatic hits and 2.5% automatic misses
     return 1.00 / (1.00 - (totreduction + 0.025))

def BlockChancePerAttack(sh, pastblocks = 0, hitdice = 0):
     totdam = 0
     totreduction = 0
     if sh == 0:
          return 0
     for roll1 in xrange(15 + hitdice * 2 / 3 + 5 * pastblocks**2):
          for roll2 in xrange(4 * sh):
               for roll3 in xrange(4 * sh + 1):
                    blockval = (roll2 + roll3) / 6 - 1
                    if roll1 <= blockval:
                         totreduction += 1 # blocked
                    totdam += 1
     return float(totreduction) / totdam

import math

# Note that there is sometimes a complex interaction between armor and shield reduction
# (not simply multiplicative)
# In the case where the attacks are not all the same, the earlier attacks will be blocked more and the later attacks will be blocked less,
# and the effect of armor on the different attacks is unequal as well
# We'll just ignore it since it's OK when the attacks are the same and these calculations don't have to be perfect
def SHReductionMultipleAttacks(sh, attacks, speed, hitdice):
     if type(attacks) is int:
          attacks = [attacks]
     bcpa = [] # bcpa[blocks] = probability of a block given previous blocks
     maxatk = len(attacks) * int(math.ceil(speed))
     for pastblocks in xrange(maxatk):
          bcpa.append(BlockChancePerAttack(sh, pastblocks, hitdice))
     bcpa.append(0) # dummy value, we'll never get this many past blocks
     # now weight by how often each number of pastblocks is happening
     # in a given turn, player will be attacked either ceil(speed) times,
     #  or floor(speed) times.
     # In the long run over N turns, player will be attacked N * speed times.
     # Therefore, a * ceil(speed) + (N - a) * floor(speed) = N * speed
     # where a is the number of times the player is attacked ceil(speed) times
     #  a * (ceil(speed) - floor(speed)) = N * speed - N * floor(speed)
     #  a/N = speed - floor(speed) assuming speed != floor(speed)
     # 
     # weights[k] = probability k'th attack will happen in a turn
     weights = [0] + [1] * (len(attacks) * int(math.floor(speed)))
     if math.floor(speed) != speed:
          weights.append([speed - math.floor(speed)] * len(attacks))
     # now calculate probability of different blocking results
     bcpa2 = [[0]*(maxatk+1) for x in xrange(maxatk+1)] # bcpa2[nBlocks][nAttacks] = P(we see the combination (nBlocks, nAttacks) at some point in a given turn)
     for x in xrange(maxatk+1): # boundary values
          bcpa2[0][x] = (1 - bcpa[0]) ** x
     for block in xrange(1, maxatk+1):  # fill in bcpa2 with dynamic programming
          for attack in xrange(block, maxatk+1):
               bcpa2[block][attack] = bcpa2[block-1][attack-1] * bcpa[block - 1] + bcpa2[block][attack-1] * (1 - bcpa[block])
     # account for the fact that the last attack may not always happen
     if math.floor(speed) != speed:
          for block in xrange(maxatk+1):
               for atk in xrange(maxatk+1 - len(attacks), maxatk+1):
                    bcpa2[block][attack] *= speed - math.floor(speed)
     # now calculate actual damage reduction
     totdmg = 0
     totreduction = 0
     for attack in xrange(1, maxatk+1):
          dmg = attacks[(attack - 1) % len(attacks)] * weights[attack] # proportional to the average damage dealt by the nth attack in a turn
          blockpr = 0
          norm = 0
          for block in xrange(0, attack+1):
               norm += bcpa2[block][attack]
               if bcpa2[block][attack] == 0: # never have this combination of blocks and attacks
                    continue
               if block == 0:  # couldn't have been blocked
                    continue
               #p = bcpa[block-1] * bcpa2[block-1][attack-1] / bcpa2[block][attack] # chance the last attack was blocked
               #blockpr += p * bcpa2[block][attack]
               blockpr += bcpa[block-1] * bcpa2[block-1][attack-1]
          blockpr = blockpr / norm
          totdmg += dmg
          totreduction += dmg * blockpr
     return totdmg / (totdmg - totreduction)

# monsters
# dam, hitdice, fighter, name
yak = [18, 7, False, "Yak"]
deathyak = [30, 14, False, "Death Yak"]
deeptroll = [[27, 20, 20], 10, False, "Deep Troll"]
veryugly = [36, 18, False, "Purple Very Ugly Thing"]
direelephant = [[40, 15], 15, False, "Dire Elephant"]
hydra6 = [[18, 18, 18, 18, 18, 18], 13, False, "Six-headed Hydra"]
orbguardian = [45, 15, True, "Orb Guardian"]
bonedragon = [[30,20,20], 20, False, "Bone Dragon"]

# total damage reduction factor
def drf(ac, ev, sh, monster, gdr=0):
     dam, hitdice, fighter, name = monster
     return ACReductionMultipleAttacks(ac, dam, gdr) * SHReductionMultipleAttacks(sh, dam, 1, hitdice) * EVReduction(ev, hitdice, fighter)
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Post Thursday, 5th February 2015, 07:51

Re: Effectiveness of one more point of ac/ev/sh (pictures!)

Duvessa are you sure monsters don't roll a zero for damage? I just had a situation where I had 0 AC (batform), was hit by a goblin wielding a +0 dagger, and got the message "the goblin hits you but does no damage".

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Post Thursday, 5th February 2015, 08:32

Re: Effectiveness of one more point of ac/ev/sh (pictures!)

Correction: it appears monsters can't roll 0 for damage unless they are wielding a weapon, in which case they can.

(What the fuck.)

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Post Thursday, 5th February 2015, 09:57

Re: Effectiveness of one more point of ac/ev/sh (pictures!)

I guess they try to hit you with the wrong end of the weapon in that case.
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Post Friday, 6th February 2015, 16:25

Re: Effectiveness of one more point of ac/ev/sh (pictures!)

gammafunk wrote:If you're just trying to show off your ability to make graphs and play with numbers, this belongs in CYC. If you don't want this moved to CYC, try to make some dungeon crawling (i.e. gameplay) advice out of it that is both useful and non-obvious.

OH SNAP

seriously if we are considering moving non-useful advice threads to cyc this one would probably not be where I would start
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Post Saturday, 7th February 2015, 14:53

Re: Effectiveness of one more point of ac/ev/sh (pictures!)

Can you make a table of the following calculation for each AC/EV value?

* Look at the difference between GDR 0 and GDR 0.34 and work out how many points of EV you'd need to get the same benefit. (linearly interpolated)

Eyeballing it, I think this is going to be worth something like 3-5 points of EV, making it a sufficiently relevant quantity to consider for characters who are trying to use the AC+EV heuristic when considering whether or not to stick with light armour or switch to heavier armour.

(I imagine for other comparisons than robe vs. chain armour, the EV equivalent would behave something like a linear dependence on the difference in GDR between the two armours?)

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Post Sunday, 8th February 2015, 04:10

Re: Effectiveness of one more point of ac/ev/sh (pictures!)

Hurkyl wrote:Can you make a table of the following calculation for each AC/EV value?

* Look at the difference between GDR 0 and GDR 0.34 and work out how many points of EV you'd need to get the same benefit. (linearly interpolated)

Eyeballing it, I think this is going to be worth something like 3-5 points of EV, making it a sufficiently relevant quantity to consider for characters who are trying to use the AC+EV heuristic when considering whether or not to stick with light armour or switch to heavier armour.

(I imagine for other comparisons than robe vs. chain armour, the EV equivalent would behave something like a linear dependence on the difference in GDR between the two armours?)

It's more like 5-6 EV, if you have enough AC that the graph stops being spiky. Just look at the difference between the GDR 0 curve and the GDR 0.34 curve in the "Defensive value" charts (not marginal defensive value). Then looking at the blue EV curve, count how many points of EV is needed to take you that vertical distance.
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Post Thursday, 12th February 2015, 21:58

Re: Effectiveness of one more point of ac/ev/sh (pictures!)

The difficulty in using the marginal defensive value is that it's rare in practice that you are looking at a 1:1 ac/ev tradeoff.

For instance right now I'm looking at "Should I wear this +2 platemail or this +2 ring mail"

Given my current training level, the plate mail is 33AC/24EV, and the ring mail is 25AC/29EV.

Now looking at the marginal values, I can easily tell that in the vaults, against deep trolls (a fairly common enemy) EV is more valuable per-point than AC, but is it 5/9ths as good? I don't know, it's close, and to make matters even harder to judge, the AC line particularly gets further from EV over the course of the graph, I can tell that at at 24/25 AC is probably better than 5/9ths, but at 33 is probably worse than 5/9ths, really what I need here is to be able to tell the difference in two marginal defensive values and compare them.

I can use the second set of graphs (the total defensive values) to get that slightly more easily, as I can look at the results going from 24->29 EV and from 25->33 AC and see that the vertical distance between the two is about the same for each change, letting me know that I should look elsewhere for "which is better".

This is also, of course, complicated by the fact that I'm changing my GDA by switching body armours, making it significantly harder to tell. (With GDA it seems probable that the plate is better, on the other hand, I have spells, and like to cast them, so there's that too)

In the long run my decision is of course not going to be decided by the total defensive value, but it's interesting to note that it's probable that in most circumstances the "net total of AC+EV" rule is probably close to optimal, although if the totals are fairly close it may be off by some margin, but in that case the difference is small enough that you should be using other criteria to make your decision.
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Post Friday, 13th February 2015, 22:51

Re: Effectiveness of one more point of ac/ev/sh (pictures!)

The curves are only drawn for a few representative unarmed melee monsters. I have in mind a more ambitious project which would largely correct this deficiency. Plan is: use sequell to look up the stats on how often each monster killed a player in a given area (e.g. pre-lair, lair, orc, vaults, depths, zot5), and what weapon, spell, or projectile was used to deal the blow. This results in a "standard assault" of all the different attacks in proportion to how often they killed players, and I could calculate how much that assault would be reduced by a player's defenses. That would let me give a rating for a player's defensive effectiveness in different areas/branches, taking into account EV/AC/SH/RMSL/DMSL/resistances/umbra. Of course, this scheme is not without flaws. Also I may not actually do it.
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Post Friday, 13th February 2015, 23:06

Re: Effectiveness of one more point of ac/ev/sh (pictures!)

Well, good luck with your project if you decide to do it. It sounds complicated enough :)

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Post Saturday, 14th February 2015, 01:31

Re: Effectiveness of one more point of ac/ev/sh (pictures!)

Now make a spreadsheet where I can enter my current skill levels, aptitudes, stats, armour, AC, and EV, and it will tell me how to get the most defensive value per skill point between dodging and armour.

Thanks for the charts.

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