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[Zivilin]

INTRODUCTION
The common dwarf is an exceedingly mortal existence, prone to a large variety of lethal incidents. A significant subset of these, namely attacks by hostile entities, can be considerably reduced by equipping dwarves with shields or bucklers. It is generally acknowledged that shields greatly increase survivability even when equipped by dwarves with little to no skill in using them, and that the benefits quickly increase along with skill level. It is also common knowledge that even attacks of relatively great power can be blocked, such as bronze colossus strikes or dragonfire [1]. However, to the best of the Author's knowledge, until now no rigorous testing was performed in order to acquire and analyze quantitative data concerning shielding.

The aim of this study was to empirically ascertain how blocking is affected by the following factors:

• Shield User skill and Weapon skill
• Using a Shield vs using a Buckler
• Shield Material
• Shield Quality and Weapon Quality
• Dual Wielding Shields

METHODOLOGY
The experimental setup used in this study is a slightly modified version of the 1vs1 ranged testing grid used in the Author's previous studies [2] [3]. For the purposes of shield testing, one or two grids of 2485 (71 by 35 - the maximum that could be squeezed into a single arena floor) individual test cells were used per experiment - one grid (2485 cells) for Skill tests, two grids (4970) for all others. A single cell consists of three tiles - two floor tiles for the test subjects to stand on and a single pit to separate them (Fig. 1). Substituting the fortification with a pit eliminates any effect that fortifications, used in the previous study, may have had on the results.

Spoiler: Fig. 1. Test cell (click to show/hide)

The test subjects used in the study were arena dwarves - a modded version of regular dwarves with no size variations and stunted skill growth. This prevents an increase in skill over the course of the experiment and eliminates the potential effects of creature size on the experiment results. The game was otherwise unmodded (vanilla). The tileset used was Phoebus'.

Spoiler: Arena dwarf modifications code (click to show/hide)

Average size enforcement:

Code: [Select]

[BODY_APPEARANCE_MODIFIER:HEIGHT:100:100:100:100:100:100:100]
[APP_MOD_IMPORTANCE:500]
[BODY_APPEARANCE_MODIFIER:BROADNESS:100:100:100:100:100:100:100]
[APP_MOD_IMPORTANCE:500]
Stunted skill growth:

Code: [Select]

[SKILL_RATES:1:8:8:16]

The arena dwarves were divided into attackers and defenders. Attackers were equipped with an oaken crossbow and a single copper bolt. Providing only one bolt, while significantly reducing the amount of experimental data to be gained, is necessary in order to eliminate unwanted variables e.g. Dwarves that lose there shields, dwarves that lose consciousness, dwarves that get tired etc. Defenders were equipped with either a single shield (varying material), single buckler or two shields, depending on experiment.

Ranged weapons (crossbows) were used in the experimental setup instead of melee weapons, because they are far more predictable. The Author's attempts at melee testing failed due to the difficulty of eliminating unnecessary variables from the experimental design (e.g wrestling, charging etc.).

A single experiment consisted of running the dwarf placement macro, one-stepping the game for 15 ticks and then copying and clearing the gamelog. The generally accepted criterion for shielding effectiveness is the probability that an attack is blocked. The Probabilities of Blocking (PoB) were obtained by counting the numbers of shots blocked (string search for "blocked") and dividing them by the number of shots fired ("the flying") minus those that missed ("missed") or were dodged ("jumps away").

PoB = (Shots Blocked)/(Shots Fired - Shots Missed - Shots Dodged)

Assuming normal distribution, using one test grid (2485 cells) should result in obtaining a PoB which is within 2% of the real value with 95% confidence. Using two test grids decreases the value of error to 1.4% with 95% confidence.

In accordance with the suggestion of Hans Lemurson, an additional parameter, the Shield Protecion Factor (SPF), is supplied where relavent. SPF is calculated from PoB using the following formula:

SPF = 1/(1-PoB)

SPF can be interpreted as the mean number of successful blocks inbetween failed blocks. For example, in conditions where PoB is 90%, the value of SPF is 10, which is the mean number of attacks one might expect to block successfully before an attack gets through. This allows for a direct and intuitive comparison of shielding equipment under different conditions, because it is directly related to survivability.

RESULTS - WEAPON SKILL AND SHIELD USER SKILL
As mentioned above, a single test grid of 2485 cells was used for this study. Attackers were given a basic quality oaken crossbow and basic quality copper bolt. Defenders were given only a basic quality oaken shield (no armor). To achieve the highest possible resolution, every combination of Weapon skill and Shield skill available in Arena Mode was tested - from Dabbling (skill level 0) to Grand Master (skill level 14). This resulted in 15x15 = 225 PoB/SPF values. These are presented in the form of a two dimensional colormap (Fig. 2).

Spoiler: Fig. 2. Probability of Blocking and Shield Protection Factor - Shields (click to show/hide)

Due to display reasons, SPF data is presented as log2(SPF). The actual value of SPF is obtained by exponentiation SPF = 2^[log2(SPF)]. Example: The (Maksdwarf = 0; Shield User = 5) datapoint is colored yellow, which corresponds to a log2(SPF) value of around 5.5. The value of SPF is thus 2^5.5 = around 45.

Observations

• An increase in the Shield User skill very quickly improves blocking probability. Proficient (level 5) Shield Users have a 90% chance or higher of blocking attacks of Marksdwarves of a similar level or lower, and a 60% chance of blocking Grand Master Marksdwarves. A Grand Master Shield User has an over 90% chance of blocking practically anything, and a 99% chance of blocking Proficient Marksdwarves and lower.
• Survivability increases much, much faster with increased Shield User skill at Marksdwarf=0 rather than Marksdwarf=14. A fairly obvious conclusion, but not as obvious when studying Probability of Blocking itself. SPF value at (Shield User=14; Marksdwarf=0) is over 500. At (Shield User=14, Marksdwarf=14), SPF is merely 16.

RESULTS - SHIELDS AND BUCKLERS
The experiments were repeated exactly as stated above, except that Defenders were given basic quality oaken bucklers instead of oaken shields. The results are presented in the form of a two dimensional colormap (Fig. 3).

Spoiler: Fig. 3. Probability of Blocking and Shield Protection Factor - Bucklers (click to show/hide)

Due to display reasons, SPF data is presented as log2(SPF). The actual value of SPF is obtained by exponentiation SPF = 2^[log2(SPF)]. Example: The (Maksdwarf = 1; Shield User = 7) datapoint is colored yellow, which corresponds to a log2(SPF) value of around 5.5. The value of SPF is thus 2^5.5 = around 45.

As the results are very similar to those obtained for shields, an additional colormap showing the ratio of Shield SPF to Buckler SPF is provided. This quantatively shows just how many times a shield is better than a buckler at different skill levels.

Spoiler: Fig. 4. Shield Protection Factor Ratio - Shields/Bucklers (click to show/hide)

Observations

• At low Shield User, high Marksdwarf skills, shields and bucklers have more or less the same effectiveness.
• As Shield User increases or Marksdwarf decreases, shields begin performing better than bucklers, by up to a factor of around 1.5 times better.
• The values of SPF ratios at high Shield User and low Marksdwarf skills must be disregarded, due to the high interference introduced by inaccuracies. Since SPF is derived from PoB using the formula SPF = 1/(1-PoB), then SPF values tend to be very large as PoB approaches unity. The 1.5-2% inaccuracies in estimating PoB have a very significant effect on the calculated value of SPF, which is very visible when comparing by ratio, as can be seen on Figure 4, where the colormap's lower left corner displays seemingly random values.

RESULTS - SHIELD MATERIAL
It is common knowledge that shield material has no effect on shield effectiveness. This experiment was devised to verify this information. Two test grids were used in the experiment, for a total of 4970 cells. Attackers were given an basic quality oaken crossbow and basic quality copper bolt. Defenders were given a basic quality shield of varying material (no armor). Skill levels were set to Marksdwarf = 10, Shield User = 3, since based on the previous experiment they result in a blocking chance slightly higher than 50% for oaken shields.Results are presented in Figure 5.

Spoiler: Fig. 5. Probability of Blocking - Shield Material (Caution, contains spoilers) (click to show/hide)

Observations

• Shield material does indeed have no bearing on blocking chance. Steel shields are just as good as wooden ones.

RESULTS - WEAPON QUALITY VS SHIELD QUALITY
It is commonly assumed that higher quality shields have better battle performance. This experiment was devised in order to see how varying Weapon and Shield quality, from basic quality (level 0) to masterwork (level 5), affects blocking chance. Two test grids were used in the experiment, for a total of 4970 cells. Attackers were given an oaken crossbow of varying quality and basic quality copper bolt. Defenders were given an oaken shield of varying quality (no armor). Item quality was manipulated by running a lua script in DFHack, courtesy of Urist Da Vinci [4]. Skill levels were set to Marksdwarf = 10, Shield User = 3, since based on the previous experiment they result in a blocking chance slightly higher than 50% for basic quality crossbows and shield. Results are presented in Figure 6.

Spoiler: Fig. 6. Probability of Blocking - Weapon and Shield Quality (click to show/hide)

Observations

• Surprisingly, neither weapon nor shield quality have any affect on blocking chance, which remained at the same level (around 52%) throughout all experiments.

RESULTS - TWO SHIELDS
Dual wielding shields is commonly acknowledged to increase blocking chance. In this experiment, two test grids were used in the experiment, for a total of 4970 cells. Attackers were given an oaken crossbow of basic quality and basic quality copper bolt. Defenders were given two oaken shields (no armor). Skill levels were set to Marksdwarf = 10, Shield User = 3, since based on the previous experiment they result in a blocking chance slightly higher than 50% for basic quality crossbows and shield.

By adding the second shield, the Probability of Block increased from 50% to 66%. This corresponds to the SPF increasing from 2 to 3.

Additional experiments resulted in an PoB increase from 10% to 15% (Marksdwarf = 10, Shield User = 0) and 75% to 90% (Marksdwarf = 10, Shield User = 5).

Observations

• Adding a second shield does indeed increase blocking chance.
• However, this increase does not correspond to the second shield having the same PoB as the first one. If that were the case, the PoB would have increased from 50% to 75%.
• Based on obtained data, it appears that the second shield suffers an approximately 50% blocking percentage penalty - that is, if the probability of blocking an attack with the first shield is 50%, then the probability of blocking an attack with the second shield (after failing the first block) is reduced to 25%.

CONCLUSIONS

• Shields tend to guarantee higher survivability than bucklers, especially at high levels of Shield User and low levels of Marksdwarf, with shield SPF reaching values up to 1.5 times higher than bucklers at the same skill levels.
• Shield material has no effect on blocking chance.
• Weapon and Shield quality have no effect on blocking chance.
• When wielding two shields, the second shield has an approximately 50% reduction in blocking chance.

Edit: Minor corrections
Edit2: Addition of the SPF parameter and related conclusions, in accordance with the suggestions of Hans Lemurson.

[Pancakes]

If someone were to equip enough shields/bucklers, would they be impervious to all forms of attack?

[Will_Tuna]

50% + 50% <> 100%
more like 75%

possible outcome with 2 shields
1 1 = block
0 1 = block
1 0 = block
0 0 = no block

= 3/4

=75% of blocking

[Walkaboutout]

Awesome science, thanks a ton for doing this work! I've been wondering lately about shields and bucklers, and I have to say the quality modifier of equipment have no bearing on any of this is a big surprise to me. I thought there would be at least a little something. Great stuff, thanks again!

[YvAd]

I love that this game is complex enough to have someone do a full scientific experiment and writeup to examine a game mechanic.
Thank dwarven scientists This is very useful information.
Related, is there a way to order your fortress mode dwarves to wield multiple shields?

[Zivilin]

If someone were to equip enough shields/bucklers, would they be impervious to all forms of attack?

I've heard anecdotal evidence that equipping multiple shields in Adventure Mode is very effective, although using the arena I can't equip more than two shields/bucklers per dwarf. I doubt whether You can become 100% impervious, more like 99.9999% impervious. However, if the the 50% reduction penalty I found holds true, then depending on the dwarf's Shield Skill the blocking series might converge a long way from 100%. With the 50% example, I'd guesstimate the most one could achieve is a 75% blocking chance, assuming each additional shield has a further 50% blocking chance reduction. On the other hand, if each additional shield has the same value of blocking chance (50% of the original) then with enough shields You would approach 100% blocking chance.

50% + 50% <> 100%
more like 75%

possible outcome with 2 shields
1 1 = block
0 1 = block
1 0 = block
0 0 = no block

= 3/4

=75% of blocking

I actually did test the two shield scenario - I know its a wall of text, so its easy to miss if You're not into the research. It turns out that, assuming single shield block chance is 50% (as in the case of a Accomplished Marksdwarf against a Competent Shield User, using a shield), then double shield block chance is around 66%. This implies there is a roughly 50% blocking chance penalty applied to the... offhand shield. This penalty might be reduced by greater Shield User skill, but its difficult to tell, since Having a high Shield User Skill already gives a blocking chance of 90% or more most of the time. More research on larger sample size could clarify this.

Awesome science, thanks a ton for doing this work! I've been wondering lately about shields and bucklers, and I have to say the quality modifier of equipment have no bearing on any of this is a big surprise to me. I thought there would be at least a little something. Great stuff, thanks again!

I also found the neutrality of quality surprising. Especially in the case of shield quality, since weapon quality is said to at least increase basic accuracy - I can imagine that the quality parameter was used in one function and not in others. But shields, well, only have blocking - if quality doesn't affect blocking, then all it affects is shield value (monetary).

I love that this game is complex enough to have someone do a full scientific experiment and writeup to examine a game mechanic.
Thank dwarven scientists This is very useful information.
Related, is there a way to order your fortress mode dwarves to wield multiple shields?

Not that I know of, but perhaps others are more knowledgeable?

[doublestrafe]

But shields, well, only have blocking - if quality doesn't affect blocking, then all it affects is shield value (monetary).

Shields also have bashing, and it is known that material weight affects bashing efficacy. I don't know if it's known whether quality affects it.

[Zivilin]

But shields, well, only have blocking - if quality doesn't affect blocking, then all it affects is shield value (monetary).

Shields also have bashing, and it is known that material weight affects bashing efficacy. I don't know if it's known whether quality affects it.

Ah, I forgot about that. True, quality may (and probably does) have an effect on accuracy/damage when bashing.

...

I wonder if any item's quality, say, a stone statue's, affects the accuracy/damage when used my a tantruming/berserking dwarf?

[doublestrafe]

I wonder if any item's quality, say, a stone statue's, affects the accuracy/damage when used my a tantruming/berserking dwarf?

You know what I've really always wondered? Whether a training weapon's quality increases or decreases the probability of causing injury. I really feel like a masterful training spear should never cause injury, while a low-quality one should. Splinters and stuff, you know.

[Quietust]

Observations

• Bucklers provide almost the same level of protection as shields. The largest differences lie at low skill levels, where shields may add up to 15% more blocking chance.
• At high levels of the Shield User skill shields are superior to bucklers by only a few percents of blocking chance.

It might be worth trying to directly modify the BLOCKCHANCE values in the raws to see what effect they have - in the stock raws, bucklers have a BLOCKCHANCE of 10 while shields have a BLOCKCHANCE of 20. It would also be useful to see how the MATERIAL_SIZE and UPSTEP values affect shield performance.

[Urist Da Vinci]

Cool stuff, I never looked into blocking/dodging/accuracy stuff much.

- If a target is being threatened in melee by a opponent (use a modded unkillable microsponge that can't hurt the target) and therefore the target's back is turned to the archer, are they able to block or dodge, and how do the percentages change?

If someone were to equip enough shields/bucklers, would they be impervious to all forms of attack?

I've heard anecdotal evidence that equipping multiple shields in Adventure Mode is very effective, although using the arena I can't equip more than two shields/bucklers per dwarf. I doubt whether You can become 100% impervious, more like 99.9999% impervious.
...

You can do this very manually by creating a "reaction" in the raw files that can be used by an adventurer, similar to knapping. The reaction gives you a backpack for free. Wearing a backpack enables you to collect more items than you can hold, and then "remove" them from the pack to equip them.

Alternate: some of the antmen have 4 arms, IIRC.

EDIT: Off-topic response:

Spoiler (click to show/hide)

I wonder if any item's quality, say, a stone statue's, affects the accuracy/damage when used my a tantruming/berserking dwarf?

You know what I've really always wondered? Whether a training weapon's quality increases or decreases the probability of causing injury. I really feel like a masterful training spear should never cause injury, while a low-quality one should. Splinters and stuff, you know.

Blunt damage isn't affected by quality, while edged weapons do benefit from high quality for damage. You don't really see the benefit in normal situations though, because low-quality edged weapons are capable of cutting unarmored people in half.

Accuracy though, that is untested. Anecdotal suggests that high quality => higher accuracy. It hasn't been proven. Remember that deflections off armor count as hits when testing for accuracy!

[MDFification]

Great to see some science with shields. So little is know about shield mechanics.

Gonna go see if I can make shield material have any affect on your protection via modding. It probably requires a massive set of raws. *sigh*

[itg]

If someone were to equip enough shields/bucklers, would they be impervious to all forms of attack?

I've heard anecdotal evidence that equipping multiple shields in Adventure Mode is very effective, although using the arena I can't equip more than two shields/bucklers per dwarf. I doubt whether You can become 100% impervious, more like 99.9999% impervious. However, if the the 50% reduction penalty I found holds true, then depending on the dwarf's Shield Skill the blocking series might converge a long way from 100%. With the 50% example, I'd guesstimate the most one could achieve is a 75% blocking chance, assuming each additional shield has a further 50% blocking chance reduction. On the other hand, if each additional shield has the same value of blocking chance (50% of the original) then with enough shields You would approach 100% blocking chance.

50% + 50% <> 100%
more like 75%

possible outcome with 2 shields
1 1 = block
0 1 = block
1 0 = block
0 0 = no block

= 3/4

=75% of blocking

I actually did test the two shield scenario - I know its a wall of text, so its easy to miss if You're not into the research. It turns out that, assuming single shield block chance is 50% (as in the case of a Accomplished Marksdwarf against a Competent Shield User, using a shield), then double shield block chance is around 66%. This implies there is a roughly 50% blocking chance penalty applied to the... offhand shield. This penalty might be reduced by greater Shield User skill, but its difficult to tell, since Having a high Shield User Skill already gives a blocking chance of 90% or more most of the time. More research on larger sample size could clarify this.

Nice work! I've got some math to compare with the empirical data. Assuming a 50% block chance, and then a 50% cumulative penalty for each additional shield equipped (so the 2nd has a 25% block chance, the third 12.5%, etc.), the best block chance you can hope to get is 71.1%. Here's how I calculate that:

• The chance of blocking an attack is the chance that at least one shield blocks the attack, so it's: 1 - P(no successful blocks)
• 1 - P(no successful blocks) for one shield is (1-50%) = 50%, for two shields is (1-50%)(1-25%) = 62.5%, etc.
• Here's the formula for infinite shields, with numerical value of about 71.1% calculated by Wolfram Alpha
  
  Of course, only further experiments will tell if this is actually what's happening.

[Hans Lemurson]

For interpreting the effectiveness of blocking with Shields vs. Bucklers, I don't think that ShieldBlock% minus BucklerBlock% is useful.

First and foremost is the fact that when you are examining the effectiveness of prevention, what you are interested in is the Chance of Failure, not success.  If a shield is blocking 95% of the blows but a buckler is blocking only 90%, the proper way to look at this is that a shield lets 5% of the blows through whereas the buckler lets through 10%.  (Furthermore, it would be erroneous to say that the shield is "5% better than the buckler" since this is a crime against the %-sign.)

The relevant difference is that 95% blocking is TWICE AS POWERFUL as 90% blocking.  A guy who is blocking 95% of shots is going to last, on average, twice as long before getting hit.  That's also something that was easy to lose track of in the data about Shield-Skill since the difficult to measure difference between 95% and 99% in fact makes one of the hugest differences in practical utility.

The relevant number to look at would be the "Shield Protection Factor", which like the SPF rating on sunscreen tells you how much more longer you can survive under assault using protection compared to being totally naked.

In summary, the valuable number is the Shield Protection Factor, which is 1/(1-"block rate"), or the inverse of the hits.

[Zivilin]

It might be worth trying to directly modify the BLOCKCHANCE values in the raws to see what effect they have - in the stock raws, bucklers have a BLOCKCHANCE of 10 while shields have a BLOCKCHANCE of 20. It would also be useful to see how the MATERIAL_SIZE and UPSTEP values affect shield performance.

I'll run a few point tests on BLOCKCHANCE, MATERIAL_SIZE and UPSTEP to see how they affect shield performance. I don't know if I'll get around to performing a full 15x15 Skill study on them though, since it is tremendously time consuming.

- If a target is being threatened in melee by a opponent (use a modded unkillable microsponge that can't hurt the target) and therefore the target's back is turned to the archer, are they able to block or dodge, and how do the percentages change?

Ah, that would be a great way to test "back attacks". After Your quote here I was under the impression that You must have two allies within melee distance to perform back- or side attacks, but I see that a single ally in melee distance and a archer might do the trick. I'll see if this is the case later on.

If someone were to equip enough shields/bucklers, would they be impervious to all forms of attack?

I've heard anecdotal evidence that equipping multiple shields in Adventure Mode is very effective, although using the arena I can't equip more than two shields/bucklers per dwarf. I doubt whether You can become 100% impervious, more like 99.9999% impervious.
...

You can do this very manually by creating a "reaction" in the raw files that can be used by an adventurer, similar to knapping. The reaction gives you a backpack for free. Wearing a backpack enables you to collect more items than you can hold, and then "remove" them from the pack to equip them.

Alternate: some of the antmen have 4 arms, IIRC.

I think I'll try the 4 arm creature version first, just because using backpacks seems like it requires an additional round of macro hell.

Nice work! I've got some math to compare with the empirical data. Assuming a 50% block chance, and then a 50% cumulative penalty for each additional shield equipped (so the 2nd has a 25% block chance, the third 12.5%, etc.), the best block chance you can hope to get is 71.1%. Here's how I calculate that:

• The chance of blocking an attack is the chance that at least one shield blocks the attack, so it's: 1 - P(no successful blocks)
• 1 - P(no successful blocks) for one shield is (1-50%) = 50%, for two shields is (1-50%)(1-25%) = 62.5%, etc.
• Here's the formula for infinite shields, with numerical value of about 71.1% calculated by Wolfram Alpha
  
  Of course, only further experiments will tell if this is actually what's happening.
  

Ah, that's the bit of infinite series math I was lacking. When I get around to those 4 armed creatures, I'll see if the reduction is cumulative or not. Also, I suspect the penalty value might not be exactly 50%, or possibly might vary with Shield User skill. I find the 66% percent I got is a little too far off from the predicted 62.5%, given that the theoretical accuracy of the test should be within 1.5% of the real value with 95% confidence (again, assuming normal probability distribution... and assuming my math is correct - always debatable).

For interpreting the effectiveness of blocking with Shields vs. Bucklers, I don't think that ShieldBlock% minus BucklerBlock% is useful.

First and foremost is the fact that when you are examining the effectiveness of prevention, what you are interested in is the Chance of Failure, not success.  If a shield is blocking 95% of the blows but a buckler is blocking only 90%, the proper way to look at this is that a shield lets 5% of the blows through whereas the buckler lets through 10%.  (Furthermore, it would be erroneous to say that the shield is "5% better than the buckler" since this is a crime against the %-sign.)

The relevant difference is that 95% blocking is TWICE AS POWERFUL as 90% blocking.  A guy who is blocking 95% of shots is going to last, on average, twice as long before getting hit.  That's also something that was easy to lose track of in the data about Shield-Skill since the difficult to measure difference between 95% and 99% in fact makes one of the hugest differences in practical utility.

The relevant number to look at would be the "Shield Protection Factor", which like the SPF rating on sunscreen tells you how much more longer you can survive under assault using protection compared to being totally naked.

In summary, the valuable number is the Shield Protection Factor, which is 1/(1-"block rate"), or the inverse of the hits.

To this... I have no defense. The Blocking Probability, while not terrible criterion, is definitely inadequate in this case. As You say, what is important is to present a shields efficiency in terms of dwarven survivability. I believe in reliability theory the relevant criterion is "Mean Time Between Failures", or at least its discrete equivalent. I am currently V. Rusty on the specifics but I believe this is what you are proposing? The SPF for a a block chance of 95% is 20 - I assume that the reliability theory interpretation of this is that a shield will fail on average once every 20 attacks.

Many thanks for this observation. I'll be making an update to the Opening Post when I figure out how to present the data. My elegant 1 to 100 percentage range has changed to an 1 to 500 MTBF range, with most values being below 50.