Active Defense

The place to discuss the new d20 dice pool mechanics.

Active Defense

Postby KingAlbert » Fri Feb 20, 2009 6:24 am

Hey, I just recently got your rules and have been looking through them. You definitely have some really good ideas, but I'm not totally sure they all work out mathematically.

In particular, the active defense seems to rarely be worth doing (not including master-cuts).

To put it in abstract terms, lets assume we have 2 characters fighting, Bob and Carol. They have exactly even stats.

Each essentially starts off with 4 potential attacks, with the probability of hitting against a passive defense as:

P[hit passive] = P[p] = x

and hitting an active defense is:

P[hit active] = P[a] = x' where x' <= x

So if Bob does 4 attacks, and Carol does 3 attacks and 1 active defense, Bob's hit probabilities adds up to:

x + x + x + x'

and Carol adds up to:

x + x + x

So unless x' is a negative value (which it can't be) then you're always better off just going for max attacks. The only exception would be when you lose the initiative and aren't sure you can survive all the enemy attacks so you might want to defend just to stay alive. The optional rule that has been suggested, "Apply active defense to all attacks made be a particular opponent" essentially fixes this problem, active defense becomes competitive.
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Re: Active Defense

Postby Galloglaich » Fri Feb 20, 2009 1:15 pm

High King Albert. Welcome to the forum.

You are perhaps forgetting that Active defense includes:

A) the possibility of an automatically generated counterattack if you roll a natural 20 or your opponent (the attacker) rolls a natural 1
B) you get to add your weapon defense bonus (passive defense doesn't get this) If you have a weapon with a good defense bonus like a staff or a longsword this is a big deal (4 or 5 more points).
C) a free dice in many circumstances (such as when using a shield, or various feats such as mastercuts which you already mentioned, as well as counterstroke, steal initiative, and etc.)

So for example, if Carol has a shield and Bob doesn't, if Bob has initiative and is attacking first Carol is likely to take a defensive strategy rolling Active Defense against each of his attacks, she will get a 'free dice' on each defense roll, and she can wait until she generates a counterattack to turn the tables, the odds are in her favor.

Similarly if Carol has Counterstroke Feat, she will take a defensive strategy for his first attack then immediately counterattack. Depending on how many hit points Bob has and whether he has armor or not she might go for a killing blow with a 3 dice attack attempting an armor bypass (all of her remaining MP) or she might be more conservative with a single die attack, retaining more dice for Active defense.

Active defense can often make the difference between being hit or not being hit, and whether or not you generate a counterattack. Though it all depends on the combatants hit points, equipment, and martial feats both parties have. In playtesting, people almost always choose Active Defense whenever they can, (if they have enough dice) the only time you see all-attack strategies is when the opponent cannot attack them back, such as when they are ganging up (this is one of the reasons why being ganged up against is very dangerous in the Codex).

Also in practice it's pretty rare to do a series of single diece attacks against an opponent who is capable of credibly fighting back. A single- die attack may not give you enough of a chance to get a hit, especially if you are trying to bypass armor, and you are at considerable risk of generating a fumble or a counterattack (5% on each roll for Bob, more like 10% if Carol is rolling Active Defense).

For example assume Carol and Bob both have Chainmail Hauberks, both are armed with arming swords and have average Strength. Bob makes 4 attacks against Carol. He hits carol three times but due to her armor DR of 6 points, his damage rolls of 3, 5, and 7 only does 0, 0, and 1 points of damage, for 1 points total. Carol instead does two Active Defense rolls and a single two-dice bypass attack (going for his unprotected arms or legs). She hits for 5 points of damage because she does not have to contend with the DR of Bobs armor. She is way ahead mathematically.

Multi-dice attacks vastly increase your odds of getting a hit. Our statistical analysis showed that a second dice is effectively a +4 to hit on average, but it also nearly eliminates the possibility of a fumble and substantially increases the chances of a Critial Hit (which in the Codex as you know is automatic, not a 'threat' like in regular 3.5).

Similar for multi-dice Active Defense.

So whether or not Active defense is a good option is largely circumstantial, and closely related to whether the strategy of mutiple single-dice attacks vs. mutli dice attacks.

There are some computer generated statistics posted here somewhere from some tests we ran a couple of months ago that can give you an idea how it all works out mathematically.

G.
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Re: Active Defense

Postby KingAlbert » Fri Feb 20, 2009 5:11 pm

High King Albert. Welcome to the forum.


Thanks for the quick response!

With regards to your response:

A) the possibility of an automatically generated counterattack if you roll a natural 20 or your opponent (the attacker) rolls a natural 1


According to the rules (p 11) Counterattacks are done with "any remaining dice in their Martial Pool". Which means it's not a free attack, you still pay for it with dice just like a regular attack. The only advantage is it can potentially occur earlier if you lost initiative and you're being attacked.

B) you get to add your weapon defense bonus (passive defense doesn't get this) If you have a weapon with a good defense bonus like a staff or a longsword this is a big deal (4 or 5 more points).
C) a free dice in many circumstances (such as when using a shield


The things you just mentioned simply reduce the value of x', but it's still >0 so it's best to just attack more.

, or various feats such as mastercuts which you already mentioned, as well as counterstroke, steal initiative, and etc.)


Mastercuts are kinda their own thing separate from Active Defenses ( I can certainly see their virtue). However, with the Counterstroke feat it states, "Benefit: May skip one normal melee attack to automatically gain an immediate special AoO". The wording is a little ambiguous, but I assume that means you must spend a Martial Die to activate it (and I believe the examples support that). The fact that you might get an extra die if you crit on your active defense isn't going to change the math of the situation.

So for example, if Carol has a shield and Bob doesn't, if Bob has initiative and is attacking first Carol is likely to take a defensive strategy rolling Active Defense against each of his attacks, she will get a 'free dice' on each defense roll, and she can wait until she generates a counterattack to turn the tables, the odds are in her favor.


It is possible to generate cases where Active Def is useful (if you have a shield & counterstroke & your opponent has few HPs left, or you don't have enough HP to survive an attack & you must Adtive Def) but in the general case of 2 chars each using 1 weapon, no shields, no special Feats, Active Defense doesn't make sense to use. You're better off hacking away.

Active defense can often make the difference between being hit or not being hit, and whether or not you generate a counterattack.


Since counterattacks require dice to activate they're not that great. Their big advantage is changing the order of attacks.

...it all depends on the combatants hit points, equipment, and martial feats both parties have.


I think there is a huge number of cases where Active Def doesn't work which is what I'm wondering about.

In playtesting, people almost always choose Active Defense whenever they can...


Well, I can't speak to that, perhaps they had their reasons.


Multi-dice attacks vastly increase your odds of getting a hit. Our statistical analysis showed that a second dice is effectively a +4 to hit on average, but it also nearly eliminates the possibility of a fumble and substantially increases the chances of a Critial Hit (which in the Codex as you know is automatic, not a 'threat' like in regular 3.5).


I think you're making a mistake here. The average for 1d20 is 10.5, the average for rolling 2d20 (take the highest) is ~14, but that's not mathematically equivalent to getting a +4 to your roll, it's actually very different.

Lets say I have a character making an attack.

Code: Select all
P[hit] = Ph


If I make a 2dice attack my expected number of hits is:

Code: Select all
Expected hits with one 2D attack == E2 = 1 - (1 - Ph)^2 = 2Ph - Ph^2


If instead I make 2 1D attacks my expected number of hits is:

Code: Select all
Expected hits with 2 1D attacks == E1 = 2Ph


Note that

Code: Select all
E1 > E2 if  Ph >0


As your probability of hitting nears 0, the two cases converge. But as you near 1 the 2 1die attacks becomes way better. If you take into account fumbles 2die attacks might be useful somewhere aroung 10% chance to hit.

However, 2 1die attacks will always produce more hits & more critical hits. The more dice you use in your attack the more lopsided it becomes
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Re: Active Defense

Postby Galloglaich » Fri Feb 20, 2009 6:03 pm

This is kind of complex so please bear with me as I dig into this with you.

KingAlbert wrote:According to the rules (p 11) Counterattacks are done with "any remaining dice in their Martial Pool". Which means it's not a free attack, you still pay for it with dice just like a regular attack. The only advantage is it can potentially occur earlier if you lost initiative and you're being attacked.


The principle advantage of a counterattack is in interrupting an enemies attacks when you have lost initiative. Whoever has won the initiative is very relevant to what strategy works best, just like in a real fight. The counterattack can trigger all kinds of other actions through Martial Feats or player tactics (more about that in a minute)

The things you just mentioned simply reduce the value of x', but it's still >0 so it's best to just attack more.


Not if you are unlikely to get a hit. If your opponent is rolling Active Defense your odds of hitting may be very low. If so you need the multi-die attack to increase your chances of a hit.

Mastercuts are kinda their own thing separate from Active Defenses ( I can certainly see their virtue). However, with the Counterstroke feat it states, "Benefit: May skip one normal melee attack to automatically gain an immediate special AoO". The wording is a little ambiguous, but I assume that means you must spend a Martial Die to activate it (and I believe the examples support that). The fact that you might get an extra die if you crit on your active defense isn't going to change the math of the situation.


You are incorrect. A counterstroke does not require an MP to activate. It requires that you let your opponent attack first. Essentially you get to interrupt your opponent after their first attack.

It is possible to generate cases where Active Def is useful (if you have a shield & counterstroke & your opponent has few HPs left,


Actually it would be more accurate to say if you have a shield, or counterstroke, or your opponent has few HP left, (or they have a good defense, or your have counterattacking Feats, or your weapon does a lot of damage, or you have a good defensive weapon, or you have the ability to rush them and close to grapple, then Active Def is useful ;) )

or you don't have enough HP to survive an attack & you must Active Def)

In your example, assuming Bob and Carol are 4th level fighters with average stats, they have an average of 20 HP each. Considering the basic damage caused by weapons in the Codex, let alone the Critical Hits, taking a sacrifice hit is a dangerous strategy, unless you have a lot of armor. Whoever takes the most damage first is likely to get in trouble quicker. If you do have armor, mult-dice attacks are necessary to get around it, since single die attacks won't give you much chance of causing damage.

but in the general case of 2 chars each using 1 weapon, no shields, no special Feats, Active Defense doesn't make sense to use. You're better off hacking away.


Not if you can't hit or can't do damage, as in the example I cited in my earlier post above where both people have 1 weapon, no shields, and no special Feats, just fairly common armor.

But it is also very important to note that "no special Feats" is an invalid assumption when using the Codex rules, becuase you get one MF per BaB, same as your MP. So if you have four MP you have four Martial Feats. Many of these are triggered by counterattacks, Active Defense and etc. and give you Free Dice circumstantially. As with a Riposte, a Nachrisen, Nukitsuke etc.

Multi-dice attacks vastly increase your odds of getting a hit. Our statistical analysis showed that a second dice is effectively a +4 to hit on average, but it also nearly eliminates the possibility of a fumble and substantially increases the chances of a Critial Hit (which in the Codex as you know is automatic, not a 'threat' like in regular 3.5).

I think you're making a mistake here. The average for 1d20 is 10.5, the average for rolling 2d20 (take the highest) is ~14, but that's not mathematically equivalent to getting a +4 to your roll, it's actually very different.


No, I don't think I'm making a mistake, I checked this very carefully before releasing the Codex. In the computer testing when we rolled 5,000 die-rolls at a time you always got a higher average die roll when you use multiple dice, and it works out that way in play. If your opponent had a very poor defense multiple single die attacks could be a good strategy, if they have a good defense and / or a good counterattacking ability it usually isn't.

Lets say I have a character making an attack.

Code: Select all
P[hit] = Ph


If I make a 2dice attack my expected number of hits is:

Code: Select all
Expected hits with one 2D attack == E2 = 1 - (1 - Ph)^2 = 2Ph - Ph^2


Your formula assumes that it is easy to get a hit, which is not necessarily the case. In the example I cited above, if Bob is doing all single-die attacks he is suffering an effective -5 to his defense because he isn't using Active Defense. His Passive Defense is 12 (8+ 0 for an average Dex and +4 for Bab).

By contrast Carols Active Defenses make her initial defense is 1d20+7 (+4 for Bab and +3 for her sword), for an average of 17.5.

Bobs average attack die roll is also 17.5 (+4Bab +3 Sword) so he only has a 50% chance of hitting Carol. Carols average die roll with her two-die attack would be 21 (14+7) compared to Bobs Passive defense of 12 so she is much more likely to hit (about 85% off the top of my head).

More importantly, Carol has a good chance of getting a hit even with a Bypass attack, meaning she could do much more damage as in my example (bypass against a Mail Hauberk would be -6 TH mod, which would drop her average roll to 15, which is still a hit.)

So actually to use your mathamatical model if Bob was doing two single-die attacks and Carol one two-dice attack, they both average one hit, but Carol will have a higher chance of a a counterattack, a better chance for a critical and almost no chance of a fumble on her attack, plus the ability to cause significantly more damage by doing a Bypass.

You are always more likely to get a higher die roll with multiple dice, the computer model shows that as does all the testing I've seen of the system. If you get hit before you can do all of your single die attacks you may not get their (statistical) benefit, and if your opponent is actively defending you are much less likely to actually hit them. Combat in the Codex is more dynamic than in canon DnD, it does not always follow the dynamic of taking turns to hit each other, because counterattacking plays a major role.

For example, Bob may have a plan to do 4 single-die attacks, but many things can happen. His remaining dice could be drawn off by Feints, or Carol could interrupt with a counterstroke, if she decided to use her Counterstroke to close to grapple for example, Bob may not get any more attacks this round. (If she has better grappling skills than Bob does or if she has a dagger bob might be in trouble ;)).

(But if Carol had only a small dagger, say, instead of a sword, the math changes quite a bit, because she would have no weapon defensive bonus to add to her Active Defense, making it less worthwhile (a +4 bonus instead of a +7 bonus). Alternately, if she had say a small staff (Defense 5), then her Active Defense would be even more valuable (+9). Same of course if she had a shield)

Everything changes depending if one person has a weapon which is good at defense or not, if one person has a weapon better at Melee vs. Onset or grapple range and etc.

However, 2 1die attacks will always produce more hits & more critical hits. The more dice you use in your attack the more lopsided it becomes


And your probability of a fumble also increases...

This has been discussed before, the multiple single-die attacks strategy doesn't really work out in a one on one fight, though it depends on the weapons and etc. like I said. I think if you will play a few sample combats you'll understand better.

G.
Last edited by Galloglaich on Fri Feb 20, 2009 10:08 pm, edited 10 times in total.
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Re: Active Defense

Postby Galloglaich » Fri Feb 20, 2009 9:26 pm

KingAlbert wrote:The optional rule that has been suggested, "Apply active defense to all attacks made be a particular opponent" essentially fixes this problem, active defense becomes competitive.


By the way what optional rule are you talking about...?

G.
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Re: Active Defense

Postby KingAlbert » Fri Feb 20, 2009 11:04 pm

You are incorrect. A counterstroke does not require an MP to activate. It requires that you let your opponent attack first. You are essentially deferring an attack until after their attack, to interrupt their attacks.


Ok, cool, that makes sense. The rules state that you "skip one normal melee attack" to get the Counterstroke. What does that actually mean? Does it mean don't get any normal attacks that round, you just get the Counterstroke attack instead?

What happens if both chars declare Counterstroke?

****************

With regards to the calculation:

Code: Select all
P[hit] = Ph
Expected hits with one 2D attack == E2 = 1 - (1 - Ph)^2 = 2Ph - Ph^2
Expected hits with 2 1D attacks == E1 = 2Ph
E1 > E2 if  Ph >0


That's Probability Theory 101 and it's true for all values of P[hit], whether high or low.
You could find that calculation in any book on probability theory and it's very relavent
to the discussion. You could ask other people on the forums to confirm it.

Now your point about fumbles is interesting and I need to take that into account. If you fumble while
attacking an opponent who is actively defending all that happens is they can counter attack immediately or
get a free die for a Counterstroke.Fumbling against Passive Defense is actually worse since it consumes an
extra Die to recover the weapon.

Let me go to a specific example fight & work through the exact results.

Bob v Carol

Both have all stats = 10, a Mail Hauberk (DR 6, Bypass 6), an arming sword, BAB 4, the Counterstroke Feat,
20 HP & nothing else (just to keep it simple). That seems like a decent example.

Offset/Melee/Defense = 7/7/7
Passive Def = 8 + 4 = 12

We'll say Bob wins initiative.

EXAMPLE #1:

Bob will do 4 1D attacks attempting to Bypass armor

Carol will elect to Counterstroke with 3D and attempt to bypass armor. She'll put 1D into Active Defense
against Bob's 1st attack.

I'm assuming a natural 20 is an auto-hit (I'm not sure if that's true).

The probability of Bob hitting =

Code: Select all
Probability of Bob hitting Carol with Carol rolling X for active defense = P[hit|X]
Total Probability of Bob hitting = P[hit] = (P[hit|1] + P[hit|2] + ... P[hit|20])/20

P[hit|X] = (20 - (- melee + defense + bypass + X))/20 = (20 - (-7+7+6+X))/20

P[hit|1] = .65
P[hit|2] = .60
P[hit|3] = .55
P[hit|4] = .45
P[hit|5] = .35
P[hit|6] = .30
P[hit|7] = .25
P[hit|8] = .20
P[hit|9] = .15
P[hit|10] =.10
P[hit|11]  .05
P[hit|12 or more] = 0

P[hit] = 3.95/20 = 0.1975

The probability of a critical hit = P[crit]

P[crit] = (20 * .05)/20 = 0.05

So the expected damage for Bob against Carol where she did a 1D active defense is

Expected Damage against 1D active def = E(1d)
= average(2d8) * P[crit] + average(1d8) * (1 - P[crit])*P[hit] 
= 9*.05 + 4.5*(1-.05)*.1975 = 1.294


So what about the rest of Bob's attacks?

Code: Select all
Probability of Bob hitting Carol's passive defense = P[hit]
Total Probability of Bob hitting = P[hit] = (P[hit|1] + P[hit|2] + ... P[hit|20])/20

P[hit] = (20 - (- melee + passive def + bypass))/20 = (20 - (-7+12+6))/20 = 9/20

P[hit] = .45

P[crit] = (20 * .05)/20 = 0.05

So the expected damage for Bob against Carol where she is passive is:

Expected Damage against passive defense = E(p)
= average(2d8) * P[crit] + average(1d8) * (1 - P[crit])*P[hit] 
= 2.374


So what's the total expected damage by Bob in 1 Round?

Code: Select all
Expected total (without fumbles) = E(1d) + 3*E(p)


But, we need to account for fumbles.

Code: Select all
The Cumulative Probability of a fumble during attacks 2,3 and 4 ~= 1 - (1 - .05)^3 = .1426

A fumbled attack will cause another attack to be missed.

Therefore

Expected total (with fumbles) = E(1d) + 3*E(p) - (.1426)*E(p)
                              = 1.294 + 3*2.374 - (.1426)*2.374
                              = 8.077


So, let's get to Carol now.

She's going to get 1 Counterstrike with of 4 dice (1 free + 3 she added from her pool), plus a possibility of an
additional free die.

What's the probability of getting a free die?

Code: Select all
If any of the 3 active defense dice roll 20 or the attacker rolls a 1
then the Counterstrike gets a free die.

Probability of a free die = P[free die] = 1 - (1 - .05)^4 = .1855

Therefore

Probability of not getting a free die = P[no free die] = .8155


Figure Carol's chance to hit without the free die

Code: Select all
Probability of a single die non-crit hit = P[hit] = .45 (using figures from above)

Probability of any one of 4 dice non-crit hitting = P[hit|4dice] = 1 - (1-P[hit])^4 = .908

Probability of a single die critting = P[crit] = (20 * .05)/20 = 0.05

Probability of any one of 4 dice critting = P[crit|4dice] = 1 - (1-P[crit])^4 = .186


Figure Carol's chance to hit with five dice

Code: Select all
Probability of any one of 5 dice hitting = P[hit|5dice] = 1 - (1-P[hit])^5 = .950
Probability of any one of 5 dice critting = P[crit|5dice] = 1 - (1-P[crit])^5 = .226


Code: Select all
Carol's expected damage = E(c)
= P[no free die]*(average(2d8) * P[crit|4dice] + average(1d8) * (1 - P[crit|4dice])*P[hit|4dice]) +
P[free die]* (average(2d8) * P[crit|5dice] + average(1d8) * (1 - P[crit|5dice])*P[hit|5dice])

= .8155 * (9*.186 * 4.5*(1-.186)*.908) +
  .1855 * (9*.226 * 4.5*(1-.226)* .950)
= 5.79


So Bob's doing about 40% more damage

That's a very precise analysis I just did for a fairly standard case and the results are pretty clear cut. Active defense seems underpowered in this case at least.
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Re: Active Defense

Postby Galloglaich » Sat Feb 21, 2009 12:54 am

King Albert,

I see that you have gone through a great deal of effort to demonstrate your theory! I think it is tricky to reduce this combat system into simple probability equations, especially since you aren't taking several factors into consideration. You can't separate out MP from MF for example (in this example Bob and Carol would actually each have FOUR Martial Feats). But I'll try to work with you on what you are doing here.

Counterstroke
The wording on Counterstroke is a little confusing because in the early editions of the Codex the MF used to automatically give you a free dice on your counterattack. I kind of nerfed it a little bit a few months ago because I felt it was a bit too powerful that way in our playtesting. I'll fix the wording on that in my next update, I'm glad you pointed it out.

If both combatants declared Counterstroke the attacker would receive little benefit from it. If you declared Counterstroke when you had the initiative it would only be triggered if you were attacked such as with an AoO. Otherwise you would make your attack, your opponent would Counterstroke, then your Counterstroke could be triggered but you would be attacking again anyway so it would have no effect.

Critical Hits and Fumbles
A natural 20 is always an automatic hit in the Codex, if the defender also rolled a natural 20, the defender may also generate a counterattack (if they have MP remaining) which is considered simultaneous.

A natural 1 is an automatic fumble, if generated during an attack against Active Defense it may generate a counterattack if the opponent has MP remaining and elects to make one. If they don't elect to make one you simply lost your attack and generated a fumble.

A natural 1 generated during an Attack vs. Passive Defense or in an Active Defense roll indicates a fumble. I left what a fumble actually means for your group or DM to decide, as most people have different house rules on this. In our game it means the weapon was dropped.

Keep in mind there is also an optional rule (Dynamic Criticals) which states that if you roll multi-dice attacks, any critical damage you roll will be multiplied by that number of dice.

Your math
It looks like in your example you are having Bob get a higher chance to hit on his first attack, during which Carol was actively defending? I am probably reading that wrong. Why is Bob going through 12 iterations? I didn't take probability in University so I'm not familiar with this shorthand. Can you break it down into math indicating percentage chances to hit followed by average damage?

If not, I can write a computer program to simulate a fight between the two say 100 or 1000 times and average the results. I might run one myself with dice in a minute and write down what I get here.

The biggest hole I noticed was that you do not seem to be including the DR of the armor in your assessment of Bobs 'expected' damage. Bob will only do damage if he successfully makes a bypass (-6 to Hit) attack or if he rolls a 7 or an 8 on a D8 for his damage roll (more likely if he rolls a Crit). Are you factoring this in above?

Also the scenario I had originally described did not have Carol making a 1 die AD followed by a 3 die bypass, Carol made two Active Defense rolls followed by a two-dice bypass attempt (which would be three dice if she managed to generate a counterattack, which, keep in mind, will happen if either she rolls a 20 on either AD roll or if he rolls a 1 on either Attack.) If she generated a counterattack, she would also have the option to attempt say a disarm, to attempt to trip, to close to grapple, to attack Bobs weapon, etc. But we'll put that aside for a moment.

Finally are you incorporating the fact that the defense bonus for your weapon only applies for active defense? This makes Carols Defense 7+ D20 for Bobs first two attacks vs. Bobs passive defense of 12. I didn't see that in your figures.

The reality of a fight using the Codex rules
To do a realistic assessment, you do have to incorporate 4 Martial Feats each, plus whatever standard Feats each combatant gets. Which makes analysis a bit more complex.

A shield of course grants an extra die for every Active Defense roll, increasing the average Active Defense roll to 14. This gives the advantage to the defender. (But you usually then have the option of cutting the shield to pieces, which is what was done in real life)

Another major factor on multiple single-die attacks is range. With an arming sword the Onset and Melee TH bonus are the same, but with say, a spear, that becomes a vital consideration. Once you do your second single-die attack, you have moved to Melee range, in which case your spear drops suddenly from a + 7 to hit to a +1, while the Defense remains at + 3. The only way to prevent this is to either do 2 two-dice attacks or a single three or four dice attack. So this also changes the alchemy quite a bit.

Have you tried running any combats yet by the way?

G.
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Re: Active Defense

Postby KingAlbert » Sun Feb 22, 2009 5:00 am

I included an updated version of my calcs below under Apendix A and you can see them at http://tinyurl.com/cyce7e


The biggest hole I noticed was that you do not seem to be including the DR of the armor in your assessment of Bobs 'expected' damage.


As it says below, all the attacks are done with bypass.

are you incorporating the fact that the defense bonus for your weapon only applies for active defense? This makes Carols Defense 7+ D20 for Bobs first two attacks vs. Bobs passive defense of 12. I didn't see that in your figures.


If you look at Apendix A, P[hit1] is the Probability for Bob to Hit Carol while she’s actively defending with 1D, P[hit2] is attacks against Passive defense. The weapon defense is only being counted in the active case.

To do a realistic assessment, you do have to incorporate 4 Martial Feats each, plus whatever standard Feats each combatant gets.


Okay, Bob & Carol have Counterstroke, Poll-Axe Fighting, Adv Poll-Axe Fighting, Cooperative Fighting, Improved Saving Throw for Reflex/Will/Fortitude. I don’t think that’ll matter. ;)

I know what you mean, I’m not using many feats in the example. But I think it’s fair to assess the use of Active Def in this case. I’m purposefully picking a simple case as it’s easier to analyze.

Your math
It looks like in your example you are having Bob get a higher chance to hit on his first attack, during which Carol was actively defending? I am probably reading that wrong. Why is Bob going through 12 iterations? I didn't take probability in University so I'm not familiar with this shorthand. Can you break it down into math indicating percentage chances to hit followed by average damage?


To explain things more:

P[hit1|1] = .60

Means “The probability of getting a non critical hit given that Carol rolled a 1 for defense”

P[hit1|2] = .55

Means “The probability of getting a non critical hit given that Carol rolled a 2 for defense”

Etc.

P[hit1] = 3.9/20 = 0.195

Means “The probability of getting a non critical hit in total”

So Bob has a 19.5% chance to hit vs Carol’s active defense.

The probability of a critical hit = P[crit1]

P[crit1] = 0.05

So the expected damage for Bob against Carol where she did a 1D active defense is

Expected Damage against 1D active def = E(1da)
= average(2d8) * P[crit1] + average(1d8) * (1 - P[crit1])*P[hit1]
= 9*.05 + 4.5*(1-.05)*.195 = 1.284


Have you tried running any combats yet by the way?


Yes, Bob beats Carol about 2:1.

If you look at Appendix B, I redid the calculations with Carol doing 2 active defenses using 1die for each plus a 2die Counterstroke. It didn’t improve things for her.

What did improve things is having the number of dice rolled for a critical hit increase with the dice in the attack. When I did that and reran Appendix 1, the damage between Bob & Carol was almost the same, Bob was only ahead by 9%. So that optional rule does a great job of equalizing their damage.

I have a spreadsheet with the calcs on it I could send you, it’s crude but usable as an example at least.
Code: Select all
APPENDIX A

Bob v Carol

Both have all stats = 10, a Mail Hauberk (DR 6, Bypass 6), an arming sword, BAB 4, the Counterstroke Feat, 20 HP & nothing else (just to keep it simple). That seems like a decent example.

Offset/Melee/Defense = 7/7/7
Passive Def = 8 + 4 = 12

Every round Carol will do a Counterstroke, just to simplify things.

EXAMPLE #1:

Bob will do 4 1D attacks attempting to Bypass armor

Carol will elect to Counterstroke with 3D and attempt to bypass armor. She'll put 1D into Active Defense against Bob's 1st attack.

I'm assuming a natural 20 is an auto-hit.

The probability of Bob hitting =

Probability of Bob hitting Carol (non-critical) with Carol rolling X for active defense
= P[hit1|X]

Total Probability of Bob hitting = P[hit1] = (P[hit1|1] + P[hit1|2] + ... P[hit1|20])/20

P[hit1|X] = (20 - (1- melee + defense + bypass + X))/20 = (20 - (-7+7+6+X))/20

P[hit1|1] = .60
P[hit1|2] = .55
P[hit1|3] = .45
P[hit1|4] = .35
P[hit1|5] = .30
P[hit1|6] = .25
P[hit1|7] = .20
P[hit1|8] = .15
P[hit1|9] =.10
P[hit1|10]  .05
P[hit1|11 or more] = 0

P[hit1] = 3.9/20 = 0.195

The probability of a critical hit = P[crit1]

P[crit1] = (20 * .05)/20 = 0.05

So the expected damage for Bob against Carol where she did a 1D active defense is

Expected Damage against 1D active def = E(1da)
= average(2d8) * P[crit1] + average(1d8) * (1 - P[crit1])*P[hit1]
= 9*.05 + 4.5*(1-.05)*.195 = 1.284

So what about the rest of Bob's attacks?

Code:
Probability of Bob hitting Carol's passive defense = P[hit2]
Total Probability of Bob hitting = P[hit2] = (P[hit2|1] + P[hit2|2] + ... P[hit2|20])/20

P[hit2] = (20 - (1- melee + passive def + bypass))/20 = (20 - (-7+12+6))/20 = 8/20

P[hit2] = .4

So the expected damage for Bob against Carol where she is passive is:

Expected Damage against passive defense = E(1dp)
= average(2d8) * P[crit] + average(1d8) * (1 - P[crit])*P[hit]
= 2.16

So what's the total expected damage by Bob in 1 Round?

Expected total (without fumbles) = E(1da) + 3*E(1dp)

But, we still need to account for fumbles.

The Cumulative Probability of a fumble during attacks
= P[cum fumble] = P[cf] ~= 1 - (1 - .05)^3 = .185

A fumbled attack will cause another attack to be missed.

Therefore

Expected total (with fumbles)
= E(wf)
= E(1da) + 3*E(1dp) – P[cf]*(3E(1dp)+E(1da))/4
= 7.487


So, let's get to Carol now.

She's going to get 1 Counterstrike with of 4 dice (1 free + 3 she added from her pool), plus a possibility of an additional free die.

What's the probability of getting a free die?

If the active defense dice roll 20 or the attacker rolls a 1
then the Counterstrike gets a free die.

Probability of a free die = P[free die] = 1 - (1 - .05)^2 = .096

Therefore

Probability of not getting a free die = P[no free die] = .903

Figure Carol's chance to hit without the free die

Probability of a single die non-crit hit = P[hit2] = .4 (using figures from above)
Probability of any one of 4 dice non-crit hitting = P[hit2|4dice] = 1 - (1-P[hit2])^4 = .87
Probability of a single die critting = P[crit]
Probability of any one of 4 dice critting = P[crit4|4dice] = 1 - (1-P[crit])^4 = .185

Figure Carol's chance to hit with five dice

Probability of any one of 5 dice hitting = P[hit2|5dice] = 1 - (1-P[hit2])^5 = .922
Probability of any one of 5 dice critting = P[crit|5dice] = 1 - (1-P[crit])^5 = .226

Carol's expected damage = E(c)
= P[no free die] *
(average(2d8) * P[crit|4dice] + average(1d8) * (1 - P[crit|4dice])*P[hit2|4dice])
+
P[free die] *
(average(2d8) * P[crit|5dice] + average(1d8) * (1 - P[crit|5dice])*P[hit2|5dice])

= 4.90

So Bob's doing about 50% more damage

 
APPENDIX B

Bob v Carol

Both have all stats = 10, a Mail Hauberk (DR 6, Bypass 6), an arming sword, BAB 4, the Counterstroke Feat, 20 HP & nothing

Offset/Melee/Defense = 7/7/7
Passive Def = 8 + 4 = 12

EXAMPLE #2:

Bob will do 4 1D attacks attempting to Bypass armor

Carol will elect to Counterstroke with 2D and attempt to bypass armor. She'll put 1D into Active Defense against Bob's 1st & 2nd attacks.

Expected total (without fumbles) = 2*E(1da) + 2*E(1dp)

The Cumulative Probability of a fumble during attacks
= P[cum fumble] = P[cf] ~= 1 - (1 - .05)^2 = .098

A fumbled attack will cause another attack to be missed.

Therefore

Expected total (with fumbles)
= E(wf)
= 2*E(1da) + 2*E(1dp) – P[cf]*(2*E(1dp)+2*E(1da))/4
= 6.719


So, let's get to Carol now.

She's going to get 1 Counterstroke with of 3 dice (1 free + 2 she added from her pool), plus a possibility of an additional free die.

Probability of a free die = P[free die] = 1 - (1 - .05)^2 = .098

Therefore

Probability of not getting a free die = P[no free die] = .903

Figure Carol's chance to hit with 3 dice

Probability of a single die non-crit hit = P[hit2] = .4 (using figures from above)
Probability of any one of 3 dice non-crit hitting = P[hit2|3dice] = 1 - (1-P[hit2])^4 = .784
Probability of a single die critting = P[crit]
Probability of any one of 3 dice critting = P[crit|3dice] = 1 - (1-P[crit])^4 = .143


Carol's expected damage = E(c)
= P[no free die] *
(average(2d8) * P[crit|4dice] + average(1d8) * (1 - P[crit|4dice])*P[hit2|4dice])
+
P[free die] *
(average(2d8) * P[crit|3dice] + average(1d8) * (1 - P[crit|3dice])*P[hit2|3dice])

= 4.36

So Bob's doing about 54% more damage

KingAlbert
 
Posts: 5
Joined: Fri Feb 20, 2009 5:44 am

Re: Active Defense

Postby Galloglaich » Sun Feb 22, 2009 4:50 pm

Okay, Bob & Carol have Counterstroke, Poll-Axe Fighting, Adv Poll-Axe Fighting, Cooperative Fighting, Improved Saving Throw for Reflex/Will/Fortitude. I don’t think that’ll matter. ;)

I know what you mean, I’m not using many feats in the example. But I think it’s fair to assess the use of Active Def in this case. I’m purposefully picking a simple case as it’s easier to analyze.


Actually, if either of them had Miesterhau, Riposte, Nukitsuke, False Edge Cutting, Lunge or Feint I am certain it would make a big difference mate ;)

Miesterhau would effectively grant 'free' Active Defense dice, Riposte would grant a 'Free dice' for every counterattack (especially useful in conjunction with Counterstroke), Nukitsuke would grant a +2 bonus to the first Active Defense (since they are using strait swords) Lunge would grant a free dice on their first attack if they didn't defend (so ideal for Bob... though you would have to keep in mind this changes the Crit damage), and Feint grants the ability to draw off enemy MP.

And Ringen could potentially throw the whole thing out the window since going to grapple in a counterstroke or automatically generated counterattack will end all the back and forth sword strikes.

But like I said, all this number crunching you are doing is very useful information, I'll go along with the experiment.

If you look at Appendix B, I redid the calculations with Carol doing 2 active defenses using 1die for each plus a 2die Counterstroke. It didn’t improve things for her.

What did improve things is having the number of dice rolled for a critical hit increase with the dice in the attack. When I did that and reran Appendix 1, the damage between Bob & Carol was almost the same, Bob was only ahead by 9%. So that optional rule does a great job of equalizing their damage.


Ok, these results are interesting. Can you run it with two slight variations:

1) each of them using a spear instead of an arming sword, and

2) with each of them using a shield.

Then maybe we can start playing around with some other Martial Feats ;).

G.
Galloglaich
 
Posts: 2010
Joined: Sat Sep 27, 2008 5:30 pm

Re: Active Defense

Postby Galloglaich » Fri Feb 27, 2009 3:45 pm

Ok King Albert since you seem to be too busy at the moment to do any more calculations, I've taken the liberty of expanding on your experiment a bit further.

Using your calculations here:

http://tinyurl.com/cyce7e

I've tested out what we would get if Bob and Carol actually had the Feats and MF they would have in Codex rules.

To keep things simple I gave them both the same feats and MF

Feats: Dodge, Weapon Focus: Arming Sword, Weapon Finesse
MF: Counterstrike (already included in your example), False Edge Cutting, Riposte, Feint

Of these only Riposte really has a major statistical effect, though Feint could effect an actual fight quite a bit. As I said before there would be a similar effect from a variety of other MF such as Nukitsuke, Miesterhau et al but I didn't want to muddy the waters to much, rather to try and keep it simple.

I didn't see any evidence of know how you are computing the effect of making multiple dice rolls in your computations, but I'm going to assume the effect is a 20% increase in the average result for a second dice, which is how it came out in our computer testing. If that is incorrect we can adjust the figures accordingly.

So with the Riposte MF this changes things quite a bit. Riposte works with Counterstroke.

Assuming as you did that Bob wins initiative (a big assumption actually in practie because who goes first changes your tactics a lot, but we'll let that slide for the moment) the flow of the first round goes like this (notation BMP#= Bobs Martial Pool, CMP#=Carols Martial Pool):

Bobs Turn
Bob attacks BMP1,
Carol Actively Defends CMP1.
Carol Counterstrikes CMP2, using Riposte (requires a piercing attack)
Carol gainst a Free Dice automatically for using Riposte. Bob does not Actively defend. (Thus she gets a two die attack but only expends one die. If she generated a counterattack she gets a three die attack instead)
Bob attacks BMP2
Carol Actively Defends CMP3
Bob attacks BMP3 (Carol Passively Defends)
Bob attacks BMP4 (Carol Passively Defends)

Carols Turn
Carol still has an MP so she attacks CMP4 (Bob passively defends)

Of course in reality this sequence would not be automatic. If Bob hit Carol and caused any serious damage she might use her last dice for Active Defense instead of another attack. If she won initiative if she realized he was only using passive defense she might do two single-die attacks, retain one Active Defense and one for her counterstroke / riposte. but again, we'll stick to this specific sequence where Bob has initiative.

Assuming the exact same criteria as in your example by my calculation this second attack raises Carols average damage to 8.5, compared to Bobs damage of 7.4. Carol wins. This is based on Carols hit probability for her 'extra' attack being 0.45

If you factor in the Dodge Feat all hit probabilities are lowered by 5%, but that is countered by the improved speed for false edge cutting in all attacks after they get to Melee range, so it's almost the same.

Now if we give both fighters a shield, as one would normally see historically when fighting with an arming sword, it gets even worse for Bob.

Assuming his hit probability is lowered by 20% for two attacks, his average damage is now 1.48+1.48+1.85+1.85 = 6.66. Compared to Carols 8.5, Bob is really in a world of hurt. With a shield Active Defense is a big advantage.


Interesting little exercise. I'm not certain if your basis of calculation is accurate or not, but assuming it is, it sheds light on how the system actually works.

Perhaps now you can see what I mean when I kept bringing up the importance of the MF. It's because they are not an "afterthought", but rather a core part of this system.

For example I could have made a counterattack automatically give you a Free Dice, which would make Active Defense more powerful, but I thought it would be more interesting to distribute those Free Dice into different MF, thus allowing Players to customize their Characters a bit more, and give the system a bit more granularity.

G.
Galloglaich
 
Posts: 2010
Joined: Sat Sep 27, 2008 5:30 pm

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