3.5 Avalanche of Blades vs. Times Stands Still

[Snowbluff]

At what attack is Avalance of Blades better than Time Stands Still for a Warblade? (Assuming level break points, like 10, 15 ,20.)

How would we begin to calculate that?

[Big Fau]

If you have a high enough attack bonus and enough rerolls that don't require a Swift action, Avalanche of Blades is better. TSS is better if you have TWFing and enough Tiger Claw support.

[Mr.Bookworm]

You can only get Time Stands Still at level 17. It simply allows you to full attack twice. That gives you eight attacks, at +17/+12/+7/+2/+17/+12/+7/+2, minimum.

With Avalanche of Blades, each attack you make suffers an ever-increasing penalty. Your attacks at level 17 are made at +17/+13/+5/-7/-23 and so on.

None of this changes as you level up, so TSS is always better than Avalanche of Blades, absent cheese.

[Snowbluff]

If you have a high enough attack bonus and enough rerolls that don't require a Swift action, Avalanche of Blades is better. TSS is better if you have TWFing and enough Tiger Claw support.
Hmm... rerolls? Goliaths get some, I think.

You can only get Time Stands Still at level 17. It simply allows you to full attack twice. That gives you eight attacks, at +17/+12/+7/+2/+17/+12/+7/+2, minimum.

With Avalanche of Blades, each attack you make suffers an ever-increasing penalty. Your attacks at level 17 are made at +17/+13/+5/-7/-23 and so on.

None of this changes as you level up, so TSS is always better than Avalanche of Blades, absent cheese.
Well, if your Attack bonus is really high for some reason or you are ignoring a portion of their AC, Avalance of Blades should pull ahead at some point.

What do you consider cheese? Should we put a Diamond Mind master together wit ha Diamond Mind Weapon, Blade Meditation, etc?

[DarkSonic1337]

How cheesy is storm guard warrior+robilar's gambit->avalanche of blades? :smallcool:

[Snowbluff]

How cheesy is storm guard warrior+robilar's gambit->avalanche of blades? :smallcool:

Less cheesy than TTS + Storm Guard Warrior (Combat Rhythm)?

[faircoin]

What Mr.Bookworm said is false. It does eventually get better, mathematically. And by eventually, I'm talking about high-epic attack bonuses.

You can use conditional probability computation (http://en.wikipedia.org/wiki/Conditional_probability) to figure out the exact attack bonus where AoB pulls ahead of TSS.

[Douglas]

With Avalanche of Blades, each attack you make suffers an ever-increasing penalty. Your attacks at level 17 are made at +17/+13/+5/-7/-23 and so on.
No, it's +17/+13/+9/+5/+1/-3/-7, etc. The additional -4 on each attack is how much the penalty increases, not how much the increase in the penalty increases.

[AmberVael]

I think Time Stands Still is going to be the better of the two most of the time.

While you can optimize your attack bonus, the penalty on Avalanche of Blades stacks up faster than you'd like. Trying to get more than 10 attacks already means coping with a -40 penalty, and it just gets worse from there... while getting 10 attacks with Time Stands Still (without such a high penalty) is incredibly simple, and it's not too hard to push it up even further. That massive attack penalty also means eschewing the damage bonus of Power Attack, which could make each of your attacks worth less if you don't have some decent extra damage source.

In short, my guess is that it'll be easier to optimize for extra attacks than it will be to optimize for the equivalent attack bonus. At least for any kind of build that would actually see play.

[faircoin]

No, it's +17/+13/+9/+5/+1/-3/-7, etc. The additional -4 on each attack is how much the penalty increases, not how much the increase in the penalty increases.

Oh, given that misquote, the breaking point for AoB to beat TSS is much sooner. With proper optimization, should be doable at level 17.

[Snowbluff]

Oh, I forgot. I want successful attacks. Just changes it a bit. What target AC should we use?

[Big Fau]

Oh, I forgot. I want successful attacks. Just changes it a bit. What target AC should we use?

Big T is a classic target.

[Nettlekid]

The thing about it is that Time Stands Still grants you a certain number of attacks no matter what. Avalanche of Blades is potentially limitless until you miss. So if you were a regular THF, Time Stands Still gives eight attacks. If you were TWF for some reason, you'll get fourteen, I think? That's a lot of attacks, with bonuses of (assuming level 20, a Str of 30, and +5 in other sundry bonuses (you'll probably have more)) 35/35/30/30/25/25/20/20, or 33/33/33/33/28/28/28/28/23/23/23/23/18/18, if I've done that right. Whereas Avalanche of Blades will have 35/31/29/25/21/17/13/9/5/1/-3, so on, assuming that you stop hitting at like 5. You're probably finding some way to use Touch AC, like with Wraithstrike or something, because it's the best way to go at high level with creatures who have high NA. I guess it looks like Time Stands Still is better in both cases, unless you can get an attack bonus so high that your Bonus-32 still hits reliably. That way, you've done 8 attacks and can still hit, so Avalanche of Blades keeps going. But that's rather high, isn't it?

Of course, if you're allowed to use slightly obscure material, then you might be able to use Rokugan's Kata: The Victory of the River, to get +1 on each attack rather than -4, and thus go forever until you roll a 1.

[Mr.Bookworm]

What Mr.Bookworm said is false. It does eventually get better, mathematically. And by eventually, I'm talking about high-epic attack bonuses.

I honestly don't care about epic levels, let alone high-epic levels. Math has long since broken down and gone home sobbing, and you might as well be playing something more mechanically rigorous, like Exalted.


You can use conditional probability computation (http://en.wikipedia.org/wiki/Conditional_probability) to figure out the exact attack bonus where AoB pulls ahead of TSS.

And? What is it?


Oh, given that misquote, the breaking point for AoB to beat TSS is much sooner. With proper optimization, should be doable at level 17.

Equal optimization being equal, what's the point at which you can hit nine times, without said optimization being cheesy?

I'm not going to dispute that AoB is better if you cheese the hell out of it, because it is. But TO is not particularly relevant if someone's asking how to make a character.


No, it's +17/+13/+9/+5/+1/-3/-7, etc. The additional -4 on each attack is how much the penalty increases, not how much the increase in the penalty increases.

Doh. True, I did completely misread that. Better, but still not as good as TSS most of the time.

[Piggy Knowles]

Oh, I forgot. I want successful attacks. Just changes it a bit. What target AC should we use?

Optimization by the Numbers (http://community.wizards.com/go/thread/view/75882/19869122/Optimization_By_The_Numbers) is always my go-to for this sort of thing. So, how about average AC of a CR 20 enemy? That would be 36.44.

As for what AB is necessary to make the one better than the other... I really, really don't know. I started trying to brute force the problem and ran into several stumbling blocks. Does Attila still have that D&D math problem thread? This would be a good one for it...

EDIT: One thing I do know is that in order to figure this out, we would need to set a fixed number of attacks in a full attack routine to compare against. I'd recommend five, so ten attacks with TSS - that's a pretty simple baseline, and any reasonable melee character should have five attacks per round by ECL 20.

[Deophaun]

At what attack is Avalance of Blades better than Time Stands Still for a Warblade? (Assuming level break points, like 10, 15 ,20.)

I'm going to say "never," and I don't think you have to go too deep in the math to say this. Avalanche of Blades has a 5% chance right out of the box to fail spectacularly if you roll a 1 on that first attack. There's about a 1 in 3 chance that you're going to roll a nat 1 before you hit your 9th attack. And let's hope that you aren't dealing with any form of concealment that you can't get around. Time Stands Still has reliability.

Avalanche of Blades is also limited to a single target. Really, at that level, if you're hitting something 8 times, it's either dead, or you need to get the caster to handle it. Time Stands Still is not limited to a single target. If your initial target dies midway through, you can switch to an adjacent target. You could also (depending on if the DM defines movement as a move action) use a swift action and Sudden Leap or Shadow Blink to another enemy if need be. So Time Stands Still is going to be better against multiple foes, and it's going to take one outlier of a singular opponent for Avalanche of Blades to have a shot at doing more damage.

[Big Fau]

Doh. True, I did completely misread that. Better, but still not as good as TSS most of the time.

Actually, for the first 4 attacks, AoB is better (Iterative attacks are made at an effective -5 penalty, while AoB ones are made at a -4). The second 4, and any additional ones from TWFing, are going to be better for TSS. As long as your attack bonuses are 16 points above the target AC AoB is going to be nearly equal to TSS 95% of the time (5% being a Nat 1).

Any higher than +16 above and AoB becomes a superior option, up until you hit 20 attacks as you are statistically going to experience a Nat 1 somewhere during that AoB. And, again, TWFing and other sources of additional attacks (Eagle's Fury, Slashing Flurry, Snap Kick, Speed weapon) make TSS better.

Assuming an equal number of attacks, and that all attacks hit, AoB may be the better choice if you are using THFing instead of TWFing simply due to a higher damage output (unless you worked Revenant Blade onto your Warblade or Swordsage, in which case you could also go Eternal Blade to get an additional Full Attack off of TSS and just wreck AoB's advantage for another 8 attacks).

Edit: Really that Natural 1 is the killer here. If you can mitigate that factor AoB becomes somewhat better the higher your attack bonus gets, but it is still possible to get incredibly unlucky on the attack roll. Aura of Perfect Order and Weapon Supremacy can both allow you to ignore one attack roll, and I mentioned several sources of rerolls earlier in the thread.

Double-Edit: And then there's always Blood in the Water and Lightning Maces, which favors AoB slightly more than TSS.

[faircoin]

I retract my former statement about conditional probability. I think this is nontrivial mathematics to give a precise answer to.

For a precise answer, I think this has to be done using Markov chains.

I would have AC be randomly sampled from positive integers, have the chain's state be sampled uniformly randomly from {x:1\leq x\leq r} representing attack rolls, with r being the value at which AoB overtakes TSS in terms of optimal output, combine the distributions for AC and attack rolls, create a directed graph of events and calculate the transition matrix from that, then compute optimal steady-state probabilities for different values of r. Finally, we compare this to TSS which has a simpler (obvious) probability calculation.

I think it'd be best if you just ballparked it intuitively, computed it, or picked sensible attack bonuses and see if it works.

[Randomguy]

It seems to me that if your attack bonus is high enough relative to your target's AC then avalanch of blades is better. Your attack routine is +X/+X-4/+X-8/+X-12/+X-16/+X-20/+X-24/+X-28, where X is your highest attack bonus, then avalanch of blades is better if X-24 or less is a guaranteed hit as long as you don't roll a 1.

This situation might come up if you cast wraithstrike against a dragon that hasn't cast Scintillating Scales, and you have a high strength.

[Metahuman1]

If I had a Nano bot swarm that I could rely upon not getting wasted, The Mage slayer line, and either a lot of luck/magic rerolls or a hard and fast house rule that you hit even on a Nat 1 + a couple of nice damage boosters, I'd go Avalanche of blades.

If I had TWF with speed weapons + Tiger Claw or a heavy on natural weapons build and some bonus damage like from bardic music or just a stupid high strength, I'd go Time Stands Still.

If I had neither, I'd go else where.

[eggynack]

It seems to me that if your attack bonus is high enough relative to your target's AC then avalanch of blades is better. Your attack routine is +X/+X-4/+X-8/+X-12/+X-16/+X-20/+X-24/+X-28, where X is your highest attack bonus, then avalanch of blades is better if X-24 or less is a guaranteed hit as long as you don't roll a 1.

This situation might come up if you cast wraithstrike against a dragon that hasn't cast Scintillating Scales, and you have a high strength.
That's actually very true. It's utterly irrelevant how much AC your enemy has, or how much attack bonus you have, at least for the purposes of calculation. All you need to know is the required difference between your attack bonus and the enemy's AC such that avalanche becomes better, and you can just fill in the blanks afterwards. I'm pretty sure that there's gotta be a single turning point where the comparison is concerned. That's a lot simpler than the alternative. After that, you just figure out the average number of attacks that are likely to hit for each one, because we can generally assume that damage is constant between avalanche hits and TSS hits. Time stands still is probably rather easy to math, particularly because you can just add the percentage chance of each hit hitting. Figuring out a good formula for how many avalanche hits hit might be more complicated.

[Douglas]

If you keep hitting, Avalanche of Blades will eventually hit a point where only the nat 20 rule matters. Call X the average number of hits from that point on. You have a 5% chance of getting 1 hit plus another roll that will average X, and the total of that will equal X. Thus, X = .05 * (1 + X). Solving produces X = 1/19.

Take that as the start of some quick spreadsheet formulas, and I get:
{table=head]Attack bonus - AC|Average hits
-20|.0526
-19|.1053
-18|.1579
-17|.2105
-16|.2632
-15|.3316
-14|.4053
-13|.4842
-12|.5684
-11|.6658
-10|.7729
-9|.8905
-8|1.0195
-7|1.1661
-6|1.3297
-5|1.5124
-4|1.7166
-3|1.9494
-2|2.2132
-1|2.3868
0|2.5807
1|2.8020
2|3.0525
3|3.2175
4|3.4017
5|3.6119
6|3.8499
7|4.0066
8|4.1816
9|4.3813
10|4.6074
11|4.7563
12|4.9225
13|5.1122
14|5.3270
15|5.4684
16|5.6264
17|5.8066
18|6.0107
19|6.1450
20|6.2951
21|6.4663
22|6.6602
23|6.7878
24|6.9303
25|7.0930
26|7.2771
27|7.3984
28|7.5338
29|7.6883
30|7.8633
31|7.9785
32|8.1071
33|8.2539
34|8.4201
35|8.5295
36|8.6518
37|8.7912
38|8.9491
39|9.0531
40|9.1692
41|9.3016
42|9.4517
43|9.5504
44|9.6607
45|9.7866
46|9.9291
47|10.0229
48|10.1277
49|10.2472
50|10.3826
51|10.4717
52|10.5713
53|10.6849
54|10.8135
55|10.8982
56|10.9927
57|11.1006
58|11.2228
59|11.3033
60|11.3931
61|11.4956
62|11.6117
63|11.6881
64|11.7734
65|11.8708
66|11.9811
67|12.0537
68|12.1348
69|12.2273
70|12.3320
71|12.4010
72|12.4780
73|12.5659
74|12.6654
75|12.7309
76|12.8041
77|12.8876
78|12.9822
79|13.0444
80|13.1139
81|13.1932
82|13.2831
83|13.3422
84|13.4082
85|13.4836
86|13.5689
87|13.6251
88|13.6878
89|13.7594
90|13.8405
91|13.8938
92|13.9534
93|14.0214
94|14.0984
95|14.1491
96|14.2058
97|14.2704
98|14.3435
99|14.3917
100|14.4455[/table]

Without a way to negate the nat 1 rule, Avalanche of Blades will never get higher than 19 average hits even with an infinite attack bonus.

I added a column for TSS average hits, and it confirmed TSS has a strong advantage until well past the point where even your last iterative hits on a 2. So, the number of hits AoB has to beat is number of TSS attacks times .95.

Assuming only 4 attacks in a full attack, TSS wins until attack bonus exceeds target AC by 29. Add Haste or the Speed property for one more, and you need 43 for AoB to pull ahead. With Greater Two Weapon Fighting and Haste, for 8 attacks per full attack, it takes 114. If you somehow have 10 or more attacks in a full attack, for 20 or more with TSS, AoB is never better no matter how high your attack bonus.

AoB will fare substantially better if you have a way to negate the nat 1 rule, but the precise math depends on the details of how the nat 1s are negated (reroll? ignore the nat 1 rule but still need to beat AC? that luck feat that treats it as a nat 20? max number of uses?).

[chaos_redefined]

To all the other posters... let's say you had an arbitrarily large attack bonus. Then, even with natural 1's as a possibility, the average number of attacks made with Avalanche of Blades would be 19, compared to Time Stands Still having an average number of attacks of 7.6.

The number of attacks you get on average with AoB is calculated as follows:
f(1) = min(0.95, max(0.05, (21+AB-AC)/20))
f(k) = f(k-1)*min(0.95, max(0.05, (21+AB-AC-4*k)/20))
average number of attacks= sum from k=1 to k=infinity of f(k)

The number of attacks you get on average with TSS is calculated as follows:
g(k) = min(0.95, max(0.05, (26+AB-AC-5*k)/20))
average number of attacks = sum from k=1 to 4 of g(k)*2

With this, you can calculate the average number of attacks with each maneuver for a give AB-AC.

With this in mind, I've got:
AB-AC = 32 -> AoB = 7.53, TSS = 7.6
AB-AC = 33 -> AoB = 7.68. TSS = 7.6

So, you need an Attack Bonus of 33 higher than the target's AC for avalanche of blades to be better tha time stands still. Someone else commented that the average AC at level 20 is 36.44. So, you need an attack bonus of around 70 at level 20.

[Douglas]

chaos_redefined, your value for f(1) is off. It's 1/19, or about .05263.

[chaos_redefined]

f(k) is the probability that the kth attack hits. The probability of the first attacking hit must be a multiple of 1/20. How can it be 0.523?

[Douglas]

Sorry, I misinterpreted it. In that case, I suspect your summation to infinite calculation is accumulating rounding errors to the point of significance. My method of calculation is much less susceptible to that, and my numbers aren't quite matching yours.

[chaos_redefined]

I used the following command in MATLAB:
f(1) = min(0.95, max(0.05, (21+AB)/20)); for k=2:10000, f(k) = f(k-1)*min(0.95, max(0.05, (21+AB-4*k)/20)); end, sum(f)

AB is 32 and 33 in the cases I provided.

Edit: Just confirmed, it treats all values after 251 as 0, and f(251) is of the order of 10^-322.

[Douglas]

Just noticed our values are exactly 4 AB apart, precisely the interval of the increasing penalty, and took a closer look at your f(k) function. F(1) is fine, but you're effectively skipping the second attack by going directly to a -8 penalty on attack 2 instead of -4.

[chaos_redefined]

That would do it. My bad.

Changing the formula to reflect that is easy.
f(k) = f(k-1)*min(0.95, max(0.05, (25+AB-AC-4*k)/20))

With this change:
AB - AC = 28 => AoB = 7.53, TSS = 7.6
AB - AC = 29 => AoB = 7.69, TSS = 7.6

That's still for just the basic iteratives. Douglas has already given all the different varieties, but as mentioned earlier, the average AC for CR 20 is 36.44, so we're talking about a bonus to hit of +65 for avalanche to be better than time stands still.

There was also some talk of touch attacks, so if you somehow get wraithstrike off first, or are comboing this with combat rhythm, the average touch AC of CR 20 creatures is 9.11, so you need somewhere around +40 to hit. A lot more manageable.

[Big Fau]

Assuming only 4 attacks in a full attack, TSS wins until attack bonus exceeds target AC by 29. Add Haste or the Speed property for one more, and you need 43 for AoB to pull ahead. With Greater Two Weapon Fighting and Haste, for 8 attacks per full attack, it takes 114. If you somehow have 10 or more attacks in a full attack, for 20 or more with TSS, AoB is never better no matter how high your attack bonus.

Again, the number of attacks isn't the only thing you need to consider when determining which is better; you also have to look at the average damage output of those attacks. If the guy using TSS is using TWFing, while the guy using AoB is using THFing, there's going to be a significant difference in damage output per attack.

[Deophaun]

Again, the number of attacks isn't the only thing you need to consider when determining which is better; you also have to look at the average damage output of those attacks. If the guy using TSS is using TWFing, while the guy using AoB is using THFing, there's going to be a significant difference in damage output per attack.
Also if they have sources of precision damage, or are under the effects of Dragonfire Inspiration.

[Snowbluff]

Yeah, this is assuming one character who is building for Attack stuff. He's using a Two Hander because he thinks TWF sucks, for the sake of the experiment.

[eggynack]

Yeah, this is assuming one character who is building for Attack stuff. He's using a Two Hander because he thinks TWF sucks, for the sake of the experiment.
Quite so. We basically have to assume all other factors being equal, or else the number of attacks isn't a workable metric for maneuver quality. However, if we're talking about ways to give the avalanche of blades an average number of attacks beyond 19, something like better lucky than good (CS, 74) would reroll the single probabilistic one, thus making the chain longer at absurd attack bonuses. Also, TWF takes feats, and feats are feats, so if we're assuming that the TSS guy is using those feats in this manner, then the avalanche guy can use his feats in a different manner.

[Chronos]

Extra damage from precision damage, dragonfire inspiration, or whatever won't change the calculation, because those are all per-hit, and we're already just counting up the number of hits. Two-handing vs. two weapons is, however, relevant, because both will gain the benefit of two-handing (higher damage per hit), but only one will gain the benefit of two-weapon (larger number of attacks in a full attack).

[Big Fau]

Yeah, this is assuming one character who is building for Attack stuff. He's using a Two Hander because he thinks TWF sucks, for the sake of the experiment.

If it's a guy going THF then AoB is better 95% of the time. That luck feat that turns a Nat 1 into a Nat 20 1/day, a Luck Blade with the Lucky enhancement from the MiC, and the Iron Heart maneuver that lets you reroll a missed attack would be enough to use the maneuver multiple times each day without fear of failure. You can even go Synad so you can take Deep Impact (for free Touch Attacks) and use an additional reroll item like the Gloves of Fortunate Striking. Maxing out your attack bonus would be fairly simple, but increasing your damage output would be even easier.

Alternatively, combine both with Stormguard Warrior and Karmic Strike/Robilar's Gambit. Murder people in 2 rounds.

Edit: Although the guy going for TSS can always go Elf Warblade 5/Eternal Blade 10/Revenant Blade 5 and still get TSS by 20th level, at which point the damage output favors TSS because of Eternal Blade and Revenant Blade screwing with it.

[Deophaun]

Extra damage from precision damage, dragonfire inspiration, or whatever won't change the calculation, because those are all per-hit, and we're already just counting up the number of hits.
It changes the proportion of damage coming from things like Power Attack. That's relevant if you're looking at whether a THF AoB that hits five times on average outperforms a TWF TSS that hits eight times on average, for example.

[Keld Denar]

One other factor to keep in mind is that TSS is a full attack while AoB is not. Haste in particular is something that should be factored into any and every level 20 assumption. It is foolish to think otherwise. Then there are simple abilities like EWM's Flurry of Strikes (Spiked Chain is a popular weapon) or the Slashing Fury feat. Haste + Flurry changes +20/+15/+10/+5 to +19/+19/+19/+14/+9/+4, all of which gets doubled with TSS but for which AoB only benefits from the +1 hit.

And also there are sometimes secondary natural attacks to factor in which benefit 1:1 with PA.

[Menzath]

I cannot remember where it was but there was an Iajutsu/katana related thing(skill, class feature I seriously cannot think of it) that with the adaption? special quality on a weapon let's you stack it with Crushing strike and Pulverize.
In this case Avalanche of blades would win hands down. Every hit would BOOST your attacks by +2 and every hit adds another D6 to the next hit.
Now you just need wraith strike to make them all touch attacks if you have trouble hitting initially, And re-rolls, you definitely need a few in case of those pesky 1's.

[Claudius Maximus]

Phaant's Luckstone from Ghostwalk is an apparently slotless 1000 gp item that gives a one-time reroll. You can afford quite a few of these at high levels. That should help extend the chain.

Also it occurs to me that an Psion (or whatever)/Warblade/Metamind could use Forced Dream and Twinned Synchronicity to restart the entire maneuver an arbitrarily large number of times, leading to a somewhat smaller arbitrarily large number of consecutive 20's and therefore additional attacks. That's kind of TO though.

[Lonely Tylenol]

What about [effective use of] Blood in the Water, ex. through crit-fishing? Functionally speaking, a crit while in Blood of the Water, using Avalanche of Blades, will only reduce your to-hit on the next attack by 3. I'm not sure how this interacts with Lightning Maces, however, if at all.

[Snowbluff]

I cannot remember where it was but there was an Iajutsu/katana related thing(skill, class feature I seriously cannot think of it) that with the adaption? special quality on a weapon let's you stack it with Crushing strike and Pulverize.
In this case Avalanche of blades would win hands down. Every hit would BOOST your attacks by +2 and every hit adds another D6 to the next hit.
Now you just need wraith strike to make them all touch attacks if you have trouble hitting initially, And re-rolls, you definitely need a few in case of those pesky 1's.


What about [effective use of] Blood in the Water, ex. through crit-fishing? Functionally speaking, a crit while in Blood of the Water, using Avalanche of Blades, will only reduce your to-hit on the next attack by 3. I'm not sure how this interacts with Lightning Maces, however, if at all.
I wish to investigate the stacking damage bonuses.

We have:
Crushing Strikes.
Blood in the Water.

[chaos_redefined]

The OP asked that what attack bonus was needed for one method to work better than the other. In most areas, this means that it is assumed all other constants are equal. So, therefore, it is only logical to exclude differences in damage. However, if damage is different, the method to take this into account is easy:

Step 1) Calculate the average damage between the two options.
Step 2) Multiply the values for TSS by the average damage per hit with the attacks made by the TSS-user. You now have the average damage per use of TSS
Step 3) Multiply the values for AoB by the average damage per hit with the attacks made by the AoB-user. You now have the average damage per use of AoB
Step 4) Compare the results of steps 2 and 3

Most crit-fishing is unaffected, with the exception of blood in the water. With that particular case, the table needs to be modified to take that into account, and that sounds like a pain.

[Darrin]

One other factor to keep in mind is that TSS is a full attack while AoB is not. Haste in particular is something that should be factored into any and every level 20 assumption. It is foolish to think otherwise. Then there are simple abilities like EWM's Flurry of Strikes (Spiked Chain is a popular weapon) or the Slashing Fury feat. Haste + Flurry changes +20/+15/+10/+5 to +19/+19/+19/+14/+9/+4, all of which gets doubled with TSS but for which AoB only benefits from the +1 hit.

And also there are sometimes secondary natural attacks to factor in which benefit 1:1 with PA.

If anyone has Time Stands Still prepared, it's usually a good bet they have Raging Mongoose ready to go as well (only works once for that round, but that's still +2 attacks per weapon).