Is "Shield" too powerful?

[Doctor Proctor]

I retrained Shield, as it had proved almost useless... I think it saved from a hit once over three levels.

Why? I would've traded it out for Expeditious Retreat. Granted, it's a Daily, but if you're not going to be able to block attacks effectively I'd rather just have a "Get out of Dodge" ability. Shifting twice your speed means that you won't provoke, and then you can follow it up with something like Ray of Frost to slow down whoever was attacking you...or just do a double move, and you would actually be 3x your speed away from whoever was messing with you.

 

[KarinsDad]

I wasn't proposing "save second chance for a critical" the entire encounter. That's a losing strategy compared to only saving it for a critical initially.

As I mention in my post, assuming that the player knows the number of attacks left against him leads to a more effective solution for how long to save Second Chance for (than if he only knows a distribution of times he'll be attacked again). The player doesn't have to be sure he'll be hit again- it's enough to know he'll be attacked a decent number of times. With n=8, and optimal k=4, the player faces 8 attacks for the encounter, and reserves Second Chance for crits on the first (8-4)=4 attacks. As I indicated above, that change in strategy makes Second Chance 13% more effective under my assumptions.

But, that's the point. 13% is the optimal savings assuming Second Wind is always used in every encounter that it can be used in. It doesn't work at all for a given encounter where the PC gets hit once, saves his Second Wind, and then never gets hit again. In that encounter, it was a 50% decrease in savings, not a 13% gain.

In order to gain the 13%, one needs to be fairly knowledgeable about whether another successful attack is going to occur.

Edit- if you believe that it's important to use a defensive power in the first half of an encounter in general, and we apply this requirement equally strictly to both Shield and Second Chance (instead of just raising it as an objection to saving Second Chance), this will favor Second Chance, because even with the "save it for a critical initially" Second Chance strategy, the rate of increase in expected damage blocked as a function of the number of attacks is higher for Shield. So anything that effectively shortens the time you have to use the defensive powers favors Second Chance.

Shield should be used the first time every time (assuming it is not a minion attack). One does not use the same potential strategy as with Second Chance.

The reason is that future successful attacks against the PC might be Will attacks, might be Fort attacks, or might be outside of Shield's 20% protection range. Hence, the best strategy for Shield is typically to use it right away against the first non-minion attack.

Note: if most or all of the enemies are minions, it might once in a while be a good strategy to use it against a minion. It's a good strategy to use it against a minion if the PC has 3 hit points remaining as well.

Note: if there is a BBEG like a Dragon, it might be a good strategy to not use it against an attack from a different opponent.

So no, waiting is typically not a good strategy for Shield. It is only slightly better for Second Wind assuming the player knows that his PC will get hit at least once more in the encounter.

 

[Elric]

But, that's the point. 13% is the optimal savings assuming Second Wind is always used in every encounter that it can be used in. It doesn't work at all for a given encounter where the PC gets hit once, saves his Second Wind, and then never gets hit again. In that encounter, it was a 50% decrease in savings, not a 13% gain.

In order to gain the 13%, one needs to be fairly knowledgeable about whether another successful attack is going to occur.

No, you don't understand the calculation I was doing. The calculation I was doing only assumes that you know the number of attacks that will be directed against you, not the number of successful hits. If you knew the number of successful hits against you, I clearly wouldn't come up with an answer of optimal k=4, since the optimal strategy would instead be "only use Second Chance on a critical hit, unless you know you won't be hit again."

It's a dynamic programming problem, where by waiting you risk the chance of never getting to use Second Chance, and gain that it's more likely to block a critical hit. Sometimes this will result in not using Second Chance at all, and sometimes it will result in using Second Chance on a critical hit instead of a regular hit. The end result of making this tradeoff optimally is that you gain 13% in expectation.

Shield should be used the first time every time (assuming it is not a minion attack). One does not use the same potential strategy as with Second Chance.
...
So no, waiting is typically not a good strategy for Shield. It is only slightly better for Second Wind assuming the player knows that his PC will get hit at least once more in the encounter.

As I indicate above, if you are solely going by expected values, waiting noticeably improves the value of Second Chance, even though all attacks are identical and all that you can wait for are critical hits to come up.

You said

The player does not have to use it on the first hit, but should use it somewhere in the first half of the encounter, if only to delay when he goes bloodied (where opponents might have other advantages against him).
But otherwise, it's a mistake to save it for too long over the long haul.

This objection is what I was referring to. If you believe that there's an intrinsic value in getting the effects of Shield/Second Chance earlier in the encounter above and beyond the fact that by getting the effects once you know the power will be of us (because you value avoiding being bloodied), this works to Second Chance's favor.

 

[AngryPurpleCyclops]

I saw your "save Second Chance for a crit" math.

Sound on the surface, but it ignores something fairly basic.

Say your PC gets hit twice in two encounters, each for normal damage (normal damage happens 90% of the time on a 50% chance to hit, critical damage happens 10% of the time on a 50% chance to hit).

The odds of 4 successful attacks hitting with normal damage (as opposed to critical damage on any of them) is 64%, a fairly common and high percentage of the time.

even if this example was correct which it's not you're ignoring more then 1/3 of the outcomes.

Using the "save Second Chance" strategy, the PC will get hit all 4 times and although he could have used Second Chance twice and only got hit twice, that did not occur. He takes 400% damage. He could have taken 200% damage. Even if the 4th attack would have been a critical, he would have taken 250% damage by using his Second Wind for the critical instead of 200%.

This is a horrible interpretation of what is being said. First of all, pc's are rarely knocked unconscious by 2 hits. So if you only get hit twice in an encounter neither power matters very much (other than a possible slight cost savings in number of healing surges spent on the short rest). Because second chance ALWAYS can block any hit the pc takes, it's easy to save it on the first few hits with the intent to block a major attack or a critical hit. It will rarely go wasted even if you save it (you can use it when you feel like the battle is nearly over and your remaining chances for use will be small if you haven't already picked a use for it) and when it does go wasted it's because you won the battle anyway. This is another weakness of shield. Because it's only available on 14.6% of attacks or 29.2% of hits saving it for something bigger runs an increasing risk of it never being used AND you can wind up losing the encounter because you didn't use it. In a nut shell, it's pretty easy to imagine a scenario where your pc is killed while shield is still available but almost impossible to imagine the same for second chance. This heavily favors second chance.

Saving a single extra 50% damage on a critical (approximately), but giving up entire encounter's worth of Second Wind is a mathematically bad strategy.

you're missing the gist of strategy entirely, second chance will almost never get wasted.

Spread over the lifetime of the PC, 10% of the times he gets hit, he will negate a critical. Since that will not happen every single encounter, he will lose a lot of Second Winds that he could have used.

once again you missed the point and made a very poor assumption.

And, this too is why APC does not get it. He too thought that saving Second Wind for a critical is important.

It's important for the PC to stay alive.

Sure, if you think your foe is a minion, a player would be smart to not use Second Wind on the off chance that a non-minion might hit him in the same encounter.

it is you that doesn't get it. saving second chance for a critical is indeed a very valid strategy that enhances it's effectiveness.

And, I read your section on k attacks remaining. I do think that a player could save Second Wind on the first few hits, waiting for that critical if he is fairly sure that he will get hit at least one more time.

But, that too is a guessing game.

The player does not have to use it on the first hit, but should use it somewhere in the first half of the encounter, if only to delay when he goes bloodied (where opponents might have other advantages against him).
But otherwise, it's a mistake to save it for too long over the long haul.

This is another very poor assumption. It's not a guessing game, you save it for a critical until you have enough damage that you feel the next hit could possibly put you down. The number of creatures who have additional affects vs bloodied is somewhat small and in those cases you could modify your behavior once you became aware of those effects. Using it because you know that a gnoll gets an additional power vs bloodied the first time your pc faces that kind of gnoll is basically cheating via metagame knowledge.

 

[Mort_Q]

Why? I would've traded it out for Expeditious Retreat. Granted, it's a Daily, but if you're not going to be able to block attacks effectively I'd rather just have a "Get out of Dodge" ability. Shifting twice your speed means that you won't provoke, and then you can follow it up with something like Ray of Frost to slow down whoever was attacking you...or just do a double move, and you would actually be 3x your speed away from whoever was messing with you.

That's precisely what I did. I swapped out Shield and took Expeditious Retreat, which proved very useful for setting up Area powers, when Fey Step doesn't have the range I need.

 

[KarinsDad]

No, you don't understand the calculation I was doing. The calculation I was doing only assumes that you know the number of attacks that will be directed against you, not the number of successful hits. If you knew the number of successful hits against you, I clearly wouldn't come up with an answer of optimal k=4, since the optimal strategy would instead be "only use Second Chance on a critical hit, unless you know you won't be hit again."

Actually, I did understand. I just ignored the "number of attacks" part of it because number of attacks is totally irrelevent to this discussion.

I hope we can agree at least upon that.

It is not the number of attacks, it's the number of successful attacks. Neither Shield or Second Chance can be used on an unsuccessful attack since they trigger on hits.

Talking about non-successful attacks muddies the water. I'm trying to lean the discussion (and the math) into what is relevant to the powers, not what is irrelevant to the powers.

It's a dynamic programming problem, where by waiting you risk the chance of never getting to use Second Chance, and gain that it's more likely to block a critical hit. Sometimes this will result in not using Second Chance at all, and sometimes it will result in using Second Chance on a critical hit instead of a regular hit. The end result of making this tradeoff optimally is that you gain 13% in expectation.

Except when it comes to discussing strategy, we are talking people, not math. There is no optimally that works every time. Everyone will make mistakes, the situations will call for different strategies, some people using it early, some people using it not at all.

There is no one equation that states that optimal useage of it that works for all encounters. The best we can do is some form of rule of thumb preferred strategy such as not using it on the first one or two non-critical hits or so, but using it early enough so that it at least gets used. That requires guesswork and intuition on the part of the player to gain anything.

You said it yourself "the calculation assumes that you know the number of attacks that will be directed against PC". People don't know that. People have to guess. And combat is fluid such that a different PC can become the focus of more attacks.

If the player avoids early Second Chance in order to use it on a critical, it takes one "used it on a critical" to make up for one "failed to use it during the encounter, but did get hit in that encounter". Actually, it is slightly worse than that.

Take the first creature in the MM. It does 2D8+8 damage.

17 average points on a hit, 24 on a critical.

The average number of points saved by using it on a critical over using it on a normal attack is 7 points (24-17). Since it is at least used in either case (where it averages 8.85 damage), the only savings is the difference between a normal hit and a critical hit or 7 points.

But, if it is never used and could have been used on a normal hit, it would have saved 8.15 points (17-8.85).

So, NOT using the Second Chance at all in an encounter is more of a loss than the gain of using Second Chance for a critical in an encounter.

As I indicate above, if you are solely going by expected values, waiting noticeably improves the value of Second Chance, even though all attacks are identical and all that you can wait for are critical hits to come up.

And that's true as long as the PC is hit during the encounter and he uses Second Chance during the encounter. If he is hit and never uses it, the expected gain evaporates and becomes a loss (percentage-wise, not necessarily in reality because the Second Chance could do nothing or even increase the damage).

This objection is what I was referring to. If you believe that there's an intrinsic value in getting the effects of Shield/Second Chance earlier in the encounter above and beyond the fact that by getting the effects once you know the power will be of us (because you value avoiding being bloodied), this works to Second Chance's favor.

For Shield, there is no doubt. Shield will never stop a critical. It should nearly always be used as early as possible during what appears to be a serious attack. Sure, there will be situations where waiting was better in hindsight, but hindsight is not something people have ahead of time.

For Second Chance, we are mostly in agreement here. One should wait some for a critical. I just think that one has to be a bit cautious and not wait too long, otherwise he runs the risk of not using it at all or of causing other problems for the party.

The PC Wizard could throw out a Fireball and suddenly instead of 4 opponents, there is only 1 and the PC Halfling Rogue will not use his Second Chance. The player of the PC Cleric asks "Why did you not use your Second Chance? I healed you because you got bloodied and then later on had to use a potion to get up the Fighter because I did not have another Healing Word."

This type of stuff happens. It's not just about the math, there are a lot of variables in it.

This is not just an optimal usage situation, it's also about resources and other "in combat" advantages. If the Halfling PC uses Second Chance, he might use one less healing surge. Or he might stay in melee longer and give flank longer. Or the Cleric might not have to heal him. Or the ongoing effect on him takes 2 more rounds to make him unconscious.

But the bottom line is that each situation is unique and saving Second Chance for the optimal point in time can easily backfire.

A more prudent strategy is to use Second Chance somewhat in the middle of each encounter (as soon as one foe in four is down or possibly when two foes are bloodied for example) so that the Halfling does not screw up the situation. Having the Cleric not have to focus on the Halfling is also a good thing.

APC will now talk about my bad assumptions, etc. zzzzzzzzzzzzzzzzz

I'm not quite sure why he re-opened this 6 months old thread just to disagree with everyone, regardless of what they write.

 

[AngryPurpleCyclops]

No. He is calculating the chance that Shield doesn’t work on each attack, so the chance it doesn’t work on one attack squared is the chance it doesn’t work on both attacks. If it does work, you decrease the damage by one attack’s worth.

This is also wrong. He’s doing a similar calculation to that in my post above, but it’s a little less accurate in a way that works in Shield's favor because it’s essentially assuming that of n attacks, exactly half will hit (rather than taking the probability distribution of those attacks into account).

It should all be pretty clear if you look at my post above.

You have misunderstood the calculation he was doing.

This wouldn’t be hard to answer, if you simply assumed no one would use Shield on minion attacks and lowered the percentage of attacks that Shield affects accordingly.

If you read further in my post I went back and read what he was saying more carefully and I figured out what he meant. I was just too lazy to go back and edit the first half of my post.

See my post above, where I do account for this. If you are targeted by 8 attacks with a 50% chance to hit (as in my example above), and crits=1.5 regular hits, optimal use of Second Chance reserving it for crits initially makes it block 13% more damage than a myopic “use Second Chance on the first hit” strategy.

The assumption that some % of attacks will target AC/Reflex is somewhat more favorable to Shield than the idea that in some combats there won’t be many enemies even targeting these defenses (and in others lots of enemies will target these defenses), but this effect is probably minor in the scheme of things. Otherwise, this is pretty much covered. See my post above.

more favorable yes, but not at all minor. If you have played a wizard with shield you'll be well aware that a significant percentage of encounters shield does not get employed. This is partly because the attacks might not be vs ac and partly because the attack rolls might not fall in the window. If you take out minion attacks this becomes even more likely. In a recent encounter with howling hags and bugbears I was only attacked vs reflex/ac 3 times out of about 9 attacks. None of them fell in the range I could affect. Trust me I would have really liked to have had second chance when the hag critical'd me for 27 with her bloodied shriek. If you take out minions, for argument sake I assume minions are about 10% of the attacks. The percentage that shield will impact an attack drops from 14.6% to 11.4%. I'll do the math if someone disputes this but that is the right number based upon 10% minions and not using shield against them.

This means that 88.6 percent of attacks will not be impacted by shield. If your wizard gets attacked 6 times in an encounter 48.3% of the time you will not be able to use shield. If he gets attacked 10 times in an encounter the percentage is still 29.8% that he will NOT get to use shield. That's a pretty significant chance in my opinion and one backed up by both personal experience and the anecdotal evidence presented in this thread by several other players. Even at 15 attacks in an encounter it's still 16.3% likely shield did not get a chance to be used. That's still 1 in 6 in an encounter which is probably at the high end of attacks on the wizard and thus pretty important that you get to use your defenses. Hardly "minor".

But, that's the point. 13% is the optimal savings assuming Second Wind is always used in every encounter that it can be used in. It doesn't work at all for a given encounter where the PC gets hit once, saves his Second Wind, and then never gets hit again. In that encounter, it was a 50% decrease in savings, not a 13% gain.

In order to gain the 13%, one needs to be fairly knowledgeable about whether another successful attack is going to occur.

this totally misses the point. again.

Shield should be used the first time every time (assuming it is not a minion attack). One does not use the same potential strategy as with Second Chance.

The reason is that future successful attacks against the PC might be Will attacks, might be Fort attacks, or might be outside of Shield's 20% protection range. Hence, the best strategy for Shield is typically to use it right away against the first non-minion attack.

Note: if most or all of the enemies are minions, it might once in a while be a good strategy to use it against a minion. It's a good strategy to use it against a minion if the PC has 3 hit points remaining as well.

you're making my case not yours here. Second chance will always get to be played or it won't matter (i.e. the combat reached a successful outcome). These are both pretty minuscule probabilities. If you have 3hp remaining the battle field will pretty likely not have any minions remaining on it. There are lots of times where it's preferable to hold shield in reserve. If a kobold hits me with a shortsword or spear but there's a kobold slinger or 3 hurling gluepots or firepots I'll probably hold off on usuing it and take my d6+3 or d8 dmg.

Note: if there is a BBEG like a Dragon, it might be a good strategy to not use it against an attack from a different opponent.

So no, waiting is typically not a good strategy for Shield. It is only slightly better for Second Wind assuming the player knows that his PC will get hit at least once more in the encounter.

I agree that waiting is generally not a good idea for shield (one more weakness compared to second chance) but there are numerous times where waiting is a good idea with shield which hurts it's overall effectiveness. Waiting with second chance is pretty much NEVER going to hurt you. You'll almost always get to use it since you can use it on ANY attack and if you don't it's because you already won the encounter.

No, you don't understand the calculation I was doing. The calculation I was doing only assumes that you know the number of attacks that will be directed against you, not the number of successful hits. If you knew the number of successful hits against you, I clearly wouldn't come up with an answer of optimal k=4, since the optimal strategy would instead be "only use Second Chance on a critical hit, unless you know you won't be hit again."

I initially skipped this because there's just no way to know the number of future attacks you'll face. Your math is a lot better than KD's but a lot of your assumptions are at least a little circumspect and ignore real world application.

It's a dynamic programming problem, where by waiting you risk the chance of never getting to use Second Chance, and gain that it's more likely to block a critical hit. Sometimes this will result in not using Second Chance at all, and sometimes it will result in using Second Chance on a critical hit instead of a regular hit. The end result of making this tradeoff optimally is that you gain 13% in expectation.

expectation of what? There are way too many variables in this equation to put a good number to it.

Bottom line, it's very easy to save second chance until you receive a crit unless you reach a point in the combat that you feel a normal hit has a significant chance of putting you down. It will almost never be wasted and in the cases where it is wasted it's moot because by default you must have won the battle. The odds that you block a crit in any major battle with second chance (by major I mean you receive 10 or more attacks) is probably in the neighborhood of 35-65% and the rest of the time you'll still have a 50% chance of blocking a normal hit. There is almost no chance of wasting it unless you're simply not paying attention. In shorter encounters, it probably doesn't even matter (though you still have a decent chance to block a crit). 13% is greatly undervalued for crit blocking. The real number is probably about an additional 40% in damage reduction.

If a crit is 150% damage and a regular hit is 100%, then we assume that 45% of that time you turn the crit into a hit and 50% into a miss
.45 x .50 = .09
and 50% of the time you turn it into no dmg at all
.50 x 1.5 = .75
and .5 of the time you have no effect
.09 + .75 = .84 or 84% average damage reduction on crits.

if you block a crit in only 20% of encounters (a very low number almost guaranteed to not be accurate) then the .84 x .2 would be 16.8% damage reduction value. If you change this to blocking a crit in 35% of encounters the value of second chance jumps by 29.4% additional damage reduction. In the encounters where it really matters (i.e. the 'hard" encounters where the party is severely challenged) then you can likely expect upwards of 8-10 attacks and the added value of second chance rises toward 50%. This is somewhat reduced by the concept that you might feel pressed to use it before you take 10-12 attacks because you hp's are getting low but you might also get a healing word from the cleric and hang in for 12-15 ATT's before you use it. The more I look at these numbers the more obvious it becomes that second chance is at least twice as effective as shield.

 

[AngryPurpleCyclops]

Actually, I did understand. I just ignored the "number of attacks" part of it because number of attacks is totally irrelevent to this discussion.

I hope we can agree at least upon that.

It is not the number of attacks, it's the number of successful attacks. Neither Shield or Second Chance can be used on an unsuccessful attack since they trigger on hits.

Talking about non-successful attacks muddies the water. I'm trying to lean the discussion (and the math) into what is relevant to the powers, not what is irrelevant to the powers.

Why is it you're so unable to grasp the concept that by ignoring attacks you increase error in you calculation? You also use this as a method to drop the probability for the attacks to be outside shields abilities out of the equation. The water is only muddied here by your desire to muddy it.

Except when it comes to discussing strategy, we are talking people, not math. There is no optimally that works every time. Everyone will make mistakes, the situations will call for different strategies, some people using it early, some people using it not at all.

There is no one equation that states that optimal useage of it that works for all encounters. The best we can do is some form of rule of thumb preferred strategy such as not using it on the first one or two non-critical hits or so, but using it early enough so that it at least gets used. That requires guesswork and intuition on the part of the player to gain anything.

First we're comparing the powers so we have to look at best usage or "optimal". Any power can be misused and decrease it's effectiveness but that isn't even close to relevant. We're discussing your belief shield is too powerful. Not only is your estimation of 1-2 VERY low but it totally flies in the face of real world application. You can hang onto second chance past the 2nd hit 99% of the time. How often are you bloodied on the first hit?

You said it yourself "the calculation assumes that you know the number of attacks that will be directed against PC". People don't know that. People have to guess. And combat is fluid such that a different PC can become the focus of more attacks.

we agree here for once. You can project the number of attacks you might still face based upon the condition of the party and the number of enemies left standing plus the tactical situation on the battle field. In any event this information hardly matters. What matters is your perception of how close to unconscious is your pc relative to the danger of each subsequent attack. If you're not bloodied, it's pretty easy.

If the player avoids early Second Chance in order to use it on a critical, it takes one "used it on a critical" to make up for one "failed to use it during the encounter, but did get hit in that encounter". Actually, it is slightly worse than that.

You keep saying this but first it's not true and second it's almost never going to happen. It's not true primarily because the objective is to survive. One blocked critical is worth 25+ missed chances to use the power because when you fail to use the power the very most it cost you is a healing surge. By definition if you fail to use the power you won the combat. Blocking a critical can save you from losing or dieing. Second because it's available on EVERY attack it's pretty easy to judge when you'll be attacked again or when the battle is nearing it's end. You are weighing two things (missed chance to use power vs block a crit) as roughly equally important when one matters almost not at all and the other could save your pc's life.

Take the first creature in the MM. It does 2D8+8 damage. 17 average points on a hit, 24 on a critical.

The average number of points saved by using it on a critical over using it on a normal attack is 7 points (24-17). Since it is at least used in either case (where it averages 8.85 damage), the only savings is the difference between a normal hit and a critical hit or 7 points.

Anecdotal and irrelevant. Take an orc raider avg dmg 9.5, critical dmg 21.5 or ~230%.
We've been using 150% dmg on a crit and that seems pretty fair.

But, if it is never used and could have been used on a normal hit, it would have saved 8.15 points (17-8.85).

Once again your hypothesis is based upon the faulty assumption that it will fail to be used because you held off using it for several rounds waiting to block a crit. TRY TO UNDERSTAND!! This will almost never happen and be irrelevant when it does. This also totally avoids the added benefit that it doesn't have to block a crit to have increased effectiveness. It can block the bbeg encounter power and gain just as much advantage.

So, NOT using the Second Chance at all in an encounter is more of a loss than the gain of using Second Chance for a critical in an encounter.

Once again, this is totally fallacious. Not using second chance in an encounter has almost no cost beyond the opportunity cost associated with using another healing surge. Blocking a crit has a huge upside because the battle is still raging and you've significantly cut down on the bad guys offense.

 

[Elric]

The assumption that some % of attacks will target AC/Reflex is somewhat more favorable to Shield than the idea that in some combats there won’t be many enemies even targeting these defenses (and in others lots of enemies will target these defenses), but this effect is probably minor in the scheme of things. Otherwise, this is pretty much covered. See my post above.

more favorable yes, but not at all minor. If you have played a wizard with shield you'll be well aware that a significant percentage of encounters shield does not get employed. This is partly because the attacks might not be vs ac and partly because the attack rolls might not fall in the window. If you take out minion attacks this becomes even more likely. In a recent encounter with howling hags and bugbears I was only attacked vs reflex/ac 3 times out of about 9 attacks. None of them fell in the range I could affect. Trust me I would have really liked to have had second chance when the hag critical'd me for 27 with her bloodied shriek. If you take out minions, for argument sake I assume minions are about 10% of the attacks. The percentage that shield will impact an attack drops from 14.6% to 11.4%. I'll do the math if someone disputes this but that is the right number based upon 10% minions and not using shield against them.

This is an example of different powers of attacks, which I’m not trying to quantify. I can quantify, say, the difference between having half of your encounters with 50% of attacks targeting AC/Reflex and half of your encounters with 100% of attacks targeting AC/Reflex, versus having every encounter with 75% of attacks targeting AC/Reflex.

I’ll assume n=8 in both cases. In the former case, Shield is 96.3% as effective in terms of attacks stopped as in the latter case (0.701 attacks stopped vs. 0.728 attacks stopped). I think this distribution across encounters leads to changes at least as extreme as what you’re likely to see in practice, and it’s still not a big difference.

I agree that waiting is generally not a good idea for shield (one more weakness compared to second chance) but there are numerous times where waiting is a good idea with shield which hurts it's overall effectiveness. Waiting with second chance is pretty much NEVER going to hurt you. You'll almost always get to use it since you can use it on ANY attack and if you don't it's because you already won the encounter.

This idea that it doesn’t matter if you don’t end up using Second Chance because then you weren’t hit again is more consistent with a “mini-max” strategy of trying to minimize the maximum damage you take, rather than what I’m doing, which is trying to minimize the average (i.e., expected) damage you take. Doing a calculation related to a minimax strategy would involves assuming players who are risk averse over the amount of damage they take, and would be quite a bit harder.

I initially skipped this because there's just no way to know the number of future attacks you'll face. Your math is a lot better than KD's but a lot of your assumptions are at least a little circumspect and ignore real world application.

Indeed, I mention this in my initial post. If you don’t know the number of attacks you’ll face, this doesn’t impact Shield at all in my example (since my assumption is that you use it as soon as possible), but does weaken Second Chance (because you don’t know if you’ll get another chance to use it, which makes optimization more difficult). I could do a calculation letting the number of future attacks be a variable, but that would be more difficult and would require stronger assumptions.

It's a dynamic programming problem, where by waiting you risk the chance of never getting to use Second Chance, and gain that it's more likely to block a critical hit. Sometimes this will result in not using Second Chance at all, and sometimes it will result in using Second Chance on a critical hit instead of a regular hit. The end result of making this tradeoff optimally is that you gain 13% in expectation.

expectation of what? There are way too many variables in this equation to put a good number to it.

Expectation means average damage blocked, since that’s the metric I’m using. I’ve made a lot of assumptions, but a nice result is that I can calculate a number.

Bottom line, it's very easy to save second chance until you receive a crit unless you reach a point in the combat that you feel a normal hit has a significant chance of putting you down. It will almost never be wasted and in the cases where it is wasted it's moot because by default you must have won the battle. The odds that you block a crit in any major battle with second chance (by major I mean you receive 10 or more attacks) is probably in the neighborhood of 35-65% and the rest of the time you'll still have a 50% chance of blocking a normal hit. There is almost no chance of wasting it unless you're simply not paying attention. In shorter encounters, it probably doesn't even matter (though you still have a decent chance to block a crit). 13% is greatly undervalued for crit blocking. The real number is probably about an additional 40% in damage reduction.

This number is the additional average damage blocked you get by saving Second Chance and not using it initially on any hit (only critical hits). The baseline calculation already takes into account that some hits will be critical hits, but assumes you use it the first time available. Your 40% statistic is the answer you get if you compare “optimal waiting to use Second Chance vs. using it on the first available hit” (with hit chance=50%, crit=1.5 regular hits, and optimal k=4) and assume n=16-17 attacks.

Assuming n=8 gives you 15.7%, actually, not 13% as I said before. See my post below. I had a minor error here- I wasn’t handling the fact that Second Chance sometimes turns a regular hit into a critical hit, which makes saving it for critical comparatively better. But Second Chance is slightly worse overall as a result.

 

[Elric]

Actually, I did understand. I just ignored the "number of attacks" part of it because number of attacks is totally irrelevent to this discussion.

I hope we can agree at least upon that.

It is not the number of attacks, it's the number of successful attacks. Neither Shield or Second Chance can be used on an unsuccessful attack since they trigger on hits.

Talking about non-successful attacks muddies the water. I'm trying to lean the discussion (and the math) into what is relevant to the powers, not what is irrelevant to the powers.

Indeed, the number of successful attacks is the metric that directly matters. What I am doing is generating a probability distribution of the number of successful attacks through a hit probability and a set number of attacks. Now, you’ve both taken issue with my assuming a set number of attacks, which could in theory be changed as well at the cost of much more complication.

By looking at successful hits directly without any probability distribution on it, you have made a more restrictive assumption than what I am assuming—essentially, you have not only assumed the number of attacks, but you have also assumed that the results of those attacks will be the average number of successful hits.

If I assumed a set number of successful hits, I couldn’t evaluate the tradeoff of not using Second Chance now on a regular hit, which lessens the chance that you’ll get to use it at all, but raises the chance you’ll get to use it on a critical. The way you do it also increases the chance Shield gets used at all.

It's a dynamic programming problem, where by waiting you risk the chance of never getting to use Second Chance, and gain that it's more likely to block a critical hit. Sometimes this will result in not using Second Chance at all, and sometimes it will result in using Second Chance on a critical hit instead of a regular hit. The end result of making this tradeoff optimally is that you gain 13% in expectation.

Except when it comes to discussing strategy, we are talking people, not math. There is no optimally that works every time. Everyone will make mistakes, the situations will call for different strategies, some people using it early, some people using it not at all.

I assume optimality because it makes calculations easier. If you would like to do calculations with a “makes mistake” term added, you can do so, but it means more work (and arguing about the manner in which people make mistakes), so I don’t do it. When you’re near optimality in general, small deviations have a small effect on the result, so nearly optimal usage will produce the nearly optimal result.

You said it yourself "the calculation assumes that you know the number of attacks that will be directed against PC". People don't know that. People have to guess. And combat is fluid such that a different PC can become the focus of more attacks.

I assume this because it’s easy to calculate and making assumptions about what PCs know about attacks remaining is much harder. As long as you have a reasonable idea when the number of attacks remaining against you drops to a very low number, you’re OK here.

One simple assumption could be that the player is 1 more conservative than my optimal k, and this accounts for the loss due to uncertainty. This doesn’t affect the “low” n=4 case, because you already use Second Chance every time you can. In the “high” n=8 case, this lowers Second Chance to 96% of the expected damage blocked that it would otherwise get. This strategy ends up being an 11% increase in average damage blocked over the “use Second Chance the first time you can” strategy.

If the player avoids early Second Chance in order to use it on a critical, it takes one "used it on a critical" to make up for one "failed to use it during the encounter, but did get hit in that encounter".

This is roughly correct, given my assumptions. Actually, now that you bring this up, I spot a small error in my calculation- I wasn’t accounting for the chance that Second Chance changes a regular hit into a critical hit properly. So this makes Second Chance a little worse in general, but increases the relative value of the “wait for a critical hit” strategy (because I was correctly accounting for the fact that on a critical hit, Second Chance can’t hurt you). This doesn’t change the optimal k.

In particular, the Low case (n=4) number for Second Chance goes from 0.53 to 0.49, and the high case number for Second Chance goes from 0.64 to 0.605. So this error had a noticeable impact on Second Chance’s average damage blocked.

As I indicate above, if you are solely going by expected values, waiting noticeably improves the value of Second Chance, even though all attacks are identical and all that you can wait for are critical hits to come up.

And that's true as long as the PC is hit during the encounter and he uses Second Chance during the encounter. If he is hit and never uses it, the expected gain evaporates and becomes a loss (percentage-wise, not necessarily in reality because the Second Chance could do nothing or even increase the damage).

No, the expected value is the average damage prevented. This accounts for the times when you don’t get to use it because of waiting and Second Chance prevents 0 damage.

For Second Chance, we are mostly in agreement here. One should wait some for a critical. I just think that one has to be a bit cautious and not wait too long, otherwise he runs the risk of not using it at all or of causing other problems for the party.

Indeed, waiting too long can cause other problems, as you suggest with the healing surge example. Knowing whether you need healing can be valuable in and of itself. To the extent that the optimal k=4 I calculated occurs very close to the end of the encounter, this could give us pause, but for n=8, it means waiting halfway through the encounter, which seems reasonable.