Agreed.
1) Second Chance is only used on a successful hit. So, 100% to start with. If the attack missed, Second Chance would not be used. It also works against all attack rolls, regardless of defense, so another 100%. With a 50% chance to hit, we have an original 90% chance of a normal hit (11 to 19) and a 10% chance of a critical (20). So the average amount it protects is calculated by:
100% * 100% * (90% normal damage * (50% of a miss - 5% chance of a critical) + 10% critical damage * (50% of a miss + 45% of normal damage))
This works out to an improvement 54.5% of the time and a worsening of the situation 4.5% of the time. 41% of the time, no change. Ironically, one could distill this down to a 50% chance of improvement (for a 50% chance to hit).
These percentages are higher if the player mostly uses the power to minimize a critical, but if he does that, there are times when he won't use it in an encounter at all. Pros and Cons.
2) Shield also is only used on a successful hit. So, 100% to start with. However, Shield works differently from game to game depending on the DM's adjudication:
2a) The DM rules that the PCs do not know anything about the attack.
73% of the time, a monster will use an AC or Reflex attack (on average in the MM).
If an attack occurs with a 50% chance to hit, Shield will nullify the attack 40% of the time (4 numbers out of the 10 possible to hit numbers).
So, 100% * 73% * 40% = 29.2%
2b) The DM rules that the PCs know the type of attack (e.g. Fort or Reflex), but they nothing about the die result or total.
100% * 100% * 40% = 40%
2c) The DM rules that the PCs do not know the type of attack, but rolls the attack die in front of the players.
If the players see an 8 through a 14 or so on the die, there is a very high chance that Shield will work (not all creatures have a 50% chance to hit). If they see a 17, Shield will almost never help (although it would have once in our game). This is also dependent on how high an AC or Reflex the PC Wizard has (higher ones bump up the range some). For the most part, the player has a high chance (80% in my estimation) of guessing correctly if he can see the die roll.
100% * 73% * 80% (a guess) = 58.4%
2d) The DM rules that the PCs know the type of attack and rolls the attack die in front of the players.
100% * 100% * 80% (a guess) = 80%
2e) The DM rules that the PCs know everything (i.e. the total attack roll and the type of the attack).
100% * 100% * 100% = 100%
Granted, 2e is not a true 100%. It is just 100% of the first attack in an encounter for which Shield will definitely work.
But, the comparison of 1 and 2a does show us that worse case scenario is 29.2% for Shield versus 50% for Second Chance. What it does not show is that Shield also can be used for multiple attacks versus one for Second Chance. That ups the 29.2% somewhat, especially in situations where the Wizard has multiple foes. Even if the first use of Shield does not stop an attack, later uses in the round might.
So, doesn't 2d and 2e seem overpowered when compared to 1 considering that Second Chance cannot stop multiple attacks and Shield can?
2a and 2b seem like the most balanced adjudications.
When one also considers how frequently people take Shield as their primary second level utility power, one should consider if 2a or 2b is the more reasonable adjudication. Personally, I think 2a is more balanced, but I can see why people would pick 2b.
2e is ridiculously powerful and useful compared to Second Chance.
"You managed to prove in your shield calculations that when shield is useful it's useful. What you didn't calculate was how likely it was to be useful. When you went from 2c to 2d you changed the last number from the chance that the number is within the range to the chance that the PCs correctly guess a number already in the range, which renders 2d and 2e totally invalid compared to the first three."
I'd go a little further. I don't think 2c is a valid calculation either. As you can see, in 2c,d,e the chance that Shield works is completely unrelated to its +4 bonus. 2c and d are hard to get a realistic calculation for, because it means specifying player knowledge of the monster's distribution of attack bonuses/what save they're attacking, versus the actual numbers. Scenario 2e is much easier.
Here's how to do a valid calculation for 2e. Let's take the simplest scenario possible: all attacks are identical in their chance to hit (hitting at least 25% of the time) and damage, with a set number
n of attacks per encounter (all this information is known to the player). We're only considering "attacks" something that targets Reflex and AC here, and expected damage prevented (==number of attacks blocked here) is the metric for effectiveness. When I run a comparison with Second Chance, assuming that all attacks are equal is definitely in Shield's favor.
Shield's chance to be effective per attack is
p= 0.2, since it's a +4 bonus.
Then the chance that Shield is effective is 1- (1-
p)^
n, and p=0.2, so this is 1-0.8^n. When you use Shield, you also get the benefit until the end of your next turn. So this means that Shield stops more than one expected attack if you’re targeted by additional attacks. Let’s assume conservatively that you’ll get targeted by 0.25 additional attacks while the Shield power lasts; at a 0.2 chance to block each, that means each time you use Shield you’re blocking 1+0.2*0.25= 1.05 attacks. This gives us an expected number of attacks blocked by Shield equal to 1.05*[1- (1-
p)^
n].
Plug in for a few different values of n:
n Number of attacks blocked
1 0.2100
2 0.3780
3 0.5124
4 0.6199
5 0.7059
6 0.7747
The more attacks there are, the better Shield gets, because the more likely it is to block an attack. Likewise, if Shield gave higher than a +4 bonus it would also be more valuable under this metric.
Second Chance: Optimal use of Second Chance, even assuming all attacks were identical, involves solving a dynamic programming problem to determine whether you should reserve it for a critical or should simply use it on any hit. Optimal strategy in this simple example would presumably be "if there will be
k or more attacks against me left after this attack, only use Second Chance against a critical hit; otherwise use it against any hit." From this you can determine the optimal
k relatively easily.
If you assume a crit is equivalent to 1.5 regular hits, and the chance to hit is 0.5 (0.45 of that non-crit, 0.05 of that a critical hit) you get optimal k=4. That is, if there will be 4 or more attacks against you after the current attack, only use Second Chance against a critical hit; if there will be fewer than 4 attacks against you after the current attack, use Second Chance against any hit (side note: assuming that the player knows exactly how many attacks there will be against him helps to use Second Chance more effectively, but as long as the player has a pretty good idea of the number of attacks he’ll face, this shouldn’t change much).
Doing some further calculations, the “regular hit equivalents” blocked by Second Chance here are a function of the number of attacks as follows:
# attacks Regular-hit equivalents blocked
1 0.2850
2 0.4275
3 0.4988
4 0.5344
5 0.5564
6 0.5934
7 0.6201
8 0.6415
9 0.6599
10 0.6765
11 0.6918
12 0.7061
Note that the number of “attacks” facing a character with Shield and Second Chance is going to be different because Shield only applies to AC/Ref, and because as a racial power, Second Chance can be used by non-wizard classes who will tend to be attacked more in combat. Let’s ignore the second factor and assume we’re only comparing them for the same wizard character. Then we can use the “75% of attacks target AC/Reflex” assumption, which means we should pick the number for Second Chance to be 4/3 of that for Shield.
For easy comparison, let’s call a number of attacks 3 (Shield) vs. 4 (Second Chance) the low case and 6 vs. 8 the high case.
Low attacks case:
Shield: 0.5124 blocked
Second Chance: 0.5344 blocked
High attacks case
Shield: 0.7747 blocked
Second Chance: 0.6415 blocked
In conclusion, from this simple calculation it should be clear that Shield and Second Chance are comparable unless these assumptions are dramatically different from actual practice. The attacks are identical assumption is the biggest simplification I’ve made, and that certainly helps Shield (though I was conservative on the number of additional attacks that Shield would apply to), but its lead in the “high case” is still under 20%.
