Comparative defence scores - explain the math to me.

[Prestidigitalis]

Two opponents each has 100 hp and do 20 damage with each attack, having a 50% chance to hit. This means they do an average of 10 damage to each other each round, scoring a mutual (average) kill on round 10.

One of these has the option of taking +2 AC or +2 to hit.

Assuming he takes +2 AC, the opponent's average damage per round goes from 10 to 8. After 10 rounds, he will have done 100 hp to his opponent, having suffered 80 hp himself.

Assuming he takes +2 to hit, his average damage goes from 10 to 12. After 9 rounds he has done 106 damage while suffering 90 damage himself. This is not quite as good a result as he got with the AC bonus, but the difference is very small.


Lets do this again, but starting with a 20% chance to hit. Each warrior will do 4 average damage and the fight will start at 25 rounds.

Assuming you take +2 to hit, average damage goes to 6. It now takes 17 rounds to do 102 damage, suffering 61 damage in return. While with +2 AC, he takes 2 damage from each blow, killing the enemy in 25 rounds after having taken 50 damage. Again, the defensive option is better, but slower.


What I did this for was to show something else, however; that the effect of the +2 bonus was significantly more important at 20% base chance to hit than it was at 50% base chance to hit. The amount of hit points the winner had at the end of the 50% chance fight was 20; the remainder in the 20% base chance fight was 50 - two and a half times as much! THIS is the most important result, and it why its called escalating returns - the same bonus is worth more as you get to more extreme values.

Okay. Two problems:

1. Math. 9 * 12 = 108, not 106
2. Presentation. Why on earth do you use 10 rounds for part of your +2 test and 9 rounds for the other???

I know it's wrong to quibble, but people do these "math" posts and half of them are far less helpful than they could be. Your point is actually dead on target, but clouded by the oddities of the presentation.

One final note: increasing the attack bonus does not increase the odds of a critical hit, so the net positive is slightly lower than it otherwise would be. The same is true for defenses. However, the effect on the calculation is pretty small.

 

[Nifft]

2. Presentation. Why on earth do you use 10 rounds for part of your +2 test and 9 rounds for the other???

First sentence: opponent has 100 hp. There's no point in continuing after the opponent is dead, and that's essential when considering damage-vs-defense trade-offs.

The Barbarian, for example, has low defenses. His opponents could kill him in less time than it would take them to kill a high-defense Paladin. The Barbarian compensates by killing his opponents faster than the Paladin would.

Cheers, -- N

 

[Prestidigitalis]

First sentence: opponent has 100 hp. There's no point in continuing after the opponent is dead, and that's essential when considering damage-vs-defense trade-offs.

Duh. I haz need learn reed gooder.

 

[Elric]

To answer my own question (slightly updated the phrasing)

Suppose you currently get hit 10% of the time by AC attacks and 80% of the time by FRW attacks.

FRW attacks and AC attacks are equally common, equally damaging/status effect inflicting, and tend to come in equally difficult encounters.

The equal FRW and AC targeting isn't because opponents are targeting you based on what your strengths and weaknesses are; it's essentially random. Likewise, you can't specifically target enemies that attack your FRWs instead of AC to kill first.

Opponents always hit you on a 19 vs AC (16 vs. FRW) and miss on a 1 without the natural 1s rule (for FRW an average enemy hits you on a 5+; this condition just ensures that your defense bonuses don't "go to waste").

Which do you prefer: +1 to AC or +4 to all FRWs?

You should take +4 to all FRWs. +1 AC negates half of the attacks that would otherwise hit you vs. AC, and +4 FRWs negates only a quarter of the attacks that would otherwise hit you vs FRWs. However, this doesn't mean that you should pick +1 AC. You need to go back to which blocks more absolute attacks (as I've assumed equal damage/status effects across AC vs. FRW attacks).

Since many more attacks would hit you on FRW than AC (equal number of attacks directed against each, but the FRW attacks are more likely to hit), the absolute fraction of attacks blocked is much higher, 0.1, for FRWs, than the fraction of attacks blocked for AC, 0.025.

As absolute attacks are what counts, you could get there directly by calculating 50% of attacks vs. FRW * 0.2 chance that each will miss b/c of +4 FRW= 0.1 fraction of attacks blocked. 50% of attacks vs. AC * 0.05 chance that each will miss b/c of +1 AC= 0.025 fraction of attacks blocked.

An analysis that gets you +1 AC as the answer here is flawed.

 

[renau1g]

I also find that the FRW attacks tend to have the icky status effects also, while the AC ones tend to be more straight damage.

 

[Starfox]

Since many more attacks would hit you on FRW than AC (equal number of attacks directed against each, but the FRW attacks are more likely to hit), the absolute fraction of attacks blocked is much higher, 0.1, for FRWs, than the fraction of attacks blocked for AC, 0.025.

While I know this was part of the issue as stated, I find its not true. About 60% of all attacks go towards AC, 20% Ref, and 10% each Wila and Fort in my experience. [Using the "out of thin air" method of statistical research]

 

[eamon]

As absolute attacks are what counts, you could get there directly by calculating 50% of attacks vs. FRW * 0.2 chance that each will miss b/c of +4 FRW= 0.1 fraction of attacks blocked. 50% of attacks vs. AC * 0.05 chance that each will miss b/c of +1 AC= 0.025 fraction of attacks blocked.

An analysis that gets you +1 AC as the answer here is flawed.

Though I don't disagree with the conclusion in your example scenario, I will nitpick: it's not the absolute number of hits prevented that count, but rather the overall number of hits before the defense boost relative to the overall number of hits after the defense boost.

So, it's entirely correct to conclude that for a person that has high defenses it's worth more to raise their defenses than for a person that has low defenses. However (as in your example) it's not correct to conclude that it's worth more to raise high defenses rather than low defenses - after all, both will impact harm prevented in the same way.

Having said that, this analysis assumes that attacks versus the various defenses are essentially random; they aren't. In fact, in any given encounter, particular attacks are likely to be dominant.

So, what happens if you're particularly poorly defended in a certain encounter and well defended in another? In practice, my experience is that particularly vulnerable characters simply fall back or fall down (and then generally don't die but rather are ignored.) That kind of thing may still matter with respect to the preferred strategy (of raising high vs. low defenses). For instance, you might image that it could be worthwhile to spread your strengths - if each party had at least 1 or 2 PC's that could "hold the tide" against any normal encounter, that might raise overall success - if only to give time to retreat.

Of course, in practice it's something of a moot point since all PC's are vulnerable on almost all NADs across all levels - so there's not likely to be an option to really raise NADs to a relevant degree.

 

[eamon]

I also find that the FRW attacks tend to have the icky status effects also, while the AC ones tend to be more straight damage.

That can definitely be true. Honestly though, this kind of high-level analysis isn't going to be that precise anyhow. I'm happy talking about "overall harm" whether that be by status effect or straight damage.

Some status effects are really nasty; those are memorable. Most are fairly harmless though. In any case, I don't think it's really critical to this issue - being the idea that raising high defenses matters more that raising low defenses.

Obviously, that concept needs to be seen in the light of not only the relative damage the hits can do, but also other harm (as you mention, status effects), likelihood, and the cost of raising the defense score, and tactical considerations (i.e. is it better to have a party with spread out defenses, or where various players have different strengths; how much does it matter if all PC's have the same weaknesses, etc.) Not to mention the reality that DM's of course tailor the story - and the combats - to PC's to varying degree.

 

[Elric]

Though I don't disagree with the conclusion in your example scenario, I will nitpick: it's not the absolute number of hits prevented that count, but rather the overall number of hits before the defense boost relative to the overall number of hits after the defense boost.

So, it's entirely correct to conclude that for a person that has high defenses it's worth more to raise their defenses than for a person that has low defenses. However (as in your example) it's not correct to conclude that it's worth more to raise high defenses rather than low defenses - after all, both will impact harm prevented in the same way.

It seems like you're saying that the amount that you want to raise defenses as opposed to some other use of the resources depends on how high your defenses are already. That's perfectly reasonable, but in the context of a particular character and which defenses to boost, it's not an issue.

The point of a thought experiment (or a model, in many cases) is to abstract away enough to clarify thought.

Having said that, this analysis assumes that attacks versus the various defenses are essentially random; they aren't. In fact, in any given encounter, particular attacks are likely to be dominant...

There's plenty of complications on this side of things, as we've discussed before.