So, I did some messing around in Excel with the various types of armor and have come up with an argument for studded leather being the uber armor for all players. I'm printing the text of my blog post about it here, but to get a look at the tables and the the graph, you'll need to visit the site directly.
Now, begin post transcription...
Today, I decided to do a little bit of musing over how armor choices affect a character in a d20 system, specifically Pathfinder. Before we get to heavy into the numbers, I want to make a few comments about the statistics that underlie any d20 game. Most obviously, you are rolling 20-sided dice. With 20 possible options, you have a 5% chance of any particular one showing up. Since PCs all start with armor class 10, they have a 50% chance of being hit by an attack before any bonuses they or their assailant may have are factored in.
A CR 1 creature should be doing 6 points of damage a round (5 is low attack damage and 7 is high attack damage). It should also be adding +1 to its hits per round, but I'm going to ignore that to make the math easier. 6 points of damage per round with a 50% hit chance (55% with the bonus) works out to 3 points of damage per round. Before CON bonuses are added in, that means a level one PC with d4 or d6 hit points can survive one hit, a creature with d8 hit points can survive two, and a creature with d10 or d12 can survive three. Assume at least a CON bonus of +1 and the d6 and d12 creatures move up to two hits and four hits, respectively.
But, increased armor significantly changes this equation. By taking a high DEX score or picking up armor and shields, characters can significantly increase their chances to avoid damage. But, both DEX and armor have tradeoffs. In the case of DEX, you are placing higher scores (for die-rolling character creation) or more points (for point-buy systems) that requires you to forego scores in other abilities. With armor, characters are limiting their maximum dexterity bonus and taking penalties on several skills checks.
Under a rational system, PCs with lower hit dice for their class would be the ones investing most heavily in armor. The characters that are the easiest to kill have the most rationale to make hitting them harder. Of course, a rational system wouldn't care much about game balance and we have a system where the classes with lower hit points per level are actually forbidden from wearing armor.
I've done a very (emphasis on very) basic analysis of the armors listed in the Core Rulebook. Following the order in which they are listed, I plugged the armor bonus, maximum Dexterity bonus, and armor check penalty for each CRB armor into an Excel spreadsheet. I then added trendlines and R-squared values for each attribute:
So, what do we notice? First of all, the R-squared values are high. That means that the trendline drawn for the series of points does a good job of predicting what the actual value of any given item will be. (Note: This analysis basically assumes we are using padded armor as '1' leather as '2,' studded leather as '3,' and so on. That basically tracks increased armor bonuses, but you could reorder the armors with the same bonuses and get slightly different R-squared values). The R-square values range from a low of 87% for maximum dexterity bonus to a high of 98% for armor bonus. Basically, this is a good way for us to see that our designers have done a good job balancing armor's effectiveness for avoiding attacks with providing limitations to dexterity and to skills checks. That's good game design!
But, they haven't completely removed any differences, otherwise the lines would be perfectly smooth and the R-squared values would always equal 100%. The other important thing that we notice is that although the R-squared for the armor bonus is almost completely explained by its level, we have a lot more variation on maximum Dexterity and armor check penalties. That differentiation means that there's room for optimization.
To figure out the optimal values, I did a very simple analysis. If the armor check or maximum Dexterity bonus for the specific type of armor was above the trendline, it got one point. I'd redone the armor check penalties as absolute values (so an armor check of '1' on the chart means a -1 penalty), so those armors received a point if they were below the trendline. If the two were evenly matched, the armor received no points. If the trendline was above the armor bonus or the maximum dexterity bonus, the armor received -1 point. If the trendline was below the armor check penalty, it received a -1. Here's what I came up with:
Note: I went by exact values, so even the tiniest difference between the trendline and the actual value resulted in a '1' or a '-1.' That said, some of them very actually really close to each other. Another analysis that discounts these minute differences may find that some armors aren't as far from optimal as discussed here.
Based on this analysis, studded leather is the optimal light armor. Chainmail or a breastplate is the optimal medium armor and banded mail or full plate is the optimal heavy armor. We also see that all light armor gives more benefits than it takes in penalties but that medium and heavy armors are mixed bags. I found this particularly interesting, because these are NOT my usual armor choices. In fact, I almost always purchase leather for characters that need high Dexterity and scale mail for characters that will wear medium armor. I generally don't run PCs that wear heavy armor, but I'd suspect that I'd lean more towards splint mail, a less than optimal choice, if I did.
That's because in my mind, the penalties to Dexterity and to armor checks far outweigh the benefits of better armor. Once you control for the differences in these things, studded leather becomes as useful as full plate. Of course, this method only considers whether or not something is different from the trendline. If I considred how far things moved from the trendline, I might end up with different results (but not very different. The high R-squared values mean that few actual numbers differ much from the trendlines anyway).
Of course, people like to play characters that can have high dexterity scores for ranged attacks and for general roguishness. I then ran the checks again, but this time, I doubled and tripled the weight for the Dexterity score trendline. Zeros stayed the same, but positives and negatives counted twice as much:
Now there's a link to a fabulous table!
Yet again, studded leather is the clear winner. Now, I'm not an optimizer in character creation. But, no one wants to pick up gear that's intentionally bad. This might explain my reticence about using medium armor. The juice doesn't isn't worth the squeeze. However, this does explain why the power gamers I ended up playing with on Friday night at GenCon were so confused that my fighter wasn't wearing heavy armor. Now that I've seen how the numbers stack up, he just might make the jump. (Actually, if he puts on heavy armor, he likely won't be making any jumps, but that's another matter!)
Now that we know that studded leather is the optimal armor choice, let's take a look at what it does for armor choices across various class hit die groups. Studded leather adds 3 to a character's armor bonus, making the default 13 before other bonuses and penalties are added in. Against the average CR 1 creature, you've moved from a 55% chance of being hit to a 40% chance of being hit. Of course, heavy armor makes more sense for warriors, so we can figure an at least +2 dexterity bonus for our level one character, reducing her chance to be hit down to 30%. Using the average damage of 6 and a 30% chance to hit, we get 1.8 damage per round. That's two hits and still alive for the d4 group, 3 hits and still alive for the d6 group, 4 for the d8 group and the d10 group and 5 for the d12 group. Constitution bonuses will only increase these numbers.
Anyway, hope that you liked it. Let me know your thoughts. Also, if anyone wants to run other numbers, I'm happy to post those or links to them. It would be interesting to look at how masterwork armor changes the equation on things as well as how adding enchantment moves the relative values between the three variables.
Honestly, I think the only character type I would ever put in heavy armour would be a Fighter, particularly of the archetype that specializes in armour. Since armour training would reduce the penalties, and you could get Adamantine Full Plate, that would stack with the DR they already get, it might seem worth it.
Otherwise, I'm with you. I've always favoured studded leather.
I am not sure I understand your logic. If the priority is AC then the best is Heavy Armor. The way you have it set up your chart it seems like you are placing the same value on ACP that you place on AC.
Light Armor maxes out at +8 (AC+Max Dex)
Medium Armor maxes out at +9 (AC+Max Dex)
Heavy Armor maxes out at +10 (AC+Max Dex)
Now, that doesn't take into account the other effects such as weight, speed reduction, ACP, etc. but if you want the best AC you can get then you clearly want Heavy Armor.
Now, where things really become tricky is when you have to buy the armor proficiency. If you have to spend a feat to get Heavy Armor AND you can max out the Dexterity of medium armor then it is better to spend the feat on "Dodge". The same holds true for spending a feat to get Medium Armor proficiency.
Light Armor +Dodge maxes out at +9 (AC+Max Dex) which is equivalent to Medium Armor.
Medium Armor + Dodge maxes out at +10 (AC+Max Dex) which is equivalent to Heavy Armor.
Of course, none of this takes into account special abilities, magic items (such as Celestial Armor), and special materials but special materials like Mithral increases the Max Dex universally so really doesn't change which armor provides the best protection if you are willing to pay the cost and have the Dex to max it out.
Summary: If your priority is Armor Class then you should go for Full Plate as it has the highest combination of AC and max dex. If your priority is something else (such as speed or low ACP) then you are sacrificing AC for that something else.
Quick side note, with an AC of 10, there is a flat 55% chance to hit with no other bonuses. Not 50%. (#10 is 5%, #11-20 is 50%)
Right? Is my math wrong there?
@Gauss, you're right--the full analysis says that if you don't care about dex bonuses or armor check penalties that full plate and banded mail are the best choices.
@Zedth - Yes-thank you. Always forget that 11-20 is 50% not 10-20. Oh well, I'm only off by 5%!
I dont know why you say Banded Mail is a good choice. It is +7 AC with a Max Dex of +1 for a total of +8.
Full Plate is clearly a better choice with a +9 AC bonus and a Max Dex of +1 for a total of +10.
As for Dex Bonuses, that is where I do not understand your position. Even if you have a +5 Dex bonus the armor that provides the best AC for you is still Full Plate.
Lets go with your Studded Leather example.
Studded Leather has +3 AC and +5 max dex for a total of +8.
Full Plate has +9 AC and +1 max dex for a total of +10.
It doesn't matter if your Dexterity is +5, the Full Plate is still a better choice for AC.
Now, if you are willing to take the -2 AC loss (in favor of speed, weight, and ACP) then go for the Studded Leather.
Now you do know that the dex bonus is only for AC right? If I had a Dex mod 5 in fullplate and using a bow I still have a +5 from my dex for the to-hit. The dex bonuses only are effecting your touch AC instead of Flat-footed. So I think the dex bonus to AC really effect anything and shouldn't really matter.
My experience is that Medium armor is essentially never worthwhile unless you're a dwarf that doesn't have Heavy armor proficiency. A character that wants a good AC due to armor should almost always either take the feat for Heavy proficiency, or dip a martial class that gets it. Wearing Medium armor sucks since it reduces your speed and generally isn't much better than the better Light armors, like studded leather and a chain shirt.
As you seem to be ignoring cost in your equations, how do you figure that a mithral chain shirt isn't strictly better than studded leather? Or (as seems to be the case) did you just not include the special material armors in the CRB in your calculations? How do things change if you throw mithral armor into the mix?
full plate and banded mail are the best choices
I dont know why you say Banded Mail is a good choice. It is +7 AC with a Max Dex of +1 for a total of +8.
Full Plate is clearly a better choice with a +9 AC bonus and a Max Dex of +1 for a total of +10.
Banded mail does not have to be better than full plate. However, of the options given, it or full plate must be the best and either equal or the other must be the second best. That is what "are the best choices" means.
As for Dex Bonuses, that is where I do not understand your position. Even if you have a +5 Dex bonus the armor that provides the best AC for you is still Full Plate.
Lets go with your Studded Leather example.
Studded Leather has +3 AC and +5 max dex for a total of +8.
Full Plate has +9 AC and +1 max dex for a total of +10.It doesn't matter if your Dexterity is +5, the Full Plate is still a better choice for AC.
Now, if you are willing to take the -2 AC loss (in favor of speed, weight, and ACP) then go for the Studded Leather.
Yes, this makes sense if you do not care about ACP. If I were only going to be concerned with Armor Class Bonus and Dex bonus, I'd run another regression looking only at those variables (and would love to see what happens if you do if you want to do so and then post results).
I'm not seeing why you value the Dex bonus at all? Why are you giving it it's own separate value in this? I just see total AC possible and ACP as what you should be working with.
Banded Mail is not even better than Breastplate and thus it is not a good choice. Banded Mail has a total of +8 (AC+max dex) while Breastplate is a total of +9 (AC +max dex).
You cannot talk about optimizing armor without discussing what quality you are optimizing for. Nowhere in your post or in your webpage did I see that you were optimizing for ACP rather than AC although it is possible I missed it. Now that is clear.
Frankly, people see optimizing armor and most of them will try to optimize the protection without sacrificing whatever they feel they need. For some that will be ACP and Speed.
In any case no regression is required for Armor Class Bonus and Dex bonus. It doesn't matter what your Dex bonus is since at any Dex bonus below +9 the best armor (for AC purposes) is going to be Full Plate.
You are really over-complicating what is a basic concept.
And there are also outliers like Celestial armor with its +8 dex penalty in chain mail.
It's always given me heartburn that such random numbers are given to armor in regards to DEX bonuses. I've beaten people in normal clothes while wearing body armor in speed and agility (of course, I also went to state in track in high school, so maybe that has something to do with it, but then, my DEX bonus shouldn't do anything for me in heavy armor, and less in medium armor, except make me slower then they are with no armor on).
One of the houserules we play with is to do away with the silly dex limitations with armor.
Interestingly enough, if you use the beginner box rules, I think you don't have to use those silly DEX limitation rules anyways.
Does it give warriors a boost...sure...but then...I don't think many are arguing that they should be nerfed compared to the casters anyways.
Interesting thoughts and charting... but I don't think your method yielded useful conclusions.
You're considering Banded Mail more optimal than Breastplate and equally optimal to Fullplate. Its stats are identical to fullplate, except fullplate get's an additional +2 to AC. An analytic method that counts that as equal is flawed. For a single point of AC advantage over breast plate, banded mail gives up 2 points of max dex and has a ACP that's 2 points more. Yet you're willing to elevate studded leather over chain shirt for giving up 1 AC and only getting 1 more max dex and 1 less ACP. I don't understand what economy your analysis is revealing to us.
Of course this is ignoring the fact that every character has a Dex score and a certain level of armor proficiency, and that plays a big part in what armor is optimal for them. Any max dex bonus beyond your dex isn't helping... and if you want to consider max-dex beyond your current for the purposes of being prepared later in your build... later in your build you should be looking at Mithral options.
I agree completely--from the beginning I said that it was very basic analysis. One of the reasons I wanted to put it up here was so that I could get an idea of what exactly the next step should be. And, this thread has given me several ideas of what are good things to consider for next steps. I doubt that I'd ever be able to get everything in there (would be very time consuming to work on multivariate regressions and I'd rather just actually play the game), so things like special materials are out. But some considerations on how the stats change when you work in considerations of magic and considerations for loss of movement squares is something interesting and worth looking into. Thanks for the feedback.
And, for the record for everyone, I agree that banded mail isn't as good as full plate and that there's little to recommend it, but it did come out as a maximal armor choice in the matrix results (demonstrating that the matrix itself has room for improvement).