Classic D&D is fast

It warms the cockles of my heart when my son says "Can we do a game with your figures?", which he did this this morning. I grabbed my current house rules and wrote up two sides.

The forces of Chaos and Darkness had discovered that the King was visiting border castles with only a light guard to defend him, and they ambushed him. The forces of Law retreated to one of the castles and prepared for a siege.

Forces of Chaos and Darkness

• Cobra commander, snake-man fighter 5
• Troll
• Dragon
• 4 rat-people
• Master archer, raven-folk fighter 4
• 4 goblin archers

Forces of Law

• Wizard-King Wilhelm
• Barnard, human fighter 7
• 6 elven archers

It took me maybe 30 minutes to write up the two forces and some rules for sieges.  My one concession to this being a war-game was that I converted hit points to hits -- every five hp was one hit, and every five damage done caused one hit -- this was largely to avoid having to track individual rat-people, goblins, and elves. Initiative was by side. The loser of the initiative roll had to move and attack first, but the winner could interrupt at any time to move and attack with a unit.

My son was in charge of the monsters, while I championed Law. This was appropriate since children are naturally Chaotic. It took my son maybe as long to triumph as it took me to write up the scenario. I was surprised by how quickly the game went. Classic D&D goes really fast, even with miniatures.

The Battle

Setup: the Forces of Chaos and Darkness set up around a ruined tower (not pictured). The elven archers arrayed themselves in the two front towers of the castle, while the king and Barnard guarded the castle gate from the walls above it.

Turn 1: The elves and goblins exchanged arrows, taking out two goblins. The raven-folk master archer winged the king. The troll, snake-man general, and rat-people headed toward the castle gate. The wizard shot magic missiles at the troll and the goblins, taking out another goblin and wounding the troll. The dragon flew behind the right wall of the castle.

Turn 2: The troll threw a rock at the gate, but it bounced off. Not wanting the troll to reach the gate, the elves concentrated their fire on the troll and reduced it down to one hit. It then was killed by a magic missile from the king. The dragon flew up to and perched on one of the towers above the gate.

Turn 3: The dragon breathed flames on the king and Barnard. Barnard made his save, but the king didn't and was burned to a crisp. In a rage, Bernard charged the dragon, and began to climb to the top of the tower where the dragon was. More effective was the volley of arrows that the elves sent at the wyrm. A couple of the arrows found weak spots in the dragon's armor.

The cobra commander discovered that he possessed a scroll of teleportation and used it to teleport next to Barnard, but failed to wound him. The goblins and raven-folk archers missed (I think they tried to shoot Barnard before the cobra commander teleported).

Turn 4: The elves shot into the melee between Barnard and the cobra commander. Getting extremely lucky, four of their arrows hit the commander and none hit Barnard. Barnard then whacked the cobra commander, bringing him down to 1 hit. The cowardly snake discovered a second scroll of teleportation (!) and teleported to safety.

However, this left the dragon not-too-injured and with an action remaining. It tried to pick up Barnard to fly off and drop him, but missed.

Turn 5: Again, the dragon tried to grab Barnard. It succeeded with both talons, and Barnard rolled two saves to grab on and avoid being carried away. The first save came up 17, so he grabbed a crenellation. The second save came up 1. He had let his guard down as the dragon circled and grabbed from behind.

We ruled Barnard was killed automatically by the fall and that this broke the elves' morale.

We both had fun playing. My son especially enjoyed incinerating my king and dropping my champion from a great height. He was not upset by the loss of his troll. When I was his age, according to my dad, I would cry any time it seemed like my dad would win. I had fun kicking the tires on my house rules and zapping some goblins.

Some Lessons Learned

(1) D&D combat can be super-fast if you are willing to hand-wave a little. (2) The initiative system we were using contributed to that speediness. Basically each character's or group's movement and actions got resolved at once, rather than having a traditional missile phase or melee phase. This system led to some strangeness on Turn 4. With a more standard initiative system Turn 4, the dragon would have joined the melee combat between Barnard and the cobra commander before the elves shot, so that the elves probably would have been unable to pull off their lucky volley against the commander, given that the dragon was a much larger target. (3) The siege environment meant that the rat-people and cobra commander did not have much to contribute to the battle. We ended up giving the commander a teleportation ability to fix this. In the future I should try to anticipate this. (4) Wizards are squishy and dragons should be taken out as quickly as possible. (5) Never mind claw attacks -- dragons should grapple!

Attrition in Blow-by-Blow Combat, Part II

This post continues my analysis of blow-versus-blow combat, begun in a previous post. The question at hand is how do damage bonuses and to-hit bonuses affect how characters fight many small monsters.

To be honest, the topic is pretty dry and the math gets more tedious, so I am going to try to wrap the topic up quickly.

This post: the art of fighting gnome hordes!

Hordes of Little Monsters

For smaller monsters kills per round (KPR) reflects combat effectiveness more accurately than damage per round. 

Kills per round is the product of kills per hit (KPH) and hits per round, which is just the probability of hitting (HITP) for characters with one attack.

Let's define smaller monsters as monsters who are with very high probability reduced to 0 HP by 1 or 2 hits. KPH depends on the probability that the first hit kills the monster outright (KILLP). 

Hits per kill (HPK) is

HPK = 1 * KILLP + 2 * (1 - KILLP) = 2 - KILLP

It ranges from 2 if KILLP=0 to 1 if KILLP=1. Inverting it gives KPH,

KPH = 1 / (2 - KILLP).

Putting these together,

KPR = KPH * HITP  = HITP / (2 - KILLP).

One can derive algebraic expressions for these in terms of more the fundamental values of attack bonus, AC, monsters' hit dice, damage dice, and damage modifier, but it is probably easier to get some insight into how an increase in attack bonus (AB) versus damage bonus (DB) affect KPH with a couple of examples.

First, unlike with big monsters where the effect of DB was linear -- each increase in DB increased damage per round by the same amount -- with small monster groups DB initially is somewhat important but after enough increases a further +1 DB does not affect KPR. For instance, consider a warrior with base d8 damage fighting orcs with d8 HP. Once her damage is d8+7, a further increase in DB does not help her KPR.

How important is DB initially? Pushing the example further, if orcs are AC 6 and the warrior has AB +3, then her HITP=9/20. Increasing AB by one brings her HITP to 10/20, or an 11% increase in HITP. Because HITP is one of the terms in KPR, this also increases KPR by 11%.

If the warrior has d8 damage which would increase by d8+1, KILLP goes from 36/64 to 43/64, so KPH goes from 64/92 to 64/85, or an 8% increase in KPH. This is also an 8% increase in KPR.

Going from d8 to d8+1 damage affects KILLP more than the move from d8+1 to d8+2. The figure below depicts this graphically. On the x-axis is every possible damage roll, and on the y-axis every possible roll for orc hp. There are 64 possible combinations, 36 of which (the green) defeat the orc if there is no damage bonus, and 43 of which (the green and the red) defeat the orc if there is a +1 damage bonus. Moving to d8+2 increases KILLP to 49/64, and so on.

Hence at least in this example, to-hit bonuses appear to be superior to damage bonuses. For most typical combinations of attack bonus and damage bonus, it is likely to be the same.

Danger, a Further Complication

A final consideration is that low AC monsters tend to be more powerful and are more likely to hit and do higher damage. With low monster AC, to-hit bonuses become relatively more important. Thus, maybe the conventional wisdom was correct all along that to-hit bonuses are superior to damage bonuses in most situations.

Summary: to-hit bonuses usually increase combat effectiveness more than damage bonuses. The one exception is large monsters with high AC (i.e., easy to hit monsters).

Dark Fantasy Basic Player's Guide (Review)

Dark Fantasy Basic Player's Guide
By Eric Diaz with cover art and design by Rick Troula
Chaos Factory Books, 2017

$4.99 (PDF) on DTRPG

Dark Fantasy Basic came out about a year ago. It's a 46-page OSR rules system written by Eric Diaz of Methods & Madness. Methods & Madness is a great blog with regular posts full of insight about OSR rule ideas (as well as nice settings material), and Dark Fantasy Basic's $4.99 (PDF) price is quite reasonable, so I decided to write up a review. I haven't play tested the rules -- these are just my impressions from a couple of read throughs.

Dark Fantasy Basic is an excellent system but that excellence is somewhat subtle. Superficially it looks like a heavily house-ruled mix of B/X D&D and modern-era D&D in a rules-light package. What distinguishes it from a typical house-rules mess is that it carefully balances the various rules and uses skills and feats in a quite ingenious way. Both skills and feats act much differently than in 3rd edition or 5th edition.

There are five basic classes: the classic four of fighter, wizard, cleric, and thief, plus the Hopeless, a generic class that starts with no attributes higher that 12, but otherwise can grab skills and feats from any class. Skills and feats are the only things that differentiate the five classes in the game -- everyone gets the same HP and can use any equipment -- so the Hopeless is a clever way to incorporate skill-based characters, like in Call of Cthulhu or Traveller, into the class-based D&D system. It's a human-only system by default, but race-as-class or race and class would easy to tack on.

Characters have one primary skill, two secondary skills, and three tertiary skills, which are chosen at character creation. They grant a level-based bonus to related actions, so that for example at level 5, the character has +5 to use a primary skill, +3 to use a secondary skills, and +2 to use a tertiary skill.

Players choose their character's skills at creation and to some extent are restricted by the character's class. Fighters must take Combat as their primary skill and Athletics as a secondary skills, but is otherwise free to take any skills. Magic-users must take Spellcasting as their primary skill, Lore as a secondary skill, and Combat as a tertiary skill. These requirements elegantly maintain the standard class archetypes: Combat is similar to base attack bonus, so a Fighter will always be much better at hitting things than a Magic-user.

Nonetheless, there is a lot of freedom in the system to make distinct, flavorful characters of the same class. A Fighter who takes Spellcasting as a secondary skill could be a good battle-mage, while a Fighter who takes Thievery as a secondary skill could be a good rogue. This way of doing skills is great!

Feats are also turned to new purposes somewhat. Each class gets a starting feat and a list of feats to choose from on leveling up. These completely replace any class-based abilities. For instance, the Thief can take the Reflex feat which gives advantage in all intelligence and dexterity checks but does not otherwise have saving-throw superiority built in.

This use of feats again allows a lot of customization, but I'm not sure it is as balanced. The Fighter can take an Extra Attack feat (once only thankfully) at any level by default, while the Magic-user has to use a feat to change Spell-casting from int-based to cha-based, replicating the sorcerer.

More novel is that spells are now feats. Clerics and magic-users each have a list of 20 spells, and start  with one. Spells themselves do not have levels, but the caster decides what level to attempt to cast the spell at, with higher level spells requiring a more difficult Spellcasting check but also being more powerful. For instance, Magic Missile generates 1 missile if cast as level 1, 3 missiles if cast at level 3, etc.

Therefore Magic-Users and Clerics will be using most of their feats to accumulate spells. Other classes can use feats for spells too, but without Spellcasting skill are going to find it difficult to use the spells.

Again, this is an inventive use for feats.

Those are the two largest system changes but the author manages to fit in a bunch of other more minor rules changes and additions as well. For example, in addition to the standard XP sources, a character surviving Death's Door (getting to 0 hit points) or "other grave dangers" adds 10% to current XP. Another nice XP touch is that when a character dies, the GM distributes up to 50% of that character's XP to people connected to the death, such as another character that the dead character lay down his or her life for.

Finally, it would be remiss not to compliment the art work in the game. It is all public domain art, mostly sourced from older books, but the selections are really well done. A lot of care and attention went into the art and layout, and it shows. The rules are clearly written for the most part, with the exception of how many feats a character has being buried in the section on Leveling Up.

Despite the minimal page length and stream-lined skill system, I wouldn't say that Dark Fantasy Basic is a simple game. My metric for simple is whether I can easily play the game with my kids with me walking them through all the mechanics, what to roll, and so on (both B/X and simpler d20 variants pass this test). The trouble is that feats are player-facing and are fundamentally kind of complicated, both if terms of making feat choices and in many feats introducing special rules that you need to keep track of. If I were going to run it for my kids, I think I would have a default feat progression for each class. You could design a bunch of subclasses and offer them to players, I guess. Come to think of that, it might be fun to design a bunch: treat Dark Fantasy Basic as a cool class-construction toolkit to play around with in addition to being a complete game in itself.

Attrition in Blow-by-Blow Combat, Part I

Famously, D&D models attacking by a two step process in which an attacker rolls to hit and then if the attacker hits, the attacker rolls for how much damage the defender takes. I used to think bonuses to hit were superior to bonuses to damage, which I think is folk-wisdom, maybe. If you don't hit all the damage bonuses that your double weapon specialization and 18/00 strength give you don't help you at all.

It turned out after writing down the math that this wisdom is incorrect or at least needs considerable caveating. Beyond providing advice to munchkin fighters, the math is also useful for thinking about how to use damage bonuses and hit bonuses to differentiate different weapons, barbarian versus monk mayhem, and so on.

Attrition

So first, what is happening in basic, boring D&D combat? The characters and monsters are exchanging blows and attempting to get the other side to 0 hit points and at the same time minimize their own hit point loss. Of course, it is a tenet of the OSR that good players will look for ways to circumvent this process, such as through ambushes, flaming oil, or sleep spells, but let's ignore that for now. The standard D&D combat process is both sides simultaneously exchanging blows and wearing down hit points. 

Attrition of monsters comes in two forms. Larger monsters, who are going to last several rounds, wear down at a rate approximately equal to characters' average damage per round. In other words, in a fight between a warrior who averages 5 damage per round and a giant with 50 hit points, the warrior on average disables the giant in 10 rounds (this calculation turns out to only be approximately correct, but for monsters with large numbers of hit points it is accurate enough; see "Are All Hit Dice Equal?" for why it is only an approximation). Disabling the giant as quickly as possible is good because it reduces the number of opportunities the giant has to squash the warrior.

Big, Tough Monster

For smaller monsters such as orcs, average damage per round is not as useful a statistic because many orcs will be cut down in one blow. It doesn't matter if the warrior does 10 damage per round on average if she only gets 1 attack on 5 hit point orcs. Usually however orcs show up in war bands that require lots of orc-killing. (Gimli: "Let the Orcs come as thick as summer-moths round a candle!"). Hence average kills per round is a better measure of combat effectiveness with smaller monsters.

Big, Tough Monsters

Average damage per round ("DPR") is the product of the probability of hitting ("HITP") and the average damage done per hit ("ADAM"). With ascending attack bonus ("AB") and descending armor class, as in Target 20, the probability of hitting is

Giants also count

HITP = (AB + AC) / 20,

I.e., someone with AB +3 attacking AC 0 has to roll an 18-20, which happens 3 times out of 20. (I am ignoring number of attacks, which just multiply DPR by some constant if they all have the same hit probability).

Damage depends on the damage die (DD) and the modifier to the damage roll (DM). It is

ADAM = (1/2 + DD/2 + DM).

Putting these together, average damage per round is

DPR = ADAM * HITP = (1/2 + DD/2 + DM) * (AB + AC) / 20.

A little algebra finds that when the attack bonus increases by 1, the increase in DPR is

ΔDPR / ΔAB = ADAM / 20 =  (1/2 + DD/2 + DM) / 20,

and when the damage modifier increases by 1, the increase in DPR is

ΔDPR / ΔDM = (AB + AC) / 20.

Hence an increase in the to hit bonus improves DPR more than an increase in the damage bonus if

ADAM / 20 > (AB + AC) / 20.

The 20's cancel, so the hit bonus increase is more beneficial if ADAM > AB + AC, and the damage modifier increase is more beneficial if ADAM < AB + AC. To be pretentious, let's say that in the first case the character is "accuracy-constrained" and in the second case the character is "damage-constrained".

Which case is relevant depends on the particulars of the character, which in turn depends on the exact form of D&D being played. On the one hand, in OD&D, neither AB nor DM increase very quickly, but my sense having never played is that characters tend to receive more bonuses to hit than bonuses to damage, and so are likely to be damage-constrained. A level 3 fighter (+2 AB) with a magic sword (+1 AB, d6) has 3.5 average damage and +3 AB, so is damage-constrained for AC's above 0.

In AD&D, on the other hand, at least for low-level fighters there are more ways to increase average damage than than average accuracy, so they would tend to be accuracy-constrained. A first level fighter with weapon specialization (+1 AB, +2 DM), 18/60 strength (+2 AB, +3 DM), and using a halberd (d10 damage) causes 10.5 damage per hit on average but only has a +4 AB. This fighter is accuracy-constrained fighting opponents with AC 6 or less. It's really noticeable fighting low-AC monsters, and that experience might be why I had thought to hit bonuses were more important.

BX probably hits a better balance, since many sources give the same bonuses to hit and to damage. A fighter with 15 strength (+1 AB, +1 DM) with a +1 sword (+1 AB, +1 DM, d8 variable weapon damage) is accuracy-constrained fighting opponents with AC 4 or less.

Now, beyond the modifiers, the character's level and class, and opponents' AC, matter for how the character is constrained. Fighters are going to have higher AB's than non-fighters as they advance in level; they're also probably going to have average damage grow at a higher rate than non-fighters, but probably not by the +1 AB every level that many systems use. Therefore fighters are going to be more often damage-constrained than other classes. Low-level characters start with some base weapon damage, so ADAM is 3.5 or 4.5, but with +0 or +1 AB, so without any modifiers tend to be more accuracy-constrained than higher-level characters. Finally, facing low-AC monsters, all characters are more likely to be accuracy-constrained.

Does which constraint limits a character's damage-dealing potential matter? Possibly not, but I tend to think it does. First, when it's way out of balance, like the AD&D fighter above, it is frustrating and may encourage min-maxing behavior. Second, it's a way to add flavor to classes and weapons. A duelist or monk should probably not be as accuracy-constrained  as a barbarian. Giving a crossbow a bonus to hit is a simple way to give its armor penetration powers, and giving daggers and knives a penalty to hit may better than reducing damage.

But all of this analysis is about DPR attrition against tough opponents, and smaller opponents complicate matters, which I will cover in Part II.

Dice step mechanics

So many cool ideas...

A dice step mechanic is one in which modifiers increase or decrease the size of the die rather add a fixed value (here's a somewhat incomplete discussion). Instead of rolling d6+1, you would roll d8, etc. The mechanic hasn't been used as much in RPG's as the fixed value modifiers has, although Earthdawn used it throughout and it shows up in DCC as well (there are useful discussions of the Earthdawn system in a RPG Stack Exchange thread and its links, but be aware that the Earthdawn system also uses exploding dice). Perhaps one reason it hasn't been used much is that to use it correctly, you ideally need a bunch of weird dice sizes like d14 and d16 which aren't as widely available as other RPG dice.

If you have these weird dice or are willing to simulate them by rolling a d20 and rerolling invalid results, then dice steps and fixed modifiers behave quite similarly. The average result on d6+1 is a 4.5, as is the average result on a d8, so fixed modifiers can be replicated by dice steps which increase or decrease the number of sides by two.

Dice steps produce a wider range of results though: neither 1 nor 8 are possible with d6+1, and so they introduce more swinginess (technically, higher variance) to the results. They are also problematic for systems where the highest and lowest roll mean something special, such as a critical success or failure. An increase in the die step reduces the probability of the critical result, which is probably undesirable.

It's purple too!

A dice step system also anchors the lowest possible at 1, but the highest outcome is exceptionally dependent on the modifier. Since the highest outcome possible moves around so much depending on the modifier, it is more difficult to implement a reasonable success / failure check using die steps because the probability of rolling more than x changes dramatically. I came up with a skill system that does OK, and Earthdawn finessed this problem through exploding dice, but it make dice steps more complicated to use for binary outcomes like success / failure.

Where dice steps tend to perform better and are easier to use in system design are in rolls for the magnitude of an effect, such as damage rolls or how long a spell lasts. Whether they offer an advantage over a fixed modifier depends on how they are used, but I think they have the potential to work better in some situations.

For example, consider a str 18 fighter (+3 modifier to damage) with a dagger (d4). With a fixed modifier the fighter is guaranteed to do 4 damage, even on a roll of 1. With a dice-step based damage roll, the fighter's player rolls d10, which might better encompass simultaneously the weakness of the weapon and the brutal strength of the warrior.

Even better, give daggers 2d6 keep lowest roll damage. With a fixed modifier, the 2d6 keep lowest choice, aimed at modeling the weakness of the weapon, reduces average damage a little bit. Average damage for str 18 fighter about 5.7, and about half of rolls result in damage 7 or higher. With a dice-step system, the fighter's player would roll 2d12 keep lowest, leading to average damage 4.5, half of damage rolls 4 or less, but still the very rare prospect of 12 damage. To me, the dice-step damage distribution feels nicer. (It also works with two-handed weapons -- a two-handed sword might be 2d8 keep higher -- reliable and better for high-strength characters).

All of the above considerations are math or system considerations, but there is also the tactile / weird aspect of dice steps. For me, rolling strange dice is fun, but I imagine it might irritate other people. I also suspect hunting around for different dice might slow the game down a little, although this problem could be mitigated by having modifiers that don't change too often and that cause dice to stay in the d6 to d12 range for most rolls.

(This many years old blog post captures some of the charm of Earthdawn).

Advantage (Math)

The publication of 5th edition D&D introduced several new ideas to the RPG community, maybe the most popular of which is advantage (and disadvantage). Advantage was an improvement over 3rd edition in a myriad number of ways: it cut down on fiddly modifiers, it encouraged DM rulings over a slavish attention to written rules, and it kept results bounded between 1 and 20 while still rewarding beneficial situations in-game. Also, it's more fun to roll more dice.

More generally, we can think of advantage-style mechanics as roll x y-sided dice, keep the highest (or lowest) n dice. Notationally, I'll write Hn:xdy, so 5th edition advantage is H1:2d20. What do advantage-style mechanisms offer for rules tinkerers? There are many, many explanations of the math behind advantage, often by more mathematically talented and clear writers than myself. I'll write down some of my thoughts / math-scribbles / insights into this mechanism, but I expect that there are other posts and forum-threads out there that you should read as well. That's what Google is for.

Success or Failure with Advantage-Style Mechanics

Say you need to roll more than z to succeed in doing something in-game. A single roll of a y-sided die is essentially a uniform random variable with a minimum of 1 and a maximum of y. The probability of getting z or less is z/y, and conversely, the probability of getting more than z is (y-z)/z.

Success on the highest of n dice is equivalent to not failing on every single die roll. The probability of failing two dice rolls is the probability that you fail the first die roll and then fail the second, so

probability of failing x rolls = (z / y) * (z / y) [n times] = (z / y)^x,

and the probability of success is

probability of success = probability of rolling more than z on 1 out of x dice = 1 - (z / y)^x.

This formula can be plugged into Excel to visualize. Another way to visualize what is happening is the example below.

The figure depicts rolling 2d10. One roll is on the x-axis, the other on the y-axis. Green squares indicate successful combinations with advantage if a 6 or higher needs to be rolled, and red squares indicate failure. One die roll would be the like the ten squares in the first row, with 5 successful roll results and 5 failing roll results. Advantage expands the possible results to two dimensions, with success proportional to area instead of length.

Advantage is really beneficial for tests that already are likely to succeed: if you need to roll a 2 or more on d10, the probability is 90%, but with advantage, the probability is 99%. Likewise, it is not very beneficial for tests that are likely to fail. If you need to roll a 10 on d10, the probability is 10%, and with advantage the probability is 19%. In both cases, though, the absolute probability only went up by 9%. Absolute probabilities increase by the most for tests that are of average difficulty. In the example above, a 50% probability jumps to 75%.

Another way to put things is that advantage makes accomplishing average-difficulty tasks more reliable. It does this better than a static 15 percentage point bonus would because in the mid-range of probabilities, it gives a higher effective bonus.Finally, there are diminishing returns to rolling additional dice with advantage. Going from 1 to 2 dice in the example above increases the probability of success by 25 percentage points, while going from 2 to 3 increases it only by 12.5 percentage points. This property is nice if you want two distinct factors that would make success more likely to matter jointly less than having one distinct factor, although the 5th edition D&D designers opted to not have multiple advantage dice, presumably to make play faster.

Magnitudes with Advantage-Style Mechanics

Now say you're rolling to determine how much you accomplish, for instance how much damage your axe crashing into a goblin did. (Note that in standard D&D combat the success roll is totally independent of the effectiveness of a hit. It's a bit of a strange choice, but it's worked for 40 years). You can use advantage-style rolls for magnitudes as well.

Damage: a question of magnitude

For these sorts of rolls, it's meaningful to talk about the average roll. Technically, we're interested in an order statistic for uniform random variables (link, technical). Conveniently, there's a simple expression for the nth highest value of a uniform variable ranging between 0 and (y-1):

Expected value of nth lowest value of x values = (y-1) * n / (x + 1).

(nth lowest means arrange the values in order from lowest to highest and take the nth result). Therefore the average highest value on xdy is about

Average H1:xdy = (y - 1) * x / (x + 1) + 1 = (y x + 1) / (x + 1).

So the average highest roll of 1d10 is (10 * 1 + 1)/(1 + 1) = 5.5 because (duh) its just one die. The average highest die roll of 2d10 is about (10 * 2 + 1)(2 + 1) = 7. As a rule of thumb, advantage increases the average result by about 30% and disadvantage reduces the average result by about 30%.

(While the uniform distribution behaves exactly like a single die roll, order statistics on two uniforms only approximate advantage or disadvantage with two rolls --  H1:2d10's exact average is 7.15).

The formula is also useful for thinking about taking multiple high dice or equivalently discarding a low die. For instance, the famous roll 4d6 discard lowest attribute rolling method is H3:4d6, and the average is approximately equal to the expected value of the 2nd lowest, 3rd lowest, and 4th lowest dice on 4d6:

Average H3:4d6 = 3 + (6 - 1) * 2 / (4 + 1) + (6 - 1) * 3 / (4 + 1) + (6 - 1) * 4 / (4 + 1) = 12

(it is exactly equal to 12.24. To mess around with these ideas, you could use anydice or the simpler one here).

So the average result is higher with advantage, unsurprisingly. A more interesting property is that results are usually higher than the average. By that I mean that most of the time your roll will be at least the average roll. 75% of H1:2d10 rolls are 7 or higher (refer to the figure above again).

This will apply, approximately, to any magnitude roll with advantage: three-quarters of rolls will be above average. Advantage-style mechanics' increased reliability extend to magnitudes as well. Of course, with disadvantage, three-quarters of rolls will be below average, reliably modeling unreliable magnitudes.

This also explains why 4d6 drop lowest is nice if you want above-average but interesting characters. More than half of attributes will end up greater than 12, but low attributes may still end up low in a meaningful way (i.e., 8 or less).

Actions and Reactions

In my previous post I listed modular, simple and elegant, and promoting rulings not rules as my goals for a skill system. I still think they're great goals, but quite abstract. At a more concrete level I want a system where

• a player making a character can write down (or roll up) a short list like "sailor, dirty tricks, goblin lore" and proceed to play, 

• a character A having a skill "riding" does not mean that character B is hopeless at staying on a horse,

• skills and combat are separate systems, but that allows skill use during combat, i.e., for grappling,

• skills and classes are separate systems, but that allows characters to pick up skills as one of the benefits of being a particular class,

• it can be be a light-weight addition to a D&D ruleset such as BX / Labyrinth Lord or OD&D / Swords & Wizardry.  

So here is my attempt (commentary follows afterward). It uses dice steps with a dice progression d3 -> d4 -> d6 -> d8 -> d10 -> d12 -> d14 (sorry) -> d16 -> d20, etc.

Basic Rules

When characters attempt an action whose success is uncertain, or to react in a way that will save themselves or prevent another’s action, the success or failure of the attempt is determined by an action check. First the player describes what the character is trying to do. The GM will consider which ability is most likely to help the character succeed (the governing ability) and whether any of the character’s skills would make the character significantly better at accomplishing the action (a relevant skill). The GM then considers how hard the action would be to perform, and assigns it a difficulty rating to reflect that. Finally, the player rolls one or more action dice, and if the result (the highest die rolled) is equal to or higher than the difficulty rating, then the action succeeds.

Yes, you do need to check to fly
standing up.

Action die: Characters have a base action die which is d8 at level 1 and improves at higher levels and for some classes. The die step is adjusted up or down by the modifier of the character’s governing attribute. Finally, the die step is adjusted upward by 1 if the character is an expert in a relevant skill or by 2 if the character is a master in a relevant skill.

Action die size = (base action die) +  (governing attribute bonus steps) + (governing skill mastery steps)

Governing ability: see ability scores for a description of each ability's applications.

Roll strength!

Relevant skill: the GM should construe skills broadly. Example: a hunting skill would be useful for successfully hiding outdoors, if that is a hunting method the character might know.

Number of action dice: characters without a relevant skill in the action, or attempting an action for which there is no governing skill, roll 1 die. Characters with a relevant skill roll 2 dice.

Difficulty rating: the standard difficulty rating is 8, which reflects a challenging but not impossible action. Most actions the characters undertake should be like that! An extremely unlikely action's difficulty can be adjusted upwards to 9 or even 10, but the GM should make these adjustments infrequently. Extra preparation on the part of the character, advantageous circumstances, or help from another character can reduce the difficulty rating to 7 or even 6. Actions that would be difficulty 5 or less should normally be treated as automatically successful.

Result: the highest number rolled on any action die is the result of the attempt. The GM gauges success or failure by the result, and ignores any other numbers rolled.

Success: if the result is at least as high as the difficulty rating, then the action succeeds. Another check is not necessary unless the situation changes dramatically.

Partial success: a result 1 to 2 less than the difficulty rating indicates success, but with some complication. The character succeeds in the action, but the success inconveniences the character somehow. The GM determines the exact complication that the character faces. Example: Bianca attempts to jump onto a ledge across a chasm and rolls a partial success. Her feet slip, but she grabs the ledge with her hands. Next round she must pull herself up.

Failure: a result of 3 or more less than the difficulty rating indicates that the action was totally unsuccessful. Another check may not be attempted unless the situation changes dramatically.

Reactions: many RPGs have saving throws to avoid or mitigate perils that the character faces. The action and reaction system can replace saving throws. When the GM would call for a saving throw, she should describe the danger to the players and ask what their characters are doing to avoid the danger. Based on each response, she should resolve the character’s action to avoid the danger. A successful reaction save may not prevent damage entirely.

Opposed rolls: when one character opposes the action of another character, as in arm-wrestling or hiding and tracking, they make opposed rolls. The GM determines the action roll for each, as above, and then both make rolls. The character with the higher result succeeds. If both results are the same, the contest deadlocks or both fail, as determined by the GM. If a character suffers a critical failure, that character suffers some additional adverse consequence beyond failure. Likewise, if a character achieves a critical success, that character’s success is exceptionally large.

Group actions: in some circumstances, a group of characters is attempting an action in which the entire group will either succeed or fail as a whole but more skilled characters might be able to help the less skilled. In this case, the group's attempt is resolved by having the character with the lowest action die and and other single character (chosen by the players) both make action checks. If neither fails, then the group has succeeded; if one fails, then the group has achieved a partial success; and if both fail, then the group has failed.

Examples

Cedrick the warrior attempts to a lift a gate that has crashed down behind the party. He has 16 strength so adjusts his d8 action die to a d12. "Hey, I have endurance skill. That should be relevant shouldn't it, because it'll help me strain to lift the gate?" asks Henry, Cedrick's player. "Sure," says the GM. Henry rolls 2d12 and gets a 1 and a 9. "Standard difficulty, right?" "Yup." Cedrick succeeds with a result of 9.

Warwolf, Cedrick, and Gesaine have been chained together by the ankles by bugbear slavers. While the bugbears are marching the characters down a forest path, Gesaine distracts the bugbears with her ventriloquism. As the bugbears investigate the suspicious noises behind them, the three characters attempt to run off. The GM calls for a group action check using dexterity. None of the characters have relevant skills, and while Cedrick and Gesaine both have dexterity 15, a plus one step modifier, Warwolf has dexterity 10. Gesaine's player rolls a d10, getting a 6, while Warwolf's player rolls a d8 and gets a 3.  One failure and one non-failure means that the group has partially succeeded. The GM rules that the characters make it to the tree line before one of the bugbears makes a hue and cry. All of the slavers begin to chase the characters and have almost caught them, but then the characters slide down a cliff to the bottom of a 50' crevice, remarkably not hurting themselves in doing so. Bugbear heads stare down at them from the top of the cliff.

Commentary

The basic system has three free parameters: the base action die size, the standard difficulty rating, and the number of dice a skilled character rolls. The current system is based on fiddling around with these parameters for a couple of hours. The numbers above represent a world in which level 1 adventurers are relatively competent even when unskilled and quickly grow to succeed more often than not. A level 1 thief with a 14 dex and skill at sneaking would succeed at moving silently at least partially 75% of the time (action die d10, 2 rolls, need 6+ for partial success so 50% chance on each die). That may be too generous; maybe a 8+ standard difficulty rating would be better.

One nice aspect of the system is how skills interact with its other components: partial success, ability modifiers, opposed rolls, and group actions.

First, skills are reliable. A character attempting an action for which she does not have an attribute bonus and who also has no relevant skill will succeed completely in 12.5% of her attempts, and partially in 25% of her attempts. If she has a relevant skill, she will succeed completely in 24% of her attempts and partially in 38% of her attempts. Total failure becomes much less likely.

Second, skills complement a character's strengths. Raw ability only gets a character so far. A character free-climbing a wall with 10 strength has a 62.5% of failure. A stronger character with 15 strength reduces her chance of failure to 50%, or by 12.5%, and even higher strength 16 reduces her chance of failure to about 42%, or by 8%. Add climbing skills to the mix, and moving from 10 strength to 15, and 15 to 16, decreases failure from about 39% to 25%. Attributes and skills are both important in ways that I find satisfying.

Skills are also especially important with opposed rolls. If two otherwise identical characters face each other in an opposed roll but one is skilled, the skilled character succeeds 75% of the time, so being skilled are very useful in opposed rolls. On the other hand, a naturally talented character can through luck overcome a skilled but less talented character. For example, character A has 11 str and is skilled at wrestling, while character B has 15 str but no skill. 1/5 of the time character B will roll a 9 or 10 and win, but the remaining 4/5 of the time both characters are equally likely to roll 1 through 8 on each of their action dice, in which case character A has an 75% chance of winning. Overall, character A has a 60% chance (4/5 * 3/4).

Finally, skills also play the natural role in group actions of making a big difference for the team. Having a skill makes it much less likely that a character will fail a check outright, so having a skilled character involved in the group check makes it likely that that group will partially succeed at least, as is likely if the characters are helping each other accomplish the task.