Creating Mathematically Average Challenges

Understanding Averages 

Helps You Build Challenges

As An Adult, I Suck At Math

    When I start talking about numbers, I like to tell people that in college I got an English degree, and that means you legally can't make me do numbers anymore nor can you make me read any more books. This is, of course, a lie, because as I was reminded last night I have been playing roleplaying games for almost two decades and have not yet found a way to escape numbers or reading books. Being trapped as a Forever GM for most of those years led me to learning very niche applications of math insofar as they help run challenges on the fly - because no matter what the system is, you can justify whatever the dice are doing with whatever fluff you need to but at the end of the day you need to know what the dice are doing so you can move on. I would like to impart some of that knowledge to you.

    A quick word of warning: The things I am about to tell you are mathematically true, but if you are a math person you will likely be unhappy with the way this information is presented because I talk about Numbers the way Word People talk about Numbers, not the way Number People talk about Numbers. I'm also not talking about ways to build narratively interesting encounters here - there are many, many more eloquent and intelligent people on the internet who've beaten me to the punch on that one. What I am to do here is to equip you with a set of tools that will hopefully allow you to improvise challenges in games that require dice-based number generation such that it can be as on-the-fly as your regular cool story improvisation is.

Finding Dice Averages Quickly

    This is not secret knowledge, but just so it's here to reference: the average roll of a single die is [biggest side]/2 + .5. The average roll of two dice can be found by adding the largest and smallest sides of that die together. So, for example, the average roll on 1d6 is 3.5, whereas the average on 2d6 is 7. My mental shorthand works like this: If I need to find the average damage on a 3rd level Fireball in D&D 5e, I look, see that it's 8d6, and that means on average it's 4x7 since [2d6] as a variable is the same as 7. Likewise, if I was playing a Paladin two-handing a longsword with 16 STR using a 2nd level spell to power a Divine Smite, then I know that the damage is 1d10+3d8+3 (assuming they're not fighting an undead or fiend) which means that the average roll is 5.5+(4.5+9)+3, or 22.  That's all well and good for practical applications for players, but where does that help you as a GM?

    (As an aside, you can use this guy to do averages on your regular D&D polyhedral dice. If you're one of those sickos out there playing Genysys with those wacky symbol dice...good luck man, this post is mostly not for you, I haven't had time to review those rules yet. Sorry to all my Campaign: Star Wars and Skyjacks homies out there.)

An Average Challenge

    So keeping with D&D for most of its iterations and derivatives, we know that a stat score of 10 is supposed to be the average - narratively, that means that anyone at a 10 is assumed to be just about as good as any random person in things related to that ability, but mathematically what that means is that there is no modifier to a roll. If we pop back to the above and see that the average roll on one die is half of its size +.5, that means that a challenge with a DC of 10.5 would be average, or in other words, if someone is assumed to be average at something, it means they're going to succeed at doing it a little over half the time. I'm sure there's a better way to put that, but this is the thesis which we're going to be using moving forward: an average challenge is one that a character is assumed to succeed at doing a little over half the time.

5e (2014 edition)

    Now, the reason I keep bringing up D&D is of course because this was a concept I needed to understand while working on One Night Strahd. My co-author, Jake, is in fact a Numbers Person, and just does all this stuff in his head. I, as we have established, am a Words Person, and so I ended up having to write this out so I could interact with it properly. Now, 5e has a lot of variables that come into play which makes people think that balancing challenges is hard - commonly, I hear stories about how "oh, this one character is OP and therefore either I make my encounters challenging for that character and that means nobody else can interact, or I make it challenging for the other players and that character can just mop the floor with it." Now, setting aside the fact that many other games solve the problem of this level of balance granularity by simply not giving a fuck about it (see: much of the OSR scene, much of the narrative game scene, etc), if you need to know if a dice roll of any kind will be challenging to a character at any given level, I have constructed this table which takes into account proficiency bonus and stat score in a given level range, but does not take into account any spells or abilities that add on an extra die - but with that said, using the theory listed above, you should be able to figure out how that changes the odds. (It also doesn't take Advantage or Disadvantage into account. Mathematically, I have always counted Advantage as a +5 bonus and Disadvantage as a -5 bonus on a roll based on this post from 2012 and this post from 2014, so while I assume that in other systems it would be a 25% change up or down as well, I don't actually know the math and I'm not confident enough to account for it in any of the following parts of this, so just...like...take care of that yourself if it matters to you.)

    The other way to say what this table does is this: if you set a target number for a challenge, consult what level the character is at, then see where that number falls - if it's closer to a number on the left side of the table, it'll be easier, and if it's on the right side of the table it'll be harder. The reason this table has Proficiency Bonus included is, of course, because if you're rolling for something you're not proficient in, your bonus to it would never change, so if you somehow had a 0 in something and were rolling for it you'd always be at a -5, if you had a 10 you'd always have no bonus, and if you had a 20 in the stat is would be at +5, meaning that the control numbers here are 5/10/15, respectively. 

    With me so far? Let's try this out with a different game.

2d6+Stat Games (PbtA, etc)

    Systems that only use 1 or 2 dice at a time make this a whole lot easier to calculate - and it isn't lost on me that many of the games that use this system to resolve mechanics are often more narrative focused and thus don't have as much emphasis on trying to Make Number Go Up. Broadly speaking, in a lot of these systems, if you know you've got a negative bonus to a stat, you know that it's something you're not as good at, you're roleplaying that, whatever. There's a lot of ludonarrative consonance there. But for the sake of math let's see what this looks like for games that only let you have a -1 to +3 to your average 2d6 roll. (Again, not taking Advantage/Disadvantage into this because I am Math Stupid.)

    Knowing that PbtA games generally follow the rule of 6- being a failure, a 7-9 being a partial success and a 10+ being a critical success, I actually find this to be more numerically interesting than in a binary pass/fail system like D&D because now you can see that on average rolls, characters will almost always succeed with a cost on average, with only the truly penalized or specialized getting into the next tiers on average. I think there's probably a lot to chew on on the idea that a binary success/fail game like D&D considers an average challenge one that a player will succeed on a little over half the time whereas a more narrative system like the Powered by the Apocalypse family of games consider an average challenge being one that the player will succeed at a cost, but this post is already getting in the weeds. Suffice it to say that you can extrapolate data like this out to other in-game meta-currencies that allow you to push success up from Failure to either Mixed or Critical successes. (This in particular is something that makes the design choice in The Between so interesting re: Masks and other abilities that function as Masks in certain situations tasty choices to consider when selecting your characters, but that is perhaps a ramble for another day.)

Applying These Concepts To Other Systems/Challenge Designs

    So, without trying to detail every possible combination of mechanics, how can you apply these principles to designing challenges in whatever game you're designing for?

1. Figure out whatever Average looks like for your game - as in, the lowest possible number that you can generate based on stat modifiers and dice you're using, the highest possible number using modifiers, and the midpoint which should be what a roll with no modifiers looks like.
   
   
2. Figure out what you want Average to mean, narratively - do you want Average to just mean success? Do you mean Average to mean success with a drawback?
   
   
3. Compare your player characters' stats against the numbers you have set for your challenges - if the challenge target numbers fall farther towards the left of the distribution, the players will be less challenged, if they fall farther to the right the players will be more challenged.
   
   
4. You can do this on a character by character basis (or even stat by stat!) in order to test how challenged by certain parts of your challenge certain parts of the party will be (say that five times fast). When getting into the nitty gritty of design, especially for Published Adventures (TM), this can really be helpful when trying to gauge what kind of power level the adventure is. I find this to be much more useful information, especially when dealing in the world of D&D because the balance of a character is often more dependent on the proficiency of the player building the character and the availability of in-game bonuses than it is on the actual numbers derived from each level.

And there you have it! So simple a particularly disgruntled baby could do it. Have fun!