Welcome to Episode 2 of Wonky and Technical. This time, I sit down with game designer and entrepreneur Heath Robinson to talk about War of Kings, Incantris, and the decisions that went into his game-design process. In this article, I explore tabletop RPG design principles, with a strong focus on the how and why of probability systems.

You’re a nerd, right? You’ve handled your fair share of d20 dice and spot checks. You’ve dealt with the frustrations of THAC0 calculations on the fly. You’ve memorized tables of spells (and forgotten the ingredients, of course). And you’ve probably complained more than once that the math is kinda screwy.

Tabletop games — role-playing games like Dungeons and Dragons, wargames like Warhammer, strategy games like Settlers of Catan, and more — have been around for decades, and they aren’t just for “nerds.” You’re more likely to find a stack of them at your local brewery than you are in someone’s basement.

Ever since Stranger Things, a lot of people have been getting into tabletop RPGs. Some are established tabletop players trying new game systems, while others are complete beginners. Most of them are having a hell of a lot of fun.

Some, though, have questions: It’s fun, yeah, but why these dice? Why this math? Why these probabilities? A lot of what we deal with in an average session is arbitrary, but it isn’t random.

The math behind tabletop gaming has never been hidden, but its relationship with the rest of the experience isn’t always clear. To munchkins and murderhobos, the math can be both the means and the end, while for others it can get in the way of storytelling or having a fun time. Sometimes it slows things down (I’m looking at you, 4e), and other times it creates a framework for freestyling. It doesn’t always work, but it’s typically intentional.

In this article, we’re taking a peek at how math, game design, and UX principles work together to make tabletop games fun.

It’s going to get weird.

The Basics

For out-of-the-loop readers, you’ll see a lot of “#d#” in this article. The number before the d represents the number of dice. The number of after the d represents the number of faces on the dice. You’ll also see things like “2d12+1d6” or “6d8–4.” You’ll get used to it.

You’ve probably played at least one game that involved rolling at least one die to produce at least one result. Yahtzee, Monopoly, Risk, GURPS—something with at least some aspect of chance.

And that’s what dice represent: probability. In Dungeons and Dragons, you roll to attack goblins and leap rivers, with a static modifier to represent your skill and the dynamic value of the 20-sided die to represent chance and circumstance.

Psst: I’m going to be describing most of this in the context of d20-based systems like Dungeons and Dragons. If you need more context for what we’re talking about, check out this free player’s guide.

That’s the big difference between board games like Chess or Go and tabletop role-playing games. RPGs use probability to add variety and create emergent experiences. The players are working together and engaging with a plot provided by the game master, and the randomness of the dice makes it feel less scripted and more deliberate. In theory.

Before game designers can dive into skills, classes, monsters, or any of the things that really make or break the player experience, they need to figure out their core “engine” for how things happen. They need to pick their dice!

How Probability Is Used in Gaming

You can summarize most rolls into a few distinct categories. Some games get really persnickety about what each roll means in each possible context, but generally speaking, rolls can be broken down into:

Rolling at least or at most a certain value

Rolling an exact value

Rolling a certain kind of number

“At Least/At Most” (AL/AM) rolls are, by far, the most common. That’s how you hit enemies with weapons in Dungeons and Dragons, successfully use skills in Pathfinder, and do just about anything in most other games.

“Exact Value” (EV) rolls are common in gambling and often come into play in games where you roll from a table of possible events or outcomes. Many games will have EV conditions in addition to AL/AM thresholds: critical hits, fumbles, etc.

“Certain Kind of Number” (CKN) rolls are interesting, because they’re often the entry point for arbitrary value sets and dice with unequal distributions. Categorize even/odd, prime number, and other mathy objectives as CKN rolls.

As a meta-category, let’s throw in opposed rolls as well. Opposed rolls are typically AL/AM rolls where the threshold is set by another roll instead of a table or a chart. Opposed rolls are most common in games where players compete against each other, instead of fictitious enemies.

All these rolls used together can represent different kinds of randomness in the world. Exact rolls can be used for random outcomes and “pure chance” situations, while AL/AM rolls tend to represent a deliberate effort to overcome a deliberate obstacle. CKN rolls add additional wild cards and extra layers of player engagement, while opposed rolls change the context of a task’s difficulty.

In broad strokes, probability is used to represent effort, difficulty, and randomness. It fills in the gap between the player and their character.

But how do game designers choose the right dice for the right situations? How do they chose the right modifiers? What makes it meaningful?

The Dice

Single-Die Distributions

Alright, alright, that’s a pretty boring graph right there. It’s flat. You roll one die, and the chance of any single result occurring is the same as any other result occurring. It’s easier to roll a specific result on a small die than it is on a larger one, and figuring out your chance of rolling at least a certain value is some easy math.

Two-Dice Distributions

Once you add a second die, everything changes. This distinct pyramid shape puts a strong bias toward that median value. That’s great when you’re trying to roll consistently (for, say, damage against an enemy), but it can make life harder when you’re trying to roll as high as possible.

N-Dice Distributions

With more dice in hand, that pyramid smooths out, and you’re left with roughly equal probabilities in the center of the range and nigh-impossible ones on either end. This is great for systems that want to incorporate highly unlikely (but awesome! or terrible!) results into a common roll without diluting the range of numbers they need to deal with in most other situations.

Exploding Dice

Alright, this is a weird one. But we’re here for the weird stuff. Exploding dice produce this wonky graph because of their rules instead of anything inherent with the dice combination. As things normally go, exploding dice are either multiplied or rerolled when a certain result is shown, creating these exponential steps in probability that you see above.

Things can get even weirder when you’re rolling multiple dice for each explosion, as then each exponential step gets its own curve, and still weirder yet when you reroll and add dice for each explosion. It’s kinda awesome.

Expanded-Range Dice

Expanded-range dice are, usually, dice with negative or null values. You’ll see them in games that use Fudge dice, and they’re almost always used in groups. The value curve is the same as any N-die distribution, but the most frequent value gets funky.

For dice with symmetrical value distributions, the most frequent value is 0. This flips the script on the usual relationship between rolled results and roll modifiers, making the modifier the more important value.

For dice with asymmetrical value distributions (more positive numbers than negative numbers, etc.), the most frequent value can be just about anything. These dice typically have close relationships with the mechanics they’re used for, and they’re pretty rare. And fun.

Arbitrary Dice

These are anything-goes dice. They can often be represented numerically, but they’re typically used by designers who want to represent specific things on the dice themselves. They’re the old-school version of an array, and the math involved can get pretty funky.

So what does this mean for your weekly game of Bunco down at the wine bar? Not much. For games of straight probability without betting or other mechanics attached, you just keep rolling. But for the wonderful world of board games, role-playing games, and just about everything else you do at the nerd shop, it’s absolutely everything.

For the sake of sanity, most games tie the majority of their mechanics to a single die or set of dice. Situational dice changes can be hard to keep track of, and they can make it harder to balance a game when you’re relying on a bunch of different distributions to make things happen.

Not to cast any shade, but d20 systems are conceptually simple. They get fiddly and top-heavy only through tedious modifiers and on-the-table situations. THAC0 was kinda wonky back in the day, sure, but it isn’t hard to grasp. And all of this is deliberate.

With a flat line of probability, modifiers that fall into consistent ranges, and a consistent roll→add→check→roll→add→sum gameplay loop, all the intellectual effort that goes into learning a d20 system is tied up in learning a particular class or set of abilities.

When you really look at it, though, the dice and numbers behind the d20 system all fit together fairly nicely. The largest die is the d20, your primary stats fall mostly into a 10–20 range, and the modifiers you add and subtract from things are often clean 10 percent increments (+2, +4, etc.). The numbers you work with are never so large that you’re forced to write out your math, and rounding down never deprives you of a significant margin of success.

D20 dice are the mascot of tabletop RPGs. Meanwhile, GURPS uses 3d6 for most things, and FATE uses 4d{-1, 0, 1}. Big difference, right? You can spend hours looking at how dice selection can influence play without touching the rest of the rules that would come into play in a finished system.

So, how do those other decisions work out?

Gameplay Scope

When you’re playing 13th Age and want to get past an obstacle, you can conceivably seduce it instead of killing it. There are rules for that.

In games that allow for creative solutions, the craziness of players’ imaginations are tempered (to some degree) by the concept of probability: Yes, you can try that, but you do have to roll for it. It’s a great way to give players a reasonable scope of agency without writing a 1,000-page rule book.

Some games get really granular with the probability of specific actions. The modifiers for two closely related actions might be totally different, depending on how a player has built their character. In other games, those actions might all belong to one large category and have the same flat bonus, while others still use opposed rolls or other situational pluses or minuses to make each circumstance statistically unique.

In most games, there will be a set of most-probable actions—things that players can rely on to get things done most of the time. The strategic bit is to choose a set of most-probable actions that will produce the best margin of success in the greatest number of circumstances, and the tactical bit is to beat and bludgeon your fictional reality into such a shape that you can continue to rely on those most-probable actions even when you shouldn’t.

Of course, it’s impossible to fit everything that might happen into the list of things they could possibly be good at. Your fighter might be strong, but that won’t help him walk a tightrope, and there’s no way that mage will ever be a frontline brawler.

That’s where the conceptual divide between simulation-oriented game design and scenario-oriented game design comes in.

Imagine you’re a game designer for a moment. You want to create a game that is:

Fun, Playable without a graphing calculator, and Mechanically symmetrical enough that a group of experienced players can play it without using a chart or reference table during every round.

What’s going to be probable, improbable, and impossible in your game? Will it need a game master of some sort to coordinate it for players? Will there be only one way to play it? Is it meant to be a critique of our modern economic system that has been slowly corrupted by a blind eye toward the past? Will there be dolphins? Most of these choices will be arbitrary.

On a different level, though, the choice between a simulation focus and a scenario focus is very meaningful. It defines the mechanical scope of the game you’re designing, and it can lead to a lot of very technical decision-making.

From 3.5 to 5e: The Evolution of D&D from a Simulator to a Scenario Game

I have a grave admission:

I like fourth edition.

I’m not joking! I started with 4e, I’ve spent the most time working with 4e, and many of my formative tabletop memories come from those high school sessions spent in a dilapidated apartment, arguing about opportunity attacks and trying to find the right errata for an obscure feat.

Sure, 4e plays slow, and the classes are systematized to a degree that the out-of-combat flexibility feels limited, but it has a very scenario-driven feel to it. The game designers were deliberately targeting a wargame–lite feel, but they flubbed it with overly large enemy health pools and a semi-deliberate cheese escalator (blood cheese > frost cheese, fight me).

Where 3.5 was a sprawling system with rules for everything from aging to crafting, 4e stripped away many of the out-of-combat features (like meaningful alignments) and simplified the skill system. They limited multiclassing to make balance more attainable, and they overhauled the encounter-building process to make it more user-friendly. Pass-or-die spells (for both players and monsters) were done away with.

Compared to fifth edition, 4e can feel glacial and fiddly. With multiple defense stats, modifiers coming from just about anything, and almost every action being open to an opportunity attack, you have to be 110 percent on top of the action to make it through your rolls in a decent time frame. Compared to 3.5, however, it’s almost like a video game.

Where 3.5 often requires the support of NPCs and towns for players to heal, craft new items, and generally manage characters “in setting,” 4e characters are largely self-sufficient. With healing surges, encounter powers, and a focus on magical loot, a 4e session feels more like a raid than a quest. That’s deliberate.

Fifth edition does away with most modifiers, most opportunity attacks, and having multiple defensive stats. It drops 4e’s systematized attack cards and returns to the visual style of 3.5, while also drastically simplifying the kinds of actions you can take. The skill list is shorter, the wargaming opportunities are fewer, but there’s a lot more flavor text and a big investment in the art of the game.

In many senses, 5e is an aesthetic return to traditional D&D trappings paired with a radical departure from the traditional gameplay loop. It looks like 3.5, it sells itself like 3.5, but it runs like a system designed for pickup-and-play sessions. It’s a continuation of 4e’s scenario focus, tempered with efforts to facilitate role-play (but not simulation).

In a simulation-focused game, the mechanical objective is to create a system that reliably simulates the abilities of a fictitious person in a particular setting. There’s a focus on classifying actions and creating a meaningful difference between winning through speed and winning through strength.

Pathfinder tries to be D&D 3.75 in many ways, serving as an extension of the 3.5 philosophies, with a steadier hand at the tiller when it comes to specific balances. It simplifies the gameplay loop a bit but still incorporates the wider scope of skills and the variety of niche builds. It’s a game that unabashedly relies on tables and reference charts, but it isn’t as systematized in-play as 4e.

On top of the game-design differences, however, there’s also something that very few people talk about:

The UX differences.

Yes, yes, I get it, this is a really big pivot in an already long article that has managed to describe design decisions at length without discussing design itself in any particular depth. But I’m serious here. System design and how it’s represented mathematically is an issue of UX first, game design second. And I will fight you on that.