10,000 random points mapped 100 at a time (100 frames taking 2 seconds to play). Genuinely random results appear to contain clusters of results and results forming straight lines, both of which we instinctively consider non-random events.
Image by CaitlinJo via Wikipedia Commons, Licensed under CC3.0 for use with attribution.

701 492 537 313 432 835.

191 489 361 702 127 659.

723 296 032 553 407 934.

Those all look like fairly random strings of digits to me. How about:

333 333 333 333 333 333?

Or
022 022 022 022 022 022?

Or
123 450 123 450 123 450?

Or
000 000 000 000 000 001?

Because the human mind detects a pattern, it rejects implicitly the possibility of achieving that pattern by random means. But all these random strings of digits are equally probable; I’ve simply cherry-picked outcomes that play to human perceptions or misperceptions, to prove a point.

In AD&D’s DMG, there was a random dungeon generator for solo play. Great for giving a new GM the chance to wander around the rules and get to know them; lousy for any other purpose. This was truly random and uncontrolled; you were just as likely to get a 40′ x 40′ room with 1 Skeleton in it as you were to get a 10′ x 10′ room with a family of Black Dragons.

The concept of randomness is fundamental to RPGs, as I have explained in past posts. The primary method by which GMs inject randomness into their games in a reasonably controlled manner is by using die rolls.

But it’s not the only way. There are mathematical functions that can be used to generate strings of random numbers on computers, and those have been adapted into various die rolling apps and contrivances over the years – starting, from memory, with Dragon Bones, long ago.

How Random is Artificial Randomness?

How random are these random numbers?

They aren’t, not really. But then, neither are any of the numbers I showed at the start of this article. They were all generated by playing around with my desktop calculator app and then throwing away any leading digits and the decimal point.

A mathematical function provides a consistent result – give it exactly the same inputs, and it will give you the same result, time after time, completely predictably. The key words here are “exactly the same inputs”. Computer random number generators all rely on a “seed value” but various values can be applied to it so that the results appear random.

If you use some sort of semi-random value as the seed, the output quickly approaches a good simulation of randomness – for example, the last 4 digits of the time since the year 1980, in seconds.

For these pseudo-random results to be of use to us in any practical sense, they usually have to be interpreted, and that’s where things once again can get sticky. There are a couple of lessons that I had to learn the hard way.

For example, let’s say we’re trying to simulate a simple d6. The random number generator spits out a value between 0 and 1 – a long string of decimal places.

The easy and obvious answer is to multiply our random number by 6, and then lop off any decimal points. But, if you do that, the first thing you’ll see is a whole heap of 0’s and no 6’s.

Okay, so you have to multiply by 5, lop off the decimal points, and add 1 to the result? Because a d6 produces results of 1 to 6, not 0 to 6, right?

Let’s say that our initial random numbers are – by pure coincidence – 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, repeat. This spreads the outcomes evenly in probability across the range 0-1, doesn’t it?

Put those into our d6 simulation, and we get 1, 1, 2, 2, 3, 4, 4, 5, 5, 6, repeat. That doesn’t look like a very even distribution at all, does it? 3 and 6 are simply not coming up often enough. And understanding why starts to get very complicated.

A fair assumption would be that it’s because of a rounding error. So, you decide that what you have to do is round the results off and not just throw away the decimal points. And this looks good at first, but that impression doesn’t survive.

It’s because we’re collapsing ranges of outcomes into a point, and our rounding-off function makes those ranges unequal. And, on top of that, because we’re mapping one scale onto another, we can still get rounding errors.

You see, people are used to thinking of numbers as this precise thing, and sometimes they aren’t. Numbers can be downright fuzzy when you get right into it. What people really mean (but don’t realize it) when they say “one” is “any number greater than the minimum threshold for an approximate value of 1 and less than the minimum threshold for an approximate value of 2”. Changing the rounding only changes the thresholds.

No, the real problem here is that we’re taking an infinite range of decimals and squeezing them into too small a set of finite values separated by thresholds – and that computers have no capacity for dealing with fuzziness.

Implications Of Fuzziness in Random Numbers

This is hugely important in computer games in which money rides on the outcome. Any sort of bias is totally unacceptable. It’s slightly less important in computer games that have in-game buy-ins, and slightly less important again with computer games that cost you nothing but time. And, at that same level of importance, we also find all the RPG applications, where these principles also apply.

Any table which does not map individual results to singular outcomes can be described as containing thresholds at which the next outcome applies and intervals between thresholds. In fact, ANY non-infinite non-recursive sequence does so, whether it be die rolls in series or randomly-generated digits in sequence, or some device intended to simulate one or the other..

I know a number of GMs who (back in the day) refused to permit Dragon Bones to be used at their tables because they didn’t trust the randomness of the outcome to be evenly distributed across all the possible results. And while this appears a somewhat paranoid perspective, to be fair (and as I hope I’ve shown), the subject can be a LOT more complicated than it first appears.

Minimizing errors

The more decimal places we carry our random number to, the smaller the resulting rounding error. This is good, because it means that we can reasonably simulate such real-world things as dice and slot machines and weather patterns and roulette wheels and what-have-you. With enough sophistication in the interpretation engine of a game, we can even simulate human behavior within a limited context – AI opponents in racing games, for example, can make decisions and even mistakes, just like a human player.

It all comes down to the interpretation of the results. And that brings us squarely back to RPGs.

The Relevance to Applications of Randomness

There are two real applications of random numbers in an RPG. The first is to select between possible outcomes of an action, i.e. to incorporate the fuzziness in outcome caused by innumerable un-enumerated variable factors. Some will bias the likely outcome one way, some the other, but the final result is a definite outcome. This is the application of die rolls to resolve attack attempts and skill rolls and the like.

And the other is as a decision-maker. If the alternatives are sensible, then this can – in theory – work. But, quite often, the alternatives are not equally-sensible, or equal in probability, and the end result is an AD&D Random Dungeon, where sometimes results are believable and sometimes they strain credibility beyond breaking point.

Yet, there can be no doubt that if we don’t throw a little randomness into the picture, the results are inherently biased. Does the villain think of the solution to the conundrum being presented by the PCs? If the GM can’t think of one, the villain obviously can’t – but what if there is a solution that’s obvious to the DM?

Well, what’s sauce for the goose, as the saying goes. Make a roll to see if the villain thinks of the way out. Or, if he’s smart enough to automatically succeed on such a roll eventually, make a roll to see how long it takes.

This is constraining the outcomes into sensible ones and then randomly selecting between them. But trying to simulate everything this way slows the game to a crawl and, worse, relies on the GM being cognizant of all the possibilities, all the time. That’s an unrealistic expectation.

One way to counter it is to roll the dice and then work out an interpretation – even, possibly, just what you were rolling dice for. In other words, roll the dice and then free-associate with the result relative to the highest and lowest possible rolls.

I’m not a big fan of this. What if you have no ideas? What if your ideas suck? What if your ideas are stuck in a rut? What if you risk becoming (gasp) predictable?

Your imagination only has to fail once and you can find yourself in big trouble. Better by far to have some notion of where the villain wants things to go, and roll for how far he is able to advance his plans – then free-associate with that.

What this all boils down to is knowing when to apply some randomness, and when not to. You could call it directed randomness, or confined randomness, or even planned randomness – but my preferred term is constrained randomness.

Why? Because that relates everything back to the skill checks that we’ve already decided are completely acceptable. And, just as you don’t require a PC roll to do up a button or tie his shoelaces, it implies control over the circumstances and restraint in the outcomes being selected between to amongst those that move the game forwards.

So, Randomness is not always a good thing?

Let’s take another example. Something is about to happen to one of the PCs, you don’t know which.

One option is to simply roll for which one it is going to be.

But a far better approach would be to consider which of the players has had the least to do so far, and which will have the least to do for the rest of the days’ play as far as you can tell, and choose the player who scores lowest in both respects. This is metagaming – but it’s metagaming to spread the spotlight a little more evenly.

A third alternative would be to choose the PC target who would interact in the most entertaining way with whatever is going to happen, enhancing the vicarious entertainment for everyone at the gaming table. Again, metagaming for a positive purpose.

In my book, choices 2 and 3 have it all over choice 1. The only thing the first option has going for it is that it looks “fair” to the players. So “roll” and collapse the outcome to your predetermined choice. This is sometimes known as a Magician’s Force.

When you play blackjack or roulette or whatever in a casino (virtual or real), you have a simple objective – end the game with more money than you started with. It won’t always happen; the odds always favor the house. But they all want you to have fun getting to whatever the outcome is so that you’ll come back and try again. And it goes without saying that the games have to be absolutely fair, or customers will walk. Even the perception of non-randomness could be a major problem for this kind of business – so they have good reason to make sure that they really understand randomness.

The analogy with roleplaying games couldn’t be clearer.

Of course, beginners tend to favor games with simple rules. That’s so that they can make mistakes that are obvious enough to learn from them, which in no way describes even a simple RPG. And, to be fair, most games these days are designed to optimize the payouts if you know what you’re doing – even slot machines such as those offered by NetBet slots.

On a multi-line slot machine, how much do the odds improve with each additional line? Which configuration of bets results in the best odds of winning more than you have spent? Or at least of minimizing losses on this pull of the handle or push of the button so that you can have another chance with what’s left?

There is also the phenomenon described as beginner’s luck. Or, to put it another way, non-random randomness.

Wikipedia’s brief article on the subject lists a number of explanations for the perceived phenomenon, some of which I must admit had never occurred to me. But RPGs add one more: the GM going easy on a new player (whether or not you should do so is a subject for another time). The key point to be made here is that randomness is inherently fuzzy unless you make it your business to delineate its significance.

The Principles Of Randomness

All this can be boiled down to a few simple Principles.

1. Only apply randomness when you know what the outcomes are that you are selecting between.
2. Only apply randomness when all possible outcomes drive the game forward.
3. Know when randomness is best applied to a PCs actions and when it is better to assume an outcome.
4. Avoid using randomness to make decisions; make decisions, then employ randomness to determine speed, or success of implementation, or overlooked factors.
5. Always know what the random numbers mean.

Randomness doesn’t have to be your enemy. In fact, it can be your friend. Letting a PC make a skill check and being ready and willing to accept whatever the outcome is because you can still navigate the game forwards regardless of that outcome, has a wonderful way of letting the players feel in control of their characters, for example, even though you may know better.

One of the most subtle lessons that Beginners have to learn is how and when to apply randomness – and how and when not to. There will be subtle nuances to the practices any given GM adopts, and this forms part of the foundation of the ephemeral but very real thing, that GM’s personal style.

But, just as with an RPG, knowing the direction in which you are heading enables you to steer your way more quickly and unerringly to that destination. Knowing that you have to learn to remove the fuzziness from your randomness makes it easier to pay attention to that, and grow in your mastery more quickly.

It’s sometimes said that probability is a statement of ignorance as to the outcome. The same is true of any black box in which initial conditions go in one end and an outcome emerges from the other. Probability is actually a set of tools for trying to guess at the inner workings of such a black box. Randomness is a black box; using it intelligently, putting it to work for you, means connecting it up to outcomes. You don’t need to understand randomness to be a good GM – but, as with most things, the more you know, the more you can tweak the results to achieve your goals. And when your goals are for everyone to have fun at the game table, that’s a good thing.

This is a suprise extra bonus post to commence the tenth anniversary of Campaign Mastery in style! There will be other surprises as the month unfolds.

I hadn’t intended to post another article on Randomness so soon, but a confluence of several different factors made it an appropriate choice. The message content is significantly different. So I don’t think anyone will have a problem with it.

And it seems somehow appropriate, given the history of the site, that I post something before the 10th anniversary officially starts. After all, for the first month, we were posting articles but not telling anyone outside of a select circle of reviewers that we existed – so that when we did go live, we appeared to hit the ground running.

Well, I can’t use that trick, this time around! But a sneaky extra post – that I can do!