### Open Ended Rolls

This is part of my RPG series of entries here at SOB. See the inaugural entry in the series for more details.

In a comment I posted in the discussion that followed my *Character Progression in Elements Eight* entry, I hinted (rather blatantly) at a future essay about open versus closed die rolls for conflict resolution. This is that essay.

#### what “open” vs. “closed” means

My use of the terms “open” and “closed”, in this case, refers to the statistical limits of one’s die rolls.

For instance, in the simplest case, a d20 roll in D&D or Pathfinder results in a strict statistical limit on the results of the roll, in the range of 1-20. The probability curve for that roll is completely flat; every single possible number in that range has a 5% chance (assuming an unweighted, random die roll) of being the result. While modifiers can change those results, the modifiers (except in cases of fudging results) exist before the roll is made — resulting in a series of twenty possible results, again in a flat curve with a 5% chance per possible result. All the modifiers do is shift the curve to the right or to the left — the curve itself is not changed at all. This is a closed die roll.

An open die roll is one like, for instance, the 7th Sea system. You roll Nd10, where N is a number determined by your character’s attributes and skills, and if you roll 10s you get to roll more dice to add onto the total. There are circumstances where you aren’t allowed to add more dice, of course, and they qualify as closed rolls (though more complex than the D&D example) — but we’ll stick with 7th Sea’s open rolls for this example. They’re “open” because there is no theoretical limit to how high your roll can go — there’s only a probabilistic limit. It’s somewhat improbable you’ll get more than a 10% boost to your total on such rolls, on average, and the probability drops off rapidly as the considered boost gets bigger.

There are other systems that offer open-ended results as well, and also systems that offer results that are superficially open but have hard limits (such as the rule in Rifts that if you get a starting attribute of 16 or more you add d6 to the total, but it stops there).

#### how a closed roll affects the game

Closed rolls inherently tend to limit the realism of a game. For instance, if someone has an AC of 30 in D&D, and two characters attack — one a sixth level fighter with a Strength of 16 and no other bonuses or penalties to the roll using a longsword, and the other a first level wizard with a Strength of 4 and no other bonuses or penalties to the roll using a dagger — the chance of hitting is exactly the same. In each case, the character has a 5% chance of hitting. Realistically, you’d expect two characters who vary so greatly in combat prowess to have differing chances of striking the enemy, but the closed roll system makes that impossible. In some systems, rather than both of them hitting on a 20 and only a 20, they might both just have zero chance of hitting at all, while in other systems they would both be able to hit within the same range (in GURPS you succeed on a low roll, so a 3-4 automatically hits, or maybe a 3-5 — I haven’t played GURPS since 2001). In each case, though, the result is an unrealistic similarity of the chances of hitting even when comparing a skilled, experienced warrior with a combat incompetent, just because the enemy is good at dodging or otherwise presents a formidable challenge for to-hit rolls.

Obviously, there comes a point at which the difference in chance of hitting between two characters is so vanishingly small that it doesn’t really matter — but that should only be the case when the chance of hitting itself is so vanishingly small that it doesn’t really matter. A skilled, experienced warrior should, up to that point, still have a better chance of hitting than the combat incompetent standing next to him.

It gets worse when you start incorporating critical hit and miss rules. For instance, in D&D, you get a critical “threat” based on your raw die roll — which means everyone using the exact same type of weapon has the exact same chance of achieving a critical threat. The actual confirmation of a critical hit might vary depending on individual character skill, of course, which helps somewhat to mitigate the unreality of it — but against high-AC opponents (such as the 30 AC character in the previous example), the chance of a critical hit is again exactly the same between the two characters.

. . . but wait, there’s more! Beyond a specific disparity in combat stats, where the target’s AC is sufficiently higher than the to-hit capabilities of the attacker, the chance of hitting and the chance of getting a critical threat are *identical*. AC doesn’t come into play in that at all — and increasing AC beyond that point doesn’t affect the chance of being hit, or the chance of being subjected to a critical hit, at all. Why should your chance of getting hit never be allowed to drop below 5%, even if you are (according to your stats) ten times as difficult to hit as another character whose chance of getting hit by a given attack is also 5%?

#### some problematic open rolling systems

Most game systems that employ an open die roll system do a piss-poor job of it. The same can be said of systems that superficially appear open, but are actually closed. This is because most systems involve rolling dice, then adding more dice based on how the original dice were rolled, and adding together the results — and such systems tend to overlook the effects their mechanics have on statistical curves.

For example, in the Rifts attribute rolling case, you simply cannot have a character with an attribute roll of 16. If you roll a 16 in the initial roll, you eliminate that number by adding another d6 roll to it. As a result, your possible attribute values, based solely on the roll, are 1-15 and 17-24. This kind of thing plays havoc with probability curves, essentially breaking them — and while many players never notice, it *does* have an effect on the flavor of the game (as does every other game mechanic used).

In the 7th Sea system, probability curves are not smooth because of the extra dice rolled in cases of rolling tens. This lends itself to min/max optimization on the part of intelligent players, as they realize that certain optimizations can be made to maximize the benefits of the expenditure of resources such as experience points, to say nothing of subtle skewing of success chances that directly result from the lack of smoothness in the probability curve (it has “bumps” in it). The old FASA Shadowrun system with its open-ended multi-d6 rolls suffered a similar problem.

If the problem is not apparent to you yet, consider that it’s similar to the effect you get from the way attribute modifiers work in D&D: for every even number above 10, you get an additional +1 to actions affected by that attribute. For even the least powergaming prone of intelligent gamers, the temptation to maximize for advancement to even attribute numbers is difficult to resist — as improving your strength from 13 to 14 gets you far more benefit (in the short term at least) than improving your intelligence from 16 to 17, even if your character is a wizard. The D&D attribute system’s modifiers comprise a problem big enough to deserve its own treatment, of course, but for now this should serve as an example of the largely psychological negative effects a “bumpy” curve can have on the gaming experience — leaving aside the purely mechanics oriented effects a discontinuous probability curve in a roll can have.

#### better open rolling systems

The challenge, then, is to come up with a system that allows for open ended rolls without hosing up your smooth probability curves. I’ve played a few games over the years that employed systems that succeeded in this regard, and I’ve created more such systems than I’ve played that were created by others.

The White Wolf system used in the original World of Darkness games (I don’t know, or much care, what new system might be used since the “reinvention” of the World of Darkness) was well-designed in this regard. Instead of rolling dice, adding up all numbers, and comparing to a target number to determine a single result, one would instead roll Nd10 where N is determined by one’s attribute and skill numbers (I’m using generic terms — don’t bother reminding me of talents and knowledges, please). Each individual die roll would be compared to a target number, and a number of successes (if any) would be determined. The general rule was that one success would be enough to achieve your end, barely, and three successes (or more) would be a more substantial overall success. Rolling 10s on those dice allowed you to roll additional dice, which could then provide additional successes.

Another option is to have a simple “critical die” system, where you make your roll as normal (e.g. a d20 to hit) and roll a second die (e.g. a d10) to determine whether you get a critical threat. What you do when you roll a 10 on that second die is up to the imagination of the person creating the system — perhaps your base attack bonus is doubled, and you roll another d10 to see if you get to double it again; perhaps you double your total roll and roll another d10 to see if you get to double it again; et cetera.

It occurred to me back in about 1991 that every time you add another die to a total for a roll, you shift the upper limit of the curve quite a bit more than you shift the center of the curve — and, as such, it makes sense to add more dice to a roll if needed to open up the roll. To get around the problem of a broken curve, you could allow extra dice to be added not when you rolled the highest number on the die, but the *lowest* instead. With a sufficiently “small” die, you could actually do this without sacrificing any reasonable chance of getting a good roll when you get additional dice. For instance, if you roll d20 to attack in D&D, adding another d20 whenever you roll a 1 is kind of pathetic: you only have a 5% chance of getting a result that adds up to nothing more than shifting the possibilities from 1-20 up to 2-21. Who cares — right? With a d6, though, you have a one in six chance of getting 2-12. Wait — you get a one in thirty-six chance of getting 3-18. No, wait — it keeps on going.

What if the base roll is 3d6? Suddenly, you have a decreasing probability over an expanding curve of getting extra dice. It’s a slow progression, but depending on how difficult you want it to be to hit someone with an AC higher than 20 or so, that might work out perfectly for you. If you want a greater chance to get higher numbers, you can use something like 4d4 or 5d4 instead of 3d6 (and keep in mind that 3d6 actually tops out at 18 instead of 20, which might imply a greater effectiveness of armor and Dexterity, perhaps offsetting the chance of getting higher numbers offered by an open ended attack roll — all consequences of changing the game system that have to be considered by a conscientious GM). With Nd4 instead of Nd6, the per-die chance of getting a 1 result on any given die is even higher, opening up the high end of the probability curve a bit more, but at the same time forcing the bulk of the probability in the total roll further toward the center of the curve.

Dealing with critical rolls in open roll systems can be interesting. Obviously, you can’t usually just say “If you roll the maximum, you get a critical hit.” There is no maximum. Something I like to use as a determiner for critical hits is a comparison of the final roll with the target number, where a roll that adds up (with all modifiers) to double the target number means double damage, triple means triple damage, ad infinitum. This especially works well with standard hit point systems, where you not only want to make it possible for an attacker to hit a higher level character without trivializing just how much higher level the defender is, but also want to give that attacker at least a snowball’s chance in hell of winning. In fact, this allows for the elusive one-hit kill — something sorely lacking in standard D&D above, say, third level — without making one-hit kills so common as to make the game almost too lethal to ever bother playing.

It even helps mitigate the problem with linearly quantified damage systems such as in Pathfinder and D&D somewhat.

#### one for the future

I have another open die roll system up my sleeve that I’ll be using in Pathfinder in the future, coupled with the vitality/wounds system for handling damage, which I’ll describe in a future essay. I came up with it more recently because of the specific needs I have for improving on the mechanics of D&D/Pathfinder right now. The multiple smaller dice idea was something I came up with to deal with shortcomings more specific to how 2nd Edition AD&D handled things, way back in the day.

I like your system a lot. These issues have been subconsciously grating on me for awhile, but I didn’t think to put them into such clear words.

Another problem I have with the d20 mechanic that you don’t really touch on – the completely flat probability curve. Considering how nearly everything in life falls along a Bell curve, 4d5 makes more sense than 1d20. You’re much more likely to roll a value closer to the center. This removes some of the gigantic aspect of luck present in D&D. Say you need to roll a 16 to hit a target: with a d20 that’s a 25% chance, with 4d5 that’s a bit less of a chance. But an increase of 2 on the roll makes a bigger impact in the 4d5 system – think of like lowering a near-heroic feat to something that’s reaching, but still occasionally possible. Now, on the low end of the scale, say when you only need a small total to make a hit, an increase of 2 on the roll isn’t a big deal, since the feat is trivial. Yet it’s still the same 10% advantage on the d20 system.

At least D&D uses multiple smaller dice for calculating damage, thus trending a lot more towards the mean than towards the extremes. I just don’t know why they don’t do the same with the fundamental mechanic. 1-20 is, quite simply, giving luck a bigger role than I am comfortable with, and not just in combat: in all manner of non-combat encounters as well. Two unlucky diplomacy checks can completely throw off an entire planned plot.

Comment by Cyde Weys — 25 June 2008 @ 02:36

You’re absolutely right about that. I decided against pursuing the flat probability curve issue in this because I felt it would throw me off my stride, as it were — but I’m beginning to think I could have incorporated it without breaking up the flow of the writing, now.

Thanks for the commentary. Even if I didn’t write about that aspect, you did, so it’s available here for other readers.

Comment by apotheon — 25 June 2008 @ 05:35

Another good article. It’s great that you take the time to explain these things so fully.

Oddly enough, I’m not sold on the multiple dice idea. This is despite my constant references to Star Wars 2nd Edition, which used multiple dice (and one of those dice was open ended).

The reason for having a bell curve and open ended dice, to my mind, deals with the goblin peon problem.

Our adventurers will fight a lot of goblin peons over their careers and so those peons need to be able to harm the adventurers. But if you have 20 as an automatic hit, then the peons are punch above their weight, and probability being what it is over the course of their careers the player characters are going to get hit a lot, no matter how powerful they are.

There are two ways to deal with this. Using multiple dice (for a bell curve) or open ended dice is one way. The effect is to reduce the probability of the goblin peon hitting without removing it entirely.

But a more effective method, I think, and one which most games use anyway, is to have some kind of buffer between getting hit and disaster (i.e. death). Hit points are one such buffer. I’m not a fan of hit points, but I am a fan of buffers. But only for adventurers (and maybe important NPCs).

The bell curve for rolling multiple dice does have some pleasant aesthetics, but I’m not convinced at its necessity. Buffer is good for player characters to have; and from the other perspective, player characters always having a chance to succeed is nice in theory, but in my experience if the chance is that small then they don’t bother to attempt it, they just come up with something else; and if they are forced to attempt it then I think I’d be doing something wrong with my GMing; i.e. I was giving them challenges that were too hard. It can be annoying if you have more than two dice and are having to manually add them up. I can see, say, 2d10 replacing 1d20… but I wouldn’t like to go back to the Star Wars 2nd Edition mechanic of rolling 8d6 and having to add it all up.

There is one other advantage of the bell curve, though. It makes having the edge very important (essentially giving diminishing returns for advantages). It encourages scraping out every bonus you can get. Which can be a good thing.

I will think long and hard over whether to do 1d20 or 2d10 for elements eight.

P.S. One other method of reducing probability that was in an earlier version of elements eight… ultimately abandoned, but I haven’t seen anything like it since so it might be of interest. That had multiple d10s being rolled, and a difficulty number, but also a precision number. Instead of just having to get over 40, say, you would have to get between 40 and 44 for a full success (if you got over 40 but outside the precision, you’d get a partial success). The precision was decided by the complexity on the action; or, for melee combat, by the armour that was worn. If you reached the difficulty and had spare dice left, you could bounce the number up and down trying to get into the precision. For example, say you have 6d10 and you’re trying to get between 20 and 23. You get 6, 8, 1, 9 giving you 24. Too much! But you have 2 dice left. So you roll one of them and this time subtract. You get a 5. Now you’re at 19. Too little! One more dice, so you roll it, adding it this time, and you get a 3. This gives you 22. Hurrah!

Ultimately though, I decided this really would slow the game down too much, which is a pity because I quite liked the maths of it.

Comment by Shackleton1 — 26 June 2008 @ 01:23

Don’t decide on a system yet — you may find that you like the additional system I’m going to describe, at which I hinted with these words:

The precision system has some interesting implications, but the complexity seems a bit much in practice — as you ultimately decided yourself. Anyway, I’ll get to describing that other system soon, probably today or tomorrow.

Comment by apotheon — 26 June 2008 @ 07:36

This talk of precision systems reminds me of the board game Risk (I’m sure everyone in here has played it; it’s a classic). Remember Risk? Remember how ridiculously large the battles would get? It could take dozens of die rolls just to resolve individual encounters. It became so tedious that I wrote a program to resolve battles automatically; just input how many troops the attacker and defender have and it would keep iterating through random number generator rolls until one side had no troops left. Thus we didn’t have to do any rolling at all, and we focused more on the tactical decisions, as it averaged out over time that whoever had more troops in an encounter tended to win. It also took a lot of fun out of the game, and we stopped playing it soon thereafter.

You definitely don’t want to end up with a combat system that becomes so complex that the players have incentive to program their way out of it. That was the major flaw in Risk, and you don’t want to duplicate it. The precision mechanic also presents a problem of false choices. There’s always an optimal, easily calculable route to take at each step (do I add or subtract the next die roll?), so it seems kind of unfair to have the players go through those same robotic motions over and over and over. It would be going through the motions exactly like Blackjack (in which, at each point, there is an optimal strategy based on which cards you have and what card the dealer has). Of course, what makes Blackjack interesting is you have money riding on the outcome. In role-playing games, well, let’s say it’s better to spend less time on the dice rolling mechanics and more time on the role-playing :-D

Comment by Cyde Weys — 26 June 2008 @ 10:52

How exactly does one make a d5?

Comment by Anthony Lozano — 27 June 2008 @ 04:42

I suspect Cyde Weys meant “5d4”, not “4d5”.

On the other hand, I’d love to have some d5 dice around here, in the form of d6s that are numbered 0-5. I haven’t found any of those yet, though.

Comment by apotheon — 27 June 2008 @ 11:34

[…] my previous essay, Open Ended Rolls, I promised I would describe the open-ended rolling system I’ll be using for the forseeable future […]

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[…] point I made in a previous SOB entry, Open Ended Rolls, is that closed progressions are often problematic from a number of perspectives. Negative effects […]

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