The Forge Archives

Symmetry is a mechanism I devised and used the very first time I ran an AD&D game back in 1979, and I've used it in various house rules and homebrews for many different genres and systems ever since. In its original form, it used a d20 or percentile die roll against a table similar to the probability table in section 5 below. Essentially, this amounted to using a table based on an exponential decay function to convert linear die rolls into the mathematical equivalent of dice pool behavior. Much later I devised the mathematically equivalent, and more convenient, table-less version using marked dice described here. (However, if you dislike dice pool rolls and don't mind a table you can still do it the old way.)

Symmetry can therefore be regarded as a slightly unorthodox modification of a linear percentile or d20 roll, or it can be regarded as a slightly unorthodox dice pool roll. It bridges the gap between the two classes of system. Also, though some addition or subtraction may be needed to determine the dice in the pool, the procedure is entirely numberless once the dice are picked up.

As the name might punnily imply, Symmetry was originally designed primarily for Simulationist play in which factors contributing to the chance of success or failure are carefully weighed and never quite add up to certainty. However, I believe it could be a good choice in other modes of play as well.

This is a mechanism, not a complete system, though I may loosely use the word "system" occasionally in the description that follows.

1. The Mechanism

The Symmetry mechanism requires dice that are modified by marking some of the sides. The markings must be easy to distinguish from any existing markings or pips on the dice. Numbers and pips on the dice are ignored.

Base Die: This is any die with half the sides marked. One is needed.

Level Die: This is any die with half the sides marked. (This can be the same as the base die, but some variations might use a different base die.) About three or four are needed.

Modifier Die: This is a six-sided die with one side marked. Three are needed. These should be a different size or shape than the level dice to avoid mistakes.

All the markings should be the same (e.g. a mark of always the same color).

The base chance of success is 50%, which is considered a "modifier total" of zero. For each point of positive modifier, add one modifier die to the roll. However, after three modifier dice are added, another positive modifier replaces the three modifier dice with a level die. In other words, each level die equals four modifier dice, and this conversion should be used so that there is never more than three modifier dice in the roll.

If there are no modifiers, or only positive modifiers, then when the dice are rolled, any roll of a marked side means success.

Negative modifiers work in exactly the same way. For each negative modifier, a modifier die is added to the roll, trading four for one with level dice as needed. The only difference is that if the modifiers are negative, when the dice are rolled any appearance of a marked side means failure.

If there are both positive and negative modifiers, they cancel each other out so that the final modifier total is either positive modified, negative modified, or at the base 50% chance. So if positive modifiers have already been added to the dice to be rolled, negative modifiers remove them again, exchanging level dice for modifier dice as needed, until there are no more positive modifier or level dice left. Any further negative modifiers would then start adding modifier dice back into the roll again, making it now a negative roll in which a marked side means failure.

Note for those who don't like marking dice: you can use d6s for the modifier dice and d10s for the base and level dice, and consider any roll of 6 or above a "marked" side.

2. What Modifiers Mean

Symmetry takes advantage of the fact that 5/6 is approximately equal to the fourth root of 1/2. In other words, the chance of rolling a 1 on a d6 given four tries is pretty close to 1/2. (There's a little error, but it's not significant, as long as you replace the modifier dice with a level die whenever their number reaches four). The mechanism embodies the following probability rules:

- When the chance of success is at or below 50%, each negative modifier reduces the chance of success by a factor of 0.84. Each four steps of negative modifier (or one level) reduce the chance of success by half. This is true regardless of what other modifiers have already been applied, as long as the starting point is below 50% chance.

- When the chance of success is at or above 50%, each positive modifier reduces the chance of failure by a factor of 0.84. Each four steps of positive modifier (or one level) reduce the chance of failure by half. This is true regardless of what other modifiers have already been applied, as long as the starting point is above 50% chance.

In either case, a positive modifier reverses the effect of a negative modifier, and vice versa.

When a modifier moves the success chance from below 50% to above 50%, or vice versa, the effect in terms of the change to the chance of success or failure per unit of modifier is not as perfectly straightforward, since at the 50% line the derivation switches from being based on powers of the chance of success to being based on powers of the chance of failure. But nothing untoward happens. The curve meets up smoothly with no discontinuities.

The consistency of this behavior makes it very easy to set an appropriate modifier for a given advantage or disadvantage, whether you're building a system ahead of time or ruling on the fly. A +4 advantage represents, to the most consistent extent possible, a doubling of effectiveness or a halving of risk. A +1 advantage is, in every meaningful respect, exactly one fourth as significant as that. A –4 disadvantage consistently represents a halving of effectiveness or a doubling of risk of failure. A –1 disadvantage is, in every meaningful respect, exactly one fourth as signficant as that. These remain consistent no matter what other modifiers they're stacked with. This is probably Symmetry's greatest strength.

3. Opposed and Unopposed Situations

In Symmetry there is no difference in dicing between opposed and unopposed actions.

When an action is unopposed, the chance of success can default to 50% (zero modifier) or an initial modifer representing the inherent difficulty of the action can be assigned. Additional specific factors such as the character's skill and circumstantial detaiils can contribute additional modifiers. All the modifiers are summed together (either by mental addition or by physically adding and removing dice from the pool to be rolled). Note that there is no difference in handling between a modifier for the inherent difficulty (traditionally called the "base chance") and other modifiers that apply.

For an action opposed by another character, the same procedure applies. Each advantage on the side of the player making the roll, and each disadvantage on the opposing side, contribute positive modifiers. Disadvantages on the side of the player making the roll, and advantages on the opposing side, contribute negative modifiers. Only one roll is needed. The odds of success for either side are unaffected by which side is making the roll. (The chance to succeed at a roll of +X is the same as the chance to fail at a roll of –X.) Hence the name "Symmetry."

4. Behavior in Comparison to a Linear Die Roll

The Symmetry base chance (when only the base die is used) is 50%.This is equivalent to an 11+ (roll eleven or higher to succeed) roll on a d20.

The effect of one unit of Symmetry modifier (e.g. one modifier die) is very similar in significance to the effect of a +1 or –1 modifier on a d20 roll, when the total modifier is between about –8 to +8. Near 50%, the effect is greater, but it decreases toward the edges. The probability curve is S-shaped; it crosses the linear d20 probability curve in the center with a steeper slope, and crosses it again between +7 and +8 (and symmetrically, between –7 and –8) with a flatter slope.


 Symmetry
modifier  -9  -8  -7  -6  -5  -4  -3  -2  -1   0  +1  +2  +3  +4  +5  +6  +7  +8  +9
% chance  10  12  14  17  21  25  29  35  42  50  58  65  71  75  79  83  86  88  90

 d20 roll
target    20  19+ 18+ 17+ 16+ 15+ 14+ 13+ 12+ 11+ 10+  9+  8+  7+  6+  5+  4+  3+  2+
% chance   5  10  15  20  25  30  35  40  45  50  55  60  65  70  75  80  85  90  95


The Symmetry probability curve exponentially decays toward 0.0 and 1.0 probability for negative and positive rolls repsectively, but never reaches these limits. Each additional 4 points of modifier (one level die) away from the center brings the remaining chance of success halfway toward zero or certainty (from any starting point on the same side of the center). Outside the range of –8 to +8, the Symmetry probability veers away from the d20 probability dramatically, and unlike the d20 roll, there are no edges requiring special handling of marginal chances of success or failure. Modifiers totalling –16 (4 level dice) still allow a respectable 3.125% chance of success, while this would be "off the table" on a regular d20 roll. Modifiers totalling +16 yield a 3.125% chance of failure. And a +1 advantage has mathematically the same effect on that failure chance (reducing it by a factor of about 0.84) when stacked onto that +16 modifier as it would if it were the only modifier in effect, modifying a 50% base chance.

You can substitute the Symmetry resolution roll into any game system that uses modifiers on a d20 roll. The effect of the change is often almost imperceptible in the short term, but makes it a whole different game in the long term. Some of the implications of the change are discussed in part 7 below. The rationale for Symmetry's different behavior is discussed in this thread.

5. Behavior in Comparison to a (d6) Dice Pool

Symmetry is, technically, a dice pool mechanism. Its most unusual feature is its explicit handling of diminishing chances when unfavorable factors accumulate, in a fashion symmetrical to the way it (and most other dice pool systems) handle increasing chances when favorable factors accumulate.

As described so far, Symmetry also lacks some of the desirable features of many dice pool systems, most notably an indication of the degree of success or failure.

The closest dice pool analogue of the basic mechanism is a d6 dice pool in which success results from rolling a specific side (e.g. one or six) on at least one die. The Symmetry base die unmodified, with its 50-50 odds, is roughly equivalent to such a roll with four dice. Each positive modifier is equivalent to an additional pool die in the roll. Each positive level die is roughly equivalent to an additional four pool dice in the roll. The comparison breaks down for Symmetry with negative modifiers and for a pool of fewer than four dice. (Just for comparison, the odds of success rolling just one pool die are about the same as Symmetry with a –6 modifier; a two-die pool has similar odds to Symmetry with a –3 modifier; a three-die pool has about the same odds as Symmetry with a –1 modifier.)

6. Transitional Directions

Increasing the Granularity. The basic mechanism as described so far has about the same granularity as d20, AD&D, or Hero Wars. The chance of a single point or die of modifier changing the outcome is 8% or less.

The simplest and most dramatic way to increase the granularity is to eliminate the use of individual modifier dice entirely. A level die becomes the smallest modifier used, so any advantageous factor yields the equivalent to what would be +4 in the basic system. Thus, the chance of a minimum modifier changing the outcome is up to 25%, comparable to a system using a single d6 roll, a pool of coin flips, or FUDGE.

An in-between choice would make half of a level die (two modifier dice) the smallest modifier used. A ten-sided die with three sides marked can become the new modifier die, and it takes only two modifier dice to equal one level die (so you only need one modifier die in play). This makes the maximum chance of a minimum modifier affecting the outcome about 16%, about the same granularity as a single d10 roll, 2d6, or 3d6, comparable to Gurps and Champions.

Degree of Success or Failure The basic mechanism yields only success or failure, with no provision for degree of success or failure or for neutral outcomes.

It's possible to simply count the number of marked sides that appear in a roll. This can be made more mathematically consistent by altering the level dice, making each level die a d8 with three sides with a single mark and one side with a double mark. (If using the half level die just described, change it too, making it a d20 with five sides with a single mark and one side with a double mark).

However, this does have a drawback. At zero modifier and above, the marked sides represent successes so the worst result that can occur is marginal failure (no marked sides rolled). At negative modifiers, the marked sides represent failures so the best result that can occur is marginal success (no marked sides rolled). That's a narrower distribution of possible outcomes than most systems using degrees of success provide.

Wider-ranging results can be arranged, but it requires a different scheme of marking the dice involving a second color, and some additional interpretation of the roll. This might not be worth the trouble.

For a lower granularity degree of success result, such as for a "no and", "no but", "yes but", or "yes and" outcome, add an additional separate marking (say, an X)  to just one of the marked sides of each level die only.  Ordinary success or failure results from a roll with no X's. An X being rolled (which can only occur if the roll is at +/-4 or greater) represents extraordinary success or failure. Two X's being rolled (which can only occur if the roll is at +/-8 or greater) represents yet a higher order of success or failure, and so forth. This can easily be made the basis for toothy fortune in the middle mechanisms, with X's tradeable for larger benefits, or requiring larger concessions to reverse, after the roll.

7. Adjustments in Play

Players and GMs accustomed to linear or additive die roll systems will have to adjust to the differences.In Symmetry, on the negative side of even odds, the effect of a +1 on a player's success rate is constant, while the chance that the bonus will affect the outcome varies. On the positive side of the table, it's the effect on a player's failure rate that's constant, while the chance of the bonus changing the outcome varies. As common sense might suggest, a bonus (or penalty) makes the most difference when the outcome is most uncertain: that as, when the chance of success is close to 50-50. In other words, a small edge (or for that matter a large one) is most likely to make a difference between failure and success when all the other factors are equally balanced. This differs from a conventional linear modifier system, where the chance of a given modifier changing the outcome is constant while the effect of the modifier on the player's success or failure rate varies widely, being greatest when the chance of success is already very low or very high.

Players and gamemasters alike are used to modifiers having disproportionately large effects when success is either very unlikely or very unlikely. Players accustomed to a +3 advantage being likely to quadruple  their effectiveness against the very toughest challenges, and almost entirely eliminate the possibility of failing against the easiest ones, may have some adjusting to do when using Symmetry. An action that has a very small chance of success to begin with, like climbing a sheer wall, will still have a small chance of success even after many units worth of advantages are stacked on. For example, if a player's chance of hitting a certain opponent with a certain attack is 2 percent, then after gaining 12 positive modifier points of ability, that player's chance will have only gone up to 17 percent, rather than to 60 or 65 percent. Yesterday's arch-enemies will not so quickly turn into tomorrow's fodder for mass slaughter. (And yet, the benefit of a +1 modifier against an even-money challenge is actually larger in Symmetry than in d20. Weird, ain't it?)

In other areas Symmetry tends to increase effectiveness. By the same token that a few positive modifiers do not dramatically escalate the chance of success when the odds are already against it, a few negative modifiers will not dramatically escalate the chance of failure when the odds are already in favor of success. This means that characters with high core skills can use those skills with more confidence in moderately adverse circumstances. (This will only work, though, if the characters' skill scores are really as high as the intended level of skill implies. This may require participants getting over the habit of being cautious with modifiers. If you're trying to perform brain surgery, being a fully trained brain surgeon should be about a +40 modifier relative to the average person's ability. If this sounds excessive, consider that even relatively simple brain surgery should have at least a -36 modifier for inherent difficulty -- that is, an average untrained person attempting it would have less than a 0.1% chance of succeeding.)

Another likely noticeable difference is that characters with lesser abilities will no longer find it a waste of time to assist specialists, even though the specialists will still be many times more effective against difficult situations. For the GM, that means when an opponent is designed to be capable of defending himself against a specialist fighter, he won't turn out to be totally immune to attacks from non-specialist-fighter player characters. In general, most characters will have better chances than they otherwise would against challenges that are above their ability level. More important, gamemasters and system designers need not be so stingy about allowing modifiers into play, allowing characters to combine them, or applying the types of situational modifiers (such as a bonus for taking extra time or a minus for distractions) that add situational verisimilitude to events. Also, and perhaps most important for some play styles, systems based on Symmetry rolls tend not to break down at higher effectiveness levels. The overall system can therefore allow, for example, very high skill modifiers when it's appropriate to a character concept.

8. A Math Footnote

On the previous theory thread it was mentioned that the correct way to calculate combined probabilities involves multiplication or division, but in play it's too much trouble to do any calculation beyond adding or subtracting. Well, remember how you learned in math class that you could multiply and divide by adding and subtracting logarithms? The steps for doing so are:

1. Take the logarithms of the numbers you're multiplying.
2. Add the logarithms.
3. Convert the total from a logarithm back into a plain number by taking its inverse log (or exponential).

Believe it or not, that's what Symmetry does. Step 1 is when you look at a situational factor and determine its modifier. Modifiers are actually logarithms of the probability of their notaffecting the outcome. This makes determining the modifier sound difficult, but it's not. It's actually easier than in a linear system. When you look at a situation and say, "this factor makes the task a little more than twice as difficult, and since a –4 modifier reduces the success chance by exactly half, I'll make it a –5 modifier," you're estimating a logarithm.

Step 2 is when you sum up all the modifiers by adding and subtracting.

Step 3 is when you use the probability table to convert the modifier score into a probability or percent chance. That table, as I mentioned before, is based on exponential functions (inverse logarithms) which convert the log sum back into a probability. Of course, you don't have to use the table; picking up the correct dice for the roll accomplishes the same thing by creating a die roll with the same odds of success as the table would yield.

- Walt

Cool. The resolution for Story Engine, and Underworld systems is the same as the more granular method that you note, or in other words only uses level dice. Also the method for the Synthesis system by myself and JB Bell is the same, but it uses only actual opposed rolls. Obviously I'm a fan of such systems.

But in relation to the previous thread on "trancparency" of die rolls, these systems are not very conducive to easy calculations of precise probabilities. It's still much easier to know that a +1 modifier adds 5% to your total chance of successraising it from, say, 50% to 55%. My system further confounds things by rolling twice (once for each side).

OTOH, I am of the opinion that players don't really need precise figures. They just need a general idea. With your system a player will have an idea of how far from 50% they are based on the number of die they roll (and can refer to a simple chart if they like). So I think such systems are reasonable. And given the other advantages, I like to use similar systems, despite any "opacity".

Mike

I really like it.  I'm strongly in favor of "design complexity, functional simplicity"(from Masamune Shirow's Appleseed).  I like taking complex and really hairy design ideas and packaging them in such a way the players or users can use it without having to know how deep it can get.

Perhaps in case of "degree of success" it can be a second roll to determine Major/Minor, or some other defining factor.

Chris

Actually, I think Symmetry is pretty good in terms of transparency. Not as transparent as a linear scale, but more so than most dice pools. This is for two reasons: first, as Mike mentioned, the success chance is "anchored" at 50% when the modifier is zero, which is a very convenient starting point for understanding one's modified chances. Second, the curve stays pretty close to 5% increments through its central range between -8 and +8. If a player assumes that the probabilty of success is 50% plus 5% for each point of modifier (minus 5% for each negative modifier point), that player won't be wrong about his chance by more than 5% anywhere in that range.

Outside that range, the success chance is either above 90% or below 10%. The level dice in the roll provide a rough idea how high or low:

-3 level dice: 1 in 16
-4 level dice: 1 in 32
-5 level dice: 1 in 64

and so forth.

Thanks, Chris, for your comments. Symmetry definitely qualifies as design complexity. Whether it measures up in functional simplicity is the open question, I think. I've been using it for years, which shows it's at least usable but also means I'm not a very good judge of whether or not someone who hasn't used it before is likely to find it simple enough compared with other mechanisms.

- Walt

Walt,
Well, I think it's pretty simple, but I don't mind math an actually appreciate well thought out systems, like this one seems to be, especially since it wouldn't seem to hurt the search/handling time much.
I think some of the marking systems could get a little complex, but that's about it.
You could also just use the value of what is rolled on the base die as the level of success/failure.  I.e. the base die is a D20. Say 11+ is a 'mark'. Thus a 19-20 or 1-2 is critical success/failure depending on if it is a positive or negative roll. (I picked both ends so that the base die could fail, but other dice succeded and still get a critical success).  You could put more levels in easily, just make sure you choose a large enough base die.
This gives a fixed probabily of each level of success/failure independent of modifiers, which may or may not be what your looking for. (You could also do this for all level dice also, thus making large amounts of modifiers have a larger chance of an extreme result).  This may have been what Walt was getting at with the 'marking successes' thing. It kinda confused me.

Just out of curiosity, what game do you use this in/how does it work?

I like it very much; mathematically beautiful, and still easy to use. However, there is one minor detail I'd change in the text:

Quote from: wfreitagWhen a modifier moves the success chance from below 50% to above 50%, or vice versa, the effect in terms of the change to the chance of success or failure per unit of modifier is not as perfectly straightforward, since at the 50% line the derivation switches from being based on powers of the chance of success to being based on powers of the chance of failure. But nothing untoward happens. The curve meets up smoothly with no discontinuities.

(Emphasis mine.) Since we're talking a discrete function here, 'with no discontinuities' is a rather meaningless notion; continuity and discontinuity are only well-defined when on continuous sets like R.

Quote from: Lord DaemonSince we're talking a discrete function here, 'with no discontinuities' is a rather meaningless notion; continuity and discontinuity are only well-defined when on continuous sets like R.

Victor,

What Walt is saying is that, unlike other systems, there are no whole number results which cannot be obtained. For instance in RoleMaster with it's Open ended die rolls, you cannot roll a 5 or a 96. Lots of sysems have these, and they are more notable in situations where, for example, you roll a D10, and on a 10 roll another die and add it to 10. Thus making 10 itself an unavailable result. So the curve is "discontiuous" there. These are often not particularly damaging to a resolution system so much as just ugly (though occasionally there are versions that are broiken by this). Walt's system safely avoids any such uglyness or damage.

Mike

I know people have been really happy with Symmetry, with comments like:

"design complexity, functional simplicity" -- Bankuei
"I think it's pretty simple" -- Daimon
"mathematically beautiful" -- Lord Daemon[/list:u]Personally, I think it is rather neat.  Unfortunately, I don't see any of the touted advantage to using it.  (Nor do I see any disadvantage or 'other system bashing' necessitated.)

Quote from: wfreitagThis will only work, though, if the characters’ skill scores are really as high as the intended level of skill implies. This may require participants getting over the habit of being cautious with modifiers. If you’re trying to perform brain surgery, being a fully trained brain surgeon should be about a +40 modifier relative to the average person’s ability. If this sounds excessive, consider that even relatively simple brain surgery should have at least a -36 modifier for inherent difficulty

This means that Symmetry users will be juggling modifiers in quantities ranging from zero to forty?  The 'Broken Linear Die Roll Modifiers' beat that at least in terms of 'where the rubber meets the road' simplicity.  However, that alone would not deter use of Symmetry.  Don't get me wrong; rolling eleven dice (one Base and ten Level) isn't that intimidating for a role-playing game.  Symmetry works fine there.

Quote from: wfreitagThat means when an opponent is designed to be capable of defending himself against a specialist fighter, he won’t turn out to be totally immune to attacks from non-specialist-fighter player characters. In general, most characters will have better chances than they otherwise would against challenges that are above their ability level. More important, gamemasters and system designers need not be so stingy about allowing modifiers into play, allowing characters to combine them, or applying the types of situational modifiers (such as a bonus for taking extra time or a minus for distractions) that add situational verisimilitude to events. Also, and perhaps most important for some play styles, systems based on Symmetry rolls tend not to break down at higher effectiveness levels. The overall system can therefore allow, for example, very high skill modifiers when it’s appropriate to a character concept.

Okay, that's about the only strength I can see listed in Symmetry over the 'Broken Linear Die Roll Modifiers.'  And for that, we have to determine the quantity of and then scoop up as many as eleven dice?  (Although you cannot beat the 'handling time' of looking for only one marked die face.)  I don't see the process you refer to as "stingy" being that at all; I attach words like 'simple,' 'quick,' or 'easy.'

Okay, it could be argued that not having to be "stingy" means less work for whomever adjudicates, but the same can be said of the relatively short list of modifiers available in a 'Broken Linear Die Roll Modifier' system.  It's really only a matter of which fruit you prefer (apples or oranges)¹.

One main problem I have with the whole essay doesn't really have to do with the Symmetry system itself, but the justification of it.  (Actually?  If I were a rabid game creator, I would grab permission to use Symmetry in a flash and make whatever gritty game idea I had next with it.  It's that good of a system.¹)  The line, "that adds situational verisimilitude to events," is right out of sense to me.  Verisimilitude, in gaming, is about a certain kind of internal consistency; often such consistency rivaling how internally consistent reality is (or games ruthlessly designed to emulate reality, I suppose).

I can't see anything more or less verisimilar about Symmetry by itself.  While I think a really good argument can be made that the relativity of equal improvements to chance brought together is more reminiscent of reality, that's not verisimilitude as in gaming, that's realism or at least the 'feel of realism.'  (I mean technically, "situational verisimilitude" is almost oxymoronic; in gaming, verisimilitude is nearly only about the relative relationship between situations, about consistency.  A single situation cannot be anything but consistent with itself.  Or are we looking into the 'realism trap' again?)

The only other problem I can see with it (and this is a tiny one) is the 'increasing the granularity' concept.  Saying that any and all modifiers are of completely equal weight seems pretty granular.  Aspiring to the 'increased granularity' of aforementioned 'Broken Linear Die Roll Modifier' systems seems contrary to the stated purpose of creating Symmetry.  Worse, when dealing with modifier schemes ranging up to potentially 40 or more, there is an unconscious sensation of the decrease in granularity.  I mean if no one needs be "stingy" with modifiers any longer, then how can these modifiers be seen with the same granularity as say a d20 system?  This leaves me with a confused sensation over Symmetry's self-identification with any level of granularity.  In use, however, this is not of any concern; it mostly has to do with Symmetry's 'explanation.'

So I give Symmetry five stars for a good, new, dice mechanic (although a little dice heavy, but it makes up for that in 'reading' time).  However, I do take exception to the lengths gone to justify its existence; is it really necessary to define 'Linear Die Roll Modifiers' as 'Broken' to suggest Symmetry?  I can't really see one as better than the other.¹

Fang Langford

¹ It still doesn't make a very solid case that 'Linear Die Roll Modifiers' are 'Broken' in and of themselves.

Quote from: Mike HolmesWhat Walt is saying is that, unlike other systems, there are no whole number results which cannot be obtained. For instance in RoleMaster with it's Open ended die rolls, you cannot roll a 5 or a 96. Lots of sysems have these, and they are more notable in situations where, for example, you roll a D10, and on a 10 roll another die and add it to 10. Thus making 10 itself an unavailable result. So the curve is "discontiuous" there. These are often not particularly damaging to a resolution system so much as just ugly (though occasionally there are versions that are broiken by this). Walt's system safely avoids any such uglyness or damage.

I understand what he's saying, I'm just stating that the word 'discontinuity' is, both in his and your post, mathematically undefined. No real problem, but I'd change it anyway. :)

[Edit]

I stand corrected - a friend of mine who studies mathematics has just started ranting about how a discrete function can be continous if you define a well-chosen topology... or something like that. :o So uh... forget what I said. ;)

The reason the discontinuities Mike describes aren't usually damaging is that they occur at the extremes. The most likely place for a discontinuity to occur in Symmetry would be where the modifier kicks over from negative to positive, which is also where the derivation of the success probability changes. Since this happens at the 50% point where the modifier is zero, such a discontinuity would be more damaging.

Actually, I did mean discontinuities in the conventional mathematical sense. The success probabilities in Symmetry are discrete in play, but they function they're derived from is continuous. The whole system would be continuous if you were to allow fractional modifiers. (Of course, the dice rolling mechanism wouldn't work for fractional modifiers, but you could still use a table, or a calculator, with a percentile roll.)

p (probability of success) as a function of m (modifier):

p = 1 - ((1 / (2 ^ (m/4))) / 2)  (m >= 0)
p = (1 / (2 ^ (-m/4))) / 2        (m < 0)

So by no discontinuities, I meant no discontinuities in the underlying function. Its first derivative is always finite.

Note that you can create variations by changing the coefficients. For example, p = (1 / (3 ^ (-m/5))) / 2 causes the chance of success to be reduced by a factor of 3 for each -5 points of modifier.

- Walt

James*, I like your idea for an approach to a degree of success mechanism. I think I was too tied up with wanting the chance of a higher degree of success or failure to be dependent on the original odds. But as you pointed out, that's not necessary. If someone succeeds against steep odds, why not have the same chance of that success being e.g. "spectacular" as when someone succeeds in an easy task? The improbability of it has already been taken into account in setting the original chance of success. I like it.

This won't please everyone, though. In some systems the higher degrees of success usually occur when the basic chance of success is high, indicating that the character found the task easy and was able to embellish -- as in, if I'm all but guaranteed to inflict a wound on an orc every time I attempt to do so, I should have a good chance of doing something spectacular like slicing its head off with one blow. In most dice pool systems, for example, added dice that make a roll more likely to succeed also make multiple successes more likely. A fixed distribution of degrees of success wouldn't do this. Instead, though, the confident character could make the more spectacular action the goal of the attempt, accepting a negative modifier for attempting a more difficult feat. If that modifier swings the result from success to failure, well, he tried to get too fancy and the orc ducked. That's a rather old school fortune-at-the-end style. It works well within that paradigm. But it's not so adaptible for fortune-in-the-middle mechanisms.

(Obviously, I haven't used the Symmetry mechanism in a system with a degree of success mechanism before. So this is all speculative for me.)

- Walt

* edited -- Sorry, I originally mis-addressed this comment to Victor, having gotten confused between damion and Daemon (can't imagine why).

I was toying around with the system and something occurred to me. Here's an alternate system for those like me who like rolling more dice, or who may not want to mark their dice, etc. Simply, the system remains the same in all ways except for the following exceptions: just use one "modifier die" per modifier, and never use any level dice or base dice. Then always roll four more modifier dice than you would have under the normal system. So for "no modifiers" roll 4d6 looking for 1s or 6s (or whatever you prefer stands out). For a +3 roll seven dice. This emulates the same curve, fairly well, never being more than a couple of percent different (caused by that difference between the 4 modifier dice and the one level die that Walt mentioned). And it hardly matters as the curve in question is arbitrary anyhow. This does have a wierd little blip in the middle where "no modifiers" are 48.2% (same problem). Which makes that first +1 a pretty big swing (+12%). So for "no mods" you may just want to subsitiute a single even/odd die or flip a coin.

Another advantage of this version is that you only need one die type, and never have to do any division to figure  of how many of what sort of dice to roll. A flatter curve can be obtained by rolling d8s instead looking for, say, ones, and adding five dice to each roll. For D10s you'd have to add seven dice, for d12 eight dice, and for d20 add fourteen dice. Another valid method is to use d5 (d10s looking for either of 2 numbers) and add three. Unfortunately the D4 and d3 options have nothing that gets close to 50% (though there are some other interestng things you can do). The d2 option is of course the simplest and already discused, though we know that the curve is pretty steep for that one (a +1 is a +25% shift).

This is a lot of dice to roll. For unskilled brain surgery with d20 you need to roll about 125 dice. But given that you're just looking for any occurance of a single number it doesn't increase the time it takes to discover if you've succeeded much after rolling. Just adds to the amount of time it takes to count up enough dice. For a large number I'd just pick up several and start rolling until I succeeded, or rolled them all. So if I had a roll of 75d20 to figure out if I hit, I'd just grab ten and roll until the number I was looking for came up, or I had rolled seven times and five more dice. Odds are that you'll not have to roll more than two sets, anyhow. Most times such attempts won't get rolled; but the system works if you want to roll for it. With d6 the same surgery only takes 44 dice, FWIW, and most reasonable actions take less than 20 dice. The reason I like D2 is that it only takes a measly ten dice or so to get the same result.

Mike "Champions 400MPH movethrough" Holmes

Hi Mike,

That works. The whole level die = 4 modifier dice thing is perhaps a rather high price to pay for being able to say that every +4 of modifier exactly doubles/halves your chance of success/failure if you're not crossing the 50% mark. However, it also cuts down on the number of dice rolled, so it may be close to an even tradeoff. I could certainly see doing it your way. Or in between: continue to use the base die but otherwise use only modifier dice, omitting the level dice.

Here's the opposite extreme, using only one d20. It uses the following table, which is small enough to memorize:


modifier   -3  -2  -1   0   1   2   3
d20 roll   15+ 14+ 13+ 11+  9+  8+  7+


If your modifier is on the table, roll the indicated number or above on the d20 to succeed; if you roll low you fail. If your modifier is off the table to the left (< -3), roll high (50-50 chance) on the d20. If you fail, your action fails. If you succeed, add 4 to your modifier and continue. If your modifier is off the table to the right (> +3), roll high on the d20. If you succeed, your action succeeds. If you fail, subtract 4 from your modifier and continue.

I knew the +40 brain surgery skill would alarm people, but it's not as bad as it sounds. You'd only be rolling the eleven dice if a player insisted on (and the GM allowed) a roll for such a less than one in a thousand chance. It usually makes little sense to make such rolls, nor to require them when the chance of success is correspondingly high, very often. Notice that in my example, a typical brain surgery roll would be -36 for difficulty, +40 for being a qualified brain surgeon, net modifier +4. The big numbers would only appear if the overall system consistently indexed all skills relative to the average person's ability. More likely, in actual practice, you'd scale the skill modifier of the brain surgeon and the difficulty modifier for the surgery relative to the ability of the average brain surgeon, and only brain surgeons would even think to attempt it.

The number of dice is likely to be most inconvenient when the chance of success is between 1% and 5%, or between 95% and 99%; that is, when modifers are between 13 and 23 (positive or negative). In this range the roll could require up to nine dice (1 base, 5 level, 3 modifier) but there's still enough uncertainty to be (at least occasionally) worth rolling for.

- Walt

Quote from: wfreitagThat works. The whole level die = 4 modifier dice thing is perhaps a rather high price to pay for being able to say that every +4 of modifier exactly doubles/halves your chance of success/failure if you're not crossing the 50% mark. However, it also cuts down on the number of dice rolled, so it may be close to an even tradeoff. I could certainly see doing it your way. Or in between: continue to use the base die but otherwise use only modifier dice, omitting the level dice.

I thought of that about ten minutes after I posted. Fixes the 48% problem with all the systems and reduces the neccessary number of dice. Also, if you go this way, you can use the four sider or three sider if that curve appeals. I know a certain somebody who should be slavering over the possibilities that the four sider option gives from both a curve and dice aspect.

QuoteHere's the opposite extreme, using only one d20. It uses the following table, which is small enough to memorize:

Code Select Expand


modifier   -3  -2  -1   0   1   2   3
d20 roll   15+ 14+ 13+ 11+  9+  8+  7+


If your modifier is on the table, roll the indicated number or above on the d20 to succeed; if you roll low you fail. If your modifier is off the table to the left (< -3), roll high (50-50 chance) on the d20. If you fail, your action fails. If you succeed, add 4 to your modifier and continue. If your modifier is off the table to the right (> +3), roll high on the d20. If you succeed, your action succeeds. If you fail, subtract 4 from your modifier and continue.

Roll the level dice until you get to the narrow range, and then roll the final odds roll.  Nifty. Would go pretty quick. You could do a pool of these for graded effects, say always roll three. One success is minor, two is major, three is great. Or something like that.

QuoteI knew the +40 brain surgery skill would alarm people, but it's not as bad as it sounds.

I only included the example so that people would see what the extremes would produce using the proposed system. As I said, even though my system uses a lot more dice, it's not a completely unmanageable amount from the Champions player's POV in the reasonable cases. And again, much quicker to handle. The standard system uses hardly any from my POV, the only downside is the division step (which isn't really bad at all).

Just looking at the permutations.

Mike

Quote
For unskilled brain surgery with d20 ...

Now thats what I call mastering your tools.

Anyway, I like it a lot.  what about marking level dice marked sides with pips?  more level dice, take highest occurring pip value.  1 pip = a slightly better than default success, up to x pips for x spectacular success / horrific failure levels.  Just a thought.