This category includes Marvel Super Heroes and a number of old TSR games (Gamma World 3rd ed., the Conan game And ZeFYRS), plus Indiana Jones, Top Secret, and DC Heroes.
There is some interesting analysis of the MSH table at darkshire here
Tables let large ranges of values be assigned a chance of success (as in MSH, where scores range from a feeble 2 to cosmic ratings of up to 5000), and/or let a designer add various result levels with exact control over likelihood of them occurring (in MSH, likelihood of “Yellow”, “Green” or “Red” results, for instance). As such this gives ideal control over both success chance and achievement level, though the game may slow down while checking a table. A table may cross-reference either difficulty and statistic, or statistic and result level; doing both requires a series of tables (e.g. ‘slices’ of a table that actually extends into three dimensions).
Tables can be used to create more 'granular' increases in ability (i.e. preventing lots of point-counting), by controlling the number of rows.
 
James Bond's system was similar to MSH (though preceding it), with a score out of 30 assigned and multiplied by a difficulty factor to get a target % . A roll determines a quality number (1-4) which then modified effect e.g. damage.
 
A table system can have bonuses that work in multiple ways. A bonus can:
-increase the statistic being used to make the check
-increase the dice roll
-directly increase the result level (often called a "column shift" or similar).
Depending on table construction, increasing roll and directly altering result level can give very different results; this is a fairly unique feature of these games. (one effect of this is that, if desired, an 'opposed roll' might be between the rolls, ignoring the 'result level', so that e.g. effects of high skills are minimized; advantages might boost roll while skill boosts result lookup, or vice versa).
Perhaps comparable to with how in dice-pool games, shifting # successes required, # dice and target number all do different things.

Column shifts can sometimes be used to gear down or reduce modifiers, as an alternative to a flat + to a roll. For instance in 0D&D having a dwarf 'save as 2 levels higher' gives a boost probably less than a +2 bonus (which could also potentially scale differently between different classes).

Historical note: in a sense I think the 'universal table' games can be thought of as descending from the early 'matrix' approach used in D&D and the like, where various situations require different lookups of results (often attacker value vs. defender value e.g. to-hit charts in D&D); the universal table is basically an attempt to compile all the matrixes into one super-matrix, with enough caveats to cover most or all potential situations. Matrixes themselves presumably derive from Combat Results Tables (CRTs) like those seen in SPI wargames. This evolution itself may be most evident in 3E gamma world, migrating from lots of individual mini-tables into one.

Tables as a supplementary mechanism: the games above use tables to set the basic probability of an action succeeding itself. As an example of a table used as an accessory only, Chill starts with a basic d100 roll-up system. If the roll success (under the %) but a level of achievement is also required the player calculates the margin of success [skill percentage minus actual roll] and looks that up on a table to get a result.

EDIT NOTE: J Arcane later on here mentions Alternate Realities' DRF system - ' an actual trigonometric function you plugged your stat value into to get the target number on d100'.

Previous edits: added James Bond mention (thanks Jibba Jibba): tables as a supplementary system

DC Heroes 2E, which came in boxed format, apparently had a 'result wheel', a physical wheel that could be turned to line up acting and opposing numbers instead of cross-referencing - so equivalent to the table but more cool.

Not to discourage, in fact I hope this would encourage and help structure your efforts, but you could do worse than to look at John Kirk's collection of RPG Design Patterns from http://legendaryquest.netfirms.com/ (Note: there's a wiki and a PDF download. I'm only familiar with the latter.)

It's somewhat influenced by and reverential towards Forge theory and games, but fortunately not in a way that impedes clarity.

Quote from: Elliot Wilen;496084Not to discourage, in fact I hope this would encourage and help structure your efforts, but you could do worse than to look at John Kirk's collection of RPG Design Patterns from http://legendaryquest.netfirms.com/ (Note: there's a wiki and a PDF download. I'm only familiar with the latter.)
 
It's somewhat influenced by and reverential towards Forge theory and games, but fortunately not in a way that impedes clarity.

Hi Elliot. I've read it, actually (probably why I described Star Wars as using a "safety value" mechanic )though I am unfamiliar with a number of the systems he references, and leery of his conclusions in places (like his discussion on how to "simplify" damage on pdf page 47, by applying to-hit overflow onto damage - since IMHO this adds a subtraction step to every hit roll). I do like the idea of design anti-patterns in an RPG context - pity that wasn't explored in more depth.
 
I'm planning on a different focus - looking more at the specific innovations of "fantasy heartbreakers"; hopefully by working through subsystem-by-subsystem I'll get a better view of context than if I were working it through pattern by pattern (divorced of context) even though some patterns may repeat themselves across different systems.  Thanks though, and I will promise to keep it in mind where relevant.

Interesting idea,  I look forward to following this thread.

Cheers : )

This section refers primarily to games which roll a number of dice and then count "successes" by finding rolls over a given target number (There are also 'die pool' systems which either take the highest roll out of a set of dice, or which are additive i.e. which just add together all the dice rolled; see below for more on these).

Shadowrun (1989) and Prince Valiant (1989; actually coin based, using total of stat+skill to determine pool) were two of the first of these, although Space 1889 also used dice pools for certain tasks (combat used target numbers, though the pool of d6s was typically used additively). EDIT : Chainmail used alot of d6s too, although its arguable if its really a dice pool, as each d6 rolled represented a separate attack. Shadowrun 1E used either attribute or skill to give number of dice (for untrained skill use or use of the wrong skill, penalties were determined using a relationship flowchart), while Prince Valiant and Storyteller use [stat+skill], giving slightly larger pools.
Systems can also have # dice determined by stat, with skill determining a base target number, or vice versa e.g.- Arrowflight 1E (2E differs) uses D6s equal to stat while skill sets the base target number to roll under.

Die pool systems are particularly good for determining how well a character succeeds, but less good at determining if a character succeeds – probabilities are less transparent and in practice designers tend to allocate # successes required for tasks in a fairly ad hoc fashion. ‘botch’ chances using 1s tend to do odd things.
Some systems independently vary both target number and # dice (oWoD – particularly Werewolf- and Shadowrun 1e/2e). This unpredictably/erratically increases the number of dice needed to get a given # successes and is an approach that should probably be used rarely. Shadowrun uses d6s and allows target numbers (TNs) over 6 (6s add and roll over).
This had a slight irregularity in that TN 7 was no more difficult than TN 6 (since there was a minimum of +1 when the d6 rerolls); shift from TN 7 to TN 8 also had much lower effect on probability of success than (say) 4 to 5, or 5 to 6. The hillbilly mecha game "Junk" was similar but rather than roll-and-add, instead required a minimum score on the reroll dice, based on the final TN e.g. TN 7 = needs a 3+ on the reroll success; equivalent to rerolling and adding using [d6-1].

oWod/Shadowrun dice pools also have some mathematical issues - e.g. see tgdmb post by sabs here on the problem that the descriptions given to skills in oWoD (3 dice = expert, 5 dice = top of field) don't match increases in dice pools, which usually are minor (or obscured by variation in attributes).

Dice roll systems integrate particularly well, with even damage etc. following the same universal system. Attack roll successes may add to Damage successes giving DEX a percentage flow-through into damage (less than Strength since successes are normally converted into extra dice, and have to be rolled again), without the math being excessively clunky.
Dice pool games are virtually never level-based: dice pools are fairly granular due to maximum pool size rapidly becoming unwieldy, making it difficult to have a reasonable number of levels. Dice pools also make "soak" rolls easy to implement, so such games are less likely to have open-ended hit point totals. Storyteller does give each of the character types a sort of 'key attribute' which is core to them and somewhat like 'level' - Generation in Vampire, Arete in Mage, Rank in Werewolf - but only somewhat.

Player errors in how many dice they should be rolling here can be fixed easily if they err on the low side (roll more dice), but tricky if they've rolled too many. This makes modifiers that subtract from # dice rolled potentially more tricky (its being easier to forget a modifier applies, than to just make a mistake in adding up skill+attribute or whatnot).

Applying Modifiers: Dice pool games may have target number shifts, bonus dice, bonus successes, conditional bonus successes (10 counts as 2 successes, 10s give a bonus dice, etc) or a total reroll which all do quite different things, and which synergize with each other to a varying extent and in different ways. "Reroll 1 die" may have little effect on success chance at TN 10 in Storyteller, but can massively decrease chance of a "botch" (excess 1s), while +1 success might guarantee success regardless of difficulty in some systems. A TN shift might have little effect or even no effect, or significantly change successes - TN 6 to TN 7 in 1E Shadowrun has no effect on successes, since 6s always rolled up by +1d6, while a shift from TN 5 to TN 6 halves expected final number of successes).
Dice pool penalties might cancel automatic successes instead (making these easier to lose) or just apply to the pool.

Critical fumble rules vary in dice pool games - generally '1s' rolled give some sort of negative, and a problem could occur if the whole pool is 1s, a majority of dice are 1s, or 1s may subtract from successes with a negative overall result being bad.

Other dice pool elaborations or examples:

*nWoD uses a fixed target number generally, however a character who would drop to a negative die pool can roll a single die (at difficulty 10 instead of the usual to succeed, known as a 'hail mary'. In a sense then, this is similar to rolling 1/3 of a die.

*Shadowrun 5E has, for reasons, rules for "limits" where a character has a maximum # of successes [hits] that can be generated regardless of the dice pool (excess successes are lost]. To be fair this lets them balance out some of the less valuable stats (Essence modifying social limit?). It does however seem to me problematic in that it is removing some of the purpose of rolling dice - it would be better to have a mechanic that only rarely generates outliers, but where those are at least possible.

*Shadowrun 4E has rules for 'glitches' - if a character succeeds but more than half their dice pool consists of 1s, they still succeed but with some drawback. This result might be comparable to 'marginal success' (roll exactly what you need and no higher) in a d20+mods, or d20 roll under system, except that chance of a glitch occurring decreases with greater skill in SR4.
Comparatively in Deadlands [though...this is a take highest step-die/dice-pool, see step dice] a character rolling a 'majority' of 1s in their dice pool 'goes bust' (Fumbles), and this overrides the normal success result.
(Characters in Deadlands can also have a fatal illness flaw [Ailin'] forcing them to roll a Vigour roll anytime they 'go bust' or die).

*Dice pools can also deliberately give negative complications e.g. from each extra '1' rolled, to discourage the maximum number of dice being rolled (e.g. area attacks in Marvel Heroic leading to collateral damage).

*The Ubiquity system (Hollow Earth Expedition) has a dice pool system where results are 0 or 1 (unlike say storyteller, 1s or 10s aren't ever specifically counted - only successes or failures). 'Ubiquity dice' let large dice pools be compressed into fewer dice e.g. a d8 labelled 0,1,1,1,1,2,2,3 (IIRC) duplicates the roll of 3 normal dice.

*Humandyne XdX (unpublished?) lets players choose # dice (1-10) and dice type (usually d6), with evens = +1, odds = -1, 1s = -2 and max roll = +2. A trait rating cancels that number of "-1s". High number of dice gives more potential variation, though is less risky for characters with a high trait.

*The Spartacus boardgame (though its a boardgame) has a sort of interesting system for opposed rolls. A character has an 'attack' or 'defense' as a number of dice e.g. 2d or 3d; rolls are opposed, with both sides pools arranged highest to lowest, and the individual dice compared to each other. For example an attack of 4-4 might be compared to a defense of 5-3; the first defense roll is higher (5 vs. 4 and so is blocked, while for the second die attack is higher and so a success (wound) is inflicted (causing loss of one die from one of the target's stats). To prevent a larger attack pool being overwhelming, an 'unopposed' dice needs to be 3+ or is ignored (e.g. an attack of 6-5-2 vs. a defense of 6-5 would deal no damage; the last attack die is too low). Characters could also have special abilities triggered by e.g. doubles or triples, or rerolls. (Compare also resource-based dice pools - Dogs in the Vineyard being similar but more complex).

*Mutant Chronicles 3 (as I understand it) uses a '2d20 roll under system where attribute + skill = TN, and a roll under skill only counts as 2 successes. Characters can sometimes get double successes for a roll ("Focus") or an extra die; some tasks can therefore require >2 successes. Failure can also be penalized separately (if any dice fails).Because two dice are rolled, attributes normally range only from 6 to 12. The other interesting thing is that 'Chronicle Points' when spent give a bonus 'roll' set to 1; hence this counts as one extra success normally or two if a character has skill.  (Post here by JoeNuttall suggests a 3-die system that's similar, useful to give steeper differential in success rates between skill levels). A similar but simpler system is also used in Ron Edwards' "Elfs" game: this uses 3d10 rolling under [attribute or sum of two attributes], giving a range of 0 to 3 successes on any roll.

*Atomic Highway uses a success-counting system where attribute = number of d6s rolled, 6s are successes, and skill adds extra points which can be spent raising rolls to 6s. Natural 6s give a bonus d6 roll (which can't be adjusted). Raw attribute checks use stat as extra points as well as dice. Opposed rolls tie on # successes, broken by highest stat, then highest skill.

*Hypothetically, I could imagine a system that would use say stat 1-10, with a pool of dice by skill and an extra single 'die result' set as equal to the stat score (i.e. Dex 7, archery 3 dice...if you rolled 4,2,6 you would get a result of 4,2,6,7). This would only work with both variable TNs and tasks requiring different #s of successes (or an opposed roll) however - it would be essentially combining 'diceless' and 'dice pool' and so could be too deterministic if 1 success is all that was needed for most tasks. It would combine well with special abilities like '+1 to a single specific dice roll' (making the 'stat die' extra success less of purely yes/no) although these could be too fiddly. Potentially this could work as a houserule for Shadowrun 1E, since it by default has no attribute bonus to skills.

*Twilight 2013 reportedly uses a d20-dice-pool (unusual, though that's in principle no different from other pools).

*games rolling stat+skill, such as Storyteller, would make it possible to work out how much of a given success is due to skill and how much to raw ability, if desired.

Combining dice pool / count successes: hypothetically these could be combined, i.e. roll a 'pool' of dice vs. target number X, rolls of X+ are added together. e.g. 3d6, TN 4+ ---a roll of 4,6,2 would give a result of 10. Generally would work much like a 'count successes' system probability-wise, but slower for players due to addition; would allow for more fine detail e.g. a character could have a rating of [3dice+2]. Rerolls could be useful even where a roll is a 'success'. A character could also have an ability to 'keep unsuccessful roll' (adding it toward their total) which would be unique to this.
Adding exact rolls could also be an optional resolution system, e.g. for damage, or for replacing a multiplier of base successes (such as x2 successes = jump distance) with just success-counts used to see if something works.


Custom Dice Dice-Pool (=essentially success-counting)
A couple of FFG games (Warhammer 3E and Star Wars: Edge of the Empire) use pools of custom dice, with varying symbols. Dice can be replaced with more awesomer dice, or negative dice (difficulty dice) can be added. Edge of the Empire apparently adds more complications/interpretable complex results rather than success/failure. Probabilities are difficult to understand by players without knowing all the dice; pools are somewhat metagameable by players (difficulty modifiers can't be easily hidden - without the GM rolling them, anyway). EotE requires more meta-game interpretation of dice results into flavour text, rather than success/failure complications appearing emergently.
More simply, Tech Noir just has dice pools which (I believe) include specifically positive dice and specifically negative dice (e.g. a negative complication adds more negative dice to the pool).

One Roll Engine: Another sub-variety of dice pool system is the “match counting” system as used in the One Roll Engine (REIGN, Godlike, etc). To succeed in ORE a character needs 2 or more dice to face up the same number. This provides two readouts a “width” (how many successes) and a “height” (what the number is); The system is particularly good for doing Hit Location (i.e. a character a rolls 3,3,4,4,4,7 might opt to target a character in their “4” location for 3 successes of damage, or in their “3” location for 2 successes of damage. The system handles differing task difficulty only with some trouble - success or failure primarily depends on character talent rather than task difficulty - cannot have “fumbles” in combat (since 1s are “left foot” shots); and at least in Godlike, the additional data available is in practice rarely used for most non-combat rolls. In combat it has the slight problem that base damage and initiative are both based off roll "width" (i.e. shots that will kill someone automatically go first).
Rolls are fairly prone to failure, although the game suggests characters only need roll for tricky situations and it is hard to adjust for difficulty (one mechanism is to also require rolls be a minimum size e.g. difficulty 4 = matched 1s, 2s or 3s couldn't be counted - which doesn't work in combat as well, except for cover, since it puts some hit locations off limits). Roll 'height' is also used in opposed rolls; opposing sets become 'gobble dice' which cancel opposing sets only as long as the value of the dice in the set is equal or less.
One ORE dice pool can potentially resolve two or more actions (the default rule being that the pool is at -1, and separate counts of matches can be made for each action) although this can't give the two actions different base chances of success / dice pool sizes.
Match-counting success systems have fairly tight constraints on dice pool size: ORE limits rolls at 10 dice for instance as a double will always occur with 11 dice. This becomes tighter again with d6s (cf. Enchanted Tales, below; it gives out rerolls sometimes rather than extra dice, increasing the odds without guaranteeing a match).
(An interesting idea for ORE might be to include 'wild dice' to randomize outputs. For instance, your pool of 8d10 could have a red die and a blue die. This could give 'sets' of matches of a different colour - like 'slashing damage' vs. 'fire damage' for a flaming sword. An ORE-type hit location system might also be added to a damage dice pool without using the full ORE mechanic - see Hybridization for ideas. ORE is also referenced in the 'constrained design spaces' post.)
ORE struggles a bit with applying 'difficulty'. Another idea for adjusting difficulty with a match-counting game might be varying die-type - increasing die size decreases # matches - although this is mathematically messy, doesn't sync with ORE hit location, and has the feature (or bug) that the threshold for 'automatic success' drifts up and down ([dice size+1] die pool means a guaranteed match, i.e. 7 dice for d6s).

"Weapons of the Gods" is similar but breaks ties based off the number showing on the dice i.e. 3 rolls of 7 is treated as a '37', two 4s is a '24', two 6s is a '26', etc.
24-hr challenge rpg "Enchanted Tales" uses a d6 dice pool with a Poker-/ Yahtzee style ranking system rather than simply counting matches. By difficulty tasks may be Easy (one pair), Simple (Two pairs), Average (Three of a Kind), Tough (Full House i.e. triple / double of a different number), Hard (4 of a kind), or Impossible (5 of a kind). Damage eschews this to just add together some d6s. Without fully analyzing this, I think most likely unlike with cards where a limited number in the deck makes higher multiples significantly rarer, the mixed-pairing results are relatively rare and increase chance only slightly.
Here someone combined 'count success' and 'count matches' with the idea that as well as a 'pair' counting as a success, so would a roll of 6.

recent edits: 3/10/2015 Spartacus note; Mutant Chronicles note; Enchanted Tales. 25/10/2015 - added ORE links
26/10 - mixed dice pool/synthetic dice, minor edits

Take-Highest Dice Pools:
...and I've completely missed systems where characters roll a number of dice, and take the highest (instead of counting them or adding them together! Thanks RobMaudib for pointing this out.
Known examples include:
Silhouette (Dream pod 9): as described by RobM here, skill level determines # dice, with highest roll taken and +1 added for each additional "6" rolled; attribute then adds to this.
Last Unicorn Games Icon system (e.g. Star Trek RPG) did the opposite, where attribute determines # of dice and skill gave a flat add; it also included a "drama die" which fumbles on a 1 or on a 6 let a character add the second-highest die, as well as the first (which would be 6).
Blue Planet also gave characters "Aptitudes" (rated from 1 die to 3 dice), with stat & skill both adding to the highest roll.
As can be seen, this sort of system is good for giving different effects to attributes and skills.
They are good for modelling situations where any character should have a chance of success, and where modelling extra benefit for higher scores isn't necessary (e.g. effect isn't a consideration). They also provide some inherent 'bonus control'.

Soothsayer (an obscure Australian RPG) uses take-highest dice pool for armour absorption, where highest value is used, with +1 for each additional roll of the same number. This was interesting in that a roll of say 4,3,3,3 (=6)gives a higher result than 5,3,3,3 (=5). (It was also interesting in that the armour roll operates differently to the damage roll).

Sorcerer (from what I hear) operates by having both sides of a conflict roll dice, with highest results on both sides cancelling out until an uncancelled roll remains.
(Altus sees to do this as well, with # dice and dice type generated from a table, based on relevant skill level).

Japanese supers RPG Double Cross uses a d10 take-highest dice pool system where any 10s rolled are re-rolled with the result added to 10 (if any are gotten in the second batch, those then are re-rolled and added to 20, etc). Some powers allow the 'critical threshold' to be lowered below 10.

Costume Fairy Adventures uses a dice pool where a pool of dice is rolled, and highest under trait score is used (i.e. a 'blackjack' system where more dice means scoring closer to the trait value).

The "Tribute" system in development has a complex setup which combines counting and take-highest.

There is also some discussion here of a combination percentile/take-highest dice pool system (perhaps similar to CoC 7E).

"Take highest" systems are fairly constrained in how many dice they can roll - chance of success rapidly approaches 100% and extra dice give little benefit (although kludges like "each maximized die rolled adds 1 to total" help). If there is no additional modifier to the dice result, chances of ties increase between opponents with large dice pools (the highest number is the most common "roll", if pool is 2+ dice). This can be moderated somewhat by taking the second highest roll, instead of the highest. The average result with 'take highest' is somewhat hard to determine - this makes it difficult to convert an opponent's roll into a fixed target number to cut down dice rolling (e.g. the way a D20 + bonuses game could just declare that a villain with +7 grapple has a 17 grapple defense). Note that unlike with 'success counting' systems, splitting a dice pool doesn't reduce chance of success (the highest result will be in either the first subpool or the second) so pool splitting can't be used for multiple attacks without additional penalties or kludges (extra 6s in a d6 pool each giving a +1, etc.).

Fantasy Dice (described under 'changing die type' games) uses a take-highest dice pool, with ties sometimes broken by second-highest dice. This has the added complication in that some cases an unopposed roll (attack) is made against fixed difficulty (i.e. no dice rolled - so no second roll). A default is used in this case.

Capes, Cowls and Villains Foul uses a mix of d12 dice pool and additive - traits have a number and an irregular (?) track which with use reduces the # d12s rolled. Penalties/overuse can drop dice pool below 1, in which case the worst die is used, much like Over The Edge (cf. multidie additive).

While its a co-op boardgame not an RPG, Legends of Andor shows an interesting way to implement a penalty here. Combat is a roll of 2-4 dice (take highest) for most characters; the Archer instead rolls one-at-a-time and can reroll until they run out of dice, meaning that a moderately high roll can get blown rerolling trying to score a 6. Some monsters / PCs with the helm can also add together doubles, while the Archer can't.

Another possible variation for a take-highest dice pool system is where dice are distinguished, for instance by colour or by position. One idea I'd had for instance, is to roll a set of attack rolls (4 dice) with the distance these land away from the player determining the hit location e.g. furthest away the head, then arms, torso, and finally legs. Simply choosing the highest result would give effectively random hit location, while higher skill lets a character perhaps still hit with a worse roll.

Yggdrasil (Cubicle7 translated) apparently uses a d10 dice pool where the best two dice are added (so not unlike Cortex+ except only d10s are used). A stat of 1 is obviously going to be disadvantaged, whereas each increase beyond 2 is less significant.

Resource-based Dice Pools
Some games also use dice pools as a resource of potential dice. Examples include Dying Earth and Dogs In the Vineyard; both handle checks as extended, opposed affairs; a conflict is continuing with the opponent putting out dice from their pool until they pool is exhausted or they cede the conflict. Dying Earth burns rolls usually one at a time, while DiTV lets characters use more than one and has multiple options (e.g. "raise", "see", and "reversing the blow" depending on how many dice are burned).
Note these are slightly different in that dogs (DiTV) uses up the whole pool in one 'conflict', while in Dying Earth a characters' dice pool is tracked in an ongoing fashion (over multiple scenes), and has varying refresh conditions. DiTV has been criticized for the way its dice pools work: d4s are described as 'complications', representing negative features, but actually do increase the chance of a character succeeding on checks.
 
Final (mathematical) note on probabilities: success-counting die pool systems have 'diminishing returns' in the increase in base odds as the dice pool increases, for 1-2 successes or so. e.g. if rolling d6s TN 4, going from 1 dice to 2 increases odds of getting 1 success by 25% (50% to 75%), from 2 to 3 by 12.5%, and from 3 to 4 by 6.25%. Hence each added dice has less effect on the final probability, despite the total number of expected successes increasing linearly. For minimum 2 or 3 successes etc. the same effect occurs beyond a certain point only - at the lower end chance of success with a small dice pool will be very low.
By contrast in a single-dice-roll game, an increase in skill level will generates a proportional increase in success chance.

Dice pools as a supplementary mechanism
Extra dice are often added via rerolls. Variants include:
*spending points of some kind to get a reroll. See 'Safety Valves'.
*D&D 5E uses a system of 'advantage/disadvantage' which lets a character roll twice and take best or worst if the task should be easier or harder than normal, according to Mearls reducing the number of possible fiddly modifiers (though potentially adding large numbers of extra rolls if you're surprised by something with lots of attacks). Barbarians of Lemuria (BoL) had this earlier with 'bonus dice' and 'penalty dice' (changing from 2d6 to 3d6). Another variation is where there are positive/negative dice and possibly multiple for either (as noted next post for Nexus).
While generally this just increases amount a roll succeeds by, a couple of experimental feats 'count successes' i.e. the 'fell handed' feat [Unearthed Arcana] lets a character with Advantage knock a target prone if both attacks would have hit.
*A system could have checks with dice rolls lower than e.g. [attribute score] being re-rolled, either just once [freerpg Incandescent example] or until a higher number or equal to than the attribute is rolled.
*Savage Worlds PCs ('wild cards') get an extra d6 to roll at the same time as their skill or stat die, but don't always get to roll this e.g. with full auto multiple shots only one wild die is rolled.
*Skills & Powers for AD&D 2E has a subsystem where high stats give extra rolls on stat checks, with some tasks potentially requiring multiple successes.
*one homebrew game I knew of, 'Brightblade', had at one stage a dice pool for determining fumbles on top of a d10 additive system. IIRC, in the event of a failed roll a '1' was a fumble, and rerolled to see if you got another 1, but tasks which the GM deemed likely to be problematic also caused extra fumble dice to be rolled and any extra 1s counted to determine fumble severity.

Lots of dice can also be rolled in extended checks such as Alternity or 4E D&D 'skill challenges'; sometimes multiple checks/dice pools could be used in less formal ways for specific subsystems - Palladium's has a rule for 'roll 2 out 3 successful coma/death rolls to recover', the GDW House System (Twilight 2000 2nd Ed, Cadillacs & Dinosaurs) uses a multiple D6 count successes dice pool for automatic weapons fire to determine number of hits, instead of a skill check (normally roll under with d10).

EDIT Notes: thanks Rob for notes here. Recent notes: 'fell handed'[5E]. Added note on Yggdrasil (*).

Kudos on starting this thread. It's a good read.

For attributes, what about fixed arrays like 3.5s standard and elite arrays?

Quote from: beejazz;496177Kudos on starting this thread. It's a good read.

For attributes, what about fixed arrays like 3.5s standard and elite arrays?

Cool, thanks.

Yes good. I'd consider these a way to speed up 'point spend' purchase systems; though it is possibly to generate a set of arrays and randomly roll which array you get, to randomly generate attributes that are still balanced by total points.

(edit note - sorry beejazz, deleted my reply to your post above to fix the page breaks between dice rolling and combat sections).

Some other exotic varieties or variants...
 
"Roll your modifier" systems - these are additive systems where the normal dice result is typically 0, so that a character can roll above or below their default rating. The interesting aspect of these is that a total die roll is theoretically on the same scale as an attribute score. 'attacking' and 'defending' values follow the same scales and can be used in unusual circumstances readily.
 
One downside here is that again, multiple rolls are difficult to make individually since dice must be paired/grouped; most of these generally assumes a group of mooks use a single die roll. Another is that with a tight range, rolling the dice may feel like a waste of time (why roll at all?).
Examples include:
*Fudge (uses 4 'Fudge dice' labelled 0,1,-1 twice) - later variants such as Fate typically use [d6-d6],
*Feng Shui (which uses d6-d6 with 6s exploding - meaning a 1 or 2 point modifier has a hugely significant effect on the probability of a task succeeding, but with explosions still occurring quite frequently that result in massive dice rolls).
*Two Fisted Tales (d10-d10)
*ShatterZone/Masterbook: this uses a roll of 2d10 and table lookup to determine the modifier, rather than a positive and negative die.
*Jasyn Jones' TORG variant (Storm Knights) (in development) apparently uses a 'hot' and 'cold' die - instead of subtracting one die from the other, the smallest of the two rolls is used with the 'hot' being positive and 'cold' negative (an identical distribution to subtracting). Babylon 5 also had the hot/cold concept, but (judging from rpgnet review) apparently used the higher of both rolls instead.
*Qin it is said uses a black (yang) d10 and a white (yin) d10. One adds and one subtracts; if both are equal and non zero a task automatically passes due to "yin and yang being in harmony" while if both are 0 the task fails due to the character being "out of touch with their chi". Which is higher may have other effects depending on the task.
 
In some of these e.g. FUDGE which is normally 4 Fudge Dice, it is possibly to easily modify the variation of a roll without changing the average result e.g. by adjusting how many dice are rolled - e.g. Mutant Bikers of the Atomic Wastelands' NPCs. Another implementation might be to replace a standard [d6-d6] with a [d8-d8] or [d4-d4], depending on the situation; also, potentially the positive and negative dice could have their sizes adjusted separately by different conditions to give a range of d4-d6, d8-d4, etc.
Feng Shui's predecessor Nexus reportedly (here) used d6-d6, with a Luck/Unluck trait that gave additional positive or negative dice respectively with the highest die in the positive or negative pool counted -an interesting variation on 5E's "advantage/disadvantage" idea, though predating 5E of course.
There could potentially be abilities or rules giving more complex manipulation of either or both the positive and negative dice pools - storing rolls for either pool, moving a roll from one side to the other, rolling-up, ignore highest result if pool is 2+, getting a more average results by adding a die to both sides, etc.
One game in development mentioned here has circumstances where a roll is made but  + treated as 0, or minus as 0.
 
Note that rolling [d6-d6] against a difficulty of opponent target number, for example, does not basically differ from rolling 2d6 against a difficulty number of [opponent target number +7] unless other elaborations appear- the distribution is identical. "Roll up" or "roll down" mechanics usually do differ though, as do operations applying to a positive or negative die specifically; Feng Shui's "6 on either die rolls up" (i.e. 1 in 6 chance for each of this occurring) is very different to rolling down on 2 and up on 12 (each of which occur on a 1-in-36 chance).
 
A positive die/negative die system is actually pretty similar to what an opposed roll looks like in a normal additive system, except "within" one character.
Potentially in a dice-dice system, one of the paired dice could be considered to be the opponent's dice and one the active characters, with operations applied to just one dice in the pair as a consequence - for instance no negative die if a roll is wholly unresisted, or with opponents able to reroll the 'negative' die only with luck points etc.
 
OTHER WEIRD SYSTEMS:
 
 
Resource Pool Systems: (Gumshoe, Dying Earth); these systems give characters a number of rolls to perform a task based on their skill/attribute - in Dying Earth a task requires a d6 roll on the resolution table, with the result applying unless the opponent likewise spends a point from their pool to resist.
 
Full Light, Full Steam; an odd system which relies on rolling 4d6 and ordering them in a stack (figuratively speaking) of lowest-to-highest. Attribute and Skill are each rated 1-4, and let the character pick the appropriate die out of the stack i.e. 1 means take lowest dice, 2 is take the second-lowest, 3 is the second-highest, 4 is the highest; the two dice are then added together. As well as being slow, the system has a built in maximum for skills/attributes (4) and is quite granular. Stat and skill the same means the same dice will be counted twice e.g. rolls of 1,2,5,6 with stat 3, skill 3 would pick the second-highest twice for 5+5 = 10.
A similar, but simpler, system is Jeff Moore's interesting free rpg Hi/Lo Heroes. It rolls 2d6, with a stat being rated as either High (take highest) or Low (take lowest). It frequently uses the same pair of rolls for multiple things, with different attributes applying - a character with High to hit and Low damage might roll [5,2] using the 5 to hit and the 2 for base damage; this same setup was mirrored for Dodge/Soak. (See also the later post on system constraints for more on Hi/Lo Heroes). As it always uses two dice there is a 'doubles' mechanic where on a double both dice add together "as both dice are High or Low".
 
Four Colours Al Fresco: this (historical superhero) system (that I first saw mentioned here) uses an ordinal changing-die-type comparison system where a characters' ratings are compared to determine success. The games' conceit is that reality is governed by five forces (Dynamic, Static, Known, Lost, and Passion). A character rates each of these with a dice (relatively lower is better). On a given task these are rolled and are either Dominant (lowest number), major (second lowest), minor (2nd highest) or weak (highest), with ties eliminating extremes. A given task will Favour or Oppose a given aspect - a character wins if the dominant Aspect is the Favoured aspect (e.g. a character with Dynamic d4, Static d6, Known d8, Passion d10 attempts to push a boulder which is a task that's Favoured-Dynamic, Opposed-Static. They roll 3,4,5,5 for the aspects in order so Dynamic is dominant and they pass. (This is a summary only - the rules are also complicated by characters being usually un-rated ('Omega'd")in one attribute i.e. partly beyond the laws of physics, and rules for extra difficulty dice.). Because the ranking on a roll is internal to the character, scores have only relative importance (d20/d20/d20/d20 is about as good as d4/d4/d4/d4; wider spreads generate an advantage on some rolls but with equal handicap to others; it can also use zocchi dice e.g. d5s or d7s, up to about d34). Difficulty of a roll affects the number of Favoured or Opposed factors; conversely the GM may pick factors which logically affect a roll and that then sets the difficulty.

Diceless systems: use straight comparison to see who wins, plus either a fair degree of either GM arbitration (Amber) or use of specific details and modifiers needed. In some systems resource pools may be used to "buy" results, but with fixed results rather than dice being added. Amber has characters taking pools of 'Bad Stuff' or 'Good Stuff' to influence GM interpretation of how much a given PC should be interfered with.
A list of several diceless systems (not comprehensive) is here:  http://forum.rpg.net/showthread.php?747321-your-opinion-history-of-diceless-RPGs

Using Weird Dice: 'Sicherman dice' - a pair of differently labelled 6-sided dice giving the same results as 2 normal d6s are discussed here:
http://www.therpgsite.com/showthread.php?t=27498. This could have some sort of application where one roll is reused for something else (with a different range). A bonus smaller die would also have less effect than a normal +d6.
(Averaging dice are mentioned in post 4.)

"D10-X" - a roll-under variant seen in Cyborg Commando; two d10s are rolled and multiplied together. This has a slight failure chance even at very high attribute scores; an 'average' roll is 25 or less, but even a character with a 99 will fail occasionally). Probability distribution of the dice roll will be weird, but this system potentially allows unusual attribute scales (i.e. average 25; max. 100).
Another game using something similar is Jeff Moore's freerpg "5x5". This uses 2d6 multiplied together; however 6s are "trumps" and automatically succeed, instead of being the worst roll possible. It has the perverse additional elaboration that characters with no skill have a -5 rating, and have to try to roll 6 to win; they however only get one d6. Rolling only one die would in theory be good for most characters, but obviously won't help roll under a negative score.
Similarly, the Zero RPG rolls (d6 x d6); a character has a 'focus' score that has to be rolled over for 'focus' (class) abilities, or less than for general or background tasks. Difficulty modifiers add +1 to +3 to the lower die rolled before multiplying. Focus rolls give 1 success level, +1 per 10 over minimum, while other tasks always get only 1 success level.
 
Multiplicative rolling (BASH, Maid); a dice roll is multiplied by an attribute score. The upside of this, is that it gives a success chance based on a consistent ratio of any two scores. For example, a giant with a Strength of 200 wrestling a giant with a Strength of 220, has the same odds in a arm-wrestle as would a human of Strength 10 wrestling a human with Strength 11.
Note that another way to get this effect, is just to have an exponential strength scale (DC Heroes for example has Strength doubling each point, so that Str 4 is twice as strong as Strength 3; D&D 3.x, +10 Strength usually quadruples lifting capacity) and use an additive system.
 
'Floating' changing-dice-type (Conflicts in Sorcery & Super Science): see http://www.therpgsite.com/showthread.php?t=19822
In this changing-dice-type variant, a character's actual dice rolled is based off the total of [their stat+opponent's stat], with their score adding to the roll. A roll greater than or over the dice maximum is a success. Again this system generates proportional success based on a ratio between opponents in a conflict.
 
Grid Rolling: a table variant sometimes used to generate treasure - not seen as a core mechanic AFAIK- a grid is made with one dice reading "across" and the other reading "down" to select a table entry.
EDIT: Old TSR wild west game Boot Hill uses this mechanic for wound rolling: ond d100 generates location, and a second d100 roll reading across gives severity e.g. head shots are more likely to be major/fatal. It is also used sometimes in Tunnels and Trolls for random treasure or monster generation - in this case usually using d6 and d6, it allows for tables with numerous, equally likely results without the user needing any polyhedral dice.
e.g. in the solo adventure Deathtrap Equalizer for T&T (Corgi Edition):

QuoteRoll 2 dice and cross-reference the result on the chart below. If you picked up more coins than the number on the chart, then you picked up a magically poisoned coin (which works even through gloves or armour). Take the difference between your total and the chart's total as hits from your totalled attributes. Divide the remaining attributes points evenly among your Prime Attributes, placing remainders where you will. You may keep as many coins as your new Strength will allow you to carry. If you die (less than 6 attribute points remain for distribution), go to 2. If you survive, go to 3.

___1_____2____3____4_____5____6____
1__1000__32____47____145___366___225___
2__82____800___333___579___1_____1515__
3__99____71____600___9_____13____111___
4__127___26____818___400___271___604___
5__8_____144___1066__1903__300___53____
6__56____4_____666___1492__446___500___

The table here could essentially be replaced with a 'd36'; it becomes more interesting if either the top/side roll are replaced with a variable like a skill roll or other modifiable roll, although that then requires the table to be logically constructed such that movement in one direction is better (cf. Harnmaster combat tables in Effect post).
The same sort of system is used in Atomic Highway random mutation and flaw rolling; this has the extra elaboration that a player can switch the two rolls. Potentially this lets the table be designed so that all the pairs of options give interesting choices, and can set up inescapable rolls (doubles like 1-1 or 4-4) to be either especially soft or especially awful.


The grid-roll table above in the Corgi edition (a 2-in-1 combined with Naked Doom) replaced a different method in the original Deathtrap Equalizer ; in the earlier Flying Buffalo edition instead had the player choose a number of coins 1-100 and triggered the same effect of stat loss for choosing either 69,77, or 100 coins). That would be one of few cases where a random number is just chosen by the player instead of being rolled. This didn't quite make sense in real-world terms (a magically poisoned coin is always the 77th coin you pick up?) but is interesting.
Player's choose naturally leads to just picking the highest number unless there's either a clear risk (e.g. spending a resource is 'player choice'), or a potential risk (in this case, that the GM might screw then over). The Kevin J. Anderson book 'Gamearth' also features a game being played by characters trying to pick the specific number(s) the GM is thinking of (awkward for complex odds which require the GM to pick multiple numbers). Guessing a number is more often used to factor into tactical decision-making (should I use power attack?) rather than being the whole resolution system. Picking numbers in chargen can also work (cf. 'Character Modelling' under generating attributes)


"Beat the Difference" - seen in the old "Time Lord" RPG. In this game, attribute was subtracted from difficulty to get a number. Then 2d6 are rolled and the difference between them is determined i.e. 1 & 2 gives a difference of 1, or 6 & 1 would give a difference of 5. If the difference on the dice is greater than the other difference, the tasks succeeds.

Roll Over: a few games use a roll over a target number, instead of roll under. This is a bit like additive except that bonuses subtract from the target number instead of adding to the die roll. Examples here include 2nd Edition AD&D THAC0, SenZar attribute checks (d20 vs. TN of [21-stat]), and MechWarrior 2nd Ed. MechWarrior uses 2d6 vs. TN of [characteristic - skill]; characteristics are derived scores where lower is better, calculated as (18 - sum of two 1-6 attributes) - build & reflexes for Athletic characteristic, reflexes & intuition for Physical, intuition & learn for Mental, intuition & charisma for Social. Attribute saves are (18 - double attribute).

Recent edits: roll over. Added 'grid rolling' example - thanks to rpg.net poster 'Cobb Webb' for Corgi Deathtrap Equalizer text. Added 'players choose a number' as a mechanic

Aside from dice, cards are a possibility. These are ideal really for generating non-numerical info (without needing a table), and may also hold some supporting rules text as a reference.
Examples:
-Savage Worlds uses ordinary playing cards for initiative determination (face cards going first, then 10s, 9s and so on). This may be intentional to reduce modifiers - unlike using opposed agility rolls, PCs don't get an edge on initiative with NPCs, despite their Wild Dice. Some Edges give a redraw, and Jokers give additional combat bonuses.
Deadlands uses playing cards as well; unlike SW the GM has his own deck so simultaneous initiatives are possible (if unlikely).

-Deadlands also uses playing cards to determine stats, which are rated for both die type and number of dice rolled (i.e. a character could have a stat of 1d12, or 4d8)
Card  Stat die
2                d4
3-8             d6
9-Jack         d8
Queen-King  d10
Ace             d12
Suit determines the # dice (Clubs=1, Diamonds=2,Hearts=3, Spades=4).
Jokers count as d12 and 'also mean that your character has a mysterious past'; they need a redraw to get a 'suit' and so # dice.

-In Everway, cards are read for GM inspiration (somewhat like fortune telling), though actual resolution is diceless - e.g. drawing the "Drowning in Armour" card means that protective measures turning against the user may be a factor influencing the outcome (however the GM wishes to interpret that).
 
-TORG and MasterBook uses a "Drama Deck" giving special results. see e.g.
http://torg.pbworks.com/w/page/22371954/Drama%20Deck
Players may use cards to gain advantages in combat such as bonuses to certain actions, second chance or removal of damage; a number of cards also allow for creation of "subplots" such as Nemesis (acquire enemy), Romance, or Mistaken Identity. Cards can be lost via enemy tricks/taunts results or gained by performing "approved actions" in combat.
 
-Castle Falkenstein uses cards directly for action resolution.
http://www.rpg.net/news+reviews/reviews/rev_1610.html
Card values vary from 2-10, with different suites mapping to different action types and the right suit needed to get the full bonus.
(I haven't seen this one, so thanks Rob for this note)
 
-TSR's old SAGA system (used in Dragonlance: the 5th Age and the Marvel Super Heroes Adventure Game) used cards 1-10 (IIRC); a player held a "hand" of these and played them periodically - effectively making them like a pool of pre-rolled dice the player could allocate as desired. (something like this this could potentially be an interesting gimmick combined with a match-counting system like Weapons of the Gods, using the cards to represent "the river" of stored dice rolls). In Saga, Hand size increased with experience and also represented the character's health, with damage forcing a character to discard cards.. Suit on the card matching action type (i.e. swords for Str-related actions) it "trumps", allowing the player to immediately draw a second card from the deck and add it.
 
-FUDGE while in development was card-based. link: http://www.panix.com/~sos/rpg/fud-des.html

Quote from: Steffan O'SullivanFudge was originally proposed with a card-based action resolution system, and I tried to make that work for a few months before scrapping it. The deciding factor, oddly enough, was shuffling. I playtested so many variants of card-based mechanisms that I came to realize that frequent shuffling was a pain compared to rolling dice over and over, and so finally scrapped the idea of using cards.

(Perhaps decks would be handy for something like Fudge; it might be possible to e.g. have a deck with cards labelled Good, Great, etc. and avoid numbers entirely).

-Other Decks are sometimes seen for specific applications (criticals, fumbles, etc); this is little different to using a table if the cards must be played immediately. 7th Ed. gamma world has tech and mutation decks - these are somewhat different from just using a table in that a player can have their own deck. Hucksters in Deadlands reportedly have a magic system based off drawing poker hands.
Compared to a table, all results with cards are equally probable (unless multiple cards are available), and the same card can't be drawn twice (or at least odds will decrease, if there are multiple copies in the deck) unless the deck is reshuffled. A card system is also customizable by adding/subtracting cards as appropriate i.e. if a card draw is used for hit determination, the GM can add more Leg cards to the location deck when the PCs are fighting a multi-legged monster.
With a large enough deck, a set of cards can have any distribution desired with a single draw, rather than a linear distribution of results. For example, Pyramid 3-34 suggests using a 216 card deck to replace GURPS' normal 3d6 roll with a card draw- possibly handy for combats with lots of rolls, though it actually also suggests sorting cards into piles to encourage card-counting of what rolls are left (!) - adding a metagame tactical level to the game and certainly slowing things down.
 
Cards will often provide results that are more difficult to modify than a die roll. If the card generates a number or [number+suite] then a bonus can be added to the final number, but it is more common for an ability to be "pick two cards" or similar. Its fairly easy to give 'pick best' of two cards (by player discretion), while choosing the worst of two cards may require adding a number or rank to all the cards, or GM arbitration.
'best of N cards' has results similar to a take-highest dice pool, although with cards with special effects rather than numbers, there may be rare massive spike results hidden in the deck (the odds of which increase slowly).
Deck-building games (non-rpg card games mainly) can include zero-value or low-value cards, which can be detrimental via pushing up deck size and making it harder to get a higher value or useful card. Cards can have rules dealing with looking ahead in a deck (see next N cards in deck, or view and rearrange before the next draw) that are distinct from what's possible with dice. Card games can also include a 'discard pile', although most manipulations of this will probably be too 'meta' for an RPG.
Once drawn, a card can also be used physically as a 'counter' of sorts, i.e. laid down with other cards with the spatial relationship meaning something. Again this is seen more in card games, e.g. the Space Crusade card game uses cards for space marines in a line showing who's close to the genestealers, exit door, or whatever (in this specific game, character cards aren't randomly generated, but it would be possible).
 
If each player has their own deck, cards can provide 'balanced randomness' since if you've drawn the 1, the 10 is still in the deck and now more likely.

Dominoes are another possibility (I believe at least one freerpg uses these; I don't remember the name.).
 
The Fastlane RPG uses a roulette wheel for randomization.

Fate of the Norns (3E onward, "Ragnarok" edition) uses viking runes for resolution.
Description here of how this handles damage by 'Raleel' on rpg.net here:

Quoteyou pull runes from a bag for powers. Each power is bound to a rune. Defenses and attacks do as well. Skill "rolls" come out of this bag. Damage ALSO comes out of the bag, so you can get hit and lose the ability to do a power or be debilitated and unable to succeed at a skill.
Runes assigned to damage move down a wound track (stun-wounds-death-drain(aka slow healing)) and the length of the track can be varied by the GM (normally, wounds is increased in length). Mental damage is handled by targeting your runes that you've drawn or played this round first, then moves into regular damage. Spiritual damage works just like physical damage, but can move things into drain (requires magical healing, or very slow). Additionally, some powers/abilities/resistances require a sacrifice to operate - that is, you move a number of wounds into various parts of the wound track on purpose to activate things. This gives the whole power management/damage management system similar to some resource management games.

The Sherpa RPG, designed to be played while hiking, assumes the GM uses a digital stopwatch, with the GM pressing the button without looking and checking the hundredths-of-a-second to get a random digit 0-9. Tokens representing luck points can be spent to give a bonus or cancel the need for a roll.
 
Coin-resolution systems are also occasionally seen (for instance, the Skyfall! series of gamebooks; Prince Valiant; Underworld).
Coins can be used as a simple "dice pool" (i.e. Prince Valiant where coins flipped =stat+skill; Gareth Michael Skarka's "Underworld"), or just used additively (i.e. base stat + number of heads rolled).
For the most part flipping a coin isn't particularly different to rolling a d2, though the joke game Combat Monster uses coin denomination as well (the players roll change with a total value equal to their attribute, counting the cash value of coins that come up heads; this lets it have a "dice pool" type system using arbitrarily large attribute scores - although the exact change chosen by the player would theoretically alter their chance of success at a given task difficulty). Skyfall adds heads and subtracts tails for some rolls (others are just e.g. +heads) making some rolls similar to FATE or Feng Shui's centered-on-zero approach (what I called a "roll your modifier" system above).
(thanks again to RobMaudib for Prince Valiant/Underworld/CF notes).
 
Player-skill-based random checks:
The game "Dread: The Impossible Dream" reputedly uses a Jenga tower, with difficult tasks requiring a pull from the tower.
Paper-Scissors-Rock (Janken) was (I think it is said) sometimes used in LARP as a convenient tie-breaker/resolver (requiring no dice, but is basically 50/50 and has no appreciable advantage over a coin flip).
Shooting hoops to win an 18 attribute (as noted in attribute generation) or having Ork miniatures that fall off the ork truck be considered killed are similar cases.

Randomization based on irrelevant details: T&T solo adventure "Hela's House of Dark Delights" features a race-change effect where a player references the monster table in the main rulebook, counting down an entry for each letter in the character's name.

Random numbers equivalent to dice rolls can also be generated without dice e.g. the Lone Wolf series of gamebooks has a table where a player can touch randomly to generate a 1-10 number, or a book may be printed with numbers at the tops of pages so opening to a random page generates a 'die roll'.

Edit history: recent 26/2/14- Fate of the Norns note. Older - added FUDGE notes; thanks to John Morrow; Fastlane note. 3/11/15-numbers equivalent to die rolls.

One of the quirks of OD&D which I've always loved is the proliferation of mini-systems within the basic rules. In modern game design parlance this is a thing of horror, much like the walking dead and upside-down crosses. Mini-systems?!? Lock the doors and get the shotgun, somebody's gotta die!! But personally I love it: it's the "spirit of adventure" in a rules system, where the creators are trying out new things, playing with options, and picking the right rules for a situation rather than maintaining consistency with a "system" at all costs. With that preface, I want to keep the class dice as a fun mini-system that makes certain classes distinct. The concept really works well with fighters, and works pretty well with thieves. I don't want to force it onto other classes where it doesn't naturally fit. And I'd rather have a distinctive feel (and a distinctive mechanic) for each class. - Joseph Goodman [Dungeon Crawl Classics designer] on subsystems , from the DCC messageboard.

(I wish I had a balancing quote from someone that loved unified systems... After thinking about it though, actually, I didn't like the fighter's extra 'deed die' in DCC that much, and thief is just the standard d20+mods for most checks. But its an interesting POV, regardless)

Many older RPGs use a variety of die rolling systems – starting with D&D – even though the idea of having more or less one, central core game mechanic dates back at least as far as RuneQuest (1978). Even the second RPG published, Tunnels and Trolls (1975), uses only about three different die rolling methods (for melee combat, missile fire, and saving rolls respectively; missile combat would later integrate into the saving roll system).    A majority of systems took their lead from D&D however, and used different mechanics for different tasks -sometimes even different mechanics for what's virtually the same thing at a different degree of difficulty e.g. Open Doors vs. Bend Bars/Lift Gates. These systems vary between the somewhat baroque (where skill checks, ability checks, and hit rolls for example may use different mechanics) to the completely disjointed; for example a number of games split off from D&D even before ideas such as "ability checks" were in vogue, meaning a variety of very specific percentages and game functions had to be evolved.  In such games values (STR, HP, etc.) are under less pressure from conflicting reuses of mechanics, with each value having a scale designed in keep with its specific assigned function, sometimes with later kludges to add on effects such as skill adjustments or the like.
They perhaps show a design trend of having an attribute range decided first, before the mechanics were decided upon, whereas in a unified system the range is decided after the core mechanic is.

Examples of games with diverse methods of dice rolling:
 
  *Palladium has situational mechanics including percentages e.g." % to charm/impress", while other actions are purely GM fiat or use different systems e.g. Arm Wrestling [Mystic China] uses a d20 with a special Strength modifier (+1 per 3 PS above 16), while escaping a Crush/Squeeze (Ninjas & Superspies, Revised) requires an opposed roll of [d20+full PS score).  Lacking an ability check system removed a constraint on the scale of attributes, letting them range from 3-24 (higher for physical stats) - possibly helping it to later function as a supers game (?).
Much like older D&D, having even modifiers calculated differently for different stats makes it harder to handle e.g. situations where an unusual attribute would apply to a task.
Having very-specific situations having their own mechanics means often things will fall through the gaps, with none of them exactly applicable.

  *Synnibarr likewise has various ability derived percentages (for shock, finding traps, determining psionics, etc) and actually evolved different maxima for each ability score; while all are initially rolled with d20 (with initial values contributing equally to the 'skill points' used to buy skills, mutations and bionics), additional bonuses can get characters up to Con 20, Wis 25, Int 30, Dexterity or Agility 35, or Strength 1000. An emergent effect of this is that a high class minimum for Strength -which uses the initial score- balances/punishes a class noticeably, as the points could have made a much larger difference on another stat.
Synnibarr displays a breathtaking disarray even within the combat subsystem. It uses an additive d100 roll for shot rolls (roll d100 + 'shot bonus' based off Dex and level), while Dodge is a normal roll-under percentage (agility x 2%) instead of being directly opposed. While weird, playing it we found this does speed up the additive attack roll slightly; usually it is enough to know that you rolled "a lot" without actually calculating an exact number. For skill rolls it also switches between roll-under-d100 for normal skill checks and rolling [skill % + d20] for opposed rolls.

  *D&Ds 3-18 ability scale turned out to be coincidentally well adapted to d20 roll under; 2nd edition AD&D frequently uses this, but in conjunction with legacy mechanics from 1E such as "% to learn spell", "% system shock" and "wisdom % spell failure" despite these being mathematically almost identical to roll under with d20. A number of core mechanics (attack rolls, saves) instead use level-based matrices; the ability check mechanic suffered from not scaling up as these did, while also defaulting to a much higher chance of success than a save or attack roll - problematic in instances where it could be unclear if a stat check or save should apply (i.e. Dex check or save to dodge a trap? Or Con check or poison save to resist getting drunk?). This can also make opposed rolls awkward, if different sides are using different mechanics. (Note: the same issues with scaling and default success chances can also apply with pure d100 systems like RQ or HarnMaster with regard to ability checks vs. skill checks).
 3.0 D&D instead transitioned to [d20+modifiers] as a system, enabling a much higher range in ability scores (e.g. Titans went from Strength 25 to Strength 37) and making it more functional for superheroic characters, but also increasing the ability scores' importance to characters by making modifiers apply to many previously unmodified checks, ranging from Listen checks to breath weapon save DCs. It also renders odd-numbered ability scores pointless except as feat prerequisites, in spite of which the point-buy system generally charges more for each of those ability points than for the (actually useful) next point. A modifiers system consistent between attributes led to increases in cases where unusual attribute modifiers applied, such as Weapon Finesse.
Modularity: Older D&Ds in particular are fairly 'modular', with it being possible to replace the roll used by a subsystem with a different roll fairly readily (e.g. Dragon magazine has any number of skill-system variants for 2E or earlier). In part this is also due to most subsystems not requiring character-design decisions. In 3E by contrast, a characters' ability at something usually requires a skill-rank (or feat) expenditure, and changing a mechanic risks invalidating a players' "investment" in some skill.

*Warhammer Fantasy 1st ed (whose mechanics were derived from the skirmish game) has stats which sometimes rated up to 10 (Strength, Toughness) and sometimes  up to 100 (Weapon Skill, Ballistic Skill), and so while it does have ability checks, it uses a d10 for some checks and d100 for others. The 2nd edition streamlines this quite deftly by having all scores range to 100 and using the 10s place as a modifier when needed (e.g. for damage).

*Dragon Warriors, while its clearly D&D inspired (possibly Holmes, given its initiative system), shows extensive divergence from it, where the miscellany of rolls and checks are replaced individually, one-by-one, with different but no more unified systems. It is good in that individual systems are quite simple - actually, working with a simplicity that could be difficult in a unified system. For instance, a Weaken spell adds a -2 attack, -1 damage penalty but doesn't note the effects on other applications of Strength. The ability doesn't require notes on a beasts' existing Strength scores or table recalculation (indeed its inconsistent with any Strength score, as no Strength score gives a damage penalty) but the downside is that its effect on the fiction and on lifting, carrying, etc. is also unclear. Most other spells are likewise simply worded but may require considerable GM extrapolation from the listed effects, or at the very least searching for prior cases and general rules that might be applied.

*Satirical quasi-system SLUG allows the GM to determine which dice are used and the target number, mainly for the purpose of slotting in characters from any system, making it wholly ruling-driven. It lets the GM use any sort of mechanic for whatever purpose.

*Hero manages to have somewhat different mechanics despite using only D6s, with separate mechanics for ability rolls (8+1/5 stat points, roll under 3d6), combat rolls (3d6 under something+OCV-opponent DCV), and damage rolls/presence attacks/opposed Str type checks (multiple dice additive). The combat subsystem being distinct causes some extra mucking around with skills, which generally run off ability checks but occasionally for combat skills, e.g. autofire, are applied to combat values.

Potentially miscellaneous mechanics might end up requiring a roll for both things (such as an attack roll + a skill roll) and so lower overall chance of success, compared to a single check working off unusual values.

Unified Systems

More recent systems strongly favour unified, cohesive core mechanics. A unified mechanic does offer a quite a few benefits i.e.
*easy for new gamers to learn
*easy for the GM to fit new situations into the framework; i.e. for 3E D&D the system is always roll d20, higher is better so the GM only has to determine which modifiers apply, and set a DC. This is particularly useful where a game is broad in scope, making it difficult to consider all situations. A notable example of this benefit would be Marvel Super Heroes: its descriptive 'Rank' system where things ranging from object Material Strengths to Resources to radiation intensity can be categorized as  "Typical" or "Good" or "Unearthly", plus a super powers system that gives examples of a wide variety of effects, let the GM fairly simply define more or less anything within the system.  (although: letting all powers scale up/down freely due to random roll, with damage etc. determined by rating rather than fluff, does you can have situations like rolling a character who does less damage with micro-missiles than they would with their fists - something that likely wouldn't happen if powers were ad hoc defined instead).

*similar resolution means a general ability can be written that affects a number of related circumstances.  For example, a -2/-2 multitasking penalty can be applied as easily to a character standing watch & also repairing repairing their armour, as it can to a character trying to fight with two weapons (since all these tasks use the same dice roll). Or a wound penalty can be applied to a variety of tasks consistently. However, the generalization can also be a potential drawback in that modifiers can unintentional affect strange things ("action at a distance", I think, in programming parlance); e.g. clerics in Castles and Crusades having better perception rolls than thieves due to having Wisdom as a "prime" stats.

Nonetheless, largely unified systems still quite often contain exceptions in how they operate since not all situations are identical, and some core mechanics struggle to cope in certain circumstances e.g.
-Storyteller at some stage used [d10+modifiers] for initiative to reduce simultaneous actions.
-D&Ds primary "effect" system uses a step-die system with d4 thru d12  (compare to True20's more integrated - if not necessarily better - "Toughness Saves" which spread around knockout/kill effects with considerable randomness).
-HarnMaster uses d100 for most checks, but with some attribute checks are done on 3d6 to skip the usual multiplication step (fumble tests) or on multiple d6s (combat results such as Stumble or Kill rolls which have a variable intensity, and where a highly random outcome is undesirable).
-MSH, despite its overall flexibility, uses a d10 roll with special modifiers for initiative.
-HERO generally uses 3d6 roll under [8+stat/5], but has a damage roll mechanic (d6 per 5 points) which is also used for grappling and Presence Attacks.
-roll-under games sometimes switch to an additive mechanic for opposed rolls - Fighting Fantasy /Advanced Fighting Fantasy used 2d6 under stat for stat checks (including ranged combat), but opposed rolls of 2d6+skill for fighting. LegendQuest used d100 under for most checks, but additive instead (d100+skill %) for opposed rolls.
-Savage Worlds uses a card draw system for initiative 'rolls', removing PCs Wild Die advantage and reducing the importance of Agility compared to the usual 'Dex check' initiative system.
Note: Attempting to use a 'universal mechanic' to cover movement as well has complications in some cases, as detailed in the 'movement' post.

Other games have a single system but have attempted to code in multiple options as a core mechanic (Interesting to me since I've noticed how specific many ideas/innovations are to only one or or a couple of the possible systems; cramming lots of them into a single RPG would probably need some method of combining mechanisms...).
-StarCluster 3E and related games uses a simple system which lets the GM/players drop in a huge variety of resolution system of their choice - whether percentile, die pool, d20, or diceless for example.
-FUDGE/FATE uses various systems to generate modifiers; as a centred-on-zero system, this generally works as it changes the range of the dice roll without changing the average (=appropriate attribute). An extreme variant here for 'polyhedral FUDGE' instead generates a wildly different result from 1 [Terrible] to [current rank + a bit], but still uses the base Fudge descriptor system.
-Tri-Stat  (I think ?) lets the group pick a resolution roll from +d4 to +d20 based on the "heroicness" of the particular universe.
-FUZION attempts to allow either a +d10 roll (from one of its precursors, the Interlok [Cyberpunk/Mekton] system) or +3d6 (related to its other precursor, HERO); the latter however works poorly since there can be a 20-point difference in possible values (cf. FrankTrollman's analysis here).          
-Space:1889 has a 1-6 stat scale, with two possible methods of rolling - a "quick roll" system (under stat on d6) for simple tasks, or by setting a difficulty (from 4 to 20) and rolling the stat or skill number in D6 to get the total.
-I've seen references to DramaScape using two separate core mechanics, but am unclear on the details.
-Stars Without Number (reportedly) handles combat and saves via d20 and skills via 2d6, but using the same bonuses for both.

Some core mechanics are slightly more intrinsically versatile than others; for instance if a designer wants Initiative to use a d10 roll and spellcasting to use a d6 roll, a "step die" system like Earthdawn's could let them do within the bounds of the system by declaring that Initiative defaults to "Step 5" for most or all characters and Spellcasting to "Step 3", or by creating new attributes specifically for those tasks. (Versions of Savage Worlds later than the "core rules revised" also declare its wild dice on/off in certain cases, such as damage rolls and running rolls; damage rolls add multiple dice instead of taking-highest)
Talislanta uses d10+modifiers for unskilled rolls and d20+modifiers for trained rolls; a system could build off this to have all raw attribute checks use d10s rather than d20s (as no skill applies) in order to weight upward the influence of attribute checks on rolls.

Even where only a particular dice is used, it may be ideal to change details of the rules connected with a roll to control uncertainty around a roll, level-scalingness, etc. For example in simple additive systems, heavier controls against gaining modifiers on tasks which are supposed to be more uncertain in outcome are reasonable; a word of warning here however is that if opposed rolls are possible between scores, a "fair" comparison between opponents requires similar scales of modifiers.
Adoption of "+level" to all dice rolls (such as in 4E D&D) lead to a "level treadmill" within the gameworld; leaving certain abilities unscaling (such as ability checks to break down doors, influence NPCs, or not fall over) produces less pressure on the GM to retrofit all the doors in the dungeon to adamantine and grease up the floors for the high level PCs.

Mechanics used consistently across the board may need adjustments to consider that different tasks need different fundamental failure rates. Stealth checks rolled repeatedly will fail -necessitating a high base chance or a 'let it ride' type rule) - conversely reactive perception checks rolled by a large party will almost certainly succeed. Hit rolls may be fine at a 60:40 or 50:50 odds; desired spell success rates may vary even between spells. A d% mechanic will usually have a varying base % allocated for different actions but opposed [d20+stat+skill] often default to the same 50/50 odds for everything, with deviations generated accidentally by scaling issues like one side being more likely to have a higher casting stat/good save/proficiency bonus. These deviations might be exploited deliberately by designers (choosing to use a skill check to increase chance of success, or make a save use a rarely improved category to reduce chance of success).

Edit: note to self- rpg.net's discussion on d20's problems as universal mechanic due to high variance here

So you succeed at forging a sword, casting a spell (if that needs a check), or whatever else you were doing. The next question beyond if a character succeeds, is how well they did (effect). How this is handled depends largely on what the core mechanic is (d20, dice pool, percentile, etc). Depending on system this might be largely a matter of GM fiat, or precisely detailed. For systems which do have defined effect, the final result might be a number (damage; distance jumped;spell duration) or be some sort of special, specific effect (Slam!; Stun!).
An "effect" might also be describable as a bonus on a second follow-up roll, or the equivalent; Fuzion "Complementary Skills" lets a character make a skill roll to modify a related skill, or 4E D&D has a "skill power" [Tumbling Dodge] that adds [roll/10] to all defenses, or there can be 'Aid Another' actions giving an ally a bonus, or a spell boosting someone's values might increase for a high spellcasting roll.


Systems for effect determination vary in how much coupling there is between dice roll for success and for effect - whether difficulty is the same as degree of success. Most d20 mechanics are tightly coupled, with high roll or skill giving a large margin of success, but damage roll is largely unrelated to how good the hit roll is (uncoupled). In between these, a system like oWoD is loosely coupled - an increase in difficulty may or may not mean less effect (a single high roll gives basic success on a difficult task, but rolling above the minimum on each die doesn't boost effect further, so some detail from the roll is lost).

A number of mechanisms for effect determination are below:
 
Non-Dice-Requiring:

No effect system (binary): a roll either succeeds or not, with no particular further detail.
(without an effect system there may be a particular emphasis on bonuses to rolls being kept small, partly as after reaching 100% there is then no further incentive to improve).
It might be argued that in some games (maybe 3E D&D attack rolls), a general absence of specific extra-effect rules built into the roll, encourages addition of lots of specific character add-ons that give 'front loaded' extra effects...instead of a particularly high roll letting you add a cool extra effect, it becomes a 'feat' that you buy. Partly so that these can be added without the rolls themselves getting complicated, partly for the opposite reason that once a high roll gives you some special effect, it can't then be 'sold separately', or at least becomes more complicated to implement (e.g. Power Attack).

GM-defined effect: the GM determines how well a task succeeds depending on whatever factor - dice roll, how the player described the attempt, etc. The GM might decide to make rolls for some specific of the situation on an ad hoc basis to see exactly how well a task succeeds. Basically will resemble some combination of below systems but via an improvised ruling. Examples: for failure  Fate Core has GM advice on letting failure become 'success at a cost', with varying complications (tough choices, foreshadowed perils, temporary aspects, plot wrinkles, new opposition, delays, damage). Assigning costs on a scale commensurate with the task being rolled for can be tricky, however (the cost should be less bad than failing the roll, so is easier the higher the stakes, and also has to make sense for the roll in question; depending on circumstances delays may not be significant, while systems with lots of PC healing including FATE make damage as a consequence less meaningful; permanent losses of equipment, etc. work for critical failures but can be too common/harsh as a consequence of fail forward).
Where the GM accepts considerable player input you can instead have 'negotiated resolution', as seen in some 'narrative' games, where stakes are set by agreement between the player and GM. Negotiated resolutions occasionally appear for normal systems in unusual circumstances, where rules are ambiguous or produce effects not desired by either players or GM - a player may then get some leeway to suggest a patch to a stupid rule.
(One homebrew game I play in (designed by someone else) deliberately runs "Partial success" results with some ambiguity - the idea being that a partial success is set up so that the GM describes it as something that could be resolved one way or the other by player skill or roleplaying, or in some cases perhaps additional rolls).

Fixed effect: whatever you rolled (assuming you passed) is irrelevant; you get a value based on the appropriate statistic or skill; being a 5th level thief gives you +3d6 sneak attack, being a 4th level wizard gives you a Rope Trick that lasts for 4 hours, or a telekinesis spell has Strength equal to the caster’s intelligence.  
Attribute-based effect can be a problem as e.g. a class ability in level-based games, since a character's ability in their class can't improve with level. The early 3.5 D&D marshal with its Cha-based 'auras' is one example.

Resource Driven Effect: e.g. in 4E D&D , the main thing determining how much damage an attack deals is what power (from a variety of one-shot power selections) is being spent. The complex 3.x variant FantasyCraft has a resource-driven setup throughout, where a high or low roll is potentially a 'critical' but must be 'activated' or 'triggered' by spending an Action Die. Likewise a fumble requires a low roll, and the GM or opponent to spend dice. Various character abilities interact with the mechanic; one thing of note is that NPCs can't score critical hits on PCs unless they burn the points to do so. In a resource driven system, characters can have modifiers to resource cost i.e. reduced cost if the character can have a particular ability, or a character might an ability that lets them recharge a power.  High results generally appear under the control of the player - making it less exciting and more metagame in nature, but also more tactical than most other options ('Going for Broke' is also somewhat player controlled, but not as predictably). Mathematically speaking, adding a resource cost can also be used to keep down the frequency of criticals which might otherwise become too common: requiring a character to spend a point to confirm a critical might be useful in e.g. a d10 system, as otherwise criticals on 10 would occur 10% of the time (a friend of mine has a homebrew simple step-wise system where a skill roll is usually just d4 to d12; it gives PCs 'catastrophe tokens' equal to 1/2 Combat Experience die which can negate 1s, among other things).
Note that although a 'safety valve' system where a player can spend points to add to their dice rolls also makes high rolls more controllable by players and so adds a resource dimension to another sort of effect determination, high rolls would still periodically occur on their own due to chance.
'Time' can also be a resource here in a way - effect may be reduced by attempting a task quickly (e.g. HERO 'casual STR', letting a character use 1/2 STR for a task but taking less time), or increased by taking longer. (It can work indirectly by giving a die roll bonus, hence better margin of success).
HERO also has a point-spend involved in effect, as Endurance can be spent to Push an ability (e.g.: Strength, Running, casting a 'spell', firing an 'energy blast') to be more powerful than normal.
Another wrinkle on this, from Marvel Heroic (MHR) is that it lets a character get 'effect' off reaction rolls by spending a plot point - so for a point a successful defense might become a riposte that deals damage.

Allocated multiple subeffects: A character using an ability gets X points to buy various sub-effects (e.g. for a spell might choose to allocate points into damage, area, or duration; LegendQuest does this with spell 'control levels'). When rolls are involved this can also be done but with the number of points varying, or even rolled after being allocated (see. HGT, below).
Note: I'm distinguishing this from 'resource' above where a character is burning an ongoing resource (powers or a point pool); here each ability use is a new 'resource pool'.

Dice-Requiring
Note that despite the below methods being classed as 'dice-requiring', occasionally a game will have diceless resolution using one of these by synthetically generating a 'die roll' number - e.g. a 3E character can 'take 10' and use that number to calculate a 'margin of success' result, or a Storyteller character can automatically pass a task with difficulty < dice pool, at one success.

Natural dice roll: particular numbers on the dice give more effect i.e. “pin on a natural 18-20” or “critical on a natural 20” (Palladium). The effect could be based off high roll (additive or 'blackjack' system), low roll (roll under), or occasionally occurs on doubles (on multiple dice or d100) e.g. IIRC 'Arms Law' for RM used to have weapon breakage on doubles, or having doubles produce 'yes, but..' partial successes for Barbarians of Lemuria (2d6+mods) is suggested in this thread. Whitehack typically rolls 1 die, but characters roll 2 dice (take best or take worst) if they have a relevant quality ('group') much like 5E D&D advantage/disadvantage, with GM-defined special success or failure on doubles. The "Metagene" RPG rolls 2d12 with a 'triumph' on doubles that succeed or a 'near triumph' if rolls are within 1 of each other (e.g. 6 and 7 rather than double-6 or double-7), giving three categories of success and making 'near' results just under twice as likely -a given result 1-12 has two adjacent numbers that are 'near' to it, aside from 1 and 12.
A natural-roll setup gives a named/specific effect, i.e. qualitative as opposed to a numerical [quantitative] output. This is similar to a table, but unlike a table, you get one or at most two specific results (e.g. for critical/fumble). In most games exceptional results are kept rare, regardless of a characters bonuses or total success chance - single die/fixed die type systems may add kludges that increase the range of critical numbers e.g. "Improved Critical" in 3.x D&D. GURPS expands its critical range based on total skill - from a natural 3-4 on 3d6 up to 3-5 at skill 15, and up to 3-6 for skill 16+.
The odds of a particular natural dice roll occurring may also vary intrinsically with character ability e.g. in a step die system, or for odds of doubles in a roll-under percentage system (1/10 of skill), or if rerolls are permitted. Some systems e.g. FantasyCraft, or D&D 3E critical hits, have more detailed subsystems for applying various modifiers to the ranges, independent of the modifier on the roll itself.

Natural dice rolls in dice pools can include e.g. 10s on d10s rerolling or counting as multiple successes (e.g. Aberrant mega-attributes), but this really works fairly similarly to a margin of success in these systems, unless a single 'wild die' is called out. Dice pools (whether count-success or multidie additive) generate numbers that are intrinsically quite 'open ended' and so it is difficult to separate off 'crits' in the same way as e.g. 20 on d20. This sometimes has the effect of requiring a separate dice roll for a special effect to be triggered (cf. 'attack rolls' in T&T, which uses lots of dice as the attack; having the entangle monster squeeze at a '1 on d6) chance rolled separately is fairer).

A natural dice roll to trigger a specific result could potentially be set to a number other than the top one or two dice values, which might be most useful if rolling multiple dice - for instance using 2d10, a critical roll of 20 would occur 1% of the time, while a roll of 16 exactly (no higher or lower) is 5% likely - I know of no systems that use that however.  

Natural dice roll has a quirk that an increase in chance of success lowers the average 'successful' roll, and vice versa (unless the roll is 'semi-independent' (see below) such as where using a wild die or 1's place die on d100. This can result in checks that automatically critical if they succeed (i.e. critical on a 20 but a 20 is needed to hit) or quirks like the 'Glass ninja' syndrome in TORG, where a target with high defense is hit only with an attack that (because of the high roll) also does a lot of damage. (Potentially this could be offset by giving a high defense a separate damage reduction also?).
Using rolling up + a margin of success mechanic can give results similar to using natural dice roll, but with more increase in success rate for characters with bonuses and possibly more useable numbers for defining effect-increase.

Margin of success: a fairly straightforward system, in the simplest version of this the character’s dice roll only determines how well they do i.e. how far they roll under the target number (roll under) or over the target number (additive). Rolling over the target number by 5 is half as good as rolling over by 10. This is fairly intuitive; it directly rewards large bonuses to checks. Minor subtraction is required. The amount of success is a number, and may be directly useful in calculating 'effect' (in damage, days of food, GP produced, or whatever). More rarely 'margin of failure' is considered rather than margin of success e.g. more damage is taken if a save is blown badly.
Effect increase under a margin of success setup has a linear return if the dice roll is linear (such as d20), while for bell curve rolls (e.g. 3d6 such as GURPS) diminishing returns is built into the margin of success - e.g. beating a TN by 6 is much less common than beating it by 4. 3d6 roll under  margins of success are generally more directly useful at a 1:1 ratio than in d20, which is more likely to use 3:1 or 5:1.
Rolling up can give blowouts in margin of success (frequently without modifying chance of success, if the max die roll would have succeeded anyway).
Systems may alter margins of success as well as bonuses; for example the Charlatan in the 2E AD&D "Complete Bard" has a Charlatanry skill whose 'margin of failure' widens as their Charisma increases; if a Pick Pockets roll fails by more than [Charisma] a critical failure occurs. Luchador: way of the mask notes that 'hasty actions' double margin of success on a failure or halve it on a success, as well as being +1 difficulty.
One friend's homebrew game, "Last Order", also has a "variable raise number" where the greater someone's skill is the larger the margin per 'raise' - needed math-wise because it uses larger dice for higher skill (...something like Savage Worlds). (see 'variable dice' post).
Complex margins of success can require a table lookup e.g. in Unisystem, the range of numbers generating success level 1 is smaller than the range of numbers generating success level 2 and it continues to widen each level to make the higher level progressively rarer. (an interesting house rule for Unisystem limits # success levels to no more than skill level despite roll).
SenZar has an odd group success system where multiple characters roll, and all separately determine margin of success and add this together to determine final margin; it also has some holes in the margin of success interpretation system in that a stat of '*' (i.e. unlimited or $TEXAS in the attribute) automatically wins opposed rolls with lower ratings without needing a roll but doesn't give a clear margin of success for system uses (such as a VoidSpawn's WILL roll to negate incoming damage, where margin of success determines maximum damage absorbed).
Dungeon World/Apocalypse World have an interesting twist on margin of success by adding player choice to what effects are generated - typically rolls are 2d6 and there's a list where 7-9 is "pick one", and 10+ gives 2 or 3 picks. So instead of a normal vanilla success and a high roll being super-success, a normal success actually comes in various "flavours". DW sometimes builds off the move structure by cross-matching lists, such as (in an early version) Sharpshooting letting a character choose 'pull a stunt' options as part of an Archery roll, or the mind-reading 'violation glaive' letting a character choose Discern Realities move options based off the targets' knowledge with a Hack and Slash move.
Teenagers From Outer Space has an 'Its Too Much' rule where a roll that's too high 'backfires' (e.g. the girl the PC asks to the movies starts stalking them) - the margin before backfire being randomly determined each session with D6. Margins of success can also just be capped e.g. in Buffy (Unisystem) a character with the 'talentless' disadvantage can't get more than level 1 on artistic rolls, as well as taking a penalty.
One other variation on the basic 'margin of success' is to replace a rounding of results with a second roll based on the remainder. The (slightly scary) Phoenix Command Amatorial Rules house rules included a roll where a result is divided by 3, with a final outcome of 4.33 then being resolved as 4 + a 33% chance of an extra success. This probably makes more sense with larger divisors and/or a linear die roll.

Base effect, plus a bonus determined by how much your total roll (dice + bonuses) exceeds the target number (for example hitting by 3 might add +3 to damage). Minor subtraction required. This is basically a hybrid of a ‘fixed’ system and the ‘margin of success’ system. It gives numerical results i.e. "damage points" or "result points". This system lets a character have an effect modifier that applies to tasks independently of the success roll e.g. a bonus to damage from Strength.
The most comprehensive and interesting system of this type is probably DC Heroes (perhaps particularly 2nd or 3rd editions). This sets separate Acting Value, Opposing Value, Effect Values and Resistance values for all rolls - e.g. to hit in combat uses a comparison of attacker DEX (acting) vs. opponent DEX (opposing), then damage would be found by comparing attacker STR (effect) and target BODY (resistance), plus column shifts from the hit roll. Parallel systems also existed for e.g. social interaction (which involves comparing Influence, Aura and Spirit) and all other tasks, though many used doubling up of the same attribute or power (for both the action roll and effect), or used points where they were likely to be unnecessary complication on rolls that should be simple pass/fail. Possibly an issue is that a 'mirror match' between characters tends to favour the defender, since equal effect stat/resistance stat gives usually only a couple of points of base effect, requiring an especially good die roll to then raise the effect more. Effect could sometimes be converted into exact distance, area, etc. via tables, or sometimes became ad hoc - e.g. 'Danger Sense' notes there's a sense of danger with 1+ effect, or exact source with 8+, after a messy process of comparing power rating as acting value/power rating as effect value vs. a table of opposing and resistance values based on danger seriousness-critical 2/2, major 4/4, minor 8/8 (though at least this is consistent for adjudicating how other powers can affect the numbers). DC Heroes also had the concept of ‘Pushing’, where an effect works at the base value automatically, but requires a dice roll to strain that limit e.g. lift more than usual. Another game using this would be FGU's Daredevils; margin of success on Athletics or Swimming rolls adds to Speed when running or Swimming, while Jumping margin of success adds to Strength for calculating distance jumped.
 
Attribute-scaled effect: in a 'roll your modifier' type system where the dice give a + or - to the base score, such as FUDGE, the final roll can give a result that's on the same scale as the stats i.e. a "Great" result. The results in this sort of game can be used to either check for success or as an effect value (although its perhaps dodgy to use one roll to give both). This is basically a variant of 'margin of success'. Note that FATE probably inflates this due to Aspects.
In some respects this is similar to what DC Heroes attempted, which also attempted to give an output in terms of 'attribute points', except that DC's output only theoretically is scaled as a stat (RAPs obtained vary from about 0 to 2x stat) and include input from multiple stats, four per check. (DC Heroes' relative Shatterzone is also an interesting development here in that it also uses an attribute-based scaling, with base attribute plus a 'bonus number' generated by a roll, but that number is used as difficulty mostly, with effect then calculated more indirectly or complexly with tables IIRC.)
It could be imagined this would combine interesting with a setup like ROAR (see advantages) where a low/high score for attributes automatically added the equivalent of flaws or merits; an effect result could then have various positive or negative traits working the same way, parallel to how attributes worked.

Independently rolled effect: the most common case of this is damage e.g. in games where hitting with your longsword deals d8 damage this is an effect largely separate to what your attack roll was, barring criticals (More on damage later). This system is readily shoe-hornable onto whatever other system is in place; it provides slightly less logical results (a huge skill check can earn minimal success) but does not excessively reward high bonuses i.e. it is ‘balanced’, and there is little extra math involved.
A separate roll is good in that its easy to vary average amount by varying what die is rolled. (If a game uses another mechanic (like pulling one die out of a dice pool for 'effect' - say damage) a separate roll could be used as an exceptional cases, e.g. for small or large weapons the roll might be separated so we could roll d4 or d8).

A roll can also be “semi independent” - for d100 systems that have special results based on the 1s die (see above), this is almost a form of built in separate roll –since the 1s place is often largely irrelevant (one-eleventh of the total dice roll on average). In a couple of systems dice are read additively but with one of the dice determining special effects e.g. in Valherjar (another Haiti disaster relief download RPG), a roll is [stat+3d6] but one of the dice is a special colour and determines effect. The Dragon Age RPG is similar - 2 dice + a wild die, with a critical occurring on any successful roll of doubles [44% chance] and giving 'stunt points' equal to the wild die which can be spend on various effects - this again gives a chance of 'critical' which is independent of character skill and can only improves only rarely/arbitrarily as a result of special abilities giving stunt cost discounts and the like. J Arcane's Drums of War uses 2d10, but also reads them as a d100 roll (as noted later in Criticals).
As another example, the "Envoy" class in Starfinder gets an 'expertise die' (+d6, etc) which normally adds to the skill result, but levels of failure can be modified by it - e.g. a Sense Motive check critical fail is reduced to a normal fail "unless the expertise die is a 1".

A simple 'semi independent' mechanism is just to count odds/evens as different results. For instance, the 5E playtest displacer beast (IIRC) incorporated a 50% miss chance into the attack roll by having odd numbered attack rolls miss, while in the solo adventure Red Circle in Tunnels and Trolls a missed save (falling into a pit) send characters to different pits depending on whether the roll is failed by an odd or even amount (using the margin here lets odds/even be used easily despite a roll using 2 dice).

The Marvel Heroic game (2012) uses a pool of dice, where 2 are selected and added to determine success, and a third selected as the "effect" die - a quirk being that the result is based on the raw die size, regardless of the actual roll. Characters may therefore sometimes lower their success roll to allocate a higher die to effect (a rules outcome similar to that of 'going for broke' -see below - though in MHR this is something that comes into play only on some die rolls in MHR since the effect die roll might not have been particularly good). A high dice could equally well contribute to 'hit' or 'damage', so there's no default distinction between effects that increase damage and that boost accuracy (reminiscent of Tunnels and Trolls). (a few SFX can step up the effect die/damage specifically e.g. Constructs, Dangerous but these are uncommon).
MHR also has a resource dimension in that 'plot points' can add an extra die, or allow a character to keep two 'effect dice' (such as dealing damage vs. two thugs).

Another more complex example of this was apparently proposed in Pyramid (noted here by Wil):

QuoteWhile not tied to any specific system, Justin Bacon wrote a very interesting article for Pyramid called "Dice of Destiny" that is usable with a dice pool system. Unfortunately you have to be a pyramid subscriber to view it (which I am not anymore). In essence, the article describes assigning attributes to each die rolled, such as time, quality, finesse, etc. When the dice are rolled, they are read normally but then the individual dice can be evaluated to unpack more information about the roll. So if, for example, you are using SilCore and roll 3 dice for results of 3,5,5 and the dice were assigned time, quality and finesse, the result would be 5 and the individual dice could be evaluated as the task taking an average amount of time, but with higher quality and finesse.

(Compare this to Heroic Golden Turbulence, below - somewhat similar outcome but without the issue of dice pool variability or needing huge dice pools ? However, it does differ in that dice are allocated after rolling, unlike HGT).

Multiple rolls: here after succeeding the initial check, the character has to make additional checks to determine exactly how well they do/or make further progress. This might appear naturally enough in play when a GM doesn’t have a system to handle effect readily. An example of this method might be running speed in 2nd ed. D&D; a character can roll a Strength check with a penalty to reach 5x normal speed, then if that fails Str checks with successively lower penalties to try for 4x speed, and finally 3x speed; 2x is automatic).
 
A rarely seen version of this that crossbreeds it with margin of success systems is to 'reroll remainder' - count the amount the roll is over the target, and then reroll against it. For example in a roll under system a character might need a 15 or less to succeed; rolling a 12 they get a reroll against 3 or less (as 15-12 = 3) for some extra benefit. (The translation of this to an additive system would be for the margin of success on a roll to be used as the bonus for a follow-up roll).
 
Going for broke: the character can get an effect bonus by taking a penalty on their dice roll (usually an add-on to another system). Examples include 3.x D&D Power Attack (i.e. take a -5 to hit, get a +5 or +10 to damage; Synchronicity (a free rpg) does the same with swashbuckling manuevers (swinging on ropes at -5 adds +5 to damage); DC Heroes has a an option called "Going for Broke" that applies to most actions.
This mechanic gives characters a chance to pull off something very risky, but very rewarding. This mechanic does help gameplay, by slowing down the rate at which dice rolls become irrelevant to characters with huge bonuses (characters are incentivized to take penalties and so continue to fail) however, some players may be deterred from doing anything cool by the chance of failure, and high rolls are less exciting if they don't provide any automatic increase in effect.
This sort of thing can in a sense occur in an informal way in many game systems/situations, often covered by GM adjudication; it can also occur on a case-by-case basis by situational rules e.g. a game might have called shots to limbs or vital locations, even if there's no general system whereby a character can trade success % for more power.
The main thing to watch is that if a character gets bonuses to effect already from the dice roll for rolling high, the effect bonus has to exceed the effect points they lose from taking the penalty. DC Heroes has a good setup for this (the effect bonus is precalculated to more than offset the penalty).
An example of what not to do is Fading Suns 1e; it uses a d20 roll under Blackjack system, where a character can voluntarily choose to add a bonus/penalty to their dice roll ("Accenting"). While this probably seemed like a good idea superficially, calculating this out shows that there is no increase in average effect - the maximum result is unchanged - just a loss in chance of success. That is a character with skill 12 generates results from 1-12 with a roll of 12, while at modifier +5 the character generates results of 6-12 at natural rolls of 1-7 but fails on a natural roll of 8+. Negative accenting in FS i.e. taking a penalty to the roll which increases success chance but decreases average victory points, fails to work for the opposite reason; it only adds to chance of success.
Feng Shui is interesting in that it has 'stunt' guidelines where extra effect has a difficulty increase based off game-level advantage, rather than real-world difficulty ('fluff') i.e. taking out 4 mooks with a trick boomerang shot is at a -4 penalty. This mirrors the sort of logic seen in super power construction in games like Hero/Champions ("effect based reasoning").
As well as a penalty to die roll, success chance can also be reduced by requiring a second die roll. One French RPG as noted elsewhere lets characters roll d20 instead of d10, increasing margin of success directly, but with autofails on odd numbers.
The 'going for broke' idea may also be applied in a margin of success system by changing dice rolled to give a greater 'spread' of results without changing the average; for instance replacing 3d6 with d20, or the mechanic seen in 'Combat Monster' (see Non-dice randomization) where a character could choose to roll either a dollar or 10 nickels, and counts the total values of heads.

Going for broke - type II: a mechanic described here for the (unpublished?) 'triune' RPG let a character increase effect by also increasing failure effect if they fail; actual success chance doesn't change. After making the roll, the character opts to roll a separate pool of 1, 2 or 3 dice (their choice) with a success being boosted by more successes and a failure worsened by more failures.

Trading down effect (for a roll bonus): The opposite of 'going for broke. "Fragged Empire" uses a 3d6+modifiers system where a character gets one "Strong Hit" per 6 rolled on the dice, letting them choose various items. This would be a straight-up natural dice roll system (in a sense combining additive and dice pool), except that the default "Strong Hit" result, "Effort" lets a character re-roll one of the other dice (that didn't come up 6). Consequently, effect is being artificially lowered for low-bonus characters because they're probably spending effect to boost up the die roll, instead of choosing other effects. This works as the opposite of the 'going for broke' option, in that it has to be an option after the roll is made, instead of before.

Extra effect, from additional actions: e.g. a character might combine a 'gather more energy' roll with a 'spellcasting' roll, to cast a spell with more oomph. This may require 'multiple rolls' (similar to the above). If a game has rules where doing extra actions gives a multitasking penalty, it is also similar to 'Going for broke' although it possibly scales differently: instead of a flat +x to effect, -y to the roll, the extra effect is generated from a whole new roll, and so could be massive if the assist task dice pool is large.

Success counting: as seen in dice pool systems (e.g. perhaps the vampire attempting to pick up in a bar rolls Appearance +Seduction and get 3 successes, so he finds a target with an Appearance of 3). Success counting systems, as with margin of success type systems and indeed most of the systems, typically give simple, numeric outputs –with a couple of exceptions e.g. *Warhammer Fantasy, 3rd edition uses custom symbol-marked dice of various kinds – some good, some bad. Individual cards/rules help interpret these custom symbols to give a variety of specific effects.
* the game Heroic Golden Turbulence uses a standard d6 dice pool which is however colour coded; the player allocates their dice pool across various colours as they wish, with the count of successes earnt in a category giving the GM info on that aspect of the conflict – an incredibly detailed result output. For example:
 
 

QuoteLau Wang is sneaking into the temple of the blood god in order to find the scripture of the armour of ancients, using his Martial Arts of 6 and the specialty stunt: stealth. He may wish to get in as quickly as possible, while avoiding death traps. This is an important crisis, requiring 2 goals. He decides on 3 dice for caution (blue), 1 dice for speed (green) and 2 dice for offsetting the death traps (grey). He scores 3 goals, 2 in caution (he doesn't get caught) and one in offsetting (he avoids the scything trap but not the secret poison spray - damn!). Because the green dice did not score a goal, he did not get out especially quickly. Now he has to contend with the imperial guard.

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Somewhat similarly, Ouroboros Engine is a dice pool game where 'dangers' and 'opportunities' each represent individual successes. Players and GM work to set stakes for rolls i.e. a character rescuing children from a burning building might need 1 success to escape damage, 1 to save a child, 2 to save an extra child - the player would choose how to allocate successes after rolling (not dissimilar to how Apocalypse World/ Dungeon World works, but using a dice pool). The key difference between this and Heroic Golden Turbulence would be that because in it, rolls are divided across multiple 'colours' before rolling, a character doesn't reliably get the specific effects they're aiming for. HGTs dice pool mechanic also makes more complex results hard to achieve for characters with low dice pools (more deterministic).
Counting successes is good for calculating damage; Twisting Tunnels (see post 13) is interesting in that expected successes work well as a 'condition', with the # dice modifying follow-up rolls.
Success-counting off a success roll can be combined with a separate roll as well, e.g. how oWoD rolls damage with base damage dice for Strength & weapon getting bonus dice from extra successes on the preceding attack roll.
ORE type systems can potentially have interesting results here if multiple matches are used to grade different aspects of success. (Legends of the Wulin?).

Counting + and - dice: reportedly, Bliss Stage rolls a pool of Fudge Dice with the + and - assigned to various categories of outcomes. This seems similar to "success levels" generally, but with negative complications occurring rather than success dimensions all being counted upward from normal or 0.

Named result levels (off a table); as seen in various universal tables these are usually non-numeric i.e. a “Great success” (Lost Souls), or a “Green” result (MSH), or a “special success” (RQ). They require referencing more rules to work out exactly what a named result does, but for clearly defined situations give perhaps more interesting results (a result might be a Slam or a Kill, instead of a +3 on damage). The most advanced system for this may be Rolemaster, which uses incredibly detailed tables giving exact descriptions of what happens for more or less any situation/dice roll. Tables can of course also just be used to get a numerical value for success in some way.

A table can also be used to cross-reference two different dice rolls/results to get a final result. 'Grid rolling' (above) is a simple case, or another example is HarnMaster combat (below) which compares two rolls' results - critical failure (CF), marginal failure (MF), marginal success (MS) or critical success (CS) for both attacker and defender - to get a result. Below; outcomes are BS (both stumble), DS (defender stumble), AS (attacker stumble), DTA (Defender Tactical Advantage i.e. free attack), nothing (dot), or a number of  impact dice for a attack that hits (A*2 = 2 dice, etc).


Advanced Concerns
Systems sometimes can add a layer of differing interpretation of effect, in addition to the raw check modifier e.g. for Marvel Super Heroes' Intensity rules applies different interpretation of effect to widen the gap between characters of different attributes, who otherwise have very similar chances of success (+5% per ability level, e.g. from Remarkable to Incredible or Incredible to Amazing).

Quote from: Dragon #187Arachne’s spikes can deliver direct injections of poisonous venom into a victim’s body. Her venom injections are of Incredible (40) intensity and can kill anyone in three to six (1d4 +2) rounds who fails an Endurance FEAT roll. Those victims with Amazing or greater Endurance need a green Endurance FEAT roll; those with Endurance ranks of In-credible need a Yellow result; and those with Endurance ranks below Incredible need a Red result to take only 10 points of damage. Failure indicates death.

In this case while 1 rank gives only +5% chance of any non-failure, shifting from requiring Red to Yellow success at Incredible boosts success chance from about 10% to 40%.

Defensive Effect: not a separate method but a principle that applies in some systems. A game may let the defender generate an 'effect' which discounts the offensive effect generated, however usually as a new separate roll, rather than following on from an existing roll, since any defense roll must already have been failed. This can be seen for instance in DC Heroes where the defense attribute directly reduces base result points (without a roll), or oWoD Storyteller, where Stamina dice are rolled to 'soak' (balancing Str successes on damage, though its slightly one-sided in that the damage roll can get bonuses from a good attack roll whereas this can't, being a standalone effect roll). Luck point (safety valve) spending can also sometimes boost 'defensive' effect.

Varying effect systems by subsystem:
Game subsystems most likely to use a different subsystem to whatever the general effect mechanism is include:
*damage (most often rolled, even if most subsystems are margin-of-result)
*magic (often somewhat resource-based when most other checks aren't)
*extended actions/group checks (accumulating multiple 'successes' and therefore discarding individual margins of success).
Of these, most can still be handled consistently in a dice pool game. but otherwise will vary.

Thoughts in closing: most systems run OK without a detailed, consistent effect-determination system, given a GM with at least a modicum of imagination.
On the one hand, systems with no coherent single effect-determination system (most of them) have evolved various different approaches to handle this - a d10 roll here; a level-based damage progression there; yet elsewhere, a direct Jump-check-DC-to-distance-jumped conversion chart. On the other hand, a built-in system for effect can be a solution in search of a problem, with unnecessarily complex subsystems built off the basic system when really, simple pass/fail checks are all that is necessary.
In a wholly integrated system, be careful that unifying major systems doesn't break them. The most important form of "effect" is the damage aspect, and while its a bonus to have other subsystems run consistently with this, make sure that sub-rules intended to facilitate other actions like running or determining spell durations don't somehow mess up the damage system.
 
Conversely, the effect mechanism used does determine how information flows through the system, and a good effect system can cover a number of things that would otherwise require additional complicating rules. For example, if amount of attack over target defense affects damage, then a good attack roll includes 'critical' effects by default, and catching an opponent with their pants down and so zero defense automatically increases damage without need for 'sneak attack' rules.

Note: see also post 136 for details on critical hit systems specifically, and magic for more details on that.

[Note to edit] Different effect systems give different degrees of 'granularity' in results. Some of these options only provide a yes/no result (i.e. 'crits' or high natural-die-roll), while others give slightly different results depending on the die type combined with i.e. 'margin' on a d100 is much larger than 'margin' on a Savage Worlds style single d8 roll, or distribution is very different for 3d6 vs. d20. Dungeon World 'moves' gives a result of basically '1 or 2 successes', similar to counting 'raises' for Savage Worlds. Similar results can be coded in different ways depending on the granularities involved i.e. a Dungeon World 'move' spending one option (out of max. 2 for a 10+ roll) to get 'your spell is not expended' is a purely yes/no equivalent of a more discrete 'subtract your margin of success from the spell point cost to cast your spell'. A finer-grain system probably provides the option to have more complex or detailed manipulation of the effect system.

('Effect' systems can sometimes be called in as part of a high-level task resolution system, letting multiple factors be considered. Say a task might add together levels of achievement for one roll, and 1/2 level of achievement off another roll so as to actually resolve a yes/no question, but stepping up or down importance of various factors (e.g. you might add together Level of Achievement for a Persuade roll, and half Level of Achievement for a Physics roll, to successfully impersonate a famous physicist).

A dice roll on a table might include 'effect' variation that's a bit of random shift, as well as improvement. An example there is 'mutation rolls' in Mutant Crawl Classics, where on a roll for extra limbs higher is generally better (add +level) but might give more arms, more legs, or both. Adding [+level] to the roll could give you more legs when you wanted more arms, however. This would be avoided if it were split into two rolls - one for bonus where level is added, and one randomizing exactly what the mutation does - though that would also mean generating a lot more results since most of the variation is in specific entries rather than effectiveness.

Treasure determination as an example of an 'effect' system:
As well as simple hit/damage, generating treasure can be a sort of 'effect' - much more complex since it isn't a result of X 'hit points' but has to include any number of specific items.
A character's attributes also aren't important here or involved in a check, though an NPC's stats might be used somehow.

Depending on the system this might be wholly freeform, involve rolling on some random tables, or be GM-determined within a budget based on e.g. an NPCs level [X GPs for a level-x NPC] or wealth rating. Occasionally a player gets some choices in treasure determination [e.g. 4E] although this is rather meta since it involves what's primarily a world detail. In a sense a player gets some input into treasure determination also by deciding who they go after.
As an example that's eerily reminiscent of doing hit/damage, here's Dragon Warriors' treasure determination rules based on a monster's treasure code for how rich it is (see also 'Monsters' post) - with roll for if you hit the jackpot and then how much is in it - some monsters may have set treasure codes while others might have the code rolled first e.g. roll d6, 1-5 meagre, 6 = poor:



Edit notes: Last order notes, Fragged Empire notes (*); varying effect systems by subsystem(*), note that yes/no systems encourage 'front-loaded' effect bonuses.(*), note on granularities. Luchador note (*), more DC Heroes notes (*), treasure determination notes.

When designing a system, when to roll dice is an important question - as it is something frequently overused. Dice rolls take time and require additional invocation of rules to resolve, and additionally generate chances of PCs failing, that may not always be appropriate. Though some players just like rolling dice. While this is partly a matter of taste, IMHO a dice roll should be a moment of drama or tension.
Dice aren't the only possible source of tension and diceless systems do exist; Amber in particular accesses fear of the unknown in players by keeping difficulty of tasks, and even a player's own character capabilities, obscure to them, rather than having uncertain dice outcomes.
 
Burning Wheel uses rolls, but generally forbids re-rolling unless a situation significantly changes ('Let It Ride'); e.g. several connected acts of sneaking requires only one sneaking roll. The justification for this is again mathematically reasonable ; that repeated rolls will eventually be the undoing of a character.
 
Similarly, people who have played sneaky rogues in 3.x should know the value of Always Taking 10 on their Hide/Move Silently, for the same reasons - sooner or later the character sneaking through a house and rolling Move Silently all the time will roll a "1" and get caught). Dungeons and Dragons 3.x calls for checks reasonably frequently, but lets characters sidestep many rolls by 'taking 10'; when not threatened or distracted a character can act as if they rolled a 10 on their d20. The positive aspect of this is that what constitutes an 'automatic' task is definable within the DC system (if my bonus is +10, I can auto-succeed on tasks with a DC of 20 or lower) rather than being set by GM fiat, as in most other systems.
The downside is that this system eliminates a number of dangerous task to cut down on die rolls - jumping chasms, climbing walls or the like can also be "take-10'd", so to be effective a terrain hazard generated by the GM must have a success chance of less than 50/50, rather than just "don't roll a 1" risky.
(A possible alternative, which cuts down fiat but would not generate this effect, is to provide bonuses on the dice roll for easy tasks, and not call for a roll when failure chance drops to 0%).
Related to Take-10, the 'Take-20' mechanic lets characters under no time pressure act as if they rolled a 20 - reasonable given situations where there is no chance of failure or time limit and multiple retries are allowed, since it avoids players rolling endlessly until they get a 20 anyway. Another alternative to this is the 'fail until you fail forever' rule, where each failure increases the target number until the check can't be passed, often seen with Open Lock in particular (or alternatively the "keep trying until you critically fail and jam your lockpicks in the lock", etc).
5E replaces the 'take-10' with a 'passive skill check' of 10+modifier; the distinction being that usually when the passive score applies is at GM discretion rather than player discretion. It sometimes becomes overly deterministic by having an opposed roll reduce to passive score for both sides, giving a set result as to whether a trap is found for instance.
Some modifiers can apply only to synthetic die roll results e.g. 3.5 includes a couple of feats/options that let characters 'Take 12' ( Int checks w/ 'Focussed Mind', Races of the Wild) or 'Take 15' (expending Psionic Focus on Concentration). These may, however, discourage dice rolling too much, and generally seem to have little justification for how they work.

Similar to Take-10, 'D6 Space' (and possibly other d6 system sourcebooks) suggests instead of rolling several d6s for rolls (particularly unimportant rolls), the gamer should just multiply the # of dice by 3 or 4. It biases this (applying an ad hoc modifier which would never have applied were the dice actually rolled) by using x3 for rolls which 'ought to fail' ("cannon fodder damage resistance checks") and x4 for actions that ought to succeed ("something the player characters are doing").
Unisystem takes a mechanically similar approach but applies 'take average roll' to NPCs only - NPCs has scores for 'Muscle', 'Combat' and 'Brains' which are base stat +6 (average for d10). Muscle being prefigured still allows randomization via the player's roll. For Combat dice are still rolled, but the average combat number is used instead to determine 'success level' i.e. damage modifier for attacks, without table checking.
BESM 3E rolls 2d6 and has a d20-inspired "Take 6" rule; it  also separately gives lots of guidelines to the GM on 'when not to roll' that are maybe redundant with this.
Damage rolls with lots of dice can sometimes also be shortcut. The Epic Level Handbook suggests using average damage for serious damaging spells (20d6 fireballs etc); the Immortals boxed set for BECMI D&D suggests a similar system, but also proposed adding an arbitrary random variable i.e. to a base of 158 average damage for a 45 die fireball, the DM could add say (2d20-20) random variation.

Though not innovative in terms of cutting down dice rolls, Unknown Armies is interesting in how it runs with the implications of the 'die rolls as dramatic" idea in setting its percentages for success: it assumes rolls will only be required in difficult situations so percentage chance of say, fixing your car, is precalculated to minimize modifying by assuming you're under (for example) time pressure from zombie attack, rather than having (as it says) the default % be the expected rating for a 'lazy Sunday'. This is probably a good idea in the context of its roll-under d% system, as psychologically there is a tendency for GMs to use the unmodified rating in roll-under (cf. John Kim's essay at Darkshire on roll-under systems).
 
A review elsewhere on the site here mentions a 'Tatzelwurm' system [2d6]; it lets PCs overcome Simple difficulty tasks by 'just roleplaying, no roll required'.

A number of games also have rolling only for the players, not the GM: these include Legendary Lives (1990), Whispering Vault (1993), and more recently Icons, Apocalypse World, Numenera, and Soylent Green's freerpg Bounty Hunters of the Atomic Wastelands (Fate). (thanks RobM for some info). Dragonlance SAGA also had a system equivalent to 'player's roll' based on a player card deck - saving the GM from needing a separate deck and giving players more control (some genuine randomness is however provided with separate 'fate' cards).
See thread here for some discussion of 'players' only roll games (includes various contentions re. immersion, overcomplexification to compensate for removal of GM rolling, philosophy of player rolls vs. GM roll [epistemiological questions vs. in character], and different style of 'interpretations of results' required in player-only vs. both-sides-rolling i.e. the GM folding description of opponent actions into the roll as well.
 
Earthdawn has a strain mechanic where using certain Talents causes the user to take damage. 3E Earthdawn sometimes uses this to discourage some skill re-rolling i.e. Search checks actually cause the user damage. "As it is, an Adept and move silently indefinitely without problem, whereas it is totally possible for that same Adept to weary himself into unconsciousness by looking for something." (quote from here).

Dragon Warriors (and the earliest versions of 5th Edition D&D) had a mechanic for automatic success on attribute checks, where no check was required if the difficulty was less than the attribute score. This gave slightly odd results in that giving an auto-success resulted in a higher success boost with lower attribute scores e.g. automatic success at a stat of 8 would boost chance of success from 40% to 100%, while for a 15 it would boost it from only 75% to 100%). 5E D&D as of the mid-2013 playtest retains the ability in a barbarian special class feature (a character can roll, substituting their Str for the result if they don't like it; which incidentally also means that the automatic success value increases twice as fast as the average value generated - 1:1 per stat point rather than 1:2 stat points. The ability also probably peaks in usefulness at a Str of 15 or 20, as these are default benchmark difficulties.)
Dragon Warriors also has a durational 'spell expiry' roll (probably to replace having to count duration of spells in combat rounds). This was normally a 2d6 roll each combat with a "12" being spell expiry, but was replaced with a 25% chance per minute of spell expiration out of combat (an equivalent percentage but with number of rolls cut down, though the need for the initial roll is maybe questionable).

Marvel Super Heroes has an 'automatic FEAT' rule, where a character succeeds automatically on a task whose Rank is 3 or more less than their attribute rank - for example, a character with Remarkable (30) Strength vs. a task of Typical (10) difficulty. This is not so much to cut down 'excessive' rolling as it is a patch on characters' relatively limited chance of success - 3 ranks increase success chance by only +15%, so without a rule such as this characters would frequently fail what should be easy tasks. Note that apart from this effect, and a rule where tasks of higher difficulty are impossible, task difficulty doesn't otherwise alter the chance of success. Conversely, a task may be 'impossible' if task difficulty is higher than the PCs rank in the ability (though it is suggested a PC be allowed to roll if an action being impossible would result in certain death).

Dice rolling is sometimes disallowed in a system because the result would be too random. D20 system for example would probably resolve an arm-wrestling contest as a 'highest wins' because opposed (d20+modifiers), its universal system, would give the higher score too little comparative advantage. In GURPS however, opposed 3d6 rolls give results that are much more predictable, so its likely to be acceptable, making GURPS paradoxically less deterministic than d20 in this instance.

The Ubiquity System (e.g. Hollow Earth Expedition) has a 'take 10' type step which instead of being optional, engages on every dice roll: if a character's average # successes from their dice pool beats the difficulty, they don't have to roll. As ubiquity dice are 50/50 likely to succeed, the average is calculated as 1/2 the full dice pool; in the case of an odd total (e.g. 5 = 2.5 average successes) the player rolls a single die as part of the 'Taking the Average' procedure, before moving on to a full dice roll if necessary (which isn't affected by this roll).

DC Heroes has 'automatic actions', generally relating to 'effect' attributes such as STR, where characters automatically get effect points equal to the attribute without having to roll on the usual success table (which could however potentially generate more action points, depending on the roll and the assigned resistance value).

One-off special situations generating 'synthetic' rolls: Apocalypse World generally just believes in not rolling unless necessary, but one interesting idea is here. "When you give 1-barter to someone but with strings attached, it counts as manipulating them and hitting the roll with a 10+ [i.e. on 2d6], no roll required" i.e. this specific circumstances is equivalent to an automatic roll of 10. Another example of the 'special situation synthetic roll' might be 3E's 'coup de grace', which gives an automatic critical (=mostly equivalent to a free '20') on a helpless foe. The same principle sometimes applies to damage (e.g. a monster taking 'minimum damage' from fire or guns or etc., instead of it being rolled). Some rolls can also be overriden by GM judgment i.e. the GM setting the result - 5E T&T has a rule where an unwounded character can lift twice 'weight possible' for 1-6 turns (which 'might' be set by roll of a die, or not). A circumstance might also generate a 'synthetic' result from a different die roll (e.g. the coup de grace 'critical' might in a sense result from a successful Move Silently roll; in a less unified system, different subsystems might need to be translated between, converting a number generated by a different die roll.

(another idea on 'synthetic' rolls: - see recently-added note under 'dice pools' for a dice pool system using a synthetic roll, added to the normally rolled pool)

It may also be worth looking at some particularly egregious instances of excessive dice rolling in games:
 
*active (defender-side) rather than passive (attacker-side) rolling for perception. A roll for a form of Perception - to search for traps, or spot opponents - can in a dungeon generate large numbers of rolls - potentially one per 10-ft square or one per room whether or not an opponent is there. The converse situation where a trap or attacker rolls to surprise a character vs. the character's relevant defense is made only when a trap or threat actually exists, cutting down potential rolling opportunities.
(Note that who rolls can be important if one side gets re-rolls for whatever reason. 5E D&D uses a rule where a character has 'advantage' gets +5 to their passive score as a fix for this).
 
*Randomized movement; some games way roll for random movement - which can be problem if taken to extremes.
(This is discussed further in the "Movement" section for combat).

*oWoD Mage can have lots of rolling since mages tend to use magic for virtually everything more complex than tying their shoelines, with the main balancing mechanic for magic really being the chance of a 'botch' that will cause paradox, making it tricky to skip over trivial rolls even when the outcome of the task itself isn't greatly important.
 
Excessive dice rolling in a system can occur for various reasons:
*when a situation originally dangerous/dramatic situation turns out, through rules drift across editions or just a misapprehension on the part of the designer as to how their game would actually be played, to be something routine and dull. For example, Cure Light Wounds in AD&D (heal d8+0 hit points, a rare effect healing a significant chunk of a targets HP) became 3E's Cure Light Wounds Wand, requiring the roll of dozens (or more) of dice between combats to determine charge depletion. Note a number of spell-point systems (T&T, SenZar) avoid similar situations by having simple 1:1 ratios of spell points-spent-to-HP-healed.
 
*when a system allows players to skip over rolls, but still gives them an incentive to roll in routine circumstances, to get higher results or extra bonuses. For example, oWoD Storyteller or Shadowrun 4E both allow "auto success" if a character has a sufficiently large dice pool (either where number of dice >difficulty rating, or at a rate of 1 success per 4 dice, respectively) but a player may frequently still wish to roll to get more successes - particularly in Storyteller, where chance of failure is probably quite low.
To address this, a designer may be able to readjust the cost/benefit ratio of this to deter extra rolling by ensuring such "voluntary rolls" retain significant failure chances, or even critical failure results (if they're a jerk).
 
*when additional rolls are call for just because a task takes a long time in-game to accomplish (extended checks in some systems). Often it may be more appropriate to roll to see how long it takes to perform a task, rather than rolling success checks over and over until a target # of successes is reached.
 
Additional Note: choosing when to roll, or not roll, can be important due to secondary rules that are invoked.
For example, if a flat roll is used for Defense, this will not be affected by other to-hit penalties. Systems where attack rolls are used to parry however, normally add an exception to applying off-hand penalties to shield parry rolls (e.g. GURPS, IIRC).
 
Extra Note: for events which have a low probability of occurring, rolls for these can sometimes be streamlined out of a system by having the roll only occur in specific circumstances (i.e. only critical hits require a hit location roll), or having something occur as a byproduct of critical success/critical failure on another roll (a 1 on your armour bypass roll results in a weapon breakage check [not from a published game; this is a Dragon Warriors house rule I've considered using]). When doing this the designer should be careful that doing so doesn't skew the likelihood of the result; e.g. in the above armour bypass example the designer would have to be careful that a larger armour bypass die for swords would result in them breaking less frequently than maces, unless the second roll to confirm the breakage was adjusted to compensate for this.
 
Similarly to this, 4E D&D streamlines out extra saving throws against effects by having one 'attack' cover both normal and 'rider' effects - e.g. an attack that hits might generate both damage and a stun. This avoids the extra dice roll, but requires a 'conflation' of defense factors - a [damage+move] effect is against the resistance score for being moved, so that defense against damage isn't considered. It also means a situation of "damage, but no move", isn't possible.
A similar case of 'conflation' of multiple rolls into one (questionable) one for simplification would be ranged attack powers in Savage Worlds i.e. Bolt. The arcane skill roll to activate the power is also the attack roll for the power: this generally works since both ranged attacks and spells have a fixed (4) target number. Complex cases can be weird e.g. " should a Bolt receive a casting roll bonus for attacking a Large monster, if yes would raises due to this still reduce the power point cost if the character has the wizard Edge?" - might lead to the same roll getting different sets of modifiers for one use for another use. The same rules quirk lets Weird Science characters just use Shooting to hit with devices and so avoid buying arcane skill in the first place if they largely intend just to Shoot things.

Occasionally an alternate system is used to shortcut what would otherwise require lots of normal rolls - for example, having a 'mass combat' subsystem instead of rolling attack rolls for a large number of participants.
Dragon...#113? has another approach; it has binomial probability tables set up using d100, which gives percentage chances of a given number of successes/failures for up to 20 d20 rolls at a given TN. That means that 20 d20 rolls can be replaced with a single d100 table check.
In other cases, rather than rolling lots of small individual probabilities for different things, its possible to have a single roll for a # of things, then a random allocation of which. For example, deadEarth rolls for ability/inability (2 or 12 on 2d6) for each of 100 individual skills in character generation - a roll of 200d6 than has to be done one at a time to pair up the dice. The end result of all this is, however, usually not much different to rolling a small variable number of abilities and inabilities (say d4 of each) and then rolling d100s on a table to determine which skills are affected.  

Another approach is prerolling, where the GM rolls lots of dice in advance then consults a list. Not something I'm terribly fond of, although its been argued that it avoids the interruption of the scene by the dice/break in immersion. Note this doesn't tend to work well with a step die system, since a different chart would need to be consulted for each die type (d6,d8, d20, etc), or for a dice pool system where number of dice varies each time.

A final note: some games are built up with lots of excessive dice rolling on the idea that its 'fun' to roll lots of dice. They can burn in hell - deliberately overengineering a complex system is insane. Situations will arise where even a simple system results in lots of rolling (like large combats), without going out of your way to add mess - and you should aim to use dice and mechanics where necessary to represent something rather than for the hell of it. Multiple dice on the same roll also give more predictable results and so can make doing checks more dull.

Rolling to determine TN.
Games also sometimes have extra unnecessary rolling, with a roll in order to set the odds of success. Because the final chance of an event occurring is the multiplication of the two probabilities, this has the same effect as just setting the final score appropriately to begin with - unless other modifiers apply at some stage, of course.
Examples here include:
World of Synnibarr luck rolls - where a d100 is rolled to find a % that is then rolled against to determine success/failure. 4d10s therefore replicating a coin flip.
In HarnMaster (3E?), the sample monster has a 1d4% chance of something (hatching?) i.e. a 2.5% chance.
In CoC, Azathoth has a roll to determine what his skill is (it might be 100%) and then a skill check against it - which could be replaced with just a single (average) attack %.
This sort of arrangement can arise naturally as a result of multiple rules interacting to simulate something; if the dice rolls together can be solved down to something similar this may be preferable, but that cause is at least some justification.

recent edits: rolls disallowed as too variable, special situations. 21/11 - Hollow Earth Expedition notes.

An interesting case of where an extra roll can't easily be streamlined out is Palladium defense numbers. Palladium gives armour a fixed AR, with a hit roll over the AR needed to damage the roll. The defender can also opt to roll a Parry or Dodge, which then is used as the target number instead of the AR (if its higher). A parry or dodge roll can't be replaced easily with a 'fixed' result since that will either be pointless, or make the AR pointless.

Extended actions

An 'extended action' can be a higher-level resolution system in a game; the hierarchy is perhaps basic action - opposed action - extended action, with extended actions being the basis of e.g. combat, and often Craft or similar tasks. The principal question is how to 'add' the results from individual dice rolls, which might be answered by the Effect subsystem- e.g. in many games beating an opponent to death requires adding together weapon damage (a sort of effect) until it exceeds the target's HP...other sorts of extended checks might simply count a 'successful' roll as 1 unit toward task completion, or a higher roll might contribute more success (SenZar). A roll might also be theoretically accomplishable in one action, with a high but not quite high enough result giving a bonus to the next check (some Rolemaster results).
# rolls permitted may be limited by time, or counts of failures may be tracked (note this discourages lower-skill involvement if multiple PCs are cooperating). Another variant is to track only critical failures.

Interesting subsystems relating to extended actions include:
*SpyCraft has an 'advisor' special ability letting them use an action die to let the advisor take-10 on that number of checks in an extended action, decreasing chances of failure.

*Dying Earth uses a table where an opponent's result (if high) forces an opponent to use more dice rolls from their resource pool.

*Rolemaster chart frequently list a % of completion, allowing continuing action across multiple rounds (barring a disastrous failure triggering a reset).

*3E D&D Craft uses the DC of the check to give GP of progress; a character can voluntarily increase the DC to work faster, but with more chance of failure.

*Dogs in the Vineyard's standard conflict resolution mechanic is extended; it has no system for simple checks however.

*DC Heroes may use cumulative result points (RAPs) for some things, but this is generally awkward in it due to the exponential nature of attributes.

*Fate Core has a contest system that looks relatively interesting - not just dice rolling since aspects/fate point costs/consequences can make the process more complicated. It does look unclear (or somewhat GM fiat) as to when a situation calls for an extended action vs. a single roll. Its abstract general systems can produce odd results when used directly - a chase for example gets a bonus to the athletics check for multiple participants, rather than considering that multiple targets won't be generally much faster, or that some may fall behind.

Some games may have specialized extended action systems of whatever kind. Palladium Arm Wrestling skill (in Mystic China) for example uses a series of rolls (d20+ 1 per 3 PS above 16) with 3 successes in a row required to succeed - first roll steadies the grip, second roll tilts their arm down, third roll slams their hand down into the table. One Marvel Super Heroes adventure, Secret Wars, represents an incredibly difficult action (persuading Galactus to help you) as needing a series of 3 Red Popularity checks, just one being not difficult enough! (perhaps fair enough considering that Karma could be spent on this).

Das Schwarze Auge (The Dark Eye) apparently uses extended rolls as the default procedure for skill checks - for instance a Climbing might be roll of d20 against STR/STR/DEX, with the difficulty of the climb giving the total margin of failure allowed. For the most part this is non-transparent, slow, and the extra rolls provide no information that couldn't have been gained from a single check, although proponents of this sort of system (such as 4E's Skill Challenge system) believe it adds more 'drama' to rolls. For games using 'safety valves', this sort of check can also require more resource burning to achieve. Requiring multiple rolls gives a more predictable outcome (despite less transparency!) but one akin to what would result from just a multidie system.
JAGS uses a very similar system to DSA for 'Drama' rolls, perhaps partly to give a more predictable outcome.

5E D&D has an interesting rule for group checks, where a group task like sneaking is successful if 50% of the party are successful. Therefore tasks aren't automatically becoming harder as number of people increase, and skill ratings for everyone involved remain relevant. Small numbers of PCs with this give wonky results - 2 characters will pass if either succeed (so its easier to sneak with 2 rogues than with one, meaning allowing a group check here is probably disallowed), and four characters have a better chance of success than three (50% of party means either three or four need 2 successes total, with four getting an extra roll to get there). 5E sneaking is thus slightly a problem in that you would expect moving an army to be more difficult than moving a single person - even if 3E dropped the odds too fast. A quick fixit would be for the GM to allow a group check, but increase the DC based on # characters.