Resolutions

I have been looking at a lot of different resolution mechanics lately, and I have three or four observations that start to muddle and blur the more I think about them, so this is my effort at sorting it out:

Tables:

• Arbitrary non-formula targets, typically with non-mathematical column and row designations. 'Weapon-type versus armor type' or early 'saving throw' tables are excellent examples of this class of tables. While attribute and level modifiers may be applied to those arbitrary targets, that functionally adds a third or even forth (imaginary) numerical axis onto an arbitrary and indispensable table. Many early games had multiple charts like this.
• Semi-arbitrary obscure, vague, or multiple formula targets, often with multiple formulas changing out at arbitrary points, in a course approximation of a curve, or actual undefined curves. Often with an upper limit. Modifiers may be applied to the roll, or affect the column or row checked, such distinctions may produce different targets. This class could be further defined, but ranking such nuances is not my concern.
• Linear- or stepped- simple formula driven targets. Modifiers applied to rolls vs shifting columns or rows may still yield inconsistent results. Stepped targets are often the result of rounding at some step of calculation.

Class modifiers:

Class may affect which table or progression on a table one uses, or give no-, flat-, arbitrary-, or linear- bonus to either the die result, or the column or row consulted.

Level modifiers:

Level is likely to be one of the axial conditions of a table, but if not, it's likely to be a linear modifier to die results.

Attribute modifiers:

• Arbitrary attribute modifiers tend to apply bonuses to die rolls at the extreme ranges of the attribute. Early games may apply a +3 to the extreme (18, typically), +2 for a 17, and +1 for a 16.  Some may grant +1 for a 15 as well. Similar penalties for the lower extant of the die range were typical. This is the most common format, and although it's typically applied to bell-curved attributes (3d6), the bonuses themselves are applied in a linear fashion at the extremes of the curve. This may not be best representative of the superior specimen an (18) represents as the best of the best, with less than 1% of the population having such capability: super-star professional athletes, and daunting intellects such as Newton, Darwin, Einstein, Tesla... they are the 18's to whom a measly +3 is granted in such a system, while a 15 still represents the top 5% of the population, but in many systems gets not advantage, and in others, a scant +1. 
• Over/Under Linear attribute modifiers typically build off a median score, and add accordingly, so if 10 is the median or baseline: 11 grants +1, 12 a +2, and so on to 18 with +8. Inversely, 9 is -1, 8 is -2, and so on. Penalties and bonuses could be ramped infinitely for super-humanly grand (or pathetic) scores. Systems such as this certainly resolve another complain often heard regarding arbitrary modifier systems: 'what is the difference between 13 and 14?' A score of 14 seems far from an 18, but it represents the top 10 percent, which may not be an olympic gold medalist or revolutionary mind, but could sill find themselves playing for a professional or regional team, or a published and well respected intellectual, and deserving of some modifier.
• Add-only Linear attribute modifiers simply add the value of the attribute as a bonus to die rolls. Has the charm of simplicity, no additional tables or lore to parse. This system requires higher targets and mandates that NPCs (and possibly objects) be given some sort of stat or built in modifiers to achieve equity with fully stat'ed characters.

Dice Check Mechanics:

Dice combinations rolled for a check, and their ramifications. I am making a general assumption that ability scores are distributed on a 3d6 bell-curve, and that the 3d6 curve is a fair representation of all functional human capability at both extremes. I am also going to assume that if skills are in play, they are on a flat or linear advancement system. While some systems plot skills on a curve, they are rare enough in my experience that this is the only mention they need for this topic. I will probably visit this concept in more detail in the future.

• Curved distributions are produced by two or more dice, 3d6 being most common. Checking against a curved target (vs attribute) using a curved check mechanism (3d6 as a check roll) makes higher attributes exceedingly easy to score successes with, and low scores abysmally difficult to overcome. Using a curved mechanic is probably best when checking against linear targets (skills) modified by curved attributes.
• Linear distributions a produced by a single die, 1d20 being the most popular for such checks. Checking against curved targets with a linear check tends to provide reasonable success/failure, even at the extremes.

To illustrate the two mechanics and their impact on play, consider:
A 15 checked against on 3d6 grants success 95.4%, while on 1d20 success is 75% likely, an 18 checked against on 3d6 is 100% success but on 1d20 grants success only 90%. An attribute score of 15 in a system favoring curved checks is superior to an 18 in a linear check system.