Design goals

1. The distribution of skill test outcomes should be a bell curve with a "nice" range such as 1-10, 1-20, -10-10, -5-5, etc.
2. As a character's skill level increases, the mean of this distribution should shift upward, and the variance should decrease; i.e., the character's results should get both better and more consistent.
3. The increase in skill from one level to the next shouldn't be too extreme.

Towards a mechanic

Considering various ranges for bell curves, I find that I prefer -10-10; this can be achieved as a nice-looking bell curve using 4d6-14 or 2d6-2d6.

I don't like the idea of either of those for a skill roll, but I like 2d6-2d6 better. Let's have the player roll 2d6 for his test, and try to beat 2d6 rolled by the GM as a sort of "noisy" difficulty.

In order to produce the required mean and variance changes with increased skill level, we will replace the simple 2d6 with the highest 2 of Nd6, where N = 2 + skill for the player, and N = 2 + difficulty or opposed skill for the GM.

A tentative mechanic

A player rolls 2d6, plus one extra die per skill level, and keeps the highest two.

The GM rolls 2d6, plus one extra die per level of difficulty or opposed skill as applicable, and keeps the highest two.

The player's "margin of success" is the difference between his roll and the GM's roll. This will be a number between -10 and 10, skewed upward for higher skill levels, and downward for higher difficulties or opposed skill levels.

Computing the margin of success can be made simple by noting that if the player rolls higher he succeeds, and if the GM rolls higher the player fails, and then just subtracting the lower from the higher; then one obtains a success or failure magnitude of 1-10, and never needs to work with negative numbers.

Merits?

The basic margin of success is distributed as a "nice" bell curve from -10 to 10 as desired, and success and failure are measured on a simple scale from 1 to 10. As the character's skill increases, his mean margin of success increases, as does his consistency by virtue of decreasing variance. An examination of the increase of the probability of success with increase of skill level looks reasonable. The three design goals seem to be satisfied.

Here are graphs of the margin of success distributions for several character skill levels against the default of two opposed dice.

And here is a graph of the probability of a successful outcome for the same skill levels and opposition.

Note the diminishing returns for increasing skill level -- this suggests that there will be no need for exponentially more difficult or expensive level increases.