Basic Role-Playing, the system which underlies Call of Cthulhu, Stormbringer, Pendragon, Runequest and a buncha other games has this mechanic: you have a stat, you try to roll equal or under it.
For head to head contests, there's this system: both parties try to roll as high as possible without going over their stat.
Here is the first point I want to make about this system: it's awesome.
1. It can be explained immediately and easily to new players.
2. It requires no charts to look anything up.
3. It requires no addition or subtraction. (Here it beats my go-to spot mechanic: both sides roll the same die and you add your whole score.)
4. (Therefore) You immediately know if you succeeded or failed.
5. It can work on ability scores scaled to any die (You could build a game where your stats ranged 1-10 and roll a d10 and it'd still work). Which means you could use it in any system that rates abilities in numbers of any kind.
6. It respects high scores in detail--for every pip your score goes up, your chances get a little better. So you can use it in a campaign-oriented game where PCs slowly improve by steps.
7. It has a degrees-of-success system built in: there's fumble (rolling the max score), failure (rolling over), weak success (rolling under but not better than the other guy or--with a slight hack--a static opposing target number), success (rolling under), mega success (rolling exactly your score).
8. It requires no derived numbers. Like a score of 17 is a score of 17, not 17 (+3) or 17 (+1). (Like, ok, a lot of non-D&D systems.)
For some reason, despite all the intertalk you hear about folks clinging to various old D&D rules purely out of nostalgia even when the rule in question is patently useful, folks rarely bring up the fact that this rule pretty much beats the hell out of D&Ds: d20-add-derived-numbers-roll-high-meet-a-target-number (and sometimes also roll under or roll a d6 trying to meet a derived number instead).
Why? Because nearly everyone who would make it is either still using the d20-roll-high system (WOTC D&Ds keep it) or is using some other less-elegant system that, if we're honest, is probably only still being used (if it was invented before BRP) out of nostalgia or (if it was invented after BRP) out of pure neophilia or just a desire to avoid a lawsuit from Chaosium.
The system has, for all the parameters above, never been beat. Lots and lots of games don't use it--very few don't use it for any kind of good reason.
It does have two drawbacks I can immediately think of:
1-You don't try to roll as high as possible. People like that 00 or 20 means something good. Or at least they seem to.
2-Not every player is trying to hit the exact same set of numbers when they roll, so you can't add bonus fun where you key the die roll to a table where, say 1 always means This and rolling your score always means That and rolling 15 always means The Other Thing.
(Here's a roll-high variant that solves both of those problems: you try to roll as high as possible. If your roll is lower than your score in the relevant stat, you get to roll again, once. It creates new problems--the most obvious one is possibly having to roll twice. It'd work well for something like DCC spell checks.)
(Some of you may be eyeballing "Roll as high as possible or take your stat, whichever is highest" but that's a drag because basically if the GM wrote the adventure, the GM knows in advance if the PC will win any contest with a static target so you're basically just deciding what will happen when you write the adventure.)
I've played games that work the BRP way, and then, hours later, played games with less elegant resolution systems. Y'know what? You don't much notice the difference in play. At a certain point these things just don't matter that much. I mean: if you were building a language from scratch you probably would spell "would" differently---but it works out alright anyway.
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