The mechanic of rolling 2d20 instead of one is very helpful in both the newest edition of D&D (where it’s used for Advantage and Disadvantage), and for other games that use an uncurved die for a single roll. By rolling 2d20 (or even more), you’re essentially adding a curve to a roll whose results would otherwise be linear. Particularly if you read the dice independently, you’ve made the results much more similar to a dice pool or iterated series of rolls. This serves to reduce swinginess, by further reducing the chance of fluke successes or failures (I suspect most players are more likely to try rolls on their high skills when given the option than their low ones, so are going to have a roll swing into a failure on a high skill more often than it swings into a success on a low skill).

Ultimately, there are a decent number of traits on a character sheet that get rolled far less often than others (e.g., you make attack rolls and perception checks all the time, but other skills maybe only a couple of times a session unless you’ve really built the character to make use of it as part of a combat mechanic). For frequently-rolled traits, averages are likely to kick in, but for something you roll once a session, you could wind up having a disappointing tally of failures over time on something that ought to regularly succeed. Particularly when something important hinges on your once-per-session roll of a high skill, it might be preferable to have some way to accentuate the curve.

This house rule adds the following options to a D20 game (particularly low-powered, high-whiff stuff like Beyond the Wall):

A player may roll a single d20 normally if not acting particularly cautiously.

A player may instead choose to act cautiously, rolling 2d20. The player can only do this in non-surprise situations (e.g., not on saves unless the source is obvious and the target is not flat-footed, or on rolls to notice something if the character isn’t actively searching).

When acting cautiously:

  • If both dice are successes, it’s a full success.
  • If both dice are failures, it’s a full failure.
  • If one die succeeds and the other fails, it’s a partial success/success with consequences (glancing blow for half damage in combat, resist the worst but not everything on a save, etc.).
  • Both dice must be a critical result for the action to count as a crit (success or failure).

Essentially, acting cautiously means that you’re lowering your chance of a crit (from 1 in 20 to 1 in 400), reducing the chance that you’ll fail outright, but adding in a decent chance of partial success. For rolls you’d normally fail 75% of the time, you drop total failure down to a 56% chance (but most of your successes are only partial). For rolls where you’d only have a 50/50 shot, you change full failure and full success to both be 25%, with partial filling up the middle 50% of results. For rolls with only a 25% chance of failure (which is still pretty risky on a roll that a lot hangs on), you lower full failure to only around 6% (but move 40% of your successes to partial ones).

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