Binary-Outcome Dice Resolution Methods: Roll-High vs Roll-Low

In many modern games, particularly RPGs and miniature wargames, randomization is represented in the form of a dice roll that has two binary outcomes. This has largely replaced the older model of wargames in which dice results are used to reference a table with a wider range of variable results. For example, many RPGs involve a "skill check" or a "to-hit check" that rolls dice, performs some fixed situational adjustments ("modifiers"), and then returns either a success or a failure for the attempted action.

The general technique is this: (1) Set a target, (2) Roll dice, (3) Perform modifications to the target value and/or the dice roll, and (4) Compare them to one another. But the comparison can be done in one of two ways. In a "roll high" system, the dice result (after modifiers) must be higher than the target. In a "roll low" system, the result must be lower. Although early games often oscillate between each method for different subsystems, modern games usually insist on enforcing one approach or the other uniformly throughout the entire game.

To further complicate the picture, note that the situational modifiers may be applied to either the target or to the roll result - or both! This effectively crates four possible approaches to choose between or combine. Let's suppose we have a 20-sided die, and we want to attempt an action that typically has a 75% chance of success, but in a situation where the odds are improved by an additional 25%. Here are the four equivalent implementations:

1. Roll-low, and modify target: The base target is 10. Any result less than or equal to the target is a success. Give a +5 modifier to the target.
•  Equation: 1d20  ≤  10 + 5
2. Roll-low, and modify result:  The base target is 10. Any result less than or equal to the target is a success. Give a -5 modifier to the result.
• Equation: 1d20 - 5  ≤  10
3. Roll-high, and modify target: The base target is 11. Any result greater than or equal to the target is a success. Give a -5 modifier to the target.
• Equation: 1d20  ≥  11 - 5
4. Roll-high, and modify result: The base target is 11. Any result greater than or equal to the target is a success. Give a +5 modifier to the result.
• Equation: 1d20 + 5  ≥  11 

It's not difficult to confirm that all of these give an identical 75% chance of success. However, that doesn't mean all of them are equally intelligible. To me, the first approach is the only one which makes the success probability obvious to me based on the way it's written. I can easily see that the target (10+5=15) is 75% of the maximum die roll of 20. This strongly predisposes me to method #1. Even method #2 isn't much worse, since it just requires a bit of algebra to slide the negative modifier to the other side of the equation.

The two roll-high methods are more opaque to me. The problem lies with the fact that a target of 6 doesn't look much like a 75% chance of success, on the face of it. Obviously it is, since there are 15 out of 20 results that succeed. But the simple "divide target by max result to find probability" trick no longer applies.

Unfortunately, this system seem to have won for psychological reasons. Rolling high feels more like a "win", since big numbers correspond to higher scores in most sports and games. So clarity is sacrificed for psychological pay-off. This occurs most dramatically in 3rd Edition D&D, where all previously existing roll-low mechanics are rewritten to be exclusively roll-high.

One additional problem with this psychological appeal is that it tends to motivate the system to grow in an unbounded way. Roll-high systems effectively can keep stacking modifiers on both the target and the result, making the situation even harder to interpret. When rolling a 1d20 for a DC32 check (i.e., target of 32) with a +14 modifier to the dice roll, you need to perform two steps of mental algebra just to extract the probability. Quick, perform that operation in your head! What are the odds? Go ahead and grab a pencil and paper, I'll wait.

OK, the result isn't quite that hard. You can subtract that +14 from the target to get 18 (switching to the slightly less confusing method #3), and then recognize that results of 18, 19, or 20 all succeed, for a probability of 15%. But why be forced to do mental subtraction, when you could use method #1 all along?

It's probably too late to eliminate the general preference for roll-high systems. Interestingly, a few games that use percentile d100 rolls have managed to retain roll-low mechanics, suggesting that the real obstacle is that few players today can even due a single division (100%*15/20 = 75%); once the system uses any other dice type, it completely gives up all interest in retaining the intelligibility of its odds.