Probabilities of the Aid move in Dungeon World

In Dungeon World (and in some other games powered by the apocalypse), a character can make a roll that, if successful, allows them to give aid to someone else’s roll. Because I mess with this move fairly substantially in Fourth World, I’ve had to analyze the probability of the Aid move and figured I’d share the results here.

Like all powered by the apocalypse (PbtA) games, Dungeon World relies on rolling 2d6, adding them up, and adding an additional modifier (typically from -1 to +3, based on a stat or something similar). On such a roll, six or below is a failure (allowing the gamemaster, who does not otherwise “get a turn”, to do something), between seven and nine is partial success (where the fun part of the game usually is) and ten or more is a full success. What happens on partial and full success depends on why the roll was made. Some rolls even have a higher level of success if you roll a 12 or more. These rolls are part of “moves” that get triggered when something happens in the fiction of the game.

The Aid or Interfere move in Dungeon World says this:

When you help or hinder someone, roll+bond with them. On a 10+, they take +1 or -2 to their roll, your choice. On a 7−9, they still get a modifier, but you also expose yourself to danger, retribution, or cost.

So, imagine you are in a game, and something is going on and you think “is it worth it to try to help out, or will I just make things worse?” How do you answer this question?

First, can you make things worse? Well, it is possible for you to fail when you Aid, and for the person you are helping to also fail. In this case, technically, the GM can now make a move of their own for each failure, where had you not tried to aid at all, the GM would have only made one. While it is not unheard of for a GM move to improve the character’s situation, this tends to be the exception. So, yes, your attempt to Aid can make things worse.

Plain Rolls

To figure out how often, lets look at a basic interaction of two 2d6 rolls without any modifiers. Let’s imagine that, for some reason, the only way to do something is to succeed on two 2d6 rolls in a row. Because each roll is an opportunity to fail, this is harder than just succeeding on one roll. The outcomes for each roll combine like so, with the number in each cell being the percentage chance of a particular combination of results:

Second 2d6 Roll
6− 7−9 10+
First
2d6
Roll
6− 17.3 17.4 6.9
7−9 17.4 17.3 6.9
10+ 7.0 6.9 2.8

So, this 3×3 table indicates that both rolls fail about nine times more often then both fully succeed. The odds of one of the rolls failing are greater than the odds of getting any kind of success (partial or full) on both. (Note that, throughout this post, percentages are rounded to the nearest tenth of a percent, so the percentages in a matrix like this may not add up to a 100%.)

But, rolls in PbtA games have bonuses added to them. Suppose the first roll gets between a +0 and +3 bonus, while the second is modified from -1 to +3. The following table shows how this pans out:

Second 2d6 Roll
-1 +0 +1 +2 +3
6− 7−9 10+ 6− 7−9 10+ 6− 7−9 10+ 6− 7−9 10+ 6− 7−9 10+
First
2d6
Roll
+0 6− 24.2 13.9 3.5 17.3 17.4 6.9 11.5 18.5 11.6 7.0 17.3 17.3 3.5 13.9 24.4
7−9 24.2 13.9 3.5 17.4 17.3 6.9 11.6 18.5 11.6 7.0 17.3 17.4 3.5 13.9 24.3
10+ 9.7 5.6 1.4 7.0 6.9 2.8 4.6 7.4 4.6 2.8 7.0 6.9 1.4 5.6 9.7
+1 6− 16.2 9.2 2.3 11.5 11.6 4.6 7.7 12.3 7.7 4.6 11.6 11.5 2.3 9.3 16.2
7−9 25.9 14.8 3.7 18.5 18.5 7.4 12.4 19.7 12.3 7.4 18.5 18.5 3.7 14.9 26.0
10+ 16.2 9.3 2.3 11.6 11.6 4.6 7.7 12.3 7.7 4.6 11.6 11.6 2.3 9.3 16.2
+2 6− 9.7 5.6 1.4 7.0 6.9 2.8 4.6 7.5 4.6 2.7 6.9 6.9 1.4 5.6 9.7
7−9 24.3 13.9 3.5 17.3 17.4 6.9 11.6 18.5 11.6 7.0 17.4 17.4 3.5 13.9 24.3
10+ 24.3 13.9 3.5 17.4 17.4 7.0 11.6 18.5 11.6 6.9 17.4 17.3 3.5 13.9 24.3
+3 6− 4.8 2.8 0.7 3.5 3.5 1.4 2.3 3.7 2.3 1.4 3.5 3.5 0.7 2.7 4.8
7−9 19.5 11.1 2.8 13.9 13.9 5.6 9.3 14.8 9.2 5.5 13.9 13.9 2.8 11.2 19.4
10+ 34.1 19.4 4.9 24.3 24.3 9.7 16.3 25.9 16.2 9.7 24.3 24.3 4.9 19.4 34.1

Clearly having bonuses helps avoid double failure. Just each roll getting a +1 cuts the chance of double failure in half. If both rolls have +2 bonuses, the chance of both rolls succeeding are nearly three in four.

Standard Aid

What does this mean for the Aid move? Based on the move text above, not only are bonuses in play, but the result of the Aid move alters the success of the test being aided by adding one to the roll, which makes success on the test more likely. This pans out like so:

Test Bonus
-1 +0 +1 +2 +3
6− 7−9 10+ 6− 7−9 10+ 6− 7−9 10+ 6− 7−9 10+ 6− 7−9 10+
Aid
Bonus
+0 6− 24.2 13.9 3.5 17.3 17.4 6.9 11.5 18.5 11.6 7.0 17.3 17.3 3.5 13.9 24.4
7−9 17.3 17.3 7.0 11.6 18.5 11.6 7.0 17.3 17.4 3.5 13.9 24.3 1.2 10.4 30.1
10+ 6.9 7.0 2.8 4.6 7.4 4.6 2.8 6.9 6.9 1.4 5.6 9.7 0.5 4.2 12.0
+1 6− 16.2 9.2 2.3 11.5 11.6 4.6 7.7 12.3 7.7 4.6 11.6 11.5 2.3 9.3 16.2
7−9 18.5 18.5 7.4 12.3 19.8 12.4 7.4 18.5 18.5 3.7 14.8 25.9 1.2 11.2 32.1
10+ 11.6 11.7 4.7 7.8 12.3 7.7 4.7 11.6 11.5 2.3 9.3 16.2 0.8 6.9 20.0
+2 6− 9.7 5.6 1.4 7.0 6.9 2.8 4.6 7.5 4.6 2.7 6.9 6.9 1.4 5.6 9.7
7−9 17.3 17.4 7.0 11.5 18.5 11.6 7.0 17.4 17.4 3.5 13.9 24.3 1.2 10.4 30.1
10+ 17.3 17.4 7.0 11.6 18.6 11.6 6.9 17.4 17.3 3.5 13.9 24.3 1.2 10.4 30.1
+3 6− 4.8 2.8 0.7 3.5 3.5 1.4 2.3 3.7 2.3 1.4 3.5 3.5 0.7 2.7 4.8
7−9 13.9 13.9 5.6 9.2 14.8 9.2 5.6 13.9 13.9 2.8 11.1 19.5 0.9 8.4 24.1
10+ 24.3 24.3 9.7 16.2 25.9 16.3 9.8 24.3 24.4 4.9 19.5 34.0 1.6 14.6 42.1

Because the Aid roll gains a bonus based on a bonds and not a stat, high bonuses to this roll are somewhat rare. Often this number of bonds will be zero, and multiple bonds with the same character are pretty rare. So in most cases, you’re looking at the +0 and +1 Aid rows.

Remember how some moves allow a higher level of success on a 12+? Given the effectiveness of high-bonus Aid rolls, a reasonable strategy might be to Aid rolls that have a high chance of success to try to push them up to the 12+ level, when such a thing is possible. To show how that works, the 3×3 matrix in each cell gets expanded into a 4×4 matrix, with the “full success” row and column get split into a 10−11 column and a 12+ column. When you break the results up like this using Aid, you get this:

Test Bonus
-1 +0 +1 +2 +3
6− 7−9 10−11 12+ 6− 7−9 10−11 12+ 6− 7−9 10−11 12+ 6− 7−9 10−11 12+ 6− 7−9 10−11 12+
Aid
Bonus
+0 6− 24.2 13.9 3.5 0.0 17.3 17.4 5.8 1.2 11.5 18.5 8.1 3.5 7.0 17.3 10.4 6.9 3.5 13.9 12.8 11.6
7−9 17.3 17.3 5.8 1.2 11.6 18.5 8.1 3.5 7.0 17.3 10.5 7.0 3.5 13.9 12.7 11.6 1.2 10.4 12.8 17.3
10−11 5.8 5.8 1.9 0.4 3.9 6.2 2.7 1.1 2.3 5.8 3.4 2.3 1.2 4.6 4.2 3.9 0.4 3.5 4.2 5.8
12+ 1.2 1.2 0.4 0.1 0.8 1.2 0.5 0.2 0.5 1.2 0.7 0.5 0.2 0.9 0.9 0.8 0.1 0.7 0.9 1.1
+1 6− 16.2 9.2 2.3 0.0 11.5 11.6 3.9 0.8 7.7 12.3 5.4 2.3 4.6 11.6 6.9 4.6 2.3 9.3 8.4 7.8
7−9 18.5 18.5 6.2 1.2 12.3 19.8 8.7 3.7 7.4 18.5 11.1 7.4 3.7 14.8 13.6 12.3 1.2 11.2 13.6 18.5
10−11 8.1 8.1 2.7 0.5 5.4 8.6 3.7 1.6 3.3 8.1 4.8 3.3 1.6 6.5 6.0 5.4 0.5 4.9 5.9 8.1
12+ 3.5 3.5 1.2 0.2 2.3 3.7 1.6 0.7 1.4 3.5 2.1 1.4 0.7 2.8 2.5 2.3 0.2 2.1 2.6 3.5
+2 6− 9.7 5.6 1.4 0.0 7.0 6.9 2.3 0.5 4.6 7.5 3.2 1.4 2.7 6.9 4.2 2.8 1.4 5.6 5.1 4.6
7−9 17.3 17.4 5.8 1.2 11.5 18.5 8.1 3.4 7.0 17.4 10.5 6.9 3.5 13.9 12.7 11.6 1.2 10.4 12.7 17.4
10−11 10.4 10.4 3.5 0.7 7.0 11.2 4.9 2.1 4.2 10.4 6.3 4.2 2.1 8.4 7.6 6.9 0.7 6.3 7.6 10.4
12+ 6.9 7.0 2.4 0.5 4.6 7.4 3.2 1.4 2.8 7.0 4.2 2.8 1.4 5.6 5.1 4.6 0.5 4.1 5.1 7.0
+3 6− 4.8 2.8 0.7 0.0 3.5 3.5 1.1 0.2 2.3 3.7 1.6 0.7 1.4 3.5 2.1 1.4 0.7 2.7 2.5 2.3
7−9 13.9 13.9 4.6 0.9 9.2 14.8 6.5 2.8 5.6 13.9 8.3 5.6 2.8 11.1 10.2 9.3 0.9 8.4 10.2 13.9
10−11 12.7 12.7 4.2 0.8 8.5 13.5 6.0 2.5 5.1 12.7 7.7 5.1 2.6 10.2 9.3 8.4 0.8 7.7 9.3 12.7
12+ 11.6 11.5 3.9 0.8 7.7 12.4 5.4 2.3 4.7 11.6 6.9 4.7 2.3 9.3 8.5 7.7 0.8 6.9 8.5 11.6

You can take these result percentages and subtract the result percentages from the corresponding results of just two rolls to see what sort of a difference Aid is making, compared to just making two rolls:

Test Bonus
-1 +0 +1 +2 +3
6− 7−9 10−11 12+ 6− 7−9 10−11 12+ 6− 7−9 10−11 12+ 6− 7−9 10−11 12+ 6− 7−9 10−11 12+
Aid
Bonus
+0 6−                                        
7−9 -6.9 +3.4 +2.3 +1.2 -5.8 +1.1 +2.3 +2.3 -4.6 -1.2 +2.3 +3.5 -3.5 -3.4 +2.3 +4.6 -2.3 -3.5   +5.8
10−11 -2.3 +1.1 +0.8 +0.4 -1.9 +0.4 +0.8 +0.8 -1.5 -0.4 +0.7 +1.2 -1.2 -1.2 +0.8 +1.5 -0.8 -1.2   +1.9
12+ -0.5 +0.2 +0.2 +0.1 -0.4 +0.1 +0.1 +0.1 -0.3 -0.1 +0.2 +0.2 -0.2 -0.2 +0.2 +0.3 -0.2 -0.2   +0.4
+1 6−                                        
7−9 -7.4 +3.7 +2.5 +1.2 -6.2 +1.3 +2.5 +2.5 -5.0 -1.2 +2.5 +3.7 -3.7 -3.7 +2.5 +4.9 -2.5 -3.7   +6.2
10−11 -3.3 +1.6 +1.1 +0.5 -2.7 +0.6 +1.1 +1.1 -2.2 -0.5 +1.0 +1.6 -1.6 -1.7 +1.1 +2.2 -1.1 -1.6   +2.7
12+ -1.4 +0.7 +0.5 +0.2 -1.1 +0.2 +0.5 +0.5 -0.9 -0.2 +0.5 +0.7 -0.7 -0.7 +0.5 +0.9 -0.5 -0.7   +1.2
+2 6−                                        
7−9 -6.9 +3.5 +2.3 +1.2 -5.7 +1.1 +2.3 +2.3 -4.6 -1.1 +2.3 +3.5 -3.5 -3.5 +2.3 +4.6 -2.3 -3.5   +5.8
10−11 -4.2 +2.1 +1.4 +0.7 -3.5 +0.7 +1.4 +1.4 -2.8 -0.7 +1.4 +2.1 -2.1 -2.1 +1.4 +2.8 -1.4 -2.1   +3.5
12+ -2.8 +1.4 +1.0 +0.5 -2.3 +0.4 +0.9 +0.9 -1.9 -0.5 +0.9 +1.4 -1.4 -1.4 +0.9 +1.8 -0.9 -1.4   +2.3
+3 6−                                        
7−9 -5.6 +2.8 +1.8 +0.9 -4.6 +1.0 +1.8 +1.9 -3.7 -0.9 +1.8 +2.8 -2.8 -2.8 +1.8 +3.7 -1.8 -2.8 +0.1 +4.6
10−11 -5.1 +2.6 +1.7 +0.8 -4.2 +0.8 +1.7 +1.7 -3.4 -0.9 +1.7 +2.6 -2.6 -2.5 +1.7 +3.4 -1.7 -2.5   +4.2
12+ -4.6 +2.3 +1.5 +0.8 -3.9 +0.8 +1.6 +1.5 -3.1 -0.8 +1.5 +2.3 -2.3 -2.3 +1.6 +3.1 -1.6 -2.3   +3.9

Fourth World Aid

In Fourth World, bonds aren’t used, so the Aid move has to change. In version 1.5 (not yet released), the move is also adjusted to have a bit more upside. It makes use of changing the result by a “step”, which means a 6− becomes a 7−9 result, a 7−9 becomes a 10+, etc.. The current text of the move is:

When you help or hinder someone, say how. You may do so either before or after they have rolled, but before the fictional outcome of their action is known. If you do it…

  • …using brute force, roll+STR
  • …with speed, agility, or physical finesse, roll+DEX
  • …with vitality, toughness, or vigor, roll+CON
  • …through emotional manipulation, roll+CHA
  • …through analysis, logic, or book-learning, roll+INT
  • …some other way, roll+WIS

On a 7–9, they take +1 or –2 to their roll, your choice. On a 10+, improve or reduce their result by one step, your choice, and choose one from the following list:

  • you do not expose yourself to danger, retribution, or cost
  • you help someone else: they take +1 forward
  • you help yourself: you take +1 forward
  • you gain a karma point

Using this approach, the exact same die rolls used in the table above turn out like this:

Test Bonus
-1 +0 +1 +2 +3
6− 7−9 10−11 12+ 6− 7−9 10−11 12+ 6− 7−9 10−11 12+ 6− 7−9 10−11 12+ 6− 7−9 10−11 12+
Aid
Bonus
+0 6− 24.2 13.9 3.5 0.0 17.3 17.4 5.8 1.2 11.5 18.5 8.1 3.5 7.0 17.3 10.4 6.9 3.5 13.9 12.8 11.6
7−9 17.3 17.3 5.8 1.2 11.6 18.5 8.1 3.5 7.0 17.3 10.5 7.0 3.5 13.9 12.7 11.6 1.2 10.4 12.8 17.3
10−11 0.0 8.1 4.7 1.2 0.0 5.8 5.8 2.3 0.0 3.9 6.1 3.9 0.0 2.3 5.8 5.8 0.0 1.2 4.7 8.1
12+ 0.0 1.6 0.9 0.2 0.0 1.1 1.2 0.5 0.0 0.8 1.2 0.8 0.0 0.5 1.2 1.2 0.0 0.2 0.9 1.6
+1 6− 16.2 9.2 2.3 0.0 11.5 11.6 3.9 0.8 7.7 12.3 5.4 2.3 4.6 11.6 6.9 4.6 2.3 9.3 8.4 7.8
7−9 18.5 18.5 6.2 1.2 12.3 19.8 8.7 3.7 7.4 18.5 11.1 7.4 3.7 14.8 13.6 12.3 1.2 11.2 13.6 18.5
10−11 0.0 11.4 6.5 1.6 0.0 8.1 8.1 3.2 0.0 5.4 8.6 5.4 0.0 3.3 8.2 8.1 0.0 1.6 6.5 11.3
12+ 0.0 4.9 2.8 0.7 0.0 3.5 3.5 1.4 0.0 2.3 3.7 2.3 0.0 1.4 3.5 3.5 0.0 0.7 2.8 4.9
+2 6− 9.7 5.6 1.4 0.0 7.0 6.9 2.3 0.5 4.6 7.5 3.2 1.4 2.7 6.9 4.2 2.8 1.4 5.6 5.1 4.6
7−9 17.3 17.4 5.8 1.2 11.5 18.5 8.1 3.4 7.0 17.4 10.5 6.9 3.5 13.9 12.7 11.6 1.2 10.4 12.7 17.4
10−11 0.0 14.6 8.3 2.1 0.0 10.5 10.5 4.2 0.0 7.0 11.1 7.0 0.0 4.2 10.5 10.4 0.0 2.1 8.3 14.6
12+ 0.0 9.7 5.6 1.4 0.0 6.9 7.0 2.8 0.0 4.6 7.4 4.6 0.0 2.8 6.9 6.9 0.0 1.4 5.5 9.7
+3 6− 4.8 2.8 0.7 0.0 3.5 3.5 1.1 0.2 2.3 3.7 1.6 0.7 1.4 3.5 2.1 1.4 0.7 2.7 2.5 2.3
7−9 13.9 13.9 4.6 0.9 9.2 14.8 6.5 2.8 5.6 13.9 8.3 5.6 2.8 11.1 10.2 9.3 0.9 8.4 10.2 13.9
10−11 0.0 17.9 10.2 2.5 0.0 12.8 12.7 5.1 0.0 8.5 13.6 8.5 0.0 5.1 12.7 12.7 0.0 2.5 10.2 17.8
12+ 0.0 16.2 9.2 2.3 0.0 11.6 11.6 4.7 0.0 7.7 12.3 7.7 0.0 4.6 11.6 11.6 0.0 2.3 9.2 16.2

How does this result compare with the standard rule for Aid? Each cell in the following table shows the result of the corresponding cell of the prior table and subtracts the same cell from the standard Aid table, showing the change in outcome for each cell (for clarity, cells with no change are shown as blank instead of zero). You can see that it only changes things when the Aid roll gets a 10+, as you’d expect from the phrasing of the change. The probability only changes a few percent either way, generally pushing to more success for the test, which was the whole idea of the change. The biggest change comes when helping someone with a penalty, which is definitately what you want from a move to help out allies.

Test Bonus
-1 +0 +1 +2 +3
6− 7−9 10−11 12+ 6− 7−9 10−11 12+ 6− 7−9 10−11 12+ 6− 7−9 10−11 12+ 6− 7−9 10−11 12+
Aid
Bonus
+0 6−                                        
7−9                                        
10−11 -5.8 +2.3 +2.7 +0.8 -3.9 -0.4 +3.1 +1.2 -2.3 -1.9 +2.7 +1.5 -1.2 -2.3 +1.5 +1.9 -0.4 -2.3 +0.4 +2.3
12+ -1.2 +0.5 +0.5 +0.2 -0.8 -0.1 +0.6 +0.2 -0.5 -0.4 +0.5 +0.3 -0.2 -0.5 +0.3 +0.4 -0.1 -0.5 +0.1 +0.5
+1 6−                                        
7−9                                        
10−11 -8.1 +3.2 +3.8 +1.1 -5.4 -0.5 +4.3 +1.6 -3.3 -2.7 +3.8 +2.2 -1.6 -3.2 +2.2 +2.7 -0.5 -3.2 +0.6 +3.2
12+ -3.5 +1.4 +1.7 +0.5 -2.3 -0.2 +1.9 +0.7 -1.4 -1.1 +1.6 +0.9 -0.7 -1.4 +0.9 +1.2 -0.2 -1.4 +0.2 +1.4
+2 6−                                        
7−9                                        
10−11 -10.4 +4.1 +4.8 +1.4 -7.0 -0.7 +5.6 +2.1 -4.2 -3.4 +4.8 +2.8 -2.1 -4.2 +2.8 +3.5 -0.7 -4.2 +0.7 +4.1
12+ -6.9 +2.8 +3.2 +0.9 -4.6 -0.5 +3.7 +1.4 -2.8 -2.3 +3.3 +1.9 -1.4 -2.8 +1.9 +2.3 -0.5 -2.8 +0.4 +2.8
+3 6−                                        
7−9                                        
10−11 -12.7 +5.1 +5.9 +1.7 -8.5 -0.8 +6.8 +2.5 -5.1 -4.2 +5.9 +3.4 -2.6 -5.1 +3.4 +4.2 -0.8 -5.1 +0.9 +5.1
12+ -11.6 +4.7 +5.4 +1.5 -7.7 -0.8 +6.2 +2.3 -4.7 -3.8 +5.4 +3.0 -2.3 -4.6 +3.1 +3.9 -0.8 -4.6 +0.7 +4.6

Summary

If you only care about the outcome of the test, and how Aid changes the likelihood of the four test outcomes, Aid clearly helps on average. This ignores the drawbacks of failing the Aid roll, though.

Test Bonus
-1 +0 +1 +2 +3
6− 7−9 10−11 12+ 6− 7−9 10−11 12+ 6− 7−9 10−11 12+ 6− 7−9 10−11 12+ 6− 7−9 10−11 12+
Without aid 58.6 33.1 8.3 0.0 41.7 41.6 14.0 2.7 27.9 44.2 19.6 8.3 16.4 41.8 24.9 16.8 8.4 33.4 30.7 27.5
Standard Aid +0 49.0 37.9 11.6 1.6 33.7 43.2 17.1 6.0 21.4 42.7 22.7 13.2 11.6 36.9 28.2 23.4 5.1 28.6 30.5 35.8
+1 46.3 39.5 12.2 2.1 31.6 43.7 17.9 6.8 19.8 42.4 23.4 14.5 10.5 36.0 28.9 24.5 4.3 27.2 30.5 38.0
+2 44.1 40.4 13.2 2.3 30.3 43.9 18.6 7.2 18.4 42.2 24.3 15.1 9.8 34.6 29.5 26.1 3.7 26.5 30.6 39.3
+3 43.0 41.1 13.4 2.6 29.1 44.4 18.8 7.7 17.7 41.8 24.6 15.9 9.2 34.2 30.1 26.5 3.3 25.9 30.6 40.2
Fourth World Aid +0 42.0 40.7 14.8 2.5 29.0 42.9 20.7 7.4 18.5 40.5 25.9 15.1 10.2 34.0 30.1 25.7 4.7 25.9 30.9 38.6
+1 34.7 44.0 17.7 3.5 23.9 42.9 24.2 9.1 15.1 38.7 28.6 17.6 8.3 31.1 32.2 28.4 3.6 22.6 31.2 42.6
+2 26.7 47.3 21.3 4.7 18.6 42.9 27.8 10.7 11.4 36.6 32.2 19.8 6.3 27.7 34.2 31.8 2.5 19.5 31.8 46.2
+3 18.9 50.8 24.5 5.8 12.7 42.8 32.0 12.5 7.9 33.9 35.8 22.5 4.1 24.5 36.7 34.7 1.6 16.0 32.4 50.0

Code

All of these numbers came from rolling a million sets of Aid/Test rolls. Some of this was also secretly an attempt to become more familiar with driving HSL color with code. The Python 3.x code used to compute these results and generate the tables can be found here: aidprob.py

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