More Task Resolution: Relative Gains

Some discussion in IRC about the merits of d20 versus 3d6 for task resolution led to some examination, and the examination led to some curious conclusions.

A +1 bonus on a d20 is seen as a 5% improvement, because it opens one more die value (a 5% chance) to be used for success.

A +1 bonus on 3d6 is more difficult to describe, because rolled values have different chances to come up (which is what a bell curve means, of course).

It was stated that if you’re up against a mob of kobolds you might not bother with flanking or other bonuses because they make so little difference (going from 5+ to 4+ to hit… you gain 3/216 more chance to hit!), but that when fighting Godsmasher the Dragon a +1 is similarly useless because going from 18+ to 17+ was still +3/216 chance to hit!

I started to dismiss this as not important because in either case you are at the edges of things you should be doing anyway.  The kobolds are so outclassed it’s likely not worth playing out except to see how badly you destroy them, the dragon so outclasses you that it’s likely not worth playing out except to see how badly you get destroyed… then Karrius suggested looking at the relative gain in each case.

With the kobolds, a +1 bonus might mean +3/216 chance of hitting, going from 212/216 to 215/216.  This is such a small gain it might not be worth worrying about.

On the other hand, with Godsmasher the Dragon that +3/216 chance means going from 1/216 to 4/216 — I quadruple my chance of hitting!  I’m still boned, of course, but I’m four times as likely to hit?  I’ll take whatever I can get here!

This becomes an important distinction — while 3/216 is ultimately a small increase in the chance of hitting, it has bigger or smaller effect depending on when it gets applied.

If you would normally succeed on 11+ on 3d6 (50%), a +1 bonus would change that to succeeding on 10+ (62.5%), which amounts to succeeding 1.25 times as often (5/4 as many times as you would otherwise).

Relative Success Increase
3d6 needed slot freq slot freq % 3d6 success 3d6 %success +1 +2 +3 +4
3 1 0.46 216 100.00
4 3 1.39 215 99.54 0.47
5 6 2.78 212 98.15 1.42 1.89
6 10 4.63 206 95.37 2.91 4.37 4.85
7 15 6.94 196 90.74 5.10 8.16 9.69 10.20
8 21 9.72 181 83.80 8.29 13.81 17.13 18.78
9 25 11.57 160 74.07 13.13 22.50 28.75 32.50
10 27 12.50 135 62.50 18.52 34.07 45.19 52.59
11 27 12.50 108 50.00 25.00 48.15 67.59 81.48
12 25 11.57 81 37.50 33.33 66.67 97.53 123.46
13 21 9.72 56 25.93 44.64 92.86 141.07 185.71
14 15 6.94 35 16.20 60.00 131.43 208.57 285.71
15 10 4.63 20 9.26 75.00 180.00 305.00 440.00
16 6 2.78 10 4.63 100.00 250.00 460.00 710.00
17 3 1.39 4 1.85 150.00 400.00 775.00 1300.00
18 1 0.46 1 0.46 300.00 900.00 1900.00 3400.00

As shown above, a +4 bonus is kind of a big deal.  If normally you would succeed on 11+ (50% of the time), a +4 bonus means you succeed on a 7+ (90.74% of the time), you succeed 81% more often (almost, but not quite, twice as often).  On the other hand, if normally you would succeed only on 18+ (0.46% of the time), a +4 bonus means you succeed on 14+ (16.20% of the time), 3400% as often (from 1/216 to 35/216).  It’s still not a great chance, but relatively speaking it is much, much better than before.  If it’s even easier (you would normally succeed on 7+ (90.74% of the time), a +4 bonus means you succeed on 3+ (100% of the time)… which is about 10% more often than normal.

This suggests that a bonus is generally nice to have, but when you significantly outmatch your challenge you might not bother trying to get all the bonuses possible, especially if it means you can’t do another things… but that when you are desperate (not likely to hit in the first place) you will likely want to do everything you can to get your bonuses up.

To go back to the initial example, you might choose to not flank the kobolds because it makes such a relatively tiny difference in your ability to hit them, instead using other abilities that cannot be used while flanking.  Against Godsmasher the Dragon, on the other hand, you’ll want to flank (or find even better options) simply so you have a chance to hit him.  The challenge you face will thus affect your tactical decisions, and I like that.

[For reference, I did the same analysis with the d20 solution and it does exhibit similar characteristics, but to a much lesser degree.  A +1 bonus provides relative benefits of +5.26% to +100%, while a +4 bonus provides relative benefits of +25% (16/20 to 20/20 -> 20/16 = 1.25 times the base, or 25% improvement) to 400% (1/20 to 5/20 -> 5/1 = 5 times the base, or 400% improvement).

This exhibits pretty much exactly the behavior I’m looking for.

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