d20 Bonus Relative Efficacy

While simulating an idea I had for speed-based initiative, I realized that the results had implications for bonuses on d20 rolls in general. When two characters roll off, how good is a relative +1? Column A is where the more capable character breaks ties, and column B is where ties are broken with more rolls.

Results A Results B Relative
+0 50% vs 50% 50% vs 50% 100%
+1 57% vs 43% 55% vs 45% 125%
+2 61% vs 39% 59% vs 41% 150%
+3 66% vs 34% 64% vs 36% 200%

Those are really intuitive odds, actually! A character whose ability bonus is 3 pips greater than another is doubly capable. This contextualizes ability scores and modifiers in 5e proper: a character with an ability score of 16 is twice as capable as a character with a score of 10 in the same ability. Compare this to B/X where you need a score of 18 to get that +3 bonus.

Then I simulated for a group of characters with bonuses ranging from [+0, +3].

Bonus Results A Results B
+0 14% 16%
+1 21% 22%
+2 28% 27%
+3 38% 35%

This aligns with the above table and makes enough sense on its own.


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