Wednesday, May 5, 2021

D20: The Fun Zone

What range of probability of success produces the most fun in an RPG? I assume over the course of a single adventure, the difficulty of monsters and encounters should vacillate, but generally trend upward as the climax approaches. Also, as PCs gain experience they generally become more capable. What was once difficult becomes easier and formerly impossible challenges become attainable. But what range of target numbers is ideal and how does that range change as the PCs improve?

To answer this question I analyzed the typical attack bonus of a human fighter and the subsequent d20 rolls needed to strike foes ranging from unarmored to full plate and shield over the various editions of D&D. This is not meant to fuel edition wars. Rather, the product of this analysis should approximate the expected probability of success for common die rolls in that edition - and lend us some insight into how often the designers of that edition thought the PCs should succeed - i.e. what probability of success is the best combination of fun and challenging? Hopefully it will help you determine your own preferences.

Assumptions

I compared levels 1-10 and assumed the fighter has a Strength of 16 at the outset. This may be uncommonly strong in the 3d6-in-order tradition of early editions but pedestrian, if not underpowered, in the 4d6, stack-with-a-racial-bonus, and arrange-to-taste preferences of modern players. But hopefully it's a fair middle point to illuminate trends. For now I also ignored magic weapons and extraordinarily armored foes. I hope to provide a follow-up analysis factoring in the recommended levels at which these are obtained and encountered.

OD&D: The 3 Little Brown Books (LBBs)

In OD&D no character class is granted any bonuses or penalties "to-hit" from Strength. The combat matrix simply specifies what each class must roll to hit each AC at a given band of levels. Fighters improve their "to-hit" by 2 or 3 pips every 3 levels. AC 9 is unarmored while AC 2 is full plate and shield.

Fighter Level D20 To-Hit AC 9-2
1-3 10-17
4-6 8-15
7-9 5-12
10-12 3-10

OD&D + Greyhawk

With the addition of the Greyhawk supplement we see the first indication of a fighter's Strength improving their likelihood to-hit, with 16 granting a +1 bonus. Thus the d20 rolls required to hit ACs 9-2 are improved by 1.

Fighter LevelD20 To-Hit AC 9-2
1-39-16
4-67-14
7-94-11
10-122-9


Basic/Expert D&D

In B/X we see the normalized -3 to +3 distribution of ability bonuses. Here a 16 Strength grants +2 to-hit. Note that at the highest tier this is the first time we see the rule "unmodified attack rolls of 1 always miss". Otherwise the final entry would read [1-8] and a 10th level fighter could never miss when attacking an AC 9 unarmored opponent.

Fighter LevelD20 To-Hit AC 9-2
1-38-15
4-66-13
7-93-10
10-122-8

AD&D 1E/2E

Advanced D&D dials back the to-hit bonus from natural abilities. Strength 16 grants +1 damage, but no bonus to-hit short of Strength 17. An unarmored opponent is AC 10 in AD&D while a fully armored foe is still AC 2. However, instead of improving every 3 levels, fighters improve their to-hit (or THAC0 in 2E) linearly by 1 pip per level. Note that a 10th level fighter only need roll a 1 to hit an unarmored foe - effectively they cannot miss.

Fighter LevelD20 To-Hit AC 10-2
1
2
3
10-18
9-17
8-16
4
5
6
7-15
6-14
5-13
7
8
4-12
3-11
9
10
2-10
1-9

3E

3rd Edition has effectively the same progression, granting fighters +1 base attack bonus per fighter level, but also +1 to-hit per 2 points of Strength above 10. Thus Strength 16 grants +3 to-hit and if ability score improvement points at levels 4 and 8 go to Strength, then by 8th level the fighter would have Strength 18 with a bonus of +4. Also note that 3rd edition uses ascending armor class where an unarmored opponent still has a base AC of 10, but a foe in full plate and shield is AC 20. Already at Level 6 we require the same rule as B/X or the fighter becomes unable to miss and at level 10 he will hit AC 20 75% of the time. We quickly see why 3rd Edition and it's descendants are considered plagued by "runaway bonuses" and high level play regularly sees ACs and target numbers well into the 20s to continue challenging the PCs.

Fighter Level Strength (modifier) Base attack bonus D20 To-Hit Ascending AC 10-20
1 16 (+3) +1 6-16
2 16 (+3) +2 5-15
3 16 (+3) +3 4-14
4 17 (+3) +4 3-13
5 17 (+3) +5 2-12
6 17 (+3) +6 2-11
7 17 (+3) +7 2-10
8 18 (+4) +8 2-8
9 18 (+4) +9 2-7
10 18 (+4) +10 2-6

5E

In 5E, Strength 16 still grants a bonus of +3. But in place of a linear base attack bonus we have a proficiency bonus that starts at +2 and increases by +1 every 4 levels as part of the "bounded accuracy" design. Also, 5E allows for regular ability score improvements as part of normal level progression - even more frequently for fighters - so we raise the fighter's Strength to 18 at 4th and to 20 at 6th level. They would gain even another improvement at 8th level but 20 is the maximum allowed for PCs. 5E seems to do an excellent balancing act, not striking the artificial bottom of the target range until level 9, and yet maintaining the need for double-digit rolls to strike a formidably armored opponent.

Fighter Level Strength (modifier) Proficiency bonus D20 To-Hit Ascending AC 10-20
1-3 16 (+3) +2 5-15
4 18 (+4) +2 4-14
5
6-8
9-10
18 (+4)
20 (+5)
20 (+5)
+3
+3
+4
3-13
2-12
2-11

Analysis

There are a few ways we can analyze this data. Averages are difficult given the varying methods of progression but the the shift in target numbers at low levels, the rate of progression from experience, and the overall upper and lower limits tell an interesting tale.

Low Levels (1-3)

Clearly at low levels the game has gotten much easier. Where OD&D and AD&D demanded d20 rolls of 10-18, each subsequent edition chipped away at a fairly consistent rate of about 1 pip each. By 5E a low level fighter need only roll a 5 to hit an unarmored commoner but still needs to roll in the mid-teens to hit a fully-armored opponent. This seems to indicate that it's not particularly fun to fail more frequently than you succeed and that even a low level PC should still be a cut above normal humans. The downside is this leaves less room for improvement before either an automatic hit or automatic miss on a natural 1.

Mid Levels (4-6)

Prior to 3E, mid-level play finds most of its target numbers between 5 and 15, perfectly centered on the D20. 3E and 5E trend slightly easier, only requiring rolls of 2 or 3 to hit unarmored opponents, and again a 6th level fighter in 3E would automatically hit if not for the rule that natural 1 is always an automatic miss.

Upper Levels (7-10)

At the upper end the game appears broken, hence the need for the rule that a natural one is always a miss. Only OD&D (pre-Greyhawk) and 5E still require double digit rolls to hit formidably armored opponents at this tier. Every other edition requires only single digit rolls, with 3E only requiring a 6 to hit a fully-armored, AC 20 foe. It may not be obvious that a 10th level 3E fighter's +14 to-hit is broken, after all 10th level should be formidable, but recall that 3E and 5E are designed to allow for play up to 20th level. Where is there room for improvement?

To be fair, TSR editions also allow for play beyond 10th level - the LBBs suggest a convoluted way to compute requirements and the BX expansion into BECMI adds levels 15-36 and beyond - but PCs beyond "name level" are considered extraordinarily powerful, likely taking many years of play to achieve. Name level - that is Lord for a 9th level Fighter, Wizard for an 11th level Magic User, and Patriarch for an 8th level Cleric - is typically where domain play is introduced. The PC can construct a stronghold, generate income, and attract followers.

All this is to say I think TSR D&D makes some assumptions about a change in the general mode of play around 10th level, where I would argue WotC D&D assumes a PC is still going on adventures, just of a larger magnitude. They're likely saving the world from hellish fiends or the universe from destruction by evil gods - opponents with ACs in the 20s or 30s.

No matter the edition, at this tier PCs succeed far more often than they fail against non-extraordinary opponents which fairly reflects the improvements gained by their experience. The question is, is it still fun? Perhaps - but the game must change in one of two ways to do so: domain-level play or extraordinary threats. Either the game shifts to focus on politics, income, raising armies, and fighting wars, or the ultra-capable, demi-god PCs require other-worldly, mythic, immortal, cosmic-level threats to fight.

Conclusions

The trends seem to show that low level play was too difficult and subsequent editions increased the baseline probabilities of success. Even a low level adventurer is more fun to play if they're a cut above the riffraff. Mid level play generally finds target numbers evenly spread around the midpoint of the D20 and only slightly trends easier against poorly armored opponents. Unarmored commoners have almost no business avoiding the business end of your blade should you set upon them. High level play finds PCs succeeding at nearly every mundane task but doesn't seem to be particularly fun without an extraordinary challenger or world-ending threat.

To me, this seems to support the informal consensus among players and polls done by WotC that mid-tier play is the most fun. If that is true, then our data indicates there could be some encounters that only require very low rolls (2-5), but the majority of target numbers after bonuses are applied should fall in the 5-15 range for a consistently fun challenge. Perhaps this should come as no surprise.

In the end, whatever your d20 game of choice, it will likely be the most fun if you manage to stay within the bounds of the d20. Too high of a probability of success and perhaps you shouldn't even be rolling. Rolling only to avoid a fumble doesn't sound particularly fun. There's little indication of any edition thoroughly exercising the upper bounds of the d20 (perhaps the saving throw charts of TSR editions) but when the target is that high, whether AC or a saving throw, the situation is bound to be interesting. Food for thought...

Other Factors: AC go down. HP go up.

One final note, while we see a trend of the game getting easier in terms of probability of success, the designers clearly recognize the need to maintain the longevity of a fight and give a consistently satisfying encounter experience. How is this addressed? Unfortunately, the answer is Hit Point inflation. Starting with Greyhawk and then B/X, fighters and monsters receive d8 hit points per HD instead of d6. Not a huge deal. But then in 3E and 5E we see even larger HD sizes and when combined with ability bonuses to Constitution, hit point totals start to skyrocket for players and monsters both. A red dragon in B/X has 45 hit points. In 3E and 5E an adult red dragon has over 250... What!?

If hitting the monsters more often is more fun, but we don't want all the monsters to die in a single round, boosting HP is the obvious solution. The unfortunate side effect is dangerous baddies become truckloads of hit points and combat devolves into a slog-fest. Note that typical weapon damage hasn't changed much since Greyhawk. Multiple attacks and bonuses from magic weapons make an effort to keep up, but hardly at the rate HP inflates across editions. Monsters especially don't seem to possess the number of attacks or damage output required to truly threaten a party. I can think of few better ways to deflate the tension in a game than taking turns reliably hitting, but only chipping away at buckets of hit points. To me, the opposite would be far more fun. Only a fool in 3E+ would wade into battle with single digit hit points, even fully armored. In OD&D or B/X though? You'd stand a chance of surviving with a high AC. Huge risk/reward.

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