d20 & uncertainty

In my previous article I touched on the topic of information entropy and explained how certain resolution mechanics can be considered random even when dice are not involved. I explained how information entropy is a measurement of the amount of information in a message. I supported this with an example of a coin and how a fair coin has an entropy of 1 and a coin with two heads has an entropy of 0. Such a coin will always land heads, we know what will happen before we flip it and such an event provides us with no information whatsoever.

Now it's time to take a look into something more interesting and commonly used in our hobby, the d20.

What's the information entropy of the d20 as the target becomes harder to hit? What does this mean to us players, GMs and designers?

First I'll build a table with the required roll to hit and the odds of failing and succeeding in an attack. The following table shows the required roll, the odds to hit, to miss and the entropy (calculated by the sum of the log of all that stuff you don't want to read about here).

d20
required roll	failure	success	entropy
8	35.00%	65.00%	0.93
9	40.00%	60.00%	0.97
10	45.00%	55.00%	0.99
11	50.00%	50.00%	1.00
12	55.00%	45.00%	0.99
13	60.00%	40.00%	0.97
14	65.00%	35.00%	0.93
15	70.00%	30.00%	0.88
16	75.00%	25.00%	0.81
17	80.00%	20.00%	0.72
18	85.00%	15.00%	0.61
19	90.00%	10.00%	0.47
20	95.00%	5.00%	0.29
21	100.00%	0.00%	0.00

Notice how the roll reaches maximum entropy at 11 when the odds of failing or succeeding are even and then begins to drop, eventually reaching zero at 21, a point in which it makes no sense to roll as failure is guaranteed. Graphing the values from 1 to 21 I get the following curve.

Since entropy is a measure of uncertainty it is easy to understand how it drops as we approach the edges. If my character needs a 2 or better to hit there is little uncertainty as to what will happen the next time it's my turn, on the other hand if I need a 20 or better it is quite certain I will miss, and if I need a 21 or better I'll surely miss! No need to even roll! Entropy is 0 and actually rolling the die provides me nothing new. As a player or GM I will discover nothing by rolling a d20 when I need a 21 to succeed.

Let me make another example to show this. Let me write out a succession of attack rolls. I'll write F for fail and S for success. When the odds are even or near even, that is anywhere between an 8 and 14 is needed to hit, the string of outcomes would look something like this:

SFSSFSFFSFSFSSFFSFFSSFSFSFSSFF.....

There's a pretty even occurrence of S and F.

If on the other hand I move to the right of the graph and require an 18 or better the string will look something like this:

FFFSFFFFSFSFFFFFFSSFFFFFFSFFSF.....

There will be considerably more Fs than Ss, there might be two successive Ss, but these will be uncommon and far in between.

In the extremes the strings will look like the following, either:

SSSSSSSSSSSSSSSSSSSSSSSSSSSSSS.....
or
FFFFFFFFFFFFFFFFFFFFFFFFFFFFFF.....

There is no uncertainty, my character will either always succeed or always fail!

Why is this important? Well it can save us a lot of time. Why make all those extra rolls if I know I will most surely succeed of fail? Obviously I can't guess the future and knowing I'm going to fail very often doesn't spare me the roll because I need to roll in the odd chance I do succeed. What I can do is compare this to other roleplaying mechanisms, for example drama which relies on the player narrating the outcome. An equivalent event is my options getting reduced more and more. More often than not I as a player take the next tunnel to the right instead of considering the other options. The lower the entropy the less options I really have available to me. I'm getting less bang for the buck as I still have to roll the dice.

So, what happens when we include more outcomes?

Lets revisit the above table and pick out a few values (11, 16 and 19) and calculate the entropy when 1 represents critical miss and 20 a critical hit. The following tables include the contribution of each individual outcome to the the total entropy of the roll. Let me explain based on an 11 required to hit when rolling a d20. The odds of hitting or missing are 50%-50% (11,12,13,14,15,16,17,18,19,20 hits, all other miss), but a 1 is a critical miss and a 20 is a critical hit. In the previous example the maximum entropy was reached at 11 and the value was 1. Now at 11 the entropy is 1.469, almost a 50% increase because we included other outcomes that add information to the roll. There are more outcomes thus there is more uncertainty as to what will happen. As the roll required to hit increases the entropy decreases; 1.257 when a 16 is needed; 0.848 when a 19 is needed.

d20 tohit 11	odds	entropy
crit miss 1	5.00%	0.216
failure	45.00%	0.518
success	45.00%	0.518
crit hit 20	5.00%	0.216
	100.00%	1.469
d20 tohit 16	odds	entropy
crit miss 1	5.00%	0.216
failure	70.00%	0.360
success	20.00%	0.464
crit hit 20	5.00%	0.216
	100.00%	1.257
d20 tohit 19	odds	entropy
crit miss 1	5.00%	0.216
failure	85.00%	0.199
success	5.00%	0.216
crit hit 20	5.00%	0.216
	100.00%	0.848

What's interesting to see is what happens with the individual elements. The critical outcomes always add .216 bits of entropy. When a 19 is required the failure outcome, which accounts for 85% of the odds, represents only 0.199 of the entropy; success and critical success have equal uncertainty (entropy). This is counter intuitive. It makes no sense to believe that an outcome which is barely possible (success) is equally important than one which should be considerably less probable. After all if my character can hardly score a hit how can I explain that one out of two hits will be critical and will have some outcome like double damage? It is this that give me the jolt feeling when I use the d20 in games. When the character is too weak to take on a challenge or too strong I get the feeling I'm rolling a lot for a whole lot of nothing and every so often I get these radical, very impacting outcomes.

Lets up the stakes a little bit and introduce a variable critical hit and miss. Instead of 1 being critical hit lets make it to hit minus something and consider two scenarios: a) tohit - 10 and b) tohit - 15, and 1 is always critical.

The following tables show the numbers for a tohit roll that requires a 19 or better to hit, but a critical is generated on a 4 (tohit-15) or a 9 (tohit-10).

d20 tohit 19	odds	entropy
crit miss 4	20.00%	0.464
failure	70.00%	0.360
success	5.00%	0.216
crit hit 20	5.00%	0.216
	100.00%	1.257
d20 tohit 19	odds	entropy
crit miss 9	45.00%	0.518
failure	45.00%	0.518
success	5.00%	0.216
crit hit 20	5.00%	0.216
	100.00%	1.469

Notice how just going from 5% odds of a critical to 15% odds of a critical has increased entropy from 0.848 to 1.257. There's a whole lot more going on in that roll now. Things get interesting when the player has to decide to roll or not against low odds of success and a lot higher odds of critical failure. Notice also how taking those same odds for a critical miss and increasing them to 45% doesn't increase entropy that much more! There's a sweet spot after which it may be just too unfair.

Now let us not forget this is symmetrical and the same happens for critical hit! Doing the numbers for a superb character that hits on a 3 or better we get the following tables, one is the plain old 20 for critical hit and the other is the tohit+14 giving the player a critical hit on a 17 or better.

d20 tohit 3	odds	entropy
crit miss 1	5.00%	0.216
failure	5.00%	0.216
success	85.00%	0.199
crit hit 20	5.00%	0.216
	100.00%	0.848
d20 tohit 3	odds	entropy
crit hit 1	5.00%	0.216
failure	5.00%	0.216
success	70.00%	0.360
crit hit 17	20.00%	0.464
	100.00%	1.257

When my character gets really good the behavior repeats itself, entropy drops drastically without the adjusted critical ranges, and I'm left rolling dice for a whole lot of nothing. My character is mowing down goblins left and right and every so often he will do a critical that will either chop three goblins with one blow or chop his leg off.

With the adjusted critical hits my character is going to be impaling three or four goblins way more often, he'll push one and throw the three behind him down a wall, he'll swing his sword around and behead two goblins at the same time, etc. You get the idea, there is more going on in the story than there was before.

Conclusion, I reviewed the d20 roll looking at the measure of its uncertainty as indicated by the entropy of the roll. I showed how outcomes become monotonous when skill is either too high or too low and thus added two more outcomes: critical hit and critical miss (as is commonly used in our games). I once again studied this as skill increased or decreased considerably and noticed how the usage of a fixed value for critical outcomes once again lead to monotonous outcomes. I adjusted this by using a variable critical, which lead to an increase of entropy once again. I compared two settings, a 10 point difference and a 15 point difference (critical miss on 4 and 9 as explained above), and noticed that the 15 point difference was good enough and the 10 point difference made it too uncertain between fail/critical fail or success/critical success, that is the odds for a normal outcome vs a critical were the same, when it's more intuitive that critical outcomes should have lower odds than normal outcomes.

After seeing these numbers it's my conclusion that using critical outcomes at -15 or +14 from the required to hit (with 1 and 20 always being critical) makes for the best d20 rolls for OSR styled games. It adds enough uncertainty that it keeps things interesting without making it so uncertain it becomes a wild guess. If I look back at the table for a tohit of 11 I see the following:

d20 tohit 11	odds	entropy
crit miss 1	5.00%	0.216
failure	45.00%	0.518
success	45.00%	0.518
crit hit 20	5.00%	0.216
	100.00%	1.469

As  my character gets better and I only require a 3 to hit I get the following

d20 tohit 3	odds	entropy
crit miss 1	5.00%	0.216
failure	5.00%	0.216
success	70.00%	0.360
crit hit 17	20.00%	0.464
	100.00%	1.257

The odds of hitting have gone up from 50% to 90%, and the odds for a critical hit has gone up to 20%, while the overall entropy has dropped only to 1.257 instead of .848 as it did before. This small drop is due to the uncertainty of hitting being removed by my character's skill. I'm more confident I'll get what I want, to win! But at the same time I'm filling the gap with more information from the critical success which is now a more important element in the story than it was before. Entropy should drop due to skill, but not so much that it steals from the story!

Well wow, just realized how long this post turned out. I hope you're still with me and that this has been helpful. I'd love to get your feedback on this and know what you're doing to get a little more from each die roll on your table.