Inspiration & Hero Points Math

[iserith]

I wonder if I can get the help of the math-inclined out there so I can make some informed decisions about Inspiration (plus a variant) and Hero Points. I'm considering some options to fit the themes of some upcoming campaigns and one-shots.

I understand that the way standard Inspiration works is the higher the AC or DC you need to hit compared to your attack or ability check bonus, the higher the effective bonus Inspiration provides. If the AC or DC is lower, it's worth less, effectively. Please correct me if I'm in error here.

How does that calculation change, if at all, if instead of declaring the use of Inspiration before the roll and rolling with advantage, it is used as a re-roll instead? You roll, you don't like the result you get (you may or may not know whether you succeeded or failed, let's say) and so you spend Inspiration and re-roll. My gut says that's better than standard Inspiration. I'm just not sure if that's true or by what degree.

Now, how about instead of getting advantage on the roll by spending Inspiration, you just get to add a d6? This would be like the Hero Points variant in the DMG. You roll, you don't like the result you get (I think you're not supposed to know if you succeed or fail, but whatever), and so you spend Inspiration and add a d6. I'm not sure how that looks maths-wise.

Would one of you kind souls with maths skills help me out here? Thanks in advance.

 

[Oofta]

Not a math geek, but here's my take.

Unless you are guaranteed to succeed (and, if you are why are you rolling?) advantage gives you roughly a +5 to your roll.

However, two rolls of a d20 are always independent of each other. So there is no numerical advantage to rolling 1 dice then another. However there is an opportunity cost to rolling both dice at a time assuming you have a limited number of chances to do so.

The average of a d6 is 3.5 so yes, it's probably worse. There are edge cases at the lower end of course if you only have a 5% chance of failure (need to roll a 1) then the d6 is better. You also have to consider critical hits, which while having a relatively minor impact in most cases is less with the d6 option.

 

[guachi]

Simple example:

You succeed on the task 50% of the time. There are four possible outcomes, each of which has the same chance of occurring.

Succeed/Succeed
Succeed/Fail
Fail/Succeed
Fail/Fail

If you must use Inspiration beforehand, it is useful 1 of 4 times used. That being the case where you Fail/Succeed.

If you use Inspiration only after you Fail then you will only use it in the last two instances. In this event, your Inspiration is useful 1 of 2 times it's used.

You'll use Inspiration 1/2 as often but it will be 2x as useful.

An extreme example where you must have a 20 to succeed. The possible outcomes are still

Succeed/Succeed
Succeed/Fail
Fail/Succeed
Fail/Fail

However, the chances of each occurring are not the same. The chances of each of the four possibilities occurring are

1/20 x 1/20 = 1/400
1/20 x 19/20 = 19/400
19/20 x 1/20 = 19/400
19/20 x 19/20 = 361/400

You will use your Inspiration 19/20 times as that's how often you'll fail but it will only benefit you 1/20 times. The net result is that you'll use the die a lot but benefit little, 4.75% of the time.

Choosing to use Inspiration after the first die is thrown will always be better than choosing beforehand. "Better" in the sense that you'll never waste a die on an already successful roll rather than "better" in the sense of having a higher chance to succeed on that particular attempt. Overall you'll have a higher success rate as you get to save dice for later.

 

[Vymair]

When I crunched through advantage and disadvantage, I came to the conclusion that it much more significant for middling target numbers but not as beneficial for high target numbers. Using inspiration after that fact is always better since you know the result of the first roll. If you have to declare in advance, a success on the first roll means you've received no benefit from the inspiration as it wasn't needed to succeed.

The tricky part of the math is figuring out the likelihood of exactly what you'll need to roll to succeed. D&D biases this higher, so it's pretty unusual to need to roll a 3 on a d20 to succeed. If all rolls were equally likely to succeed the benefits of advantage are about 3.3, but clearly the low end of the rolls are not going to occur very often which means the benefit is greater.

The other factor to consider is the d6 would only be rolled when the player felt they had a chance to succeed, they won't be using it when they roll a 2. I suspect the impact on the game would be that characters would succeed more often on really tough target DCs as they could roll the d6 when they got close enough.

 

[guachi]

As to the d6, it's a bit like Precision on the Battlemaster (just with a d6 instead of a d8).

Let's assume you actually do know exactly how much you failed by. Let's assume you have infinite uses of Inspiration so you'll use it whenever you fail by six or less. Let's further assume that you can actually fail a roll by up to six (I mean, it's possible there are rolls where a four will succeed and, therefore, can't fail by any more than three but that makes math messy so we'll skip it).

You'll end up with 1 of 20 rolls failed by each of 1, 2, 3, 4, 5, and 6. The chance of succeeding after rolling a d6 if you fail by...
1: 100% (6/6)
2: 83.3% (5/6)
3: 66.7% (4/6)
4: 50.0% (3/6)
5: 33.3% (2/6)
6: 16.7% (1/6)

The net result is that with an infinite supply of Inspiration you always use on a roll failed by 1 to 6 your total chance of success increases by 17.5%. (I've skipped a math step here).

Let's use our previous example of succeeding 50% of the time (needing an 11+ to succeed). Under the "roll another d20" example you use your Inspiration 50% of the time and your total success rate increases from 50% to 75%. Under the "add a d6" example you use your Inspiration 30% of the time (6/20. You'd never use the die if you succeeded or if you failed by 7+) and increase your success rate from 50% to 67.5%.

Let's use our previous example of succeeding 5% of the time (needing a 20 to succeed). Under the "roll another d20" example you use your Inspiration 95% of the time (19/20) and your total success rate increases from 5% to 9.75%. Under the "add a d6" example you use your Inspiration 30% of the time and increase your success rate from 5% to 22.5%.

As you can see, the change to your success rate and the use rate of Inspiration varies wildly rolling another d20 but is constant using a d6.

 

[iserith]

That's very interesting - thanks for the analysis!

From a player's perspective, if you were limited to say up to 4 Inspiration per session (one per personal characteristic category), would you prefer to have the option to reroll the d20 or add a d6? Assume that you know what the AC and DC are because I'm pretty transparent with that in my game.

Would you have the same opinion if you could have more Inspiration per session because other players could apply their Inspiration to your rolls?

 

[Vymair]

I'd go with the d6, but I value control. If you flub your roll badly, you'll never bother with the d6, but if you miss a little, the d6 is almost a guaranteed success.

 

[Bawylie]

That's very interesting - thanks for the analysis!

From a player's perspective, if you were limited to say up to 4 Inspiration per session (one per personal characteristic category), would you prefer to have the option to reroll the d20 or add a d6? Assume that you know what the AC and DC are because I'm pretty transparent with that in my game.

Would you have the same opinion if you could have more Inspiration per session because other players could apply their Inspiration to your rolls?

I’d take the d6s because I’d feel better turning edge case failures into scraped-by successes. Whole rolling double failures on 2d20s sucks hard, choosing to accept a failure (when I’m 1-6 points away from success) or risk it and go for it, feels like a more weighty and cool choice. And it also feels less bad when you’re 7 points away from success because you can just shrug and say “well, nothing I can do here” thereby avoiding that double-failure feeling.

That’s not a math answer, but there it is.

 

[Satyrn]

That's very interesting - thanks for the analysis!

From a player's perspective, if you were limited to say up to 4 Inspiration per session (one per personal characteristic category), would you prefer to have the option to reroll the d20 or add a d6? Assume that you know what the AC and DC are because I'm pretty transparent with that in my game.

Would you have the same opinion if you could have more Inspiration per session because other players could apply their Inspiration to your rolls?

I prefer rerolls . . . assuming the ACs and DCs are reachable. Like, if I need to roll a 15 or less I'd find the reroll to be far more fun. While if I need to roll a 18 or more, the reroll would feel like a pointless waste

16-17 is a gray area . . .



Nah, that doesn't sound horribly scuzzy.

 

[Satyrn]

oh, wait.

I’d take the d6s because I’d feel better turning edge case failures into scraped-by successes.

Yeah, if I could choose to roll the d6 after seeing the d20 result, I'd definitely find that a much more satisfying choice. I was assuming not in my first answer.