Inspiration & Hero Points Math
I'd take the d6 if I knew how much I failed by. I might take a reroll if I only knew I failed but not by how much. Though most people generally have a reasonable idea how much they failed by even if all the know is if they failed.
So... I think I'd take a d6. It's not so massively better than another d20 that it's really broken but its greatest benefit is on the rolls that are really hard and you just must succeed on. I'd take more risks as a PC if I had an Inspiration die that took my 5% chance to succeed and made it 22.5%! That seems fun!
On the other hand... reroll is best in a situation like Death Saves. That's one time where a reroll comes in REALLY handy.
So... I think I'd take a d6. It's not so massively better than another d20 that it's really broken but its greatest benefit is on the rolls that are really hard and you just must succeed on. I'd take more risks as a PC if I had an Inspiration die that took my 5% chance to succeed and made it 22.5%! That seems fun!
On the other hand... reroll is best in a situation like Death Saves. That's one time where a reroll comes in REALLY handy.
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iserith
Magic Wordsmith
Great, thanks all. The general consensus here so far and in some Twitter and Discord chats I'm in says d6 is the way to go, so that's what I'll do.
I'm designing something to work with Tales from the Yawning Portal where there are adventurers in the tavern telling a story about the current adventure as if it took place in the past. So there will be some mechanics that those storytellers can use to influence the tale. Among them will be a version of my Claiming Inspiration mechanic which earns you d6s (up to four times per session per player). If someone other than your character fails at something, you can say "That's Not How I Remember It..." then add a d6 to the roll and we'll see who's right!
I'm designing something to work with Tales from the Yawning Portal where there are adventurers in the tavern telling a story about the current adventure as if it took place in the past. So there will be some mechanics that those storytellers can use to influence the tale. Among them will be a version of my Claiming Inspiration mechanic which earns you d6s (up to four times per session per player). If someone other than your character fails at something, you can say "That's Not How I Remember It..." then add a d6 to the roll and we'll see who's right!
Harzel
Adventurer
As hinted at by [MENTION=6785802]guachi[/MENTION], one needs to add quite a few assumptions to the OP to get to something quantifiable, particularly around opportunity cost (that is, "Do I use the Inspiration Die now or save it for later?"). I guess it's not all that useful to just say "It's hard."; if I get inspired, I'll work on it.
Ovinomancer
No flips for you!
Being able to use the advantage mechanic before or after the roll is known doesn't affect probabilities. It does, however, affect efficiency of use, so you're get better use efficiency out of rolling after, but you really won't change the probabilities of success from rolling with.
The probabliities involved are heavily dependent on the roll needed. As such, there's not general solution. Instead, you have to look at use cases to get a feel for how each works at different break points. Let's assume the easy/medium/hard (DC 10/15/20) breakdown. Let's also assume level 5 characters with a +6 proficient skill and a non-proficient +0 skill and compare the two at the above DCs.
DC.|.Pro....+d20.....+d6.........Non.....+d20....+d6
10...85%...97.8%...94.2%.....55%....79.8%....72.5%
15...60.......84.......77.5..........30.......51.........47.5
20...35.......57.8.....52.5.........5.........9.8.........22.5
As you can see, the margins matter. +d6 is the better option when you need to roll 17 or better, otherwise, rolling a second d20 beats the odds every time. This is because, over time, rolling the d6 every time you're within 6 of success (it doesn't matter otherwise and doesn't add) nets (as stated previously) a 17.5% increased chance of success. When rolling a second d20, the odds are increased by 18.8% at 16 and 16% at 17. So, 17 is the break point where the d6 will improve chances over the second d20. At level 5, NONE of the DCs meet this for the proficient -- they're always strictly better off with the second d20. However, for the non-proficient, there's huge increase in chances as they approach a more difficult task. This may be a behavior you wish to encourage -- the non-proficient have a improved chance to succeed at progressively harder DCs, but I find it a bit counter-intuitive.
In short, the +d6 has very distorted impacts at the very low and very high ends of the scale, and moderate effects in the middle. The +d20 has a fairly smooth curve of effectiveness, very high at low rolls, to good in the middle, to a low impact at the high end of the scale. This is for the d20 roll, though, you need to factor in what roll bonuses look like at your chosen play level because that will definitely impact the use of these. High bonuses make the d20 very much the best choice, Low (or negative) bonuses gain a great deal from the +d6 at the high end, but not so much in the middle.
The probabliities involved are heavily dependent on the roll needed. As such, there's not general solution. Instead, you have to look at use cases to get a feel for how each works at different break points. Let's assume the easy/medium/hard (DC 10/15/20) breakdown. Let's also assume level 5 characters with a +6 proficient skill and a non-proficient +0 skill and compare the two at the above DCs.
DC.|.Pro....+d20.....+d6.........Non.....+d20....+d6
10...85%...97.8%...94.2%.....55%....79.8%....72.5%
15...60.......84.......77.5..........30.......51.........47.5
20...35.......57.8.....52.5.........5.........9.8.........22.5
As you can see, the margins matter. +d6 is the better option when you need to roll 17 or better, otherwise, rolling a second d20 beats the odds every time. This is because, over time, rolling the d6 every time you're within 6 of success (it doesn't matter otherwise and doesn't add) nets (as stated previously) a 17.5% increased chance of success. When rolling a second d20, the odds are increased by 18.8% at 16 and 16% at 17. So, 17 is the break point where the d6 will improve chances over the second d20. At level 5, NONE of the DCs meet this for the proficient -- they're always strictly better off with the second d20. However, for the non-proficient, there's huge increase in chances as they approach a more difficult task. This may be a behavior you wish to encourage -- the non-proficient have a improved chance to succeed at progressively harder DCs, but I find it a bit counter-intuitive.
In short, the +d6 has very distorted impacts at the very low and very high ends of the scale, and moderate effects in the middle. The +d20 has a fairly smooth curve of effectiveness, very high at low rolls, to good in the middle, to a low impact at the high end of the scale. This is for the d20 roll, though, you need to factor in what roll bonuses look like at your chosen play level because that will definitely impact the use of these. High bonuses make the d20 very much the best choice, Low (or negative) bonuses gain a great deal from the +d6 at the high end, but not so much in the middle.
Here. Have a d6.
TwoSix
Dirty, realism-hating munchkin powergamer
If the inspiration is a limited resource, and I know the DC/AC targets, than I'd rather have the d6, since I can choose to deploy it in a situation when I have a 5/6 or 6/6 chance of the bonus shifting the failure to a success.
I'd want the d20 option for when I screw up a save really bad, and the d6 option isn't going to save me.
I'd want the d20 option for when I screw up a save really bad, and the d6 option isn't going to save me.
GlassJaw
Hero
If the d6 can be added after my roll, I'd always take the d6 over advantage, even if I don't know the target number. The biggest "disadvantage" of advantage (see what I did there?) is that per RAW, you have to decide to use it BEFORE you roll. And as the math shows, it becomes a lot more effective if you use it after you see your roll (again, not allowed per RAW).
I like the concept of Inspiration in theory but definitely agree that it needs something to make it more engaging. Most of the time my players completely forget about it.
I've experimented with a Hero Point system that I ported over from Trailblazer. I used the Hero Point system in the DMG and revised it for 5E. It worked really well but was a bit too complex for my group of mostly inexperienced gamers. It also doesn't play very well alongside Inspiration (I would use one or the other in future experiments).
Complexity aside, it was a lot more engaging for the players but it's definitely more powerful than Inspiration so keep that in mind. It probably would work best in a high-action/low-magic campaign like Conan where there are fewer modifiers in the mix but you want the players to punch up fairly often.
A friend of mine uses a modified Inspiration system that I really like: each player can have up to 3 Inspiration points. Gaining advantage on a roll costs 1 point (standard Inspiration), rerolling (after your initial roll but before you know the result) costs 2. There are some additional options but that's the meat of it.
I like the concept of Inspiration in theory but definitely agree that it needs something to make it more engaging. Most of the time my players completely forget about it.
I've experimented with a Hero Point system that I ported over from Trailblazer. I used the Hero Point system in the DMG and revised it for 5E. It worked really well but was a bit too complex for my group of mostly inexperienced gamers. It also doesn't play very well alongside Inspiration (I would use one or the other in future experiments).
Complexity aside, it was a lot more engaging for the players but it's definitely more powerful than Inspiration so keep that in mind. It probably would work best in a high-action/low-magic campaign like Conan where there are fewer modifiers in the mix but you want the players to punch up fairly often.
A friend of mine uses a modified Inspiration system that I really like: each player can have up to 3 Inspiration points. Gaining advantage on a roll costs 1 point (standard Inspiration), rerolling (after your initial roll but before you know the result) costs 2. There are some additional options but that's the meat of it.
iserith
Magic Wordsmith
It is kind of nice to have to offset disadvantage though. That's one thing a d6 won't get you.
Check this out: The Case for Inspiration.
GlassJaw
Hero
Yes I've seen that. It's a great idea. I also like that it gives the players incentive to focus on their Backgrounds (or come up with them in the first place). Sometimes just that is a struggle.
I'd even go a step further and allow players to award Inspiration to each other. This would probably work well in the system I outlined above where players have up to 3 Inspiration "points".
I'm probably going to run Curse of Strahd next and I'm thinking about using the Tarroka deck when a player gains Inspiration. There is a DMs Guild product called Tarroka Deck Unleashed. It needs some tweaking but the core concept is sound.
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