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I'm looking at D&D 5e to homebrew a specific item and I'm debating what would be better: advantage or raw bonus? The item is a coin that you toss before an ability check or an attack roll. If the coin lands heads, you gain a boon. If the coin lands on tails, you get a penalty.

I was considering making the boon a raw bonus of +3 to the subsequent roll if it's a boon or -3 if it is a penalty. But it was brought to my attention that giving advantage/disadvantage could potentially be more balanced. So here's my question, what would be better mechanically for the player?

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    \$\begingroup\$ Related: How does rolling two d20 and taking the higher affect the average outcome? \$\endgroup\$
    – Someone_Evil
    Aug 17 '20 at 20:27
  • \$\begingroup\$ Then assume +3 but take in mind just the single use. Not sure if the multiple uses would affect it. \$\endgroup\$ Aug 17 '20 at 20:31
  • \$\begingroup\$ What do you mean by single use vs multiple use? Like you can flip more than once? Or should we just assume that this is a single toss? \$\endgroup\$
    – NautArch
    Aug 17 '20 at 20:33
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    \$\begingroup\$ Oh, and Welcome to RPG.SE! Take the tour if you haven't already and see the help center or ask us here in the comments (use @ to ping someone) if you need more guidance. Good Luck and Happy Gaming! \$\endgroup\$
    – Someone_Evil
    Aug 17 '20 at 20:33
  • \$\begingroup\$ Clarification: you say you'd flip the coin "before" an ability check or attack. Once you've flipped it, do you still have to go through with the action you were attempting? \$\endgroup\$
    – Mark Wells
    Aug 17 '20 at 20:38
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I'm going to assume that your coin only affects ability checks because it doesn't seem to be the sort of thing that, thematically, you could whip out in the middle of combat or when you trigger a trap. It makes no difference if applied to saving throws but if applied to attacks the possibility of critical hits and automatic misses makes things more complex and requires consideration of damage dice and modifiers for completeness.

Let's assume that your target number to pass the ability check is \$t\$ (with \$1 \le t \le 20\$); which is calculated by subtracting the PC's bonuses (ability, proficiency etc.) from the DC. Ignoring the coin for the moment, they have the following chances to succeed:

$$ \begin{align} P(s)&= \begin{cases} {1-\left(t-1\over20\right)^2}, &\text{advantage}\\ {21-t\over20}, &\text{normal}\\ {\left(21-t\over20\right)^2}, &\text{disadvantage}\\ \end{cases}\\ &= \begin{cases} {0.0025(-t^2+2t+399)}, &\text{advantage}\\ {0.05(21-t)}, &\text{normal}\\ {0.0025(t^2-42t+441)}, &\text{disadvantage}\\ \end{cases} \end{align} $$

+3/-3 Coin

If you use this coin, the chances of success are:

$$ \begin{align} P_\pm(s)&= \begin{cases} {.5\left(1-\left((t-4)\over20\right)^2\right)+.5\left(1-\left((t+2)\over20\right)^2\right)}, &\text{advantage}\\ {.5\left((18-t)\over20\right)+.5\left((24-t)\over20\right)}, &\text{normal}\\ {.5\left((18-t)\over20\right)^2+.5\left((24-t)\over20\right)^2}, &\text{disadvantage}\\ \end{cases}\\ &=\begin{cases} {0.025(-t^2+2t+390)}, &\text{advantage}\\ {0.05(21-t)}, &\text{normal}\\ {0.0025(t^2-42t+450)}, &\text{disadvantage}\\ \end{cases}\\ \end{align} $$

So, we can determine when it is better to use the coin and when it isn't and when it makes no difference.

It's obvious that if you have neither (or both) advantage or disadvantage using the coin makes no difference because the probabilities are the same either way.

For advantage, you should never toss the coin because for all values of \$t\$ not tossing has a higher chance of success.

For disadvantage, you should always toss the coin because for all values of \$t\$ tossing has a higher chance of success.

So, this method gives the player no interesting choices to make because it's completely irrelevant.

One tiny edge case is if the target is 21 to 23 - then the coin has a 50% chance of making the impossible possible but that is a sufficiently rare situation that it hardly seems relevant.

Advantage/Disadvantage Coin

I will first point out that if the players have the time and inclination, they should always be able to find some method of getting advantage on an ability check - the Help action, for example.

Leaving that aside, there are 4 possible situations the player considering using this coin can be in:

  • Advantage without disadvantage: never flip the coin; you can only make things worse.
  • Disadvantage without advantage: always flip the coin; you can only make things better.
  • Both advantage and disadvantage: doesn't matter; the coin can't change anything.
  • Neither advantage nor disadvantage: doesn't matter; you can do the math yourself but a 50% chance of advantage or disadvantage gives you exactly the same chance as a normal roll.

Conclusion

This is a thematically cool and interesting magic item.

However, the mechanics just make it boring because the coin is either useless or obviously worth using or not. There is no significant choice to be made by the player here. Even the mathematically illiterate player will use this a few times and then just forget about it.

How to fix it

Do away with the negative on a fail and give it 3 charges/day with the usual 1 in 20 chance of losing its magic if you use the last charge.

I'd go with advantage on success but a +3 is OK too - its more powerful than a Bless spell but less powerful than Bardic inspiration. Or you could use those as a model and roll a die on success - possibly more in keeping with the luck feel of a coin toss.

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  • \$\begingroup\$ For clarification on your suggestion: the suggestion is that, on a fail, nothing happens (rather than a negative outcome), right? \$\endgroup\$
    – HellSaint
    Aug 18 '20 at 6:00
  • \$\begingroup\$ And for further clarification: as you mentioned, the use case seems to be outside of combat, and outside of combat players will usually have advantage through Help or something else (in scenarios where they can realistically take a coin out of their pocket and throw it, I guess), so why are you suggesting advantage is better than the +3? It feels like advantage would be more meaningless more frequently. (note: I +1'd the answer anyway I'm just interested in improving it a little bit from my POV at least haha) \$\endgroup\$
    – HellSaint
    Aug 18 '20 at 6:04
  • \$\begingroup\$ @HellSaint Correct. On a fail, a charge is expended but nothing else happens. So flipping can only benefit you, but may only be done up to 3 times a day. As for your second question, you can usually get advantage out of combat but not always. Sometimes you don't have party members nearby, and sometimes your DM will deem that the action can't be helped with (I often see argument this with lockpicking). Advantage is roughly equivalent to a +5 (give or take depending on target), so it's technically more powerful even if less frequently necessary via the item. \$\endgroup\$
    – Doc
    Aug 18 '20 at 19:02
  • \$\begingroup\$ If they still want something negative gave for tails coming up, maybe 3 charges a day but loses all remaining charges if it comes up tails? Or 2 charges get used up for a tail? \$\endgroup\$
    – Shane
    Aug 18 '20 at 21:21
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    \$\begingroup\$ or you can use it any number of times per day UNTIL you get a tails, at which point it stops working until the next day \$\endgroup\$ Aug 19 '20 at 0:13
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An important distinction about advantage / disadvantage is that it is part of 5e general concept of bounded accuracy.

You cannot pass a skill check you could not otherwise pass before with advantage. For example, if you have +5 Athletics, advantage will help you pass any DC 5-25 ability check. But it cannot allow you to pass a DC 26 check. A +3 bonus however allows you to actually change the outcome of that higher DC check.

Just as importantly, there are a LOT of ways to grant advantage. And they do not stack. Again, this is by design with bounded accuracy and simplicity.

So if you intent is for this item to almost always have a chance of positively or negatively affecting the outcome. A -3/+3 modifier will apply in more situations than an advantage/disadvantage flip would.

Because if you were at disadvantage anyways, the coin is no risk. And if you were advantage anyways, there would be no possible upside.

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  • \$\begingroup\$ Does advantage stack in way of if you have two conditions giving advantage and one giving disadvantage, advantage is still the result? \$\endgroup\$
    – lukethecat
    Aug 17 '20 at 22:50
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    \$\begingroup\$ No. Any number of advantages and disadvantages results in a normal roll. So 10 advantages are countered by a single disadvantage. \$\endgroup\$
    – Daveman
    Aug 17 '20 at 22:53
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The coin that grants advantage/disadvantage is stronger

Hereafter I'll refer to the coin that grants advantage/disadvantage as the Roll Coin and I'll refer to the coin that grants a flat bonus/penalty as the Flat Coin.

Either coin you propose can be flipped under certain circumstances for a net benefit, and can be flipped under other circumstances for a net detriment. Naturally, a savvy player will always choose to flip the coin for a net benefit and will never choose to flip the coin for a net detriment. Therefore, the stronger coin is the one that can be flipped for a net benefit under the most circumstances.

Disadvantage and advantage cancel out and never stack. Consider a character who already has disadvantage on their roll. In this circumstance, a character can choose to flip the Roll Coin for a net benefit: heads is beneficial because it cancels out the disadvantage; tails does nothing because it doesn't stack with the preexisting disadvantage.

Advantage and disadvantage are not always equivalent, because there are abilities that increase the benefits of advantage and abilities that decrease the detriments of disadvantage. Two examples of such abilities include the Sneak Attack feature (in some circumstances) and the Lucky feat. Characters with such abilities can choose to flip the Roll Coin for a net benefit when those abilities (and circumstances, if any) apply.

Flat bonuses and penalties are not always equivalent. Let's refer to the number you need to roll on the d20 to succeed as the TDC (true difficulty class). For example, if the TDC without the Flat Coin is 21 then success is impossible, and in this case flipping the Flat Coin is a net benefit: heads makes success possible because you can roll 1d20+bonus; tail does nothing, because success is impossible regardless. The same applies to any TDC from 21 to 20+bonus. If the player can predict the TDCs over 20 with some accuracy, then they can flip the Flat Coin for a net benefit.

As you can see, the Roll Coin can be used for net benefit under many more circumstances than the Flat Coin, therefore the Roll Coin is stronger than the Flat Coin.

Advantage/disadvantage is more impactful than ±3

The mean of a 1d20 roll is 10.50, whereas the mean of advantage and disadvantage are 13.82 and 7.17 respectively. At face value, this means that advantage/disadvantage is similar to ±3.32, but in reality this depends on the TDC, which is not fixed. For example when the TDC is 11, advantage/disadvantage is equivalent to ±5, and when the TDC is less than 1 or greater than 20, then advantage/disadvantage is equivalent to ±0. However, due to a series of rules and design constraints known as bounded accuracy, we know that the TDC tends towards 11 and almost never lies outside of 1 and 20. As such, the true worth of advantage/disadvantage across a normal 5e game, lies somewhere between ±3.32 and ±5.

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  • \$\begingroup\$ For Sneak Attack, keep in mind that the adjacent-enemy way of getting Sneak Attack requires that you don't have disadvantage. So if you're about to make a straight attack roll where you would already get sneak attack, the downside of disadvantage is huge: even if you hit, you do much less damage. (Opposite of the case you're considering, when you're about to make a straight roll against an isolated target where you wouldn't get Sneak Attack, so gaining advantage would give you extra damage dice.) \$\endgroup\$ Aug 18 '20 at 6:48
  • \$\begingroup\$ @PeterCordes yes, naturally the rogue won't flip the coin if an ally is already granting sneak attack, as that would result in a net detriment. I feel like I should go into more details with sneak attack, elven accuracy, and lucky because it's not immediately obvious how these abilities turn the coin in a net benefit, but at the same time it would be a massive digression that bogs down the rest of the answer. Do you think it would be worth explaining an example for each of those abilities in an addendum? \$\endgroup\$
    – Ruse
    Aug 18 '20 at 7:22
  • \$\begingroup\$ Against a target with high enough AC, a low to mid level rogue might consider losing sneak attack for a higher chance of hitting. Probably only worth it with a multi-class rogue + something with lots of damage per hit other than sneak attack, though, or for a target that's almost dead. Anyway yes, that was my point, that the coin isn't worth the risk when it might lose you Sneak Attack. If you're going to mention gaining it, you should mention losing it as an extra downside. On an easy target where you probably hit anyway even with disadvantage, you'd otherwise use it to fish for crits. \$\endgroup\$ Aug 18 '20 at 8:10
  • \$\begingroup\$ IDK if it's worth spending a lot of extra time explaining the details, though. I tend to agree with Dale's answer that neither of these are the best choices of mechanic. Also that most people who'd find the min-maxing interactions with Lucky and so on interesting can already work it out for themselves. (I think?) If you think it would be fun to write up, maybe a quick note on one of those. \$\endgroup\$ Aug 18 '20 at 8:15
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A flat bonus or penalty to the roll is a static effect, while advantage and disadvantage are more dynamic. Depending on the feel you want for the coin you should choose one or the other accordingly.

Advantage is worth a bit over +3, but can be worth more than that depending on the target you're attempting to achieve.

If you want the coin to be "swingy" (basically thematically having a lot of pull one way or the other) then I strongly suggest going with advantage/disadvantage. If you want it to be a bit more grounded of a modifier, the +3/-3 would be a better option.

(You have to remember that your coin is equally "useless" (flip tails, it's a negative) so the question you're asking isn't worded as precisely as it could be.)

The other thing you need to consider is that multiple sources of advantage/disadvantage don't stack, and do cancel each other out.

This fact has a major impact on how you can "game" the item. If you know that you have disadvantage and are using the +3/-3 coin, it's a risk. If you're using the advantage/disadvantage version, it's not a risk at all.

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personally I would prefer the advantage/disadvantage coin, especially if your intent is to make a useful item for your players. If I already had advantage, I would not roll it, and if I already had disadvantage, I would always roll it. This would effectively give me a way to cancel disadvantage. I would put a limit on the number of times/day you could use the coin, since always having a 50% chance of ignoring disadvantage is pretty strong. The +3/-3 coin I would only ever use if I knew it was impossible to make the roll even on a 20 and thought it was possible with an extra +3. It is much more limited usefulness.

That being said, if the point is to make a useless/cursed item, the +3/-3 is better, as it has very limited benefit to players as it will screw them over as often as it helps them. I kind of like the idea of it being a luck coin that has been cursed and if you find the right magic items/ingredients you can lift or reduce the curse, reducing the -3 to a -2, -1, or even a flat +0, making it into a useful item after they have earned it through adventuring.

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On average, over time, advantage is worth approximately the same as a flat +3 1/3.

If you average all the numbers on a d20, the average is 10.5. If you work out the 400 possible rolls you can get with 2d20 and average the results, it comes out to 13.8333..., for a difference of 3.3333...

The difference is that the distribution of values will be different. Your die result with advantage is still a range of 1-20, but weighted heavily towards the upper end, while d20+3 gives you a range of 4-23 but it's flat all the way across. You're as likely to roll a 4 as a 23 with the +3, but you're vastly less likely to roll a 1 and much more likely to crit with advantage.

The philosophy of Bounded Accuracy means the devs don't like having lots of added plusses, because you wind up in a scenario where you can win any fight by stacking up enough bonuses (and conversely, if you fail to do so, you're not playing the game well). See 4th edition (or World of Warcraft) for an example of the kind of gameplay that leads to: you constantly need a bigger sword and better armor or you're falling behind the curve, becoming less capable even as you level up.

So while it doesn't make a huge difference in balance for this one item, I would probably advise you to use advantage/disadvantage for that reason.

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