D&D 5E On Die Averages and Hit Points in 5e

[clearstream]

2 of my 5 regular players prefer to roll for HP even knowing the average is less than the fixed value - they both tend to take the fixed value their first few level ups so they have a decent base of HP and then start rolling at 4th or 5th level. A third would prefer to roll for HP but doesn’t because she knows the average is lower than the fixed value. One never likes to roll for things if there’s an option to take a fixed number instead, and would continue always taking the fixed value if I implemented this house rule. And the fifth is pretty indifferent to rolling vs fixed value. So, I’m pretty sure this house rule would be nothing but a positive for my players.

Yup, for sure groups differ. That is why I included my sample size and period. A dozen players? As cohorts go, that's tiny.

 

[Charlaquin]

I might not read too much into the 20th-level case. Assume most characters top out and 9th, and some make it to 15th. We didn't roll at 1st, so we are going to roll 8-15 times per character. More is better. For the low value - which is the most common case - a bad early streak will feel horrible and will not correct (there will not be sufficient additional rolls to reinforce the average.)

Sure. The 20th level case really has very little to do with my decision making here. The issue is, players who like to roll dice feel punished for doing so because the fixed value is higher than the average. When you have one no-risk option and one risk-reward option, but the potential reward is actually lower than the fixed value on average, the risk-reward option is just a trap for folks who like to gamble. I don’t like trap options. Rerolling 1s fixes that problem. How many times you roll HP over the course of your adventuring career really doesn’t matter, the point is to make both options feel valid.

 

[clearstream]

Sure.

Will you go with infinite reroll-1s? Also, what do you think of also reroll-2s on first die?

The underlying thesis is that a bad roll is more bad for a character than a good roll is more good. Say I start as a cleric with 8 HP. Rolling 2 puts me at 10 at 2nd-level instead of 13. Rolling 2 again puts me at 12 instead of 18. Say I instead rolled two 8s, so had 24 HP. I suspect being down 6 HP is more bad than being up 6HP is good.

 

[Charlaquin]

Will you go with infinite reroll-1s?

Yes

Also, what do you think of also reroll-2s on first die?

Seems fine I guess. Depends on what the goal is, really.

The underlying thesis is that a bad roll is more bad for a character than a good roll is more good. Say I start as a cleric with 8 HP. Rolling 2 puts me at 10 at 2nd-level instead of 13. Rolling 2 again puts me at 12 instead of 18. Say I instead rolled two 8s, so had 24 HP. I suspect being down 6 HP is more bad than being up 6HP is good.

That’s the risk you take by choosing to roll for HP instead of taking the fixed value. I’d rather leave it up to the player to decide if the risk-reward proposition is worth it to them or not. Like I said, I have two players who prefer to roll even though they know the average result is lower than the fixed value and one who would prefer to roll but doesn’t because she knows the average result is lower than the fixed value. Adjusting the probability so the average is the same as the fixed value will make all three feel better about rolling, without making the player who prefers not to roll unless they have to feel like the rule favors the folks who roll.

 

[NotAYakk]

Reroll 1 on first roll makes 1dX become:
1/X chance of 1dX (average (X+1)/2 = X/2 + 0.5)
(X-1)/X chance of 1d(X-1)+1; average 1 + (X)/2 = X/2 + 0.5 + 0.5
So you get X/2+0.5 + 1/2 (X-2)/(X)

For 1d6 this is +0.33
For 1d8 this is +0.38
For 1d10 this is +0.4
For 1d12 this is +0.42

Reroll 1 always adds +1/2 to the average.

Reroll 1 or 2 on first roll makes 1dX become:
2/X chance of 1dX (average (X+1)/2 = X/2 + 0.5)
(X-2)/X chance of 1d(X-2)+2. (average 2 + (X-1)/2 = X/2+0.5+1
So you get X/2+0.5 + (X-2)/X

For 1d6 this is +0.67
For 1d8 this is +0.75
For 1d10 this is +0.8
For 1d12 this is +0.83

Reroll 1 always and reroll 2 on first roll is 1+1d(X-1) with reroll 1 on first roll.

Which is (X-1)/2 + 0.5 + 1/2 (X-3)/(X-1) + 1
= X/2 + 0.5 + 0.5 + 1/2 (X-3)/(X-1)

For a d6 this is +0.8.
For a d8 this is +0.85
For a d10 this is +0.89
For a d12 this is +0.91

Reroll 1s and 2s always adds +1 to your average in every case (it is 2+1d(X-2)).

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The alternate-system where Con bonus gives you rerolls might work like this.

A bonus of +X con means you always reroll HD rolls of X or below. If your bonus exceeds the die size, just use max.

A penalty of -1 means you reroll max HD rolls; each additional penalty means you also reroll the next highest roll. Never reroll 1s.

In this system, a +1/-1 con is worth 1/2 of a HP/level, about half the impact it has normally. It doesn't work with non-rolling HP however.

If you mix it with the "reroll all of your HD" I think it becomes a bit fun. I'd do the "if your con bonus changes, you gain or lose (level * con bonus change)/2 immediately for the corner case, which works unless you have a con of +6 on a d6 HD.

 

[clearstream]

Seems fine I guess. Depends on what the goal is, really.

To compensate for bad rolls being experientially more bad than good rolls are good. Were I persisting with something like this method, I might go with infinite-reroll-1s-and-2s. Mooting that the half point on offer will compensate the risk of early low rolls.

That’s the risk you take by choosing to roll for HP instead of taking the fixed value. I’d rather leave it up to the player to decide if the risk-reward proposition is worth it to them or not. Like I said, I have two players who prefer to roll even though they know the average result is lower than the fixed value and one who would prefer to roll but doesn’t because she knows the average result is lower than the fixed value. Adjusting the probably so the average is the same as the fixed value will make all three feel better about rolling, without making the player who prefers not to roll unless they have to feel like the rule favors the folks who roll.

And thus laid bare a distinction between a-priori and a-posteriori

 

[Charlaquin]

And thus laid bare a distinction between a-priori and a-posteriori

But which one’s which in this case?

 

[clearstream]

Reroll 1 on first roll makes 1dX become:
1/X chance of 1dX (average (X+1)/2 = X/2 + 0.5)
(X-1)/X chance of 1d(X-1)+1; average 1 + (X)/2 = X/2 + 0.5 + 0.5
So you get X/2+0.5 + 1/2 (X-2)/(X)

For 1d6 this is +0.33
For 1d8 this is +0.38
For 1d10 this is +0.4
For 1d12 this is +0.42

Reroll 1 always adds +1/2 to the average.

Reroll 1 or 2 on first roll makes 1dX become:
2/X chance of 1dX (average (X+1)/2 = X/2 + 0.5)
(X-2)/X chance of 1d(X-2)+2. (average 2 + (X-1)/2 = X/2+0.5+1
So you get X/2+0.5 + (X-2)/X

For 1d6 this is +0.67
For 1d8 this is +0.75
For 1d10 this is +0.8
For 1d12 this is +0.83

Reroll 1 always and reroll 2 on first roll is 1+1d(X-1) with reroll 1 on first roll.

Which is (X-1)/2 + 0.5 + 1/2 (X-3)/(X-1) + 1
= X/2 + 0.5 + 0.5 + 1/2 (X-3)/(X-1)

For a d6 this is +0.8.
For a d8 this is +0.85
For a d10 this is +0.89
For a d12 this is +0.91

Reroll 1s and 2s always adds +1 to your average in every case (it is 2+1d(X-2)).

---

The alternate-system where Con bonus gives you rerolls might work like this.

A bonus of +X con means you always reroll HD rolls of X or below. If your bonus exceeds the die size, just use max.

A penalty of -1 means you reroll max HD rolls; each additional penalty means you also reroll the next highest roll. Never reroll 1s.

In this system, a +1/-1 con is worth 1/2 of a HP/level, about half the impact it has normally. It doesn't work with non-rolling HP however.

If you mix it with the "reroll all of your HD" I think it becomes a bit fun. I'd do the "if your con bonus changes, you gain or lose (level * con bonus change)/2 immediately for the corner case, which works unless you have a con of +6 on a d6 HD.

Where I have landed though, is that after 1st-level, you roll all your HD to get your new max HP; or +1 if new would be < current.

So at 1st-level as a wizard I have 6 max-HP. (Max is still taken at 1st.)
At 2nd-level I roll 2d6, say getting 5? My max-HP becomes 7.
At 3rd-level I roll 3d6, say getting 16? My max-HP becomes 16.
At 4th-level I roll 4d6, say getting 16 again? My max-HP becomes 17.

My aim is lower HP totals - nearer the averages - without locking in bad rolls. That is based on my observation that - to my taste - high-level characters have too much HP. I am seeking a game with monsters per CR by DMG guidelines are on average more challenging.

 

[Mannahnin]

Where I have landed though, is that after 1st-level, you roll all your HD to get your new max HP; or +1 if new would be < current.

So at 1st-level as a wizard I have 6 max-HP. (Max is still taken at 1st.)
At 2nd-level I roll 2d6, say getting 5? My max-HP becomes 7.
At 3rd-level I roll 3d6, say getting 16? My max-HP becomes 16.
At 4th-level I roll 4d6, say getting 16 again? My max-HP becomes 17.

My aim is lower HP totals - nearer the averages - without locking in bad rolls. That is based on my observation that - to my taste - high-level characters have too much HP. I am seeking a game with monsters per CR by DMG guidelines are on average more challenging.

I'm also a fan of this concept and have tried it out in a short campaign. I think overall it tends to inflate HP a bit over an always-roll scheme, because lower rolls are not locked in, but higher rolls are.

Right now I'm also playing in an OD&D game where each time we level the DM lets us BOTH roll another die AND re-roll the lot, and take the best total. That's been a nice little bump compensating a bit for general PC fragility in 1974-days.

 

[NotAYakk]

Where I have landed though, is that after 1st-level, you roll all your HD to get your new max HP; or +1 if new would be < current.

So at 1st-level as a wizard I have 6 max-HP. (Max is still taken at 1st.)
At 2nd-level I roll 2d6, say getting 5? My max-HP becomes 7.
At 3rd-level I roll 3d6, say getting 16? My max-HP becomes 16.
At 4th-level I roll 4d6, say getting 16 again? My max-HP becomes 17.

My aim is lower HP totals - nearer the averages - without locking in bad rolls. That is based on my observation that - to my taste - high-level characters have too much HP. I am seeking a game with monsters per CR by DMG guidelines are on average more challenging.

Sure. But are you adding con bonus?

If you are, then the roll system you have has HP about as high as the baseline one at medium high levels.

Ignoring "keep the lower level version", at level X you have one less than (HD/2) + (Level/2) fewer HP on average. Relative to baseline, this has the largest impact at low levels, and is smaller at higher levels.

The ability to use lower rolls +1 gives you a larger boost at higher levels I think? Basically, the chance that 7d8 > 8d8 is higher than the chance that 3d8 > 2d8. It also is significant in T1, where the +1 over a max die is a reasonable possibility to help.

This is probably worth simulating, as the math gets annoying. It is small enough that a probability based simulation isn't needed; just do every die combination maybe with a bit of dynamic programming, and you can solve it completely.