Mathematical Analysis of "Long Live HD"

There was a recent blog post by Undead Waffle that offers an alternative to the typical D&D hit point system [1]. Rather than rolling your hit dice to determine your hit points, just hold the dice. You won’t need them until you get hit.

When you are attacked, your enemy rolls a six-sided die. The result indicates how hurt you are. You get 1 wound point per pip and can sort of sustain up to 4 wounds, but you start bleeding out at 3 wounds.

Your hit dice allow you to try to soak up those wounds, to better sustain or even negate the hit if you’re lucky. Suppose that your attacker rolls a 3, which means you’re not dead yet but it’s coming up. If you have a hit die, you can spend it and roll it. If you rolled a 2, you would ultimately gain just 3 - 2 = 1 wound. You can spend as many hit dice as you like, but each one is a gamble.

And I’ve done the math: toe to tip, it’s a gamble. My friends asked me to figure out the mathematical effects of the rule on player longevity. Being bored and depressed, I did just that! The hit-dice-only system differs significantly from the hit-point system in terms of lethality and player knowledge (or lack thereof). In fact, its a lack of player knowledge which in part makes the hit-dice-only system more lethal.

P.S. My computer is officially a goner, so I am going to buy a new one next month. Still can’t reply because my blog is weird about mobile :(

Isolated Cases

At a first glance, the system does not differ much from standard fare hit points except for when you roll your hit dice. This is especially because you would be screwed not to spend your hit die unless the damage was really low, such as a 1. Let’s consider two cases.

• Case A: You have 1-6 hit points. Hit points are determined to be 3. Later, someone attacks you for 5 damage. You spend 3 hit points to gain just 2 wound points.

• Case B: You have 1 hit die. Someone attacks you for 5 damage. You roll your hit die and it turns up a 3. You only gain 2 wounds.

The cases above, despite not saving you from all the damage attempted, protect you from the brunt of it. You won’t die until you take another hit (and, with zero hit dice or points, you will die since 3 wounds causes you to bleed out). But everyone is used to that. One hit die means if you’re hit once, you’re a goner.

The issue is when you roll higher than you need. Suppose in the above cases, you rolled up 6 hit points (Case A), or rolled a 6 on your hit die (Case B). In Case A, you would still have 1 hit point leftover. In Case B, the extra pip is lost. It does not roll over onto subsequent hits. When you roll a hit die, you make the gamble that you need it now and not later.

This is more apparent with more than one hit die. Here is an extreme case:

• Case A: You have 2 hit dice, which become 6 + 1 = 7 hit points. An enemy attacks you for 3 damage, so you have 7 - 3 = 4 hit points left over. You get attacked again for 4 damage, so you lose your remaining hit points but take no wounds.

• Case B: You have 2 hit dice. An enemy attacks you for 3 damage, which you negate by spending 1 hit die and rolling a 6. An enemy attacks you again for 4 damage, but you only roll a 1 on your last hit die. You take 3 wounds and bleed out.

In Case A, having all your hit points in one pool allowed you to survive both of those attacks without a scratch. In Case B, however, spending hit dice by the roll means that you lose the benefit of high rolls because those points are lost. You die because your second roll did not cut it, despite having rolled the same numbers as in the other case where you survived. The best case scenario is always to roll exactly the number you need because, if you roll higher, it’s a disappointment.

The Element of Choice

The central question of this rule’s gamble is, “Should I spend hit dice now or later?” With such a low number of wounds before you start dying (and even at 2 wounds you’ve broken a bone or lost a limb), you’d be screwed not to spend one or more hit dice so long as you have at least one.

So, I tried to figure out how many hit dice someone is likely to spend in order to reduce damage to the point where they will not die (minimum 0, maximum 2 wounds). This math assumes that you have no existing wounds. If you have any wounds, then the chance of death is so much worse. Having 1 wound means that gaining 2 more will bleed you to death.

Here is how many dice you will spend when hit for 1-6 damage, if the goal is to end up with no more than 2 wounds.

• 0 HD: 432/1296 (33.3%)
• 1 HD: 648/1296 (50.0%)
• 2 HD: 192/1296 (14.8%)
• 3 HD: 23/1296 (1.78%)
• 4 HD: 1/1296 (0.077%)

See the picture below for my work. I cheated to determine the values for 3 and 4 HD by determining that the only way that you’re spending 4 HD is if the attacker rolled a 6 and your first three rolls all turned up 1, so you’re just 1 pip short. That is the outcome of four previous dice rolls, so its likelihood was one out of 6^4 = 1296. Then, to find the chance of 3 HD, I subtracted 1 plus the likelihoods of the other outcomes from 1296.

Each time you’re hit, you are likely to spend 0.856 hit dice if you have no wounds and want to avoid more than 2 wounds.

Accumulating Wounds

The above ignores, basically, final results of 1 or 2 where you gain that number of wounds. So, let’s ask: how often will you end up with 1 or 2 at the end? This was a headache to figure out, but it also helps us figure out how a character’s chance of survival worsens as they accumulate wounds.

When you decide to spend no hit dice on an attack, you will end up taking either 1 or 2 hits. This is a 50-50 chance.

When you spend one hit die, you will end up with 0 to 2 hits. Each of these outcomes has a 1-in-3 (33.3%) chance.

Things get weirder at two hit dice. There is a 192-in-1296 chance that the buck stops there, because there is a 1-in-6 chance that you will spend more than one hit die but whether you spend two, three, or four depends on increasingly unlikely chances. Really, there are thirty-two outcomes where the buck stops at 2. Six of these turn out as 1, and six turn out as 2. Each has a ~18.75% chance, meaning there is a ~62.5% chance of no wounds at this point.

There is then a 23-in-1296 chance for stopping at three hit dice, since the twenty-fourth pip would lead to you spending just one more to avoid death. There are also actually twenty-three unique outcomes at this point; no funky division here on out. Of these, four result in wounds of 1, and four result in wounds of 2. That’s a ~17.4% chance for each, or a ~65.2% chance of 0 wounds.

Finally, our 1-in-1296. Our cursed result. Keep in mind that I didn’t calculate results past that since, initially, I was just looking for final results of 0-2. Now, take the previous numerators and denominators and multiply both by 6 (so 1296 becomes 7776). We have our final six outcomes. Of these, one result is 1 wound and one 1 result is 2 wounds. That’s a chance of 1-in-6 each, or a total 4-in-6 chance of not taking any more wounds at all.

So we not only know the chances of getting 0, 1, or 2 wounds, but we can use this knowledge to figure out your chance of survival if you start with 1 or 2 wounds instead of 0. If you already have 1 wound, convert all the results of 2 into failures instead of successes to find your new chance of survival. If you have 2 wounds, your only chance of survival is by getting zeros. The likelihood therefore decreases severely the more wounded you are, but that’s already pretty typical of D&D.

Lost Hit Points

We can sort of calculate how many hit points you lose out on via this method instead of counting hit points as a pool. The distribution of one hit die versus one damage die (d6 - d6) is the same thing as 2d6. There is a 1-in-36 chance of overshooting by 5 pips, a 2-in-36 chance of overshooting by 4, and so on until there is a 5-in-36 chance of overshooting by just 1.

On average and without further parameters, you will lose out on ~2.33 hit points if you overshoot. Since you overshoot ~41.67% of the time, that’s overall ~0.97 hit points. In other words, your 1-6 hit points in typical D&D have become 0-5 hit points, plus a helpful 3 wounds before you start bleeding out. So, at one hit die, that’s like 3-8 or d6+2 hit points, except you roll it when you’re hit. Hey, it’s kinda better odds than usual!

You might point out that the flip non-zero ~41.67% cancels out the negative effect above, having a positive +0.97 impact on the average for an overall zero. Although that is true with respect to the outcome of a single die roll, what I am trying to measure instead is the loss of points from a cumulative sum of dice rolls. In this sense, there is a ~41.67% chance that the dice will roll so high that pips are wasted on the outcome.

To-Hit Rolls & More

The above is extremely harsh if you just roll damage and that’s that. However, I realized that the lethality of this rule is likely offset with an implicit to-hit roll that determines if an attack lands at all. This is a very easy fix: multiply the number of hit dice statistically lost (0.856) by the percent chance of being hit (say, 50%). Then, we’d end up with a more conservative 0.428 hit dice lost per hit which not only aligns well with the usual hit die loss rate of 58.3%, but is even more generous.

But this brings to mind another issue: how long it takes to resolve one hit. Standard D&D has you roll to attack and then, if successful, to reduce hit points. That is up to two rolls per hit. This method, however, will often result in three rolls per hit (if the target has at least 1 hit die), and even more if the target has them and is willing to spend them (which is likely because of the effects of having even 3 wounds or leftover damage). Keep in mind too that hit dice are not gambled blindly, i.e. rolled at the same time as the attacker, but rolled one after the other as the defender sees fit. This is a lot of extra complexity for just a slight change in likelihood of survival or for some smaller numbers.

Conclusion

I think this method should be most attractive to people who want fights to be extra risky, so that players cannot waltz in and think they can get out of it with like 7 hit points. If you told me I had 2 hit dice that I rolled on the spot, and then spend upon rolling, instead of 7 hit points that I know I have, I’d be sweating! It’s sort of like Schrodinger’s hit points.

But I don’t think it really simplifies the math except for smaller numbers. It makes the order of operations to resolve a single hit much more involved, which might not be desirable in encounters with more than four people or something like that. But some people also want that since it makes combat that much more stressful and desirable to avoid. It's 100% a matter of taste!

My favorite alternatives to typical D&D combat are the various other methods I’ve seen to convert 1 HD into 1 point. Some examples:

• (The Scones Alone). 2020-03-12. “Ideas for speeding up D&D combat”, The Scones Alone. All characters have “hits” equal to their HD that are reduced initially upon taking an attack, and then have 1-8 “hit points” (modified by constitution) that are lost as per usual (with damage dice) upon running out of “hits”.

• Benete, Jackson. 2021-03-06. Classic Fantasy (Chainmail + OD&D House Rules). Player-characters have 1-6 hit points, but monsters have hit points equal to their HD value. Attacks usually deal 1 hit point of damage, but may deal 2 points when rolling a 6 on a die.

What I’d lean towards myself is giving characters something like 3-5 hit points (by type), giving monsters 1 hit point per HD, and maybe allowing a saving throw against death for players. A successful attack deals 1 hit or, maybe on a natural 20, double that (rather than on a separate d6 roll like Benete’s house rules). But I’m actually averse to death, or combat in general, than most other people. I just like math problems.

Finally, I don’t want this to come across like I’m ragging on the idea. It’s simply not for me, but it took all of this work for me to feel like I could have an informed opinion about it, and I did all of that just because it sounded interesting! I think it should work really well for people who want combat to be an uncertain gamble, just less for me who doesn’t want combat to linger.

Endnotes

[1] (Undead Waffle). 2022-10-19. “Death to HP – Long Live HD”, The Breakfast Ossuary.