Wednesday, June 8, 2016

Math: How To Not Die - Spending Points on HT or HP

First good search result for:
"too many hit points"
So, I was doing some thinking, and an idea occurred to me to graph the relative values of HT versus HP to get a kinda picture of whether one is worth more attention or not when trying to create a character that doesn't die. I don't think the graphs are especially surprising, but hey, graphs is graphs, and who doesn't like looking at numbers turned into lines and/or analyzing if I'm taking a game a bit too seriously?

Method of Calculation

So, in GURPS, you do not die at 0 HP, on the contrary, you can go several multiples of Max Hit Points into the negatives before sweet release is yours. But, each struggling shuffle towards the end of the mortal coil also gives you a small chance of dying before reaching this assured decimation at -5 x Max Hit Points. Salvation is given if one succeeds at an HT roll at each of 4 checkpoints on the highway to the afterlife.
This means that sheer grit and staying-aliveness is a function of HP and HT. HP increases the number of absolute possible steps before the final destination, and HT increases your chances of not dying before you reach that point.
Mathematically, we call this a binomial probability, because we have two outcomes at each fatality check: we succeed or fail (there are some optional rules for slightly more sophisticated and varied outcomes, but they are still mostly degrees of failure and success than a total paradigm shift. So, some Hit Points are locked behind the chance that you might die, and sometimes, the chance that you might die. You are (mostly) guaranteed that you can go from HP to -HP without dying, so everyone has 2xHP. After those HP though, you have to stay alive to get the next HP Hit Points, after that, you need to stay alive again, and after that, you need to stay alive again, so your chances of getting the next bit of hit points is cumulatively dependent on your odds of getting all the HP before it.
So, "Effective HP" is:
2 x HP + (HT Success Odds) x HP + (HT Success Odds)^2 x HP + (HT Success Odds)^3 x HP + (HT Success Odds)^4 x HP.
I graphed these values, and got the charts below:

Health held constant with HP as the Independent Variable

The vertical axis is "Effective HP" and the horizontal axis is the number of real HP. Each line corresponds to an HT value of 10 to 16, and HP goes from 10 to 20.
The improvements are literally linear (look at the trendline equations, and the R squared value of 1, which means a 100% perfect match.) the lines are a little wobbly because I rounded them to integers before graphing, but this means, point for point spent on HP, the "return on investment" so to speak is always the same. At higher levels of HT, each HP is more valuable, and at 16 HT a single HP is worth almost twice as much as it is worth at 10 HT, as we can see by the increasingly steep slope of the equations.

HP held constant with Health as the Independent Variable

This graph is a little more interesting and follows a more polynomial plot, almost a 4th order equation (not important, but an observation.) We see that with this graph value rises a lot faster per each HT added than in the previous graph, but not all HT are created equal, as beyond 16, the graph completely flatlines (because 17 and 18 are always a failure.) Not that going beyond 16 HT is worthless, but it loses value in terms of simply staying alive. Looking at the trendline equations and looking at the line, the fits are obviously not as close as those for the previous graph, but to kinda get an idea of how much each point of HT helps, take a look at the constant in the equation being multiplied by x. We are getting, in the best situation, an improvement of survivability almost 6 times what we get from increasing HP. The constant next to x^2 however, is negative, indicating diminishing returns; each increase in HT is worth less than the previous, as opposed to HP where increasing HP by one when you are at 10, or when you are at 20 still "extends the lifebar" the same amount.

Final Notes

At the low end, increasing HT does a lot more to help than increasing HP. If I knew how to draw a 3d graph, that'd be nice, but I don't, I don't have a license for Matlab either, but taking a look at these charts, if you ask yourself, "Should I get another HT or another HP?" You can answer that by looking at the graph you like the most, let's say the first one because it's linear and easier to eyeball at a glance. If the dot above your dot (on a different line) is better then the next dot on your line, it is better to improve HT (if you have the choice) If the next dot on your line is better, then it is better to increase HP. Of course, this isn't always accurate because really, one point of HP costs a lot less than one level of HT. If you like the other graph, just swap HP and HT in that statement. That being said, one level of Hard to Kill is closer in price (I think the same price) so you can use that, and some advantages like Fit improve HT for the sake of staying alive, and come with a slew of other helpful benefits.
Also, a lot of this is hypothetical, and depends on certain assumptions like, "being -20 HP with a lot of HT is no worse than being at 10 HP with a lot of HP," and that is not exactly true either. Move and Active Defenses will be penalized, and you might need to roll often to stay conscious... but at least you aren't dead!
Edit: Thought about it, and here's a reasonable approximation of what the graph might look like according to Wolfram Alpha if it were in 3d, note HT is from 0.5 to 1, which is 50% odds of surviving (or, HT 10) to 100% chance of surviving (HT 16 is like 98.5%, technically, not possible in GURPS except for maybe with some "no nuisance roll perk" at 18 HT.)


  1. Interesting. And improved HT will protect you against disease and such, improve your fatigue, and give you additional Basic Speed, while HP will give you additional slam damage. Your analysis does seem to tilt things in favor of HT, that's for sure.

    1. One thing I didn't include, which might also be interesting information is a matrix that shows at each step which is better, more HP or HT.
      Given a choice, if survival is your primary concern, one should get to 14 HT, then after that, it gets a bit more shaky, I think something like "take 4 HP, then one more HT, a few more HP, and finally, 16 HT."

      And yeah, this analysis is way oversimplified, because, like you said, there are tons of benefits of HP and HT that I'm ignoring.


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