GURPS Supers, Hit Point and DR values.
Bryan J. Maloney
The following is some work I did on re-aligning the values of DR and HP for
GURPS Supers.
Balancing Act:
A Modeling Approach towards Reassigning Values to DR and Extra Hit Points in
GURPS Supers
Copyright 1991 Bryan J. Maloney
The second edition of GURPS Supers deserves a great deal of praise. The
powers have been expanded, and many of the old ones rewritten. Furthermore,
they did away with the "Power Groups", which virtually forced a GM to play
with one particular vision of how super powers work rather than giving her own
ideas precedence. Martial Arts and Psionics were moved into their own books
(rightly so, at least considering the wealth of detail in GURPS Martial Arts).
To be succinct, the changes were numerous, and are appreciated.
However, nothing is ever completely perfect, no matter how good it gets. The
second edition fixed many problems, but left one unresolved: Damage
Resistance vs. Extra Hit Points. The problem is quite simple--HP are too
expensive vs. DR, and both are too expensive vs. the new, lower cost of attack
powers.
To demonstrate this, I will need to briefly explain mathematical modeling.
Short and sweet, it is reducing a system to a set of equations, plugging in
known values, and looking at the results. One type of system that lends
itself rather well to modelling is the combat system from most rpgs. The
parameter we wish to observe for our purposes can be called "relative staying
time", which is the amount of times a character could take an attack of
arbitrary power without dying, divided by the amount of time a "zero"
character (10 HT, 0 DR, and 0 Extra HP, in our case) would last. The basic
equation for this is:
R = (H / (P x D)) / C
where:
R = relative staying time
H = hits to kill
P = probability for an attack to hit
D = average damage for that attack
C = the constant staying time for the "zeroed" character.
(Historical note: This equation was first introduced to gamers many years ago
in by Don Turnbull.)
GURPS's combat system is somewhat more complex than this simple equation would
allow. H, in GURPS could be set at many points, but the point at which HT
rolls to avoid death begin is a good start; therefore, H=HT x 2. P varies
greatly--we can set it arbitrarily at the highest possible chance in GURPS, a
roll of 16 on 3d6 (equal to 0.981481...). D is also arbitrary, and actually
only matters for our purposes when dealing with DR. We will set it at 52.5,
about 2-3 dice above the typical attack of the sample GURPS Supers characters.
Why this high? I wish to model using a "high-average" approach rather than a
"dead-average" approach--primarily for genre considerations, after all, how
many supers are knocked dead from one "average" damage shot? A good rule of
thumb to adjust for this is that DR becomes more valuable the lower your
average damage.
Unfortunately, this would still not model GURPS combat accurately enough. A
character in GURPS can, if nothing else, dodge an attack she is aware of. To
take account of this, the (P x D) expression must be multiplied by a factor we
will call F, which is the probability of a dodge failing.
One more thing and we're through. HT has a further effect in that it not only
acts as HP in its own right, but modifies the chance of dying once a character
has reached -HT hit points. I decided to use a 50% chance of death as my
constant (this being the chance for a 10 HT character at -HT hit points). The
question here is how many HT rolls does a character with HT greater than 10
have to attempt to get a cumulative 50% chance of failure? This is answered
by the function 0.5=S^x where S is the probability of a HT roll succeeding and
x is the number of rolls necessary to bring the cumulative chance of success
down to 50%. Solving for x gives us x=log(S)0.5 (log of 0.5 to the base S).
Subtracting one from this gives us the number of additional HT rolls (over the
first) that a character of that HT would theoretically have to make. We can
then convert this into a "bonus" to HT x 2 by multiplying it by 5 (since a
second roll is at -HT + 5, then -HT + 10, etc.). This gives us an expression
for H:
((HT x 2) + (((log(S)0.5) - 1) x 5))
Dividing this by the expression derived above for PD, we get the following:
T = ((HT x 2) + (((log(S)0.5) - 1) x 5)) / (P x D x F).
If we substitute 10 for HT and the appropriate values for S, P, D, and F, we
can calculate C, making our final equation thus:
R = (((HT x 2) + (((log(S)0.5) - 1) x 5)) / (P x D x F)) / C.
Now we need to take the effects of extra HP and/or DR into account.
Extra Hit Points are simple. Add the HP to the ((HT x 2) + (((log(S)0.5) - 1)
x 5)) expression. (For our purposes we will assume that HT would be 10 in all
such cases.)
DR is more difficult. We could just subtract DR from the average value of the
attack but that only works when the DR is below (very roughly) 80% of the
average damage. After this point, the effects of DR are increasingly non-
linear. This is because dice do not generate "average" results, rather they
generate a range, with different frequencies for each result. Therefore, the
most accurate way to get a DR effect is to subtract the DR from each possible
result, treating less than zero sums as zero, multiply these by the relative
frequency of the corresponding result and sum the products--a tedious process.
Quite a bit of work, no? This probably explains why little apparent modelling
went into determining the values of HP or DR for Supers.
In any case, on to the example:
Assume the following: A character has 125 points to spend on either DR, HP or
HT, how long would she last relative the "zero" character vs. a 15-die non-
modified attack? Such a character could either have 18 HT, 41 DR, or 25 HP.
Defense Relative Staying Time
18 HT 12.3
41 DR 5.2
25 HP 2.0
None 1.0
Do the results surprise you? The 41 DR character lasts more than twice as
long as does the 25 HP character, but the 18 HT character outlasts the DR
character by nearly the same margin!
This leads me to conclude that the costs of DR and HP both need to be
adjusted.
Before we continue, a few arguments have been raised against altering the cost
of HP vs. DR, to wit:
"DR's advantage is 'cancelled' by the Armor-Piercing enhancement."
This doesn't really hold water, taking enough Armor-Piercing to halve DR (the
usual example given) doubles the price of an attack, but Impaling only
multiplies the attack cost by 1.5! Furthermore, while there is the Hardened
enhancement for DR, no such counter for Impaling exists for HP. HP are still
on the short end of the stick. In addition, how common is Armor-Piercing?
The argument only gains any small validity when Armor-Piercing becomes quite
common.
"It isn't in genre for characters to have high HP, so the costs should reflect
this."
This would work if and only if every person on earth who wanted to run GURPS
Supers agreed 100% with a single interpretation of "the genre". Wasn't this
why power groups got dropped? Furthermore, many comic book characters do
exist who can't bounce an attack but can really soak up the damage. Captain
Britain, Spiderman, Nightcrawler (all properties of Marvel Comics, use is in
no way intended to infringe on any trademark or copyright owned thereby),
Martian Manhunter (property of DC comics, use is in no way intended to
infringe on any trademark or copyright owned thereby), and Titan (from the
Justice Machine--property of Mike Gustovich, use is in no way intended to
infringe on any trademark or copyright owned thereby), to name a few.
Therefore, HP are not balanced vs. DR, but they should be. Why? Part of the
beauty of GURPS is that it gives players and GMs the choice as to how their
campaigns should be run. Altering the costs of these two Advantages would
restore more of that freedom to the gamers, rather than lock it up in a game-
system imbalance.
How to reassign the costs, then? I calculated relative survival rates for
hypothetical charcters of HT 10 to HT 57. Then, solving my above equations
for HP or DR, I calculated an Extra Hit Point or DR total which would be
closest to the relative survival time for each HT. A partial example of my
results is in table II.
Table II: Increasing HT, Survival Time Relative to HT 10
HT Relative Time Hit Pionts DR
10 1.000 0 0
11 1.219 4 9
12 1.527 11 18
13 2.030 21 26
14 3.083 42 35
15 5.156 83 43
16 11.162 203 49
17 11.268 205
18 12.315 226
19 12.429 229
20 12.543 231 50
25 14.833 277 51
30 22.928 439 53
45 79.578 1572 59
50 288.824 5756 62
55 747.810 14936 65
The above table gives guidelines to assign point values to DR and HP.
However, translating them into useful costs will require some loss of
accuracy. First, we need "clean" numbers--by this I mean that whenever
possible, cutoffs should fall in areas that "feel intuitive" to most people.
For most of us this means numbers ending either in 5 or in 0. While we lose
accuracy by this convention, we render the system more readily comprehensible.
Second, and more importantly, we need to "flatten" any curves or fluctuations
both along the continuum and especially "flatten" out the end--otherwise we'd
be dealing with ridiculously cheap HP and expensive DR. For our purposes, the
useful portion of the curve lies between 10 HT and 37 HT (0 to 600 character
points) since this should fully address any variation in the vast majority of
Supers campaigns. Flattening these curves gives the following results:
Table III: Revised Cost of Extra Hit Points
Hit Points Cost per Hit Point
1 - 5 2
6 - 35 1
36 - 199 1/2
200 - 300 1
301 + 2
Table IV: Revised Cost of DR (Adjusted for Attack Frequency)
DR Cost per Point, Versus
Everything Common Occaisional Rare Very Rare
1 - 15 1 1/2 2/5 1/5 1/10
16 - 55 2 1 1/2 1/4 1/5
56 + 3 2 1 1/2 1/4
You may note that my choices for "flattening" were somewhat arbitrary,
especially at the high end. My reasons are as follows: I decided to raise
the cost of HP per point after a large extent since this seemed to balance a
bit better over 125 character points vs. the cost for HT. I drastically
lowered the apparent cost for DR since to keep it on the curve, characters
would have to pay 100+ character points per DR!
The DR table is the one most likely to need adjustment, since it was designed
against a 15-die attack. (The relative value of HP do not change, regardless
of attack power.) One suggestion would be to lower the beginning of the 3-
point cost to 51 or 46. Using the table above, a character would have to
spend a 72 point attack (12 dice) would require 69 character points worth of
DR to counter on average. Lowering the upper threshold to would require a
character to spend the same amount, but she would have much greater difficulty
building up her defenses beyond this point. However, the lower threshold
could also be lowered to 11. At this point a character would have to spend 74
points to defend against an attack worth 72 character points--an option that
may balance better, but only if very high damage attacks (20+ dice) are rare.
Why allow a character to have such "high" defenses so cheaply? "High" depends
on your perspective. The above numbers were calculated assuming that a GM
might not decide to use stun, damage reduction, etc. In other words, attacks
can kill. These points are designed to give some parity to a character's
defense vs. the cost of attacks. As they stand, attacks are too highly
favored vs. defense. This system not only brings DR and HP in line with the
costs of HT, but also manages to balance DR against damage in a way that gives
a Supers character a fighting chance.
What to do about "realistic" campaigns, where the attacks might average as low
as six dice? I would recommend that, whatever the thresholds chosen, a GM be
suggested to double the prices as given for a low-power campaign, both to
reflect the greater relative value of DR vs. the lower damage, and increase
the risk so much a part of gritty comics.
Appendix: Alteration of Defense-related powers.
Some Super Advantages confer some sort of damage reduction. Of them, the
following would be affected by these changes. They are listed along with
proposed new costs.
Body of Fire 2.5 points/level
Body of Ice 15 points
Body of Metal 8 points/level
Body of Stone 6 points/level
Increased Density 2.5 points/100 lbs
mr. news
>
>The following is some work I did on re-aligning the values of DR and HP for
>GURPS Supers.
Comment: it is unimportant to calculate how many _attacks_ a given set of
defenses can take; what is important it how many _hits_ it can take.
Dodge should be entirely ignored for calculation of the effects of defenses,
since it affects everyone equally.
From: dem...@watnxt1.ucr.edu (Anthony Jackson)
Path: watnxt1!demian
Now, I found some possible problems with math; wasn't sure exactly what you
were doing, but I though I would provide people with correct figures for how
many HT rolls one tends to make (I think you forgot about -HTx5).
First, the maximum number of HT rolls one must make is 0.8xHT.
Each of those rolls adds hp to your average survival equal to (5*(chance to
reach this point and make this roll.)
chance = (chance to make 1 roll)^(rolls to make)
thus, with 10 HT, you have 5*(.5^1+.5^2...+.5^8)--slightly less than 5
There is a general formula for series in this format:
(first term)*(1-chance^(terms-1))/(1-chance); this particular case is:
5*(chance-chance^terms); the average number of rolls made is (chance-chance^ter)
HT chance terms rolls made HT to die rolls, w/o HT*5 limit
10 .500 8 .992 25 1
11 .625 8.8 1.62 30 1.67
12 .741 9.6 2.63 37 2.86
13 .838 10.4 4.18 47 5.17
14 .907 11.2 6.15 59 9.8
15 .954 12 8.37 72 20.6
16 .981 12.8 10.48 84 53
17 .981 13.6 11.11 90 53
18 .981 14.4 11.74 95 53
>Assume the following: A character has 125 points to spend on either DR, HP or
>HT, how long would she last relative the "zero" character vs. a 15-die non-
>modified attack? Such a character could either have 18 HT, 41 DR, or 25 HP.
>
>Defense Relative Staying Time
> 18 HT 12.3
> 41 DR 5.2
> 25 HP 2.0
My calculations ignore dodge, etc, but still disagree
defense hits to average death (52.5 each)
18 HT 1.81
18 HT* 5.73 *ignoring HTx5 limit
41 DR 2.17
25 HP 1.05
Obviously, DR is slightly superior to HT, and vastly superior to HP. Further,
this was the case for a relatively _high_ attack; against low attacks the
difference is even more obvious; for HT to be better requires hits of
56 pts or better, and HP are _never_ superior to _either_.
>Do the results surprise you? The 41 DR character lasts more than twice as
>long as does the 25 HP character, but the 18 HT character outlasts the DR
>character by nearly the same margin!
Yes, they do surprise me. As I believe I show above, they are wrong (note:
if you ignore HTx5 limit your results are quite similar to mine)
>Table II: Increasing HT, Survival Time Relative to HT 10
Table II is incorrect.
>Hit Points Cost per Hit Point
> 1 - 5 2
> 6 - 35 1
> 36 - 199 1/2
>200 - 300 1
>301 + 2
Hah. Right. Cost per Hit Point should not go up, once it hits .5
>DR Cost per Point, Versus
There is some logic to having non-linear cost for DR, but there is not good
reason to have it cost at less than the current 3 pts (as it is, if you put
the same number of points in defense as you do in attack, it works out to
2 DR per die--which I can live with)
Bryan J. Maloney
You CANNOT ignore Dodge when modelling GURPS!!!!!!
This is because, if you are using HT as a referent, DODGE GOES UP ONE FOR EVERY
FOUR POINTS OF HT. THIS MAKES A BIG DIFFERENCE. LEARN THE GAME SYSTEM.
mr. news
>
>You CANNOT ignore Dodge when modelling GURPS!!!!!!
>
>
>This is because, if you are using HT as a referent, DODGE GOES UP ONE FOR EVERY
>FOUR POINTS OF HT. THIS MAKES A BIG DIFFERENCE. LEARN THE GAME SYSTEM.
Path: watnxt2!demian
I am not modeling GURPS. I am modeling damage absorbtion in GURPS. As such,
I am perfectly free to ignore dodge. And thank you, I know the game system
perfectly well.
Think before you flame. Consider that the person you are flaming may know
what he/she is talking about. In any case, you cannot usefully model the
effect of dodge from HT, mainly because the importance of an extra point of
dodge is _completely_ dependent on what your dodge was before. If the extra
HT increases your dodge from 5 to 6, that is basically insignificant; if
it increases your dodge from 15 to 16 that is a huge difference.