RogerBW's Blog

Logarithmic scales for RPGs 23 May 2021

Over the years quite a few RPGs have used logarithmic scales for various purposes. Why aren't they more popular?

The basic idea is quite simple: each +1 on the game scale represents a multiplication by a set value. Particularly in a superheroic game, this lets you rate massively powerful characters on the same scale as normal people, without losing all resolution at the low end.

This gets more interesting when one realises one can also use them as log tables. For the sort of gamer who is resistant to mathematics at the table, it may be easier to say "I need to go distance-value 15 and I can travel at speed-value 10, so it'll take me time-value 5" than to try to divide a distance by a time (especially if they're using a mixed unit system so that they're trying to wrangle yards, miles per hour, and seconds).

More generally, this becomes a universal system for calculating the effects of powers: for example, if you're improvising a spell, its range, duration and time can all be worked out in terms of game measure and added up to get an overall difficulty. All of a sudden your spell list doesn't need to mention individual ranges or durations any more; and this doesn't add any complexity to the system beyond learning the universal scale in the first place.

The earliest example I've found of this is Jeff Dee's Villains and Vigilantes from 1982, which uses a straightforward ×2 per +1. Greg Gorden's DC Heroes, which was the first one I played, did the same thing in 1985 (a rating of 0 is 50 lb, 10 feet, or 4 seconds).

GURPS does this a bit; the Size and Speed/Range Table, which seems to have come in with third edition in 1988, goes up at ×10 per +6 (roughly 1.5 per +1). But while this is used for range penalties in combat, and the difficulty of spotting something at a distance, it's never been broadened into a single universal mechanic.

I don't know just when GURPS long-range modifiers came in (with the magic system in original Fantasy maybe?) but they're ×10 per +2, about 3.2 per +1.

At some point the Hero system adopted a mass scale of ×2 per +5 (1.15 per +1), but I don't know when this came in or which of its authors may be responsible.

Greg Gorden extended his DC Heroes system for Torg (1990): distance, time and weight all scale at ×10 per +5 (about 1.6 per +1). This time the zero values are 1 kg, 1 second, 1 metre, which is nice… but speeds are inconsistent, because they're calculated as metres per ten-second combat round rather than per second. Looked at another way, a mile (16) per hour (18) should be speed -2, but is actually listed on the table as +3.

Ray Winninger's Underground (1993) (which I think may be back in print?) used a universal units system similar to the idea in DC Heroes. I've never played this one, but I think it's ×2 per +3 (1.25 per +1). Apparently there's a volume measure, and one might reasonably argue whether volume should be on the same scale as linear distance… I think I would have to say that no, it should be on a tripled scale, so whatever the extra measure is for going from 1m to 2m in a straight like, going from a 1m cube to a 2m cube should be three times as much.

Greg Porter's CORPS (1994/1998) appears to have carrying capacity proportional to strength squared, which GURPS would adopt in its fourth edition (2004).

Greg's EABA (2001/2013), which I can't really use because its PDFs require Acrobat Reader which hasn't been released for Linux this century, is sadly inconsistent; mass, money and quantity are ×10 for +10 (0 = 13kg), distance/speed/time is ×2 for +2 (0 = 3-5m, 1s) but time has different multipliers at differnt points on the scale… why did you DO it Greg?

I had a quick look at Jeff Dee's Living Legends (2005), and it's using a universal table for converting real values to game measure, but the progression is weird - there are blocks of 3, 4 or 10 where the difference between one value to the next stays the same, some blocks where the differences are randomly 438 or 439, then suddenly the next set of differences is 819-820… I'm not really sure what's going on here.

Why hasn't this sort of thing become more popular? They don't use scary words like "logarithm" in the game books; they just tell you to look up the value on the table, do the calculation more easily than you would have before, look up the result. Still, many modern systems don't seem to want even this level of detail.

All right, it's a bit odd if a good die roll can make you ten times as effective as a bad one…

Tags: rpgs

  1. Posted by Jon Hancock at 10:08am on 23 May 2021

    "why did you DO it Greg?"

    You should ask him. He's always struck me as a designer who is happy to show his working and who usually has a reason for the choices he makes.

  2. Posted by Peter C at 11:17am on 23 May 2021

    We spend perhaps too much time with smart people who spend a lot of time fiddling with computers and numbers and find logarithmic scales perfectly intuitive. However, the vast majority of people can barely do basic arithmetic and are utterly baffled by logarithms.

    Log tables were already a historical curiosity by the 1980s and few would recognise them now. The game rules would have to explain how to use them and this might still be one level of indirection too many for "the sort of gamer who is resistant to mathematics at the table".

    So I guess it comes down to whether the game is being designed for smart people, or whether they want it to have more general appeal. Apparently the market has spoken.

  3. Posted by David Pulver at 07:09pm on 23 May 2021

    Traveller's STRIKER miniature rules (1982) adopted a logarithmic system for weapons and armor damage, though was not used across the rest of the Traveller game system.

    Log systems offer a great deal of utility in games where high power differences exist, e.g., superhero or sci-fi, but do have some game design issues.

    First off is stacking of cumulative effects, especially damage resistance, defense, or whatever. In GURPS if you stack DR 10 armor with DR 2 Toughness while hiding behind DR 10 cover you know you have DR 22. In a log system that (say) adds +1 to defense for each (increment) you have to consult a table or formula.

    Second, in point-based games, you have the extra complexity of how point costs produce escalating effects. In a log system, if you use linear point costs but log-based effects (as in Hero) you can get rapid escalations of power, which can be good or bad depending on what you want to simulate. More perniciously, it means that the same (say) 5 points might give you a minor power if you buy level 1, but give a dramatic effect if you're boosting a high level. It's also possible to make the point costs increase at the same ratio as the effect, but then you get really high numbers. Of course, some games do that: D&D experience levels often double in cost for each +1 level of effect.

    Third, the Greg Gordon-style systems such as Torg and DC Heroes and those that derive elements from them (for instance, some parts of Mutants and Mastermind do so) are vulnerable to huge shifts of effect. Batman can lift 500 lbs. (say) but when he pushes his strength he gets a +5 column shift which suddenly lets him lift 5 tons or whatever.

    Fourth, the log-based tables where you add numbers together can be poorly used. DC Heroes and Mutants and Masterminds both had rules for throwing that were derived from their lifting log tables. Add ST log number to weight log number , look up on the range log table, voila, you get the distance or speed. Great, simple... except that looking closely it let you get ridiculous results when you moved outside whatever narrow range was examined (an average man tossing a marble several miles at supersonic speed or whatever; I forget the specific error, but the throwing rules were broken.) The problem in this case was a neglect of basic physics: in the real world, air resistance exists, and increases with the square of speed, so applying linear lifting-force ST to determine throwing speed and range produced a wrong result. This could have been corrected by something like "for every TWO levels of ST, go up one step" but it serves to demonstrate that log systems aren't as simple or easy as they might appear.

  4. Posted by DP at 07:11pm on 23 May 2021

    I'd also point out that many game systems are what one might call "stealth logarithmic." Some of the more complex implementations of FATE, e.g., Mindjammer, use log-like progressions for certain parts of the game, e.g., spaceship/vehicle designs or effectiveness of influence social abilities.

  5. Posted by RogerBW at 07:21pm on 23 May 2021

    David - thanks, excellent points. I never played Striker back in the day…

    Come to think of it, GURPS' radiation PF system is multiplicative, and it doesn't half confuse people when one tries to explain that this sort of armour gets multiplied rather than added together. (Even worse when something's "PF 20 per inch" and you have to say how much PF six inches gives you.)

  6. Posted by David Pulver at 08:26pm on 23 May 2021

    The radiation PF system is certainly confusing.

    I never actually played Striker, but it made a fascinating replacement combat system for Traveller. The key difference from Traveller was that it replaced the combat system with an early example of a non-HP based wound level system applicable to both vehicles and living beings. Weapon damage was logarithmic (9mm pistol = 1, assault rifle = 3, .50 MG or gauss rifle 6, a light anti-tank weapon or plasma gun in the 18-34 range depending on TL, a tac nuke was 80-ish, etc. Armor matched damage, e.g., a flak jacket 3-5, a powered battledress (powered armor) 10 to 18, etc. For humans you rolled 2d6 + damage - armor; a 3- bounced off, a 4-7 was a light wound (-1 penalty, 1 in 6 chance of KO), a 8-11 a SW (usually a KO), 12+ dead. For vehicles, the various injury levels translated to component damage hit location results. The interface between striker and the core RPG was an afterthought; in my own games, I used my own variant.

    Wound-level ish systems often seem to work well for log damage games because hit points (despite Hero's best efforts) have some issues for cumulative effects when each extra few points means a doubling of effect.

    Re. your recent Call of Cthulhu autofire post, autofire and the damage issues it imposes is often an interesting test of log systems. Systems like GURPS or Hero or CoC where they combine hit point values with "in theory, tracking every bullet/shell" autofire can cause problems that are magnified when logarithmic values are calculated for hit points (as in Hero and a lesser extent 4e GURPS) yet bullets or other strikes are tracked on a 1-1 ratio (so 20 bullet hits means 20x times the damage).

    A non-cumulative or semi-cumulative wound level based system is thus probably optimum if you're doing logarithmic damage, especially in a modern setting; Mutants and Masterminds et al is perhaps the best known recent example of many such implementations, although some of them are just "wound boxes" standing in for "we have a few hit points". Of course this in turn creates a bit of a problem since (a) it's boring to hit someone multiple times and produce no extra effect as you wait for that high roll that DOES take him down and (b) a lot of damage in the real world is somewhat cumulative (due to blood loss, bruising - subdermal blood loss, etc.) I thus favor a semi-cumulative system (each light wound might also reduce your fatigue by 1, or whatever). Still

  7. Posted by RogerBW at 09:05pm on 23 May 2021

    Hit points may well have been an accidental approximation in the first place (the Hero in Chainmail took damage like four fighting-men, i.e. had 4 hits until they died) and they don't seem to be anything like an accurate model of the way human bodies take damage. On the other hand they're easy to understand.

    I think one might say that the most minor wounds worth tracking at all could accumulate to a certain level of impairment, but not beyond that; at some point along that track there'll be so much endorphin washing around the system that one doesn't even notice further minor injury. But 20 one-point wounds over the space of an hour should never, I think, be as damaging as one 20-point wound.

  8. Posted by David Pulver at 09:36pm on 23 May 2021

    Indeed. I suppose it depends on what 1-point represents, though. D&D hit points are an abstraction of fighting prowess, fatigue, luck, and injury. GURPS hit points are perhaps best seen as a cumulative statistical representation of the probability that an injury will be serious or that shock effects will incapacitate someone.

    In GURPS or other linear systems you have the gamist advantage that multiple 1 point hits from large rats or fists or whatever can take someone down over time, thus making fists or rats or whatever a credible threat rather than a waste of time. In a log system, where a person 10 or maybe even 100 times tougher might only be 3-6 points different in stats there tends to be a greater likelihood the game mechanics will set things up so that minor attacks against a big thing won't do ANYTHING; this can be good or bad, but is yet another thing complicating game design (do I want lots of rat bites to bother Conan, or SMG pistol-caliber fire to have an effect on a large but unarmored dinosaur or vehicle?) In a GURPS-style linear system, you can ignore this problem as it's a just a matter of setting up a linear DR and deciding what number of HPs is realistic.

    GURPS 4e is unfortunately broken in my eyes in that it should have differentiated the ST-based "damage to disable/serious wound/kill" in one shot from the "cumulative injury" rules. I think you could fix a lot of GURPS problems by giving people more Hit Points for CUMULATIVE damage (set it so that the average person has about 20 hit points to start KO checks; scale it from two-thirds root of mass (i.e., sscaling with surface area) while retaining the cube root of mass factor for "damage required to stun/major injury/disable limb" (half ST) and "damage required to force a HT save to avoid KO or death (1x or 2x ST). But this is digressing a lot from your initial topic; my apologies.

  9. Posted by RogerBW at 10:01pm on 23 May 2021

    Alas, the experiments to work this out in realistic detail are unlikely get past the ethics committee, even if we use graduate students rather than rats.

    It does seem that for example you can empty SMGs at a main battle tank all day and not achieve very much in terms of damage… except that you'll scar up the vision blocks, sand off the antennae, and so on. That doesn't blow it up, but it does make progress towards a mission kill. (And if the occupants are anything like normal people, they'll be really annoyed.) That's something that I think a game system ought to be able to model.

    I think log-scale systems may be more than usually prone to Murphy's Rules, in part because they're easy to use: they tempt the player to step outside the standard human-scale range of activities because the numbers seem so easy, and then they find they're doing implausibly good or bad things.

  10. Posted by DP at 05:27am on 24 May 2021

    Interesting point on Murphy; I think you're right.

    Striker actually did this reasonably well back in the day: it had a Surface damage table for very minor effects that specifically was limited to damage to sensors, vision systems, radios, and exposed weapons, etc.

  11. Posted by Paul Blackwell at 06:00pm on 27 May 2021

    Perhaps you're understating GURPS' use of the speed/range table as a general mechanism? There seems to be LOTS of uses of the size modifier as a way of, well, modifying things (generally die rolls or costs) based on (log of) size. There are also lots of other uses of the speed/range table for general-purpose logarithmic scaling (without using the term, as you say). As a mathematician, I cringe a bit when a rule involves something like "take the age in months, treat it as if it was a distance in miles, look it up on the speed/range table, and add the resulting SM to the FP cost". I can see that explicitly talking about logarithms would probably be a marketing disaster, but it would be nice - and potentially easier - if there was a slightly more elegant way to use the general mechanism. On the whole, GURPS 4e does well in unifying such things elsewhere.

  12. Posted by RogerBW at 06:18pm on 27 May 2021

    I went looking for that kind of thing for my GURPS Timescales article in The Path of Cunning #1, where I tried to formalise a GURPS-ish approach to logarithmic time, and didn't find as many as I'd expected.

    But generally I don't like that kind of case by case thing; that's why in effect I invented the "time" column for the Size and Speed/Range Table in that article, and I think one could do similar things for mass. Though as David pointed out above, if you say things like "throwing range-value equals strength-value minus minus-value" you start getting problems quite quickly.

  13. Posted by Douglas at 03:21am on 29 May 2021

    Mutants & Masterminds had an interesting approach to log damage. Damage was resisted with a damage save but with some powers it was possible to ignore damage up to a certain level. It help distinguish characters who could soak up a lot of damage but were vulnerable to a lucky shot vs those who could bounce off some damage (e.g., Spiderman vs Powerman.)

    I've noticed the problem in GURPS with hit point scaling, especially if you keep hit points strictly in line with mass. It's too easy to whittle down large creatures and vehicles with low damage attacks. A wound system might work or something like allowing a HT roll to shrug off damage below a certain threshold. But allowing higher HP that doesn't affect wound or pain thresholds might be simpler. Not sure what the appropriate limitation on HPs would be for that. -50%?

    One other log system - Fudge rpg. Though the scale is a little odd.

  14. Posted by RogerBW at 09:53am on 29 May 2021

    Douglas (welcome!) – in JGD's and my statting up of WWII bombers for The Path of Cunning, where they might have ST in the 100-200 range, we ended up using a suggestion from David to make hit points equal ST²/20 (but the thresholds for major damage from a single attack are still the normal fractions of ST). That gave much better results in terms of what sorts of damage aircrat can survive and what will shoot them down.

    For those who haven't met it, it's worth mentioning a trap that Torg fell into by over-using this system: the "glass-jaw ninja". To hit someone you compare your weapon skill plus die roll bonus against their evasiveness; to damage them once you've hit them, it's weapon damage plus the same die roll bonus against their toughness. So if you needed a large bonus to hit someone at all (because they're very evasive), they'll take a huge amount of damage when that same bonus is added to your damage; it's very hard to do a little bit of damage to someone evasive. Which Torg fans write off as a genre convention, but it doesn't sit well with me.

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