Log scales and resolution mechanics

[wild_card2007]

Hello The Forge,

I'm working on my Fantasy Heartbreaker (is that TM Ron? and while I'm happy plugging away on all the color and fluff and such on my own, I'm getting stuck on the mechanics.  I recently discovered log scales, and the math is simple enough but I really don't understand how to use them in resolution mechanics.  Here's a scenario based on old-school linear mechanics:

Let's say Joe wants to use his Acrobatics skill to clamber up a wall and run across it.  He's pretty good at Acrobatics, skill rating of 5 (on a scale of 1..9).  He also has good Coordination (16 on a 1..20 scale).  The GM thinks this is a pretty tough feat, however -- the wall is narrow and in poor condition.  So he gives it a difficulty rating of 8 (1..10 scale).  Now, let's say the resolution mechanic says to roll d10 X times where X is the relevant skill rating, and add the relevant attribute bonus.  In this case, 5d10+2.  Joe is successful if he rolls at least one die above the TN of 8.

Can someone rewrite this example and show me where and how a log scale would apply?  How does Joe's attempt, using Acrobatics and Coordination, at a given difficulty, translate to a die roll in "log terms" instead of "linear terms"?  (Does that question even make sense?)  Where does the log scale fit in in the mechanics?

Thanks,
Thomas

[J. Scott Timmerman]

Now, let's say the resolution mechanic says to roll d10 X times where X is the relevant skill rating, and add the relevant attribute bonus.  In this case, 5d10+2.  Joe is successful if he rolls at least one die above the TN of 8.

I can't tell where you're adding in Attribute here.  If the only measure of success is that "one of Joe's 5 Skill dice come up TN or higher", then Attribute doesn't figure in.  Unless by 5d10+2, you mean you're adding 2 to each die, in which case any die only needs to be 6 or higher.

More importantly, what kind of explanation are you looking for?  I could work out the math on this, find out the probability curve, and perhaps give you an equation involving logarithms that has something to do with the results.  I figure this is possible since I've contrived such equations with White Wolf games that have d10 pools with floating TNs; similar to what you're working with.  But instead of doing something which will, in all likeliness, not have anything to do with what you're asking, and not help you out at all, could you tell me what your goals are in using logarithms in the first place?  What benefits do you wish to achieve in using logarithms?

For my goals, I personally find that old-school mechanics don't work the best when trying to implement a good, simple scale involving logs.  But once you let me know what purposes you've been throwing around, I'd be glad to help you out a little.

-Jason T.

[wild_card2007]

Hi Jason, thanks for the response.

I see logs coming into play in two respects.  One, let's say a human and a giant both want to bash in a door.  The giant is a lot stronger.  So it should be considerably easier for him to bash it in.  The giant ought to be a higher unit value on the strength scale, and as I see it that should affect his chance of success.  And maybe the effect.  (Could that be based on his scale as well?  So maybe effect is another aspect where the log comes into play?

Second, contests between opponents at different scales.  Let's say a human and a kobold in combat.  The human is a lot stronger (say, two units higher), but the kobold is extremely skilled in combat.  In this case, I would guess the human's strength advantage should give him an advantage, and then the difference in skills is applied separately.

I can "visualize" how it works in terms of size, strength, area, etc.  I just can't translate it to a mechanic.

Thomas

[wild_card2007]

Jason, I just re-read the beginning of your recent thread.  I think I may be starting to get an idea of how it ties in.  Going back to the scenario of bashing in a door, the GM assigns a probability of 35% to the human who wants to bash it in.  If giants are 2 scale units higher in strength, increase the probability of success based on the exponent.  So if I say each unit is 1.25, that 35% chance becomes a 55% chance.  (35*1.25*1.25).

Am I anywhere near the ballpark?

Thomas

[J. Scott Timmerman]

Thomas,

What do you mean by, "In the ballpark"?  As far as your math, it's functional.  As far as your goals, only you can tell yourself that.  From comments like:

I think I may be starting to get an idea of how it ties in.

I'm not so sure.

As I understand it, your goal is to create a system where an additive increase by a certain amount in a stat multiplies the chances of success.  If that is your goal, my suggestion is to use a factor that simplifies this process.  If you're using 1.25 (=5/4) I suggest you round things off a bit for convenience.  Here's what I mean.

Let's say s is the relative difference in a stat (strength, size, skill, whatever) from another creature.  Then let's say c is a multiplication factor for the chance of success of any given action, relative to that other creature.  Then let's say f is the chance of success, where the difficulty for that "other creature" would be 40% (conveniently chosen...).

s           c                                               f
-4          2/5                                           16%
-3          1/2                                           20%
-2          2/3                                           27%
-1          4/5                                           32%
(0)         (1)                                           (40%)
+1         5/4  (=1.25x)                             50%
+2         3/2  (=1.5x)                               60%
+3         2                                              80%
+4         5/2  (=2.5x)                               100%

If you round things this way, it might be easier.  The reason you can do this is that (5/4)^2 is 25/16, which is close 'nuff to 3/2.  Likewise, (5/4)^3 is 125/64, which is close 'nuff to 2.

As for my interpretation of your math, you want a system where, if an expert has double the chance a novice does of succeeding at an easy action, then that same expert should also have double the chance a novice does of succeeding at a hard action.  It makes sense, when you look at things that way.

One issue I can foresee, is that a difference of only 6 can turn a 25% into a 100% (no-brainer).  You may want to keep your scales fairly restricted for PCs.  Maybe, 0-5 for attack/combat actions.  Knowledges and things could certainly have bigger ranges, meaning an elementary-school student may have a 1% chance of solving a calculus problem that's a no-brainer (100%) to a high-school student.  But these suggetions are just based on assumtions, since I don't know your real goals.

-Jason T.

[J. Scott Timmerman]

Thomas,

How is this for a dice-mechanic, so you don't have to think about multiplying percentages?

Roll a pool of dice (d10s) equal to (Difficulty - Stat).  If any of them come up as a 1 or 2, you fail.

If "Stat = 3" is the average character, then you could make "6" the average difficulty (51% success rate).  Diff 5 would be 64%, Diff 4 would be 80%, and any difficulty lower than your Stat does not need to be rolled.

There is an issue of granularity with selectable difficulties, but since it puts that on the same granularity as your Stat scales, I figured it's okay.  If you really want to smooth out your granularity, you could change "1.25" to "1.11", and just treat 1s as failures.  Otherwise, this fits the math you're looking for.

-Jason T.

[wild_card2007]

Whew.  You gave me something to think about, so I went off and thunk for a while.  I wound up clarifying some things about what I want to do with scaling.  Here's my current thinking:

1. Character's stats are measured as values on an exponential scale.  A stat value is applied to a base and scalar, resulting in a value in some measurable unit.  For example a "strength rating" of 5u, when applied to carrying capacity, might mean the maximum weight the character can lift is 100lbs x 1.25^5, or about 300lbs.
2. Skill is measurable, either in absolute or relative terms.  "Archer 1" means I can hit a 5" target at 50 yards in 8 seconds 50% of the time.  "Sword 1" means when up against a similarly trained opponent with similar strength and coordination, I can hit him 50% of the time in 8 seconds.  (This may be more realism than I want.)
3. The base chance to perform a skill under the "default" conditions is 50%, which in exponential scale terms is 0u.  An exponential change in a unit relevant to that skill increases/decreases that chance.  If the target is 1u farther away (50x1.25^1), the chance of hitting is reduced by 1u.  If I'm fighting a more skilled opponent (Sword 4), my chance of hitting is reduced by 3u.  If my strength is 5u my chance of hitting him is increased by 5u.
4. Applying the relevant modifiers to the 0u base for a skill check results in an adjusted value.

This is as far as I've gone.  My next step is to come up with a formula that uses the adjusted value and results in a percent chance to succeed.  To simplify game play I'm tempted to make it a simple scalar where each 1u is 5% or something.  So 50%+3u = 65% chance to succeed.  My other thought is a log scale; this would then require either a lookup or remembering what dice to roll for a given "x"u value.

This is all speculative and explorative at this point for me.  I welcome your thoughts.

Thomas

[J. Scott Timmerman]

Thomas,

I remember my first thread on this site.  Ron Edwards was emphasizing to me what a major issue handling time is in task resolution.  You'll want to think hard about how much you want charts to be part of your game, and how much you want calculation to be part of your game.  The main question is, what steps in your handling process are meaningful to having fun in your game?  My opinion is that addition is are preferable to multiplication.  The only reason you wish to throw out your logarithmic math is to simplify gameplay?  Why not just use a different die mechanic, which would simplify things even further than your example, and keep your logarithmic math?

-Jason T.

[wild_card2007]

Handling time is a bit of an issue, yes.  I struggle with it; there's a gamist/simulationist in me that enjoys dealing with things like weapon reach and fatigue.  But I don't want to get caught up in it, either... I want to be able to get through a gaming session and do more than a few "rounds" of combat.  I'll figure it out as I playtest I suppose.

I haven't a clue how to come up with a dice mechanic that implements what I've described.  How do I take a given curve (formula) and fit it to a mechanic?  Especially one that starts at 50% and has two log curves going towards 100% and 0%?

[J. Scott Timmerman]

Aloha Thomas,

The mechanic I posted about 4 posts back is exactly that, except that it starts at about 51% (at 3 dice) instead of 50%.  It follows your math perfectly.  I bolded it in big letters so it's easy to find.

But that's only one example.  If you don't like this idea, where higher dice pools mean lower chances of success, then I'm sure there's something else out there.

Other creators on the site have accused me of going a little too far in the simulationist department. I've tried to lean toward abstraction, trying my hardest not to annihilate the concept I'm working with (whether it be reach, fatigue, or something else).  My troubles in simplification mostly come from creating things that players aren't used to.

-Jason Timmerman

[wild_card2007]

Oh.  Right.  I'll have a re-look at that mechanic.  In the meantime I've thought of another.  I haven't done the math yet (sitting down and playing with probabilities after a couple glasses of wine just didn't seem the thing to do) but I like the feel.

Roll Nd10 where N is your skill level in the relevant skill.  Take the highest die roll.  Add or subtract the adjusted difficulty.  If the adjusted value is 6 or better you succeed.  (And when relevant, this also gives the degree of success.)  Here's what I like about it:

(1) At low skill levels your performance is generally poor and variance is high, whereas at high skill levels performance is high and variance is low.  This matches my own experience with skills I've learned.

(2) If the action you're attempting is really easy or really hard (difficulty modifier <=-5 or >=+5), there's no point in rolling: you either automatically succeed or fail.  I don't want players to have to roll for everything just on the wild chance they'll succeed or fail.

(3) I just like rolling more dice as your skill increases .  And the handling time here is minimal even if you have to throw a dozen dice.

Regarding goals, I'm trying to design a system that supports "heroic realism".  My characters are better-than-human but still operate within the laws of (in-game) physics.  I'm aiming for game mechanics that will support coarse-grained realism across a wide spectrum of skills, abilities, and scales.  From magic to cooking, rodent-sized to dragon-sized.  Hmm.  This-paragraph needs more-dashes.

[J. Scott Timmerman]

I haven't done the math yet (sitting down and playing with probabilities after a couple glasses of wine just didn't seem the thing to do) but I like the feel.

The aesthetics of a die mechanic can be just as important as getting the math right as well.  That's something I've had to bow to in the development of my own mechanics.

Saying that the target number is 6, but you adjust the highest die by +/-4 is the same as saying that you have a floating target number 2 to 10.  So if you just measure difficulties as being 2 to 10, with 6 the default, wouldn't it just be easier for the player to check to see if any of the dice came up the Target Number or higher?

Now, back to the math, for any given difficulty, each die reduces the chance of failure by a certain percentage.  Say the Difficulty is +0 (which would be Difficulty 6 in the alternate version I presented).  Then each additional die reduces the chance of failure by 50%.  That means at standard difficulty, your curve looks like 1d=50%, 2d=75%, 3d=87.5% etc.

For the difficulty -4 (Diff 10 in the alternate version I presented), the curve looks more like this: 1d=10%, 2d=19%, 3d=27.1% etc.

Basically, you can still think of the math as having a logarithmic/exponential nature:
Failure% = [50% - (Difficulty x 10%)] ^ (# of dice)

Or in the variant I offered:
Failure% = [(Difficulty - 1) x 10%] ^ (# of dice)

It doesn't fit your old mathematical design principle, but it seems to fit your new design principle of realism, when looked at from a couple angles.  For instance, no matter how many dice you have, you still have a chance at failure.  As the saying goes, "Even the monkey falls from the tree."

-Jason T.

[wild_card2007]

Sorry Jason, I didn't explain my thoughts as to how the difficulty modifier would affect the success rate (in a log fashion).  My bad.  And I can see now the mechanic I suggested last night doesn't have that characteristic to it (although in my defense I did say I hadn't worked out the math...).

My understanding from a few posts ago (where I made points 1..4) still stands in basic form.  To restate myself: The base chance to perform a skill under the "default" conditions is 50%, which in logarithmic scale terms is 0u.  Here's the key: A logarithmic change in a unit relevant to that skill increases/decreases the difficulty modifier by +1/-1.  (Originally I said affects the chance of success.  Which is true, but not completely accurate.)  So my archer wants to hit a target that's 1u farther away (1.25x), but also 1u larger (1.25x).  The disadvantage of increased range is countered by the larger size; so the two modifiers cancel out and his effective chance will be the same.  (It's been a long time since I've pulled a bow, but this makes sense to me.)  Or, my swordsman at Sword 2 is up against a swordsman at Sword 4.  All else being equal, he's at a penalty of -2.  (And that difference between their skill levels is 2u.)

This system gives me a modifier range of -4..+4 to arrive at a final TN of 2..10.  Anything outside that range is either an automatic success or automatic failure; you can't roll over 10 or under 1.  Chance of success for a given difficulty level increases in a log fashion with skill level.  As I see it, scaling units (distance, time, size, etc.) such that a logarithmic increase/decrease affects the modifier by 1 will give me the ability to handle a wide range of skill levels, situational modifiers, and creature ability modifiers.  Does this seem reasonable to you?  Can you see anything that seems Just Plain Broken here?  Drastically unrealistic, bad math, ...?  (In other words can you poke a hole in my thinking big enough to drive a truck through?)

Regards,
Thomas

[J. Scott Timmerman]

As I see it, scaling units (distance, time, size, etc.) such that a logarithmic increase/decrease affects the modifier by 1 will give me the ability to handle a wide range of skill levels, situational modifiers, and creature ability modifiers. 

Yes, that it would.

Does this seem reasonable to you?

Not only reasonable, it seems achievable.

Can you see anything that seems Just Plain Broken here? 

The only thing broken is the connection between your original math and the recent mechanic.  Either will work on its own.  They just simply aren't congruent with each other.  I, personally, can't think offhand of a way to make your math mesh with the mechanic you've presented, at least not without a complete revision of the mechanic.

Drastically unrealistic,

Certainly not unrealistic, at least not more than other dice mechanics.  Your mechanic has its own advantages, depending on how you envision bonuses and penalties.  You can explain, logically, how the math works.  It's a realistic philosophy to think that a certain amount of skill difference affects the Chance of Success of tasks of different difficulty similarly.  The bigger issue here is how you're defining your units, and how they fit into your mechanic. 

bad math, ...?

This is the problem.  Incongruent math.  If you want the exact equation for how this happens, since Chance of Success is just (100% - Chance of Failure), then you could rewrite my earlier equation on your mechanic like this:

(Chance of Success, Skill +1u) = 100% - {(100% - {Chance of Success, Skill +0u}) x (5 - Difficulty) x 10%}

which is not the same as:

(Chance of Success for Skill +1) = (Chance of Success for Skill +0) x (1.25)

which was your original design goal.  If that original mathematical design goal, along with your concept of units, is fundamental to your game, you may wish to revise your ideas about the mechanic.  Then again, if you like the mechanic, you may wish to revise your math and unit system.

(In other words can you poke a hole in my thinking big enough to drive a truck through?)

Just the disconnect between your math and the mechanic as presented.  Lastly, if I do think of a mechanic, where dice pools increase with skill level, that fits your math, I will let you know.  If the mechanic I've already offered you is not thematically appropriate for your game, I understand.

-Jason T.

[wild_card2007]

Jason, thanks for all your comments, explanations, and suggestions over the past week.  You've helped me a lot!  I'll respond to a couple comments in closing, and then I'm going to go try to break the system/mechanic I've arrived at.  I may be back....

The bigger issue here is how you're defining your units, and how they fit into your mechanic.

Yep, I agree.  At the moment I don't have a clear idea of exactly how I'm going to handle the actual scaling of units/measurements and the mechanics thereof, and exactly how that fits in with stats and skills.  Exploration of this will be a key element in deciding whether this system is going to work for me.

If that original mathematical design goal, along with your concept of units, is fundamental to your game, you may wish to revise your ideas about the mechanic.

My original formula was just an early attempt at creating a mechanic to handle logarithmic scales.  What's fundamental to my game is being able to handle little guys like kobolds, and big guys like dragons, with a mechanic that (1) doesn't break down horribly from a perspective of realism and (2) doesn't break down horribly from a perspective of mechanics or handling time.

Again, thanks a bunch.  And best of luck to you in your own efforts!

Regards,
Thomas