Interesting and possibly unique probability mechanic

[ffilz]

Chris Weeks has suggested I post this here to offer it up for thought.

A friend in college had an interesting fortune mechanic. He came up with a simple method to use the normal distribution. Gamers have long used multiple dice mechanics to generate bell curves and open ended results. Why not use the real thing?

What my friend did was instead of using the dice to generate the end result, he used them to generate the probability. What is the difference? Well, if you are using a flat percentile probability (d100), then they are the same. If you use a flat 1d6, then the probability of a 6 or higher is 16.667%, a 5 or higher is 33.33% etc.

My friend produced a chart that converted the probability into a modifier for use in resolution (so for example, if an attacker had an offensive value of 20 and the defender had a defensive value of 24, the attacker would need a +4 or better to hit. The particular chart my friend created was derrived from the normal distribution, setting the mean to a modifier of 0, and setting it up such that one standard deviation was equivalent to a +6.6667 modifier.

To provide an example, that isn't too big, I'll provide a chart where +1 is one standard deviation:

Code Select Expand

result probability  fraction
-3       0.14%       .0014
-2       2.28%       .0228
-1      15.88%       .1588
0       50.00%       .5000
+1      84.13%       .8413
+2      97.72%       .9772
+3      99.86%       .9986

Then the way use the table to generate results is pretty simple. You use d10s to generate the digits of the fraction. My friend set up his chart so you would have two "significant" digits of the fraction, where a leading string of 0s or 9s are insignificant digits (so .000000853, a .506, and a .9999712 all have 3 significant digits).

I think the result is really cool. No weird step functions with open ended die systems where you re-roll and add if you get the maximum result.

The table is daunting at first glance (we had tables that ranged from -40 to +40 though his original table ranged from -25 to +25), however, it turns out that in most situations, you have less than a 50% chance of success, so you just ignore a low roll (unless it's really low, in which case you might have fumbled). Above 50%, most of the rolls are within a standard deviation or so, which means that if you memorize 6-8 numbers off the chart, you hardly ever have to look at the chart.

The primary use of the open endedness was in determining the effects of critical hits. If an attack succeeded by 7 or more, double damage was scored. For each additional +2, an additional multiple of damage was scored.

I haven't seen a similar mechanic, and perhaps it's simply too much work for the benefit, but to a geeky engineer, it was a pretty cool concept. I'm not sure what tools my friend used to generate his original table, but a siimple Excel spreadsheet will let one play with the tables.

So, I don't know if this is of any interest to anyone, but I offer the idea up for discussion.

Frank

[rafial]

Then the way use the table to generate results is pretty simple. You use d10s to generate the digits of the fraction. My friend set up his chart so you would have two "significant" digits of the fraction, where a leading string of 0s or 9s are insignificant digits (so .000000853, a .506, and a .9999712 all have 3 significant digits).

Could you explain the actual die mechanic a little more?  I'm imagining what you did is roll one d10, look up the most significant digit, and if that was ambiguous, roll another d10, and so on?  And I'm not sure what the singificance of the "significant" vs. "insignificant" digits are.

The notion of have the die roll generate the modifier is similar to Fudge (where you generate a bell curved -4 to +4) and Feng Shui (where you roll d6 - d6 and use the result as a modifier).  But Fudge is not open ended like the mechanic you describe, and Feng Shui uses rolls and adds (or subtracts) creating the step function issue you allude to.

Neat.

[ffilz]

The way we normally make the rolls is to roll two d10 (basically d100). That helps people with the familiarity. Then, if you roll a 90-98, you need to roll one more die. If you roll a 99, you need to roll two more dice (and perhaps more if you roll a 90-99 on those dice). I should have noted that a 00 is read as 00, not 100. If you roll an 01-09, you roll an additional die also, and if you roll an 00, you roll two additional dice (and again, may need to roll even more if you roll an 00-09).

Here's some numbers from the top of the chart to help:

Code Select Expand


+9   .912
+10 .933
+11 .951
+12 .964
+13 .975
+14 .982
+15 .988
+16 .9918
+17 .9946
+18 .9965
+19 .9978
+20 .9987
+25 .999912
+30 .9999966
+35 .999999924
+40 .99999999901

So if you roll a 99, you roll again, that comes up 99 again, so you roll again, and get a 12. Your final result is .999912 or +25.

It is different from Fudge in that if you have actual Fudge dice, you can read the modifier directly off the dice (or you can roll 4d3-8 and get the same modifiers).

I just came up with my meaning of significant digit on the spot. It doesn't have quite the same meaning of significant digit in science, well, it does if you look at the values near 0 instead of numbers near 1, the number .000000012 has 2 significant digits in scientific terms. The harderst part always seems to be explaining when you roll extra digits. The problem is that we aren't really used to reading the dice as digits (even though that is in fact what we do when we roll a pair of d10s as percentile dice).

Frank

[M. J. Young]

So you're not saying that the zeroes and nines are insignificant, but rather that if you roll them as leading digits you have to keep rolling until you get two digits that are neither.

It's kind of the same as saying that you can roll the digits one at a time (in this case two at a time) until you've hit one that produces a unique return. Thus on your +9 to +40 chart, if I roll 97 I'm really done, because that's a unique result (+13), but if I roll 98 I need another die to distinguish 982 from 988.

What's not clear is what happens if I roll 9850. Since it was done by engineers, I'd wager that I round up at half (since somewhere beyond that final digit I'd eventually break to "more than half") but if that was stated in the description I missed it.

--M. J. Young

[Vaxalon]

Either that, or .985 doesn't make the .988 mark, so it gives you +14.

There seems to be extra handling time in this system; in order to resolve a check, you have to 1> Roll a die, 2> Check a table, 3> Interpret the result, and that's if the first roll wasn't ambiguous.

In a normal system, you would 1> Roll a die, 2> interpret the result.

Now this may not seem like much, but over and over in a dice-heavy session it adds up.  I would only recommend using this mechanic for systems where the dice are not intended to be rolled a great deal.

[Christopher Weeks]

I agree on the time thing, but also note that Frank said that you quickly come to remember the common threshold numbers, which makes sense since they're the same for every roll of every kind across the game.  And aren't there lots of games where you compare your roll to a table for interpretation?

On my first thinking about it I was with Vaxalon (rather than M. J.) on resolving which tier a roll came in at.  But on reflection, doesn't that system arbitrarily favor lower results?  (i.e. your chance of getting a -1 would be 34.12% while your chance of getting a +1 would be 13.59%)

So it seems like the "right" thing to do is calculate the half-way point between the two numbers and see how your number compared to that.  But that sounds awful.  Am I missing something?  "OK, I rolled a 9982, is that +19 or +20?  Let's see, 9987-9978 is 9 so the midpoint is 99825..."  Do you then have to roll another digit to see if it's over or below the five?  That can't be right.  What's the deal, Frank?  I don't recall if we discussed this bit in the past.

Chris

[ffilz]

Ok, some clarifications. If you roll a .985, that's a +14, you need to roll at least the fraction on the table. It does mean the table isn't exactly symetrical even though it looks symetrical (since a .99999999901 is a +40, while a .00000000098 is a -41). But it also means that when attack equals defence, there is a 50-50 chance of success (whereas Fudge has nearly a 70% chance of success in such situations - except that in Fudge, on an opposed roll you need to EXCEED your target).

The reason I mention the bit about significant digits (which as was mentioned basically means roll two more digits after a leading string of 0s or 9s) is so that you just make the roll, AND THEN do the lookup. If you roll a 9x you probably need another digit, so just roll it. Of course once you remember that a .94 is a +10 and a .92 is a +9, you might start to skip rolling the extra digit (but it probably generally saves handling time to not think about it and just roll).

You could write a chart which was more like Fudge where +0 filled the center (for example .44 to .55, with +1 being .56 to .61 and -1 being .38 to .43 [using the numbers from the +6.667 per standard deviation instead of the +1 per standard deviation]).

The system definitely does have a higher handling time than other systems, but it does actually run pretty quick. The main thing that slows down combat is that you usually need double damage or better to actually effectively injure someone. That could be fixed by tuning the system (increasing the chance of a crit, or increasing base weapon damage). The system also has a handling time issue in that a good crit requires rolling a fistfull of dice, but then you don't do that very often (and if you do, the number of rounds of combat will be small). The fistfulls of dice do have a good thing though. In D&D, the Wizard with his Fireball is the only one who gets to roll fistfulls of damage. Fighters almost never do.

But I didn't want to get too much into the rest of the system (unless folks want to - with the understanding that it's not my system, though it's close to an indie system [never distributed for money, but used by more than just the author and his own gaming group - I don't think I've ever played in the author's game]).

One final thought I have on time to resolve combat. My experiece is that once you start using a battle mat and miniatures, that handling the map dominates the time to resolve combat. Of course a system where you had to make a roll, lookup on a table, make another roll, lookup yet again, and then roll three more times and record 5 different types of damage would still be dominated by mechanics handling.

Frank

[Walt Freitag]

Yeah, Christopher is right. The probabilities are asymmetrical if you read the result as always the next higher probability listed than the die roll, or as always the next lower probability listed than the die roll.

If you read it as next lower when you roll above 50.00, and as next higher when you roll below 50.00, then it's symmetrical (equal chance of getting e.g. either +1 or -1) but you'd get 0 as the result 68.25% of the time. It's also symmetrical if you read it as next higher when you roll above 50.00 and next lower when you roll below 50.00, but then you'd have little or no chance of a 0 result. (Little chance, if you counted an initial roll of 50 as a 0 result; no chance, if you assume indefinite reroll of additional digits after an initial roll of 50, inevitably producing an "above 50.00" result once something other than 00 came up in the roll.)

Using the means between the table entries instead, so that you'd be reading (on the current table) the "closest" rather than the "next highest" or "next lowest" entry to the die roll, would make it symmetrical. But that would also skew the normal distribution just a little (for instance, the 0 outcome range would end up a bit wider than it should be). Best bet would be to recalculate the table plugging values/bins of ... -2.5, -1.5, -.5, .5, 1.5, 2.5... instead of ...-2, -1, 0, 1, 2... into the normal distribution function.

- Walt

EDIT: Cross-posted with Frank. True, the asymmetry doesn't matter if all rolls are opposed by other rolls from the same table, or against target number that were set taking the asymmetry into account. It's an issue only if you're reading the result from the table in a "how was my performance, compared with my average" sort of way.

[Lee Short]

I actually did this many years ago.  It worked pretty well with my then-current crop of players.  We were all at Harvey Mudd College, where you've got to be pretty good at math and such to get in, much less survive.  I made games around "Gaussian percentiles" and "Gaussian d20s".  

In practice, it was really pretty easy to use the charts.  There were some really easy heuristics that would get you more or less the right value 90% of the time, and you only had to look at the chart 10% of the time.  They went something like this:  
 --  if you roll between 25 and 75, move the result 3 points closer to the middle (50)
 --  if you roll 75-85, or 15-25, use the result
 --  if you roll over 85 or under 15, you probably know if you've succeeded without even looking at the chart
 -- if none of the cases above handle it, use the chart

You can see how this would easily adapt to d20s as well.

[ffilz]

The symetry issues are complex. If you want to have a +0 result (or center result, for example, 2d6 has 7 as a center result, 3d6 doesn't have a single center result), you either don't have a way to produce a 50-50 probability, or you have an asymetrical distribution.

I think the key to the way the mechanic was used in my friends system is that he was focusing on the probabilities, and then mapping the probabilities into something useful. Fudge focuses on the result values, and then fiddles with the number of dice and the use of the values to make use of the probabilities.

When designing a mechanic for a game, I think you need to understand the geeky details of the mechanic, but then you need to step away from those details and decide how the mechanic feels. Fudge dice certainly have a good feel even though there is no way to generate a 50-50 probability. This normal distribution chart has a good feel even though it's asymetrical, in part because it's easy to understand that you have a 50-50 chance of success if your offense equals the defense.

Of course what's cool is that this particular mechanic can be mapped to produce either type of curve (symetrical probabilities, or 50-50 split).

Frank