Sorcerer Dice Mechanic, Examined

[Paiku]

Let's say you're Binding an enormous demon.  You have 7 dice for the Binding roll, and the hulking fiend has 15.  We're using d10s.  What would you say is the most likely outcome of this roll?  Obviously the Demon is favoured, but what is the expected value of the Binding?  In other words, if we did this roll 1000 times, what would be the most common outcome?  How about the average out come?  +8 in favour of the Demon?

I did an informal test with two handfuls of dice and about two dozen trials.  To my surprise, the most common outcome by a long shot was: +1 in favour of the demon.  There were a couple of +2's, a +7, and a few Successes for the sorcerer.  Really not what I was expecting!  So I modeled this roll in a spreadsheet, copied it 1000 times, and took some statistics. 

Here is the probability distribution of 15 d10s vs 7 d10s :



Mode (most common outcome): 1 Victory for the Demon
Mean (average outcome):           1.38 in favour of the Demon

65% of the time : Demon Success
18% of the time : Sorcerer Success
(why don't these add up to 100%: see "Errors, Limitations" below)

58.1% of the time : Binding roll is 1 or 2, for either party
36.0% of the time : Binding roll is 1 for either party

24.8% of the time : Binding roll is 3 or greater, for either party
13.4% of the time : Binding roll is 4 or greater, for either party

Errors, Limitations:
17% of the results are "Zero."  This is a limitation of my simple spreadsheet.  It doesn't know what to do when both characters have the same number of highest results, e.g. both chrs roll two 10's.  I can't think of a quick way to sort out these cases automatically, and I don't want to spend all night on this.

Conclusions:
This tells me that, in general, most opposed rolls in Sorcerer are going to end up with a low number of victories, just 1 or 2 - no matter how asymmetric the dice pools are.  Highly asymmetric Bindings (4 or greater Victories) are expected to be rare, even with enormous demons (Will 15!).

This was significant because I'm working on developing some rules specifically for Demon Lords (a-la Elric/S&S).  I had an idea that depended on highly asymmetric binding rolls, but now I see that I can't count on these occurring.  Hmmm.

Was anyone else surprised by this?  Does it seem "right" that a highly unbalanced Binding roll should end up with just 1 or 2 victories most of the time?

-John

[jburneko]

I've never seen the analysis but I've known for a long time that there was a diminishing return for adding more dice.

It keeps uncertainty high in the game which is extremely frustrating for some players.  Sudden reversals and unexpected outcomes against seemingly incredible odds happen all the time.  I love the volatile nature of the Sorcerer die mechanic.

As a side note you'll notice that this is almost the complete opposite of many other "story oriented" designs.  Many, many games which purport to focus on "story" in play include lots and lots mechanics that enable more predictable and directly controlled outcomes presumably so the group can ensure that they get the outcome that's "best" for "the" story.

This is a design feature totally absent in ALL of Ron's designs.  There are very few "escalation" or "pacing" mechanism in any of Ron's games.  Every conflict is statistically equivalent to every other conflict.  Which is counter-intuitive from a "story" perspective.  It flys in the face of the most frequent question I've seen asked about "story" focused design: "How do you get the characters to fail in the beginning and triumph in the end."

I consider this cold and uncaring approach to conflict resolution a unique feature of Ron's work.

That was kind of a long aside.

Jesse

[Eero Tuovinen]

Was anyone else surprised by this?  Does it seem "right" that a highly unbalanced Binding roll should end up with just 1 or 2 victories most of the time?

It's not surprising - what you're witnessing is not so much that unbalanced rolls have low margins, but that rolls with large pools have low margins. This effect is caused by the way victories are calculated: the higher the highest loser's die is, the greater proportion of the winner's dice pool is disqualified from counting as victories. For example, if the loser only has one die in his pool, then the winner's margin will be one half of his dice pool on average - the loser's result being 5.5 on average, and half of winner's dice being higher than that. However, adding just one die to the loser's pool ups his average result to what - a bit over seven if I remember correctly; this causes the winner to only count one third of his dice as victories on average, as opposed to the half he got when the loser just had one die. It takes four or five dice to get your expected best die to the 9-10 range if I remember correctly - there's a diminishing return in this, although the cancellation of tied dice ensures that extra dice are always useful even if they don't actually improve your highest die.

Take that principle up into high ranges like you did, and the result will asymptotically approach the situation where the loser's best die after cancellations is 9, while the winner's best dice will be tens, and only tens will contribute to the margin of victory; therefore at arbitrarily high pools the margin of victory will be 10% of the winner's pool minus 10% of the loser's pool (to account for the cancelled dice) on average. This won't be a clean result on normal numbers of dice, but even 7 vs. 15 like you rolled already causes the phenomenon to appear strongly. Roll a thousand dice vs. a hundred dice, and the expected margin will still be merely 90 or so - the thousand dice will score a hundred 10s, while the hundred dice will score ten 10s that deduct from the winner's margin; the hundred dice will almost certainly also include a nine, which makes all the other dice in the winning pool irrelevant. In general, getting a +1 to margin with high dice pools requires a 10 dice difference in the size of the pools due to how only one in ten dice on average will give the dominant pool that necessary '10' that actually matters in improving margin.

(Note that the above examination implies that using smaller dice would increase the margins on uneven checks - d4s with high pools would give 25% of the high pool minus 25% of the low pool as expected margin; with thousand vs. hundred dice that would be +225 instead of the +90 you'd get with d10s, and improving the margin would take only four dice in difference for a +1 to margin. Interesting, that. Even more interesting is that larger dice actually also increase the margins at mid-size pools due to how much more absolute difference in expected results you get per die, and how much larger the pool needs to be for maximum results to be obtained; for example, with d20s five dice won't be enough to cap the roll into the 19-20 range the same way five d10s likely give you a 9-10, and thus a larger percentage of the winner's dice will become victory margin with larger dice. This effect disappears with "large" pools no matter the dice size unless you use d100s or such, though, so that first phenomenon will be the more importance once you're manipulating 10+ dice per pool.)

Examining unbalanced dice pools where the lower pool is small gives very different results. For example, reducing your 7 vs. 15 to the equal difference of 1 vs. 8 gives an expected margin of +4 due to how the expected losing result is no longer a 9, but rather just 5.5. I won't comment on whether this feels "right", but it does mean that when penalties start accruing, more drastic outcomes become more likely. Perhaps this would help you with your design - cause certain conditions to cancel dice from the opposing pools to increase the odds in the favour of the dominant pool while also improving the expected margin.

It's also interesting to look at how Sorcerer approaches very small dice pools. The canonical rule is that a "-3" dice pool is actually just a one die vs. a +3 to the opponent's dice. This means that very large pool penalties first drop the loser's expected result to that 5.5 average and then start adding dice to the opposing pool, which therefore gains a half point of expected margin per added die. A simple variant mechanic for the Sorcerer dice, one that I've used when I've stolen the dice mechanics, is to roll a "-3" pool as "three dice, pick lowest" instead of giving the extras to the opponent. This solution is somewhat more drastic than the canonical mechanic with significant pool differences due to how it can drag the expected loser outcome below 5.5 - extreme negative pools would ultimately cause the winner's margin to be 90% of his own pool size instead of the 50% + half of the loser's negative pool given by the canonical handling of negative pools. If you want larger margins, perhaps that's something to consider, especially combined with some player options for manipulating penalties to drag the pool sizes to the sweet small range.

In general I like the Sorcerer dice mechanics a lot, it's very beautiful mathematically. The game itself benefits in subtle ways from the math, and I understand that Ron chose the mechanic organically instead of deriving it mathematically, so it's a happy accident of design in many ways.

[Paiku]

Thanks Eero, fascinating stuff!  Always loved stats and probability, I wish my undergrad hadn't been so long ago.  Thanks for the analytical solution which had evaded me.

Rather than try to tweak Sorcerer's fundamental dice mechanic, I'll try to think up another way to address the situation (I hesitate to call it a "problem" just yet...).  When I'm ready to ask for suggestions I'll start a new thread specifically for that.  Soon.

By the way, in my numerical solution, I've figured out a way to handle the "zero" results.  Canceling out the matching highest dice on each side and considering the remaining dice is simple to do with real dice, but a tricky thing to make a spreadsheet do!  The solution (I think) is to assume that the zero results, if the matching highest dice could be removed, would have approximately the same probability distribution as the whole set.  As you pointed out Eero, this becomes less true as the dice pools become smaller than the number of die faces, but the error would be small.  One could recursively replace the zero results with a normalized PDF until the zeroes become vanishingly small, but the simple way to accomplish the same thing is to simply remove the zero results, and re-normalize the rest of the PDF.



1.24   Mean
1   Median

22.49%   Sorcerer Successes
77.51%   Demon Successes

52.57%   Binding >=2
27.51%   Binding >=3
16.26%   Binding >=4

(this won't exactly match up with the previous results, as the random component was recalculated as well).

This doesn't change our previous observations much.  The character with the larger dice pool is likely to win, but is not likely to score a large number of victories.

As for the effect of all this on the game in general: I agree Jesse, it keeps things lively and makes for interesting narratives!  Sorcery is truly dangerous, no matter how prepared one is.

Cheers,
-John

[Ron Edwards]

The long right-hand tail is very real. I recommend thinking in terms of playing through many, many rolls during the course of a session. The rolls in which you have more dice will display that tail in that context, especially in contrast with the rolls for which you do not. This is a key feature of play in which system is wholly integrated with the fiction.

Therefore it is not correct to think of Sorcerer dice as being only a small variant away from an essentially 50-50 system. The long tail does show up as a way to distinguish one character with more dice in X from another with less dice in X. Nor is it correct to think of more dice on the table as being irrelevant. It would be useful for you to run the same exercise for 10 vs. 5 dice (or 11 vs. 5 if you want to preserve your 15/7 asymmetry) and for 4 vs. 2 (or 5 vs. 2, ditto). This will show the different length of the tail in each case, obviously, and I'm a little curious about what else it might show, if anything.

I fully endorse the comments in this thread which praise the uncertainty in the fiction, even if one has much higher dice than one's opponent. My points above are there to stress that this uncertainty does not mean that, over time, the dice system is merely fully uncertain and the numbers of dice rolled are not important.

I'm quite pleased to see that the including or not including the ties does not change the probabilities (barring the obvious and trivial loss of the 0 result). It would be nice to see an exercise in which, applying the rules for ties correctly, the tied scores are converted into their final results, as follows:

A: 9, 9, 5, 4, 2
B: 9, 8, 1, 1, 1

= A wins with one victory

Best, Ron

[Paiku]

Thanks everyone for your comments.

Ron, that the ties would have the same distribution of results as the set of all rolls was an assumption of mine, a simplification really.  I haven't proven it.  Come to think of it, the distribution would have the same shape, but the tails would be shorter.  Oh well, like I said, an approximation.  I'd like to see the ties resolved properly too, but I'm approaching the limits of my analytical probability and numerical analysis chops!  Eero?  Anyone else?  If anyone wants my spreadsheet to work with, let me know and I'll post it.

Here are the results of the simulation run again for 11 vs 5 dice and 5 vs 2 dice.

11 vs 5 dice:

1.25 Mean
24.38%   Sorcerer Successes
75.62%   Demon Successes
16.71%   Binding >=4
27.97%   Binding >=3
53.59%   Binding >=2



5 vs 2 dice:

1.18 Mean
24.72%   Sorcerer Successes
75.28%   Demon Successes
10.93%   Binding >=4
25.97%   Binding >=3
49.54%   Binding >=2



I should say that 1000 rolls isn't really enough.  The results above change significantly every time I re-calculate.  I'd extend it to 10,000 rolls, but I need to sleep...

Oh, and I've started a separate thread to discuss the problem that led me to examine large asymmetric Binding rolls in the first place...  Elric, Demon Lords and Sorcery. Modeling it all.

Cheers,
-John