From: pmwg1641@uxa.cso.uiuc.edu (Peter M. White)
Subject: SRUN:  A Better Way to Roll the Dice
Summary: a smoother die rolling method
Organization: University of Illinois at Urbana
Distribution: usa
Date: Mon, 31 Aug 1992 19:51:32 GMT
Keywords: srun


Well, I heard a few grumblings about the dice probabilities, so I thought
that reposting these, they ran about 11 months ago, might be of interest
to some of you out there.


--Peter



______________________________________________________________________________
						|
     "Guards!  Let me draw their fire!"		|	p-white@uiuc.edu
		   --Scapegoat,_Marshal Law_	|
______________________________________________________________________________




SRUN:  A Better Way to Roll Dice


(If you want to cut to the chase, skip to The New Rule section.)

Before considering a change in die rolling, the first question to as
is:  What is wrong with the SR die rolling?

Actually, there is a lot going for it.  It is simple.  Fast.  And it
blends (almost) smoothly with the SR scheme of target number penalties
without resorting to various sided dice.  I find particular pleasure in 
shaking a pile of dice to prepare for an important roll.

Unfortunately,  SR die rolling is not perfect.  The essential failure is
probability "kink" at target numbers 5-6-7, with numerous annoying play
results.  This kink also severely aggravates rough spots in the rules.
(Especially apparent in the Matrix.)  I will elaborate with a single
simple example:  All Physical Adepts have strength 6.

Why?  In the important sense, people are born and not made.  Even PCs.
Characters possess physical and mental attributes which they have little
control over.  (I am not considering the miraculous ability of the GM
and players to manipulate the superiorly endowed.)  However, PAs
effectively _choose_ their physical attributes since they have the
ability to raise them.  Certainly, they have a plethora of excellent
powers to spend their Magic points on; increasing strength is only
one of them.  Most will be melee experts.  Therefore they _should_ 
choose to have a 6 strength.  Why?  They are foolish to only have a 5
strength, since with merely an expenditure of .5 Magic points they
could cut their opponents effective body in half.  (Think about the
probabilities of resisting power level 6 vs. power 5.)  7 strength is
silly, too.  Don't bother.  It is no more difficult to roll a 7 than a 6, 
unless injured.  The injured never live long around hostile PAs anyway.
8?  A body attribute is 80% as effective against a power level of 8
as power 6.  Really, there is no worthwhille advantage in raising 
strength above 6 unless the character goes as high as 9.  Strength 6
provides the optimal power level for the cost.  The principle
applies equally well to decking, magic and other combat power levels.

But is this a problem?  Definitely yes.  The underlying point is that
a penalty of +1 from a previous target number of 5 is a BIG DEAL.  Once
at target 6, +1, or even +2, is not at all a significant penalty.
The result is a lot of unnecessary game play distortions and aggravations.
The above example is but one of many instances.

Another way of looking at the problem:  +1 = +1  or  +1 = +2   or  +1 = +0.
All three are _effectively_ true depending on the base target number.  
(Consider target #s 4,5 & 6.)  This is what I intend to fix.  A +1
penalty, the smallest penalty the GM can impose, SHOULD ALWAYS HAVE
APPROXIMATELY THE SAME MEANING.


The New Rule:

I propose a new way of rolling the dice.  I call it the (6)5/4 algorithm.
A generalized algorithm for generating numbers from 1 to infinity, which
we can call algorithm (X)Y/Z, may be stated this way:

	(Take an X sided die and roll it.)
	If the number is Y or higher, add Z to the running total & repeat.
	Otherwise you are finished.

By this description, the SR standard is the (6)6/6 algorithm.

(We will only concern ourselves with six-sided dice and drop the (6).)

Consider the following chart:



			Difficulty		Marginal Proportional
Target Number		Level:			Difficulty Increase:

			6/6	5/4		6/6	5/4
-------------		---	---		---	---
	1		1	1		0.2	0.2
	2		1.2	1.2		0.25	0.25
	3		1.5	1.5		0.33	0.33
	4		2.0	2.0		0.5	0.5
	5	  -->	3.0	3.0		1.0	0.2
	6	  -->	6.0	3.6		0.0	0.25
	7	  -->	6.0	4.5		0.2	0.33
	8		7.2	6.0		0.25	0.5
	9		9.0	9.0		0.33	0.2
	10		12.0	10.8		0.5	0.25
	11		18.0	13.5		1.0	0.33
	12		36.0	18.0		0.0	0.5

"Difficulty" is defined as the # of dice needed to roll for average of
one success.  (Example:  if a want to average four successes at target #
3 with SR rolling, I will need 4*(1.5)=6 dice.)  Observe the kink at
5-6-7 for 6/6.

"Marginal Proportional Difficulty Increase" indicates what fractional
increase in dice is required to maintain the same average number of
successes for a +1 target # penalty.  Essentially, it is the degree
that +1 penalizes, where 0.0 is no penalty. (Calculated as
[Diff(n+1)/Diff(n) - 1.0].)

(If you graph out the above chart, you can really easily compare.)

The MPDI _should_ be roughly constant, the quantitative corollary to
+1 penalties always having approximately the same meaning.  The MPDI
illustrates quantitatively what is wrong with the 6/6 (SR) algorithm. 
Instead of being reasonably constant, it varies wildly between 0 and 1.0.  
Now look at the 5/4 algorithm results.  MPDI is between 0.2 and 0.5,
a variation less than a third as great as 6/6.  Notice no kink at
5-6-7.

The pros and cons are...

Advantages:
	
	i)  Greater uniformity of penalties.  A +1 targetting penalty
	is +1.  Not +1 = +2 or +0, as it sometimes does for 6/6 (SR).
	(This is the major improvement.)

	ii)  No obvious optimum power level (i.e. PAs will not all
	have strength 6.)  If you look at the chart, one sees the best
	probability payoff is at 5.  Successively higher power levels
	give modest, but useful, improvements.  PAs will typically have
	strength 5 or greater.


Disadvantages:

	i)  More die rolling required.  Really not much of an issue.
	Most rolls are against target numbers 2-5.  The number of
	times needed to roll a die is unaffected for target numbers
	2-5.  Die rolling for targets of 6 & 7, 8 & 9 increase a 
	whopping 17% & 14%, respectively.

	ii)  Changes effective target numbers and power levels.
	Sure, a little finageling may be in order, but, overall,
	the change is pretty small.  Most target numbers range
	from 2 to 10.  2-5 & 9 are completely unchanged in their
	effectiveness.  8 & 10 are changed little.  If you squint
	your eyes and look at the big picture, its the same minus
	an annoying kink.

In conclusion, I would say the advantages of 5/4 far outweigh the disadvantages
while solving a serious problem.  I personally believe in this new die
rolling algorithm 200% and give it the highest recommendation :)

Have fun with it.  And I'd love to hear what you think.


--Peter

P.S.  If you wanted perfectly constant +1 penalties, i.e. constant MPDI,
you could try a (6)2/1 algorithm.  It is, by some stretch of the imagination,
playable, although it requires *lots* of picking up and rerolling.

______________________________________________________________________________
						|
     "Guards!  Let me draw their fire!"		|	p-white@uiuc.edu
		   --Scapegoat,_Marshal Law_	|
______________________________________________________________________________
Newsgroups: rec.games.frp
Subject: Re: SRUN: A Better Way to Roll the Dice
Keywords: SRUN, shadowrun

rgt@kepler.unh.edu (Richard G Tomasso) writes:

>Nice work. Have you tried using the (X)Y/Z form on (6)6/5? I believe that
>would model the system used by some groups that rolls after the first act
>as 1d6-1, so target #'s of 7 actually mean something.

Indeed!  I should have mentioned that 6/5 is the first solution I considered.
Once I had thought up 5/4, I decided 6/5 was less desirable because:

	1)  It only partially fixes the probability kink.  Although I did
	not say so explicitly, the kink has two features:  6 = 7, and
	a punishing jump from 5 to 6.  6/5 solves the first, but not the
	second.  It is preferable, I believe, for a +1 penalty to "always
	have approximately the same meaning".  Which implies not having
	wild MPDI swings.  6/5 allows variations from 0.2 to 1.0, still
	very large.  A MPDI of 1.0 is brutal: consider what happens
	to deckers in Red-5 nodes after they take their first wound, for
	instance.  (This is the main shortcoming.)

	2)  How 6/5 alters probabilites.  The higher target numbers are
	more difficult to roll.  Personally, I find this pretty detrimental;
	if anything, I prefer to see the difficulties for the higher numbers
	to drop down a bit.  5/4 changes the _overall_ probabilites for
	2 through 10 surprisingly little compared to 6/6.  6/5 is actually
	quite vicious above 10 compared to both 6/6 & 5/4.

	3)  The difference in volume of die rolling is negligible.  The
	vasty majority of dice are rolled vs. target numbers 2-5 where there
	is no change.  The overall change in dies rolling is going to be
	no more than a couple percent, tops.  Really, this is a neutral
	point between 6/6, 6/5 and 5/4.

Glad you liked the posting :)


--Peter

______________________________________________________________________________
						|
     "Guards!  Let me draw their fire!"		|	p-white@uiuc.edu
		   --Scapegoat,_Marshal Law_	|
______________________________________________________________________________