Conference rusure::math

301.0. "Simulation Problem" by NOVA::RAVAN () Tue Jun 04 1985 19:05

I have a problem in probability I would like some help on.  Let me preface my
description of the problem with a caveat:  I am not trained as a mathematician,
so if this description sounds like it's coming from a naive novice, it is. 

I'm a game master in a role-playing game and I am trying to model something
near and dear to everyone that plays these types of games - the fact that a
player will roll a sequence of numbers from 1-100 on percentile dice and say
something like "yes, I know I rolled a 13 there in the middle, but right before
it I rolled a 68 and right after it I rolled a 74, so that bad 13 roll doesn't
effect the overall outcome as much as it would otherwise." 

Well in my specific case, my players are always trying to learn new things
(e.g., spells, skills), trying to adapt to new situations (e.g., interacting
with one of my characters, like a shop keeper), or trying to exercise some
skill over a long period of time (e.g., trying to build or make something, like
a boat or a house).  In order to do this, I have them roll a sequence of
numbers from 1-100, then I take a look at the sequence and intuit how well the
task at hand has been accomplished.  I would like to eliminate this intuition
and try replacing it with a simulation program. 

My current model is this:

Consider a sphere of mass m in a frictionless environment.  This sphere can be
accelerated with probability p and decelerated with probability 1-p. I call
this probability the 'breakpoint' of the situation.  Any roll above p
accelerates, any roll below p decelerates the ball.  The force applied at each
point is directly proportional to the distance between the roll r and the
breakpoint.  Further, after every roll the mass of the ball increases by some
percentage I call the 'mass accretion constant', MA. After n rolls, the sphere
will have some velocity v which is a measure of the goodness of the sequence of
rolls.  A large velocity means that the player rolled well in general, a small
positive or negative velocity means the player rolled poorly.  The mass
accretion constant MA attempts to model the fact that 'initial impressions are
lasting' or 'in a learning situation, habits are hard to break, whether good or
bad'. 

There are two other variables, PK and NK, by which the difference (r-p) is
multiplied before the force is applied to the mass.  This allows me to model
situations where bad rolls (rolls less than p) may have a lesser or greater
effect than good rolls (rolls greater than p). 

My problem is this: "How do I measure how good a final velocity v is?" I came
up with a brute force bucket counting program which runs the algorithm with one
setting of p (.5), NP, PK (1), and MA (.05, 5 percent change in mass after
every roll). n in these tests was 13.  One fellow's rolls causes the ball to
achieve a final velocity of 88, while another causes the ball to accelerate to
241.  I think this means (given the output of the bucket counting program) that
88 puts player 1 in the 69th percentile, while 241 puts player 2 in the 99th
percentile.  What I need is a general way of computing "For any breakpoint p,
mass accretion constant MA, and values PK and NK, what is the probability
distribution of final velocities after n rolls?" 

Other questions I have about the function are:

    Does the number of rolls (n), affect the final output distribution?

    What would happen if I added another constant on applied force, I,
    which would vary directly as a result of the player's intelligence?
    Thus, the force applied by every roll of a more intelligent player
    would be larger than the force applied by every roll of a less
    intelligent player.

If this matters, the ranges for the variables are:

    p	0-1
    MA	-infinity to +infinity (usually 0-1)
    PK	-infinity to +infinity (usually 0-1)
    NK	-infinity to +infinity (usually 0-1)
    r	1-100

Thanks in advance for any help and vast apologies for ineptness of description,
-jim