From: daveb@hpgrla.gr.hp.com (Dave Boyd)
Newsgroups: sci.math
Subject: More Chutes and Ladders
Date: 21 Jan 92 00:29:11 GMT

     This is regarding the discussion of a few days ago about the game
Chutes and Ladders.  I've had a little more time to tinker with this
problem, and have some more results.

     Plotting the results of my simulations showed that the game length
distributions looked suspiciously like graphs of the gamma distribution
in my old statistics book.  Doing some curve fits showed that the game
length is indeed very close to being gamma-distributed (with an offset
because no game can be less than 7 turns).  The form of the gamma
distribution I used is:


   probability of (game length - 6) = 

                        1
             -------------------------- * x^(alpha-1) * exp(-x/beta)
            (beta^alpha * gamma(alpha))

   where gamma is the gamma function, and exp is the exponential function.


     Using a simplex curve fitting algorithm, the best fit alpha and beta
for different numbers of players are found to be:


         Number of players            alpha           beta
       ------------------------------------------------------------

                 1                    2.124          14.53
                 2                    2.542           7.768
                 3                    2.829           5.486
                 4                    2.990           4.455
                10                    3.467           2.402


     For example, the formula above predicts that the probability of a
3-player game lasting exactly 23 turns is 0.03787.  In fact, in my
simulation of 100,000 games, 3826 of them lasted exactly 23 turns.  For 1
player and 50 turns, the formula predicts a probability of 0.01093 and the
simulation shows 0.01028.  For 4 players and 18 turns, the formula predicts
0.05504 and the simulation shows 0.05541.

     The  curve fit for 1 player is fairly good (except in the region around
100 turns), while the fits for 2, 3 and 4 players are progressively better.
The fit for 4 players is amazingly good.  For 10 players, the fit starts
to degrade slightly again.

     I'm sure it's possible to derive approximations for the other
distributions from the one-player distribution, but I don't know how to
to do it.  I suspect it involves a convolution.  I also suspect the
answer wouldn't be as accurate as my simulations because of end-of-game
effects, but that's only speculation.

     Now, if this information were only good for something....

  Dave Boyd
  A humble math hobbyist
  Hewlett-Packard, Greeley, Colorado
  Standard Disclaimers Apply