So?

After “what?”, I suppose the next nagging question to address is “why?”. Why am I doing all of this mathematics on a board game like Chutes and Ladders? Can any of this be applied to other scenarios?

            

Well, in truth, the results of this investigation will be mostly self-contained. The rather specific configuration of the Chutes and Ladders game board makes its “mathematical model” unique and largely inapplicable to other events. While the results are very specific, the process used to obtain these results can be more easily generalized. Part of my task this week was to explore some other Markov processes which might be “modeled” in a similar fashion to Chutes and Ladders. Specifically, I was looking for a class of Markov Chains called “absorbing Markov Chains”. An absorbing Markov Chain consists of one or more states from which you cannot transfer into another state; such a state is called an “absorbing state”. Chutes and Ladders has one absorbing state – the very last square – and may therefore be classified as an absorbing Markov Chain.

            

One of the more interesting examples of a “real-life” absorbing Markov Chain I could find (courtesy of Mrs. Bailey) was the process of a fire spreading about in a forest. In this somewhat simplified model, a fire on one tree has a certain probability of leaping onto a neighboring tree based on their proximity. This process continues until all of the trees in the forest are consumed or the fire goes out on its own. We may consider this an absorbing Markov Chain, as all the states that can exist after the fire has gone out represent the absorbing states (clearly if there is no fire, no trees can burn).

A "forest" in the process of being burned. See citation for more information on this applet.

My hope is that some of the techniques used to model Chutes and Ladders can be helpful for modeling other   Markov Chains. I'm going to be trying as much as possible to consider the "bigger picture" when working on this project and will focus more on approaches that can be broadly applied to other other Markov Chains. Perhaps if I exhaust Chutes and Ladders, I can work on other models by recycling some of the techniques I already used.

* Wilensky, U. (1997). NetLogo Fire model.http://ccl.northwestern.edu/netlogo/models/Fire. Center for Connected Learning and Computer-Based Modeling, Northwestern University, Evanston, IL.
* Wilensky, U. (1999). NetLogo. http://ccl.northwestern.edu/netlogo/. Center for Connected Learning and Computer-Based Modeling, Northwestern University, Evanston, IL.