## *
How fair is ** Monopoly*?

The game of * Monopoly * is one in which chance plays an
important role and so, an analysis of the game using tools from
probability is likely to provide some information on the nature of the
game. Such an analysis is carried out in
an article by Ian Stewart
on page 104 of the April 1996 issue of Scientific American .
The general question discussed is whether or not the game is fair in the sense that
each square is equally likely to be landed upon, in the long run. The
April article shows that the game is fair but,
a subsequent article by Ian Stewart
on page 116 of the October 1996 issue provides a more detailed
analysis showing that there might be a slight deviation from fairness.
Using the ideas
discussed in these two articles, calculate the probability, for each
square, that it will be landed upon during the course of a long game.
In your analysis you may ignore the rules of the game pertaining to
rolls of doubles; in other words, each players moves only once each
turn, regardless of whether or not doubles have been thrown. However,
your analysis should take into account the possibility of landing on a
"Chance" or "Community Chest" square. In such cases, the player is
required to draw a card from a shuffled deck and follow the
instructions on the card. In some cases these cards instruct the
player to move to some other square. Here are the relevant data on
these cards.
There are 16 "Community Chest" cards and, of these, 2 instruct the
player to move to another square:

- There is 1 card instructing the player to move to "Go".
- There is 1 card instructing the player to move to "Jail".

There are 16 "Chance" cards and, of these, 10 instruct the
player to move to another square:

- There is 1 card instructing the player to move to "Go".
- There is 1 card instructing the player to move to "Jail".
- There is 1 card instructing the player to move to "St. Charles".
- There is 1 card instructing the player to move to "Illinois".
- There is 1 card instructing the player to advance to the nearest
utility.
- There are 2 cards instructing the player to advance to the nearest railroad.
- There is 1 card instructing the player to move to "Reading Railroad".
- There is 1 card instructing the player to move to "Boardwalk".
- There is 1 card instructing the player to move back 3 squares.

For those who are truly ambitious, a second
follow-up article provides some ideas on how to create an even
more realistic analysis of the game.

In doing this this assignment, keep in mind that there there is an
intelligent way of creating the 1600 entries of the transition matrix
and a not so intelligent way of doing this. By no means should you try
to do this by hand! Instead, notice that the probabilities in the
transition matrix come from various sources: a roll of the dice, a
"Chance" card, a "Community Chest" card and so on. Each of these has a
corresponding transition matrix which can be quite easily created
using Maple's built in functions such as "seq",
"piecewise", etc. Afterwords, the individual matrices can be combined using
matrix operations. (The commands "mulcol" and "addcol" from Maple's
"linalg" package should prove to be useful in this context.) In creating your transition matrix you should be
using exact fractions for the probabilities. However, in finding
eigenvalues and eigenvectors you will find that you will have to convert, using
"evalf", to floating point entries. Why? Give some reasons why you
should not use decimal expressions for the probabilities at the outset.

## Instructor

Juris Steprans

email address: steprans@mathstat.yorku.ca

Department of Mathematics and Statistics.

Ross 536 North, ext. 33921

York University

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