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 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.

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.

Once you have this matrix, you will then need to get some information out of it. What is the best single piece of property to have? What is the best collection of properties to have? Explain clearly what criterion you are using to make these judgements. Is it better to own the two blue properties or the three red? If two players have split up the properties and we play a simplified game for 10,000 rolls of the dice, is there a way of determining which player will (probably) be ahead at the end of the game? Try to simulate a game. Is there a way of coming up with the answers to these questions just by examining the matrix?

A champion Monopoly player has given some strategies for playing the game. Can you justify the properties that he says are true with your analysis?

The first question that was asked with this article is 'Is Monopoly Fair?'

Mike Zabrocki

Math 2042

zabrocki@yorku.ca