## 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.

Juris Steprans