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### Most Desirable Properties in Monopoly

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Date: 02/16/2001 at 12:06:44
From: David Price
Subject: Probability

Dr. Math,

(NJ) High School, and I'm responsible for the math/science end of
things. I'm not a math teacher, so I need a little assistance with a
probability problem I dreamed up: Which are the most desirable
properties on a Monopoly board?

We will assume that the most desirable properties are the ones upon
which players are most likely to land, and we'll consider properties
in groups - purple, light blue, magenta, etc. We will also assume
that after the first trip or so around the board the probability of
landing on any one square is equal to that of landing on any other,
if only the roll of the dice is considered. This probability is
changed by two considerations - Chance and Community Chest, which
direct players to certain squares, and the fact that periodic
involuntary relocation of players (i.e. "Go to Jail") increases the
probability of visiting certain spaces (i.e. those immediately
following the "Jail" space).

I believe I have a handle on the Chance and Community Chest aspect of
the problem - I just redistribute a portion of the probability of
landing on those squares to the probability largesse of the target
squares. I'm not sure how to handle the second part, though - it
would seem to depend on the number of times one circled the board.

I'll probably leave the Railroads (as a group) out of the problem,
since there are four of them and at most three of any other type of
property, or I may rephrase the problem to require the most desirable
single property.

Do you think you could be of any assistance? Thanks for your
consideration.

David Price
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Date: 02/17/2001 at 23:56:14
From: Doctor Douglas
Subject: Re: Probability

Hi David, and thanks for writing.

To do this problem properly is not easy, but if we make some
simplifying assumptions, we can make some progress. We can make a
large matrix that shows the relative probabilities of moving from any
space to any other space (including the same one, e.g. for failing to
get out of Jail) on any roll of the dice. Many of the entries (e.g.
from Baltic Ave. to Marvin Gardens) will be very small or negligible.
We could handle the railroads and the utilities in this way, if we
wished.

We start by making the assumption that the distribution of the
occupancy is nearly uniform, and then looking for perturbations that
change this assumption (obviously the biggest ones are "JAIL,"
"Advance to Go," and so on).

One good approximation method is the following: imagine that each
square is occupied by many tokens, and see where (according to dice
rolls and chance/community chest cards, and jail, etc.) these tokens
end up, according to the possible die rolls and the possible cards
(you could put in the "advance to nearest railroad" cards too here,
for example).

You'll end up with a list of probabilities that deviate somewhat from
the uniform distribution. I imagine JAIL and GO will be among the
spaces with the highest probability. Now iterate the process, by
imagining the number of tokens proportional to the probability
computed in the first step above. It is at this point that your
students will appreciate the value of the Orange group (Tennessee, St
James, and New York Avenues).

We keep on iterating, until we get a "self-consistent" solution. In
mathematical jargon, this solution will be a "fixed point" of the
Markov model that describes the flow on the board.

An alternative method is the following: start with one token, and keep
rolling dice while tracking its movement around the board. After many
many rolls/turns, the token will presumably visit the squares with the
appropriate self-consistent distribution that we would have calculated
above. Whether or not this is in fact true is a delicate issue, but
the two solutions ought to be close enough to make an interesting

Finally, I know that there are people who have done these iteration
and simulation calculations. I think that their result is that
Illinois Avenue is the "most desirable" property (Jail is the most
visited space), because it receives a strong boost from freed Jail
prisoners, as well as the "Advance to Illinois Ave" chance card.
Other desirable properties are St. Charles Place (it gets a somewhat
smaller boost from Jail, plus its own card), the entire orange group
(Jail boost), and Boardwalk.

Probabilities in the Game of Monopoly - Truman Collins
http://www.teleport.com/~tcollins/monopoly.shtml

Monopoly Probabilities - Mandy and Jim D'Ambrosia
http://home.att.net/~dambrosia/programming/games/monopoly/

Monopoly - the Invariant Distribution - Allan Evans
http://www.cms.dmu.ac.uk/~ake/monopoly.html

There is also a very good article by Ian Stewart in the May 1996
issue of Scientific American. It is mentioned by some of these sites.

I hope you and your students have fun with this problem. Please write

- Doctor Douglas, The Math Forum
http://mathforum.org/dr.math/
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Associated Topics:
High School Probability

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