Most Desirable Properties in Monopoly

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

Dr. Math,

I co-advise the Rutgers Academic Challenge team here at Riverside 
(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

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 
probability exercise for your students.

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.

Here are some sites with more information about this problem:

   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 
back if you have more questions about this, or anything else.

- Doctor Douglas, The Math Forum
  http://mathforum.org/dr.math/