Most Desirable Properties in MonopolyDate: 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/ |
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