Ivars Peterson's MathTrek

June 9, 1997

# Monopoly Dollars and Sense

Recently, my son Eric received as a birthday gift the Star Wars edition of the real estate trading game known as Monopoly. Players can choose to be members of the Rebel Alliance or battle for the Galactic Empire in an effort to accumulate properties scattered throughout the galaxy. They can then build up fleets of X-wing or TIE fighters to earn a Corellian freighter or Star Destroyer to defend an acquired site.

Though the setting is dramatically different from the streets and sites of Atlantic City, the rules of Star Wars Monopoly are virtually identical to those that accompany the standard version, which I have had since high school and which still affords hours of entertainment for the whole family. Indeed, the same rules underlie all of the varied editions of Monopoly, which feature themes ranging from golf to your favorite city.

Monopoly's popularity and longevity testify to the immense appeal of the rules and of the game board invented by Charles B. Darrow in 1933. His particular blend of structure and chance created an involving, entertaining game enjoyed throughout the world. The game has also attracted the attention of mathematicians and others determined to understand how it all works and to develop strategies for winning play.

In Monopoly, players buy, sell, rent, and trade real estate in a cutthroat competition to bankrupt their opponents. They take turns rolling a pair of dice, with the totals indicating how many spaces to proceed along an outside track that includes 22 properties, four railroads, two utilities, a Luxury Tax square, an Income Tax square, three Chance squares, and three Community Chest squares. Corner squares are marked Go, Just Visiting/Jail, Free Parking, and Go to Jail.

Players start at Go. A double warrants a second throw, but three consecutive doubles sends a player directly to the "In Jail" square. To get out of jail, the player must throw a double. If he succeeds, whatever sum he gets decides how many spaces he can advance along the board.

It's possible to describe the game in terms of a mathematical construction called a Markov chain, named for the Russian mathematician Andrey A. Markov (1856-1922). Markov studied systems of objects that change from one state to another, according to specified probabilities. When an initial state can lead to subsequent states, and these states, in turn, can lead to additional states, the result is a Markov chain.

Suppose we examine a game of Monopoly in which there is only one player. Initially, the player's token is on Go, and we call this condition state 0. The first roll of the dice produces either 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, or 12. The probability of getting 7, for example, is 6/36, or 1/6. So, the player has a probability of 1/6 of landing on the seventh square (Chance). There is no way for the player to roll 1, so the probability of landing on the first square (Mediterranean Avenue) is zero. Although the most likely distance traveled on the first turn is seven squares, you can go as many as 35 squares by throwing 6,6; 6,6; and 6,5. With each succeeding turn, you calculate the probability of landing on any square. In the long run, the likelihood of visiting any given site is simply the total of the accumulated probabilities for that square.

A number of people have done the required calculations, taking into account as much as possible the rules of the game. That gets a little tricky when it comes to including the consequences of certain Chance or Community Chest cards and such player activities as auctions, so these details are sometimes omitted.

Here's the analysis done by Stephen Abbott and Matt Richey of St. Olaf College in Northfield, Minn. The table below lists the squares in descending order of the relative frequency of visits.

RankSquare #NameRelative Frequency
110Jail11.724
224Illinois 2.990
340 (or 0)Go 2.907
520Free Parking 2.826
618Tennessee 2.822
719New York 2.809
916St. James Place 2.681
1028Water Works 2.650
1221Kentucky 2.611
1312Electric Company 2.610
1423Indiana 2.563
1511St. Charles Place 2.550
1626Atlantic 2.536
1731Pacific 2.519
1827Ventnor 2.515
1939Boardwalk 2.480
2032North Carolina 2.468
2129Marvin Gardens 2.434
2214Virginia 2.424
2334Pennsylvania Avenue 2.349
2417Community Chest #2 2.295
2535Short Line 2.287
2633Community Chest #3 2.224
27 4Income Tax 2.187
28 8Vermont 2.179
2913States 2.171
30 9Connecticut 2.163
31 6Oriental 2.124
3237Park Place 2.057
3338Luxury Tax 2.047
34 3Baltic 2.034
35 1Mediterranean 2.005
36 2Community Chest #1 1.769
3722Chance #2 1.045
3836Chance #3 0.815
39 7Chance #1 0.814
4030Go to Jail 0.000

Clearly, the long-term frequency distribution is not uniform. It probably won't surprise aficionados of the game that players spend a lot of time just visiting or in jail. There are lots of different ways to get there. The jail square also influences visits to properties on succeeding squares, especially those reachable by throwing doubles. So, St. James and Tennessee rank high.

Long-term frequency distribution for Monopoly, displayed in the order in which the squares appear on the board.

You can also calculate the expected values -- in effect, the probable return on your investment -- of the various color groups, once they have hotels. Abbott and Richey worked out the number of rolls required to recoup the cost of a group of properties and reach the break-even point.

Color GroupBreak-EvenValue per RollTotal Cost
Purple4414.17 620
Light Blue3036.641070
Magenta3457.221940
Orange2680.372060
Red3487.212930
Yellow3587.293050
Green4196.473920
Dark Blue3580.452750

Notice that the owner of the orange group reaches the break-even point fastest, by far. However, the green group has the highest overall value per roll. "Which is better?" Richey asks. "You decide."

Once you have a computer program for calculating Monopoly probabilities, it's also possible to investigate the effect of making changes in the rules or the board. Is there an even more appealing game possible, or did Darrow happen to hit upon the ideal combination?

### References:

Ash, Robert B., and Richard L. Bishop. 1972. Monopoly as a Markov process. Mathematics Magazine 45(January):26-29.

Brady, Maxine. 1976. The Monopoly Book: Strategy and Tactics of the World's Most Popular Game. New York: David McKay.

Stewart, Ian. 1996. Monopoly revisited. Scientific American 275(October):116-119.

______. 1996. How fair is Monopoly? Scientific American 274(April):104-105.

Bar chart created using Mathematica 3.0 ( http://www.wolfram.com/).

Monopoly, the distinctive design of the game board, the four corner squares, as well as each of the distinctive elements of the board and playing pieces are trademarks of Hasbro, Inc. for its real estate trading game and game equipment. Website at http://www.monopoly.com.