Ivars Peterson's MathTrek

June 9, 1997

Monopoly Dollars and Sense

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.

Rank	Square #	Name	Relative Frequency
1	10	Jail	11.724
2	24	Illinois	2.990
3	40 (or 0)	Go	2.907
4	25	B&O Railroad	2.889
5	20	Free Parking	2.826
6	18	Tennessee	2.822
7	19	New York	2.809
8	5	Reading Railroad	2.797
9	16	St. James Place	2.681
10	28	Water Works	2.650
11	15	Pennsylvania Railroad	2.633
12	21	Kentucky	2.611
13	12	Electric Company	2.610
14	23	Indiana	2.563
15	11	St. Charles Place	2.550
16	26	Atlantic	2.536
17	31	Pacific	2.519
18	27	Ventnor	2.515
19	39	Boardwalk	2.480
20	32	North Carolina	2.468
21	29	Marvin Gardens	2.434
22	14	Virginia	2.424
23	34	Pennsylvania Avenue	2.349
24	17	Community Chest #2	2.295
25	35	Short Line	2.287
26	33	Community Chest #3	2.224
27	4	Income Tax	2.187
28	8	Vermont	2.179
29	13	States	2.171
30	9	Connecticut	2.163
31	6	Oriental	2.124
32	37	Park Place	2.057
33	38	Luxury Tax	2.047
34	3	Baltic	2.034
35	1	Mediterranean	2.005
36	2	Community Chest #1	1.769
37	22	Chance #2	1.045
38	36	Chance #3	0.815
39	7	Chance #1	0.814
40	30	Go 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 Group	Break-Even	Value per Roll	Total Cost
Purple	44	14.17	620
Light Blue	30	36.64	1070
Magenta	34	57.22	1940
Orange	26	80.37	2060
Red	34	87.21	2930
Yellow	35	87.29	3050
Green	41	96.47	3920
Dark Blue	35	80.45	2750

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?

Copyright © 1997 by Ivars Peterson.

References:

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.