Your Monopoly Game May Really Be Endless
Since the pioneering 1944 work of von Neuman and Morgenstern, Theory of Games and Economic Behavior, the study of strategy games has matured into a mainstay of economics and business. However even today some games do not yield easily to analysis of the sort pioneered by von Neuman and Morgenstern, in part because the expected outcomes of strategic choices may be hard to measure, especially if chance is involved in the course of play.
|Professors Henderson (left, rear) and Friedman watch as Gutiérrez (left) and Byuen simulate the play of Monopoly without benefit of the computer.
Professors Eric Friedman and Shane Henderson, working with a group of students, have attempted to analyze strategies for the game of Monopoly, using simulation techniques to determine outcomes for particular strategies. Computer simulations are well-suited for dealing with such complex situations involving chance, since games can be 'played' repeatedly with outcomes dependent on the toss of computerized dice. In the course of this work they observed that under some strategies the games showed a tendency to go on a very long time, potentially forever. As a precursor to the analysis of strategies, they attempted to find circumstances under which a game might in fact go on forever.
Together with Thomas Byuen, an ORIE undergrad, and Germán Gutiérrez Gallardo, an ORIE Master of Engineering student, Friedman and Henderson have now computed the odds that a Monopoly game played with a simple strategy will go on, and on, and on, forever. Their results, presented at a recent meeting of INFORMS in San Diego, have been published in a conference volume and have garnered press attention. They show that nearly one in eight games of Monopoly with two players and no trading will go on forever.
While the ORIE work on Monopoly was done somewhat in a spirit of fun, the effort involved insights that may be applicable to a broad range of more serious situations. "If we cannot analyze the relatively simple situation of competitive strategy for a board game that proceeds sequentially under uncertainty, how can we hope to contribute to the analysis of the broad range of real life situations having such characteristics?" said Henderson.
Previous Research on Monopoly
In surveying the limited academic literature on the game of Monopoly, the team of authors found that in 1975 two Cornell undergraduates studied strategy for the game "using what were, at the time, state of the art simulation techniques - playing many monopoly games," according to the paper. Back then, the two published a book, 1000 ways to win at Monopoly. (Parker Brothers, who owned the game then, sued to stop publication, but lost the suit. Proceeds from sales of the book went to pay legal fees).
One of the 1975 authors, Jay Walker, went on to found Priceline, the internet company that employs reverse auctions (a topic Friedman teaches) to sell travel and other goods. The other student, Jeff Lehman, went on to become President of Cornell from 2003 to 2005 and is now the chancellor and founding dean of the School of Transnational Law at Peking University's Shenzhen campus.
Simulating the Play of Monopoly
The primary tool employed by the current ORIE team was a computer simulation that captured the properties of a (slightly simplified) two-player version of the game in the Java computer language. At this early stage of the research, no trading of properties was incorporated. The simulation was run for 3100 games, until each game ended or 1000 rounds had been played. Byuen, who is from Woodside, NY and was recently a co-op student at General Electric in Erie, PA, commented that "the simulations were entertaining, especially since we had a graphical user inteface that displayed the real-time progress of each game."
However, to use simulation alone to prove that a game can go on forever would seem to require that the simulation itself go on forever. Even though the simulation showed that the fraction of games still going on after n rounds settled down (as n increased) to 12% plus or minus 1%, there is "no way of knowing whether the curve would drop to an even lower level" were the simulations to be continued beyond n rounds, according to the paper.
Motivated by this question, the team employed three separate analytical methods to mathematically confirm the simulation-based estimate. To do this they first divided the game into two stages, in the first of which properties are bought and houses built. After a while, if the game hasn't ended (the odds of which can be computed) all properties have been purchased and both players have at least $5000 in Monopoly money. After that point, for the remaining period, stage two, they estimate the probability that one of the players subsequently goes bankrupt. This probability relates to the difference in the amount of rent collectible by the players.
Computing the Probability of Unending Play
The probability that the game reaches stage two and neither goes bankrupt is the probability that the game goes on forever. Drawing on the separate but related theories of Brownian motion, Markov chains (also used by Lehman and Walker, who conjectured that some plays of the game might be endless), and importance sampling, the team used each approach to independently derive a 95% confidence interval for the probability of going on forever to be .12 +/- .01. "We had fun computing them," the authors note, of the process to derive the three confirming estimators. Gutiérrez, who is from Guadalajara, Mexico, commented that the work "required me to learn and apply concepts that I had barely seen before. It was more of a challenge than classes, where the application of concepts is limited to the material covered in class."
The team, together with students Alexander Aidun, Raghavendar Chandrasekaren, Sean Choi and Dennis Li who worked with them and then graduated, has become the Cornell University Monopoly Strategy Research Group. Looking forward, they hope to use the tools they have developed to identify highly effective player strategies, and have some promising ones under investigation. The MonStR web site also includes calculators that determine whether your game will ever end.
Since the rules of Monopoly permit players to trade properties and Monopoly money among themselves, the team is investigating strategies that incorporate trading. "When trading does occur, it allows greater development of the board, and the probability that the game goes on forever drops significantly," says Henderson. He reports that, for some more realistic strategies, the probability has been seen to be on the order of one percent. "I feel that there is so much room for our 'Monopoly-bots' to grow, especially when we delve into trading," said Byuen.
"An important question is how to use some kind of automated learning process to identify the best of a family of strategies," the authors wrote in the published paper. "The reason this is important is not just that we are enthusiasts and see it that way, but also because Monopoly is a microcosm of many important systems in real life, where sequential decisions under uncertainty are made in the face of competition from other entities."
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