Location, location, location! The path of the Monopoly hat

December 22, 2009

I used to play Monopoly all the time when I was a tapering age group swimmer. In fact, our group of friends used to have the equivalent of a Monopoly championship each year, the winner of which would store the coveted Monopoly board until the game the following year. Needless to say, when my in-laws suggested that we play Monopoly last night, I was all too eager to relive the memories.

During the game play last night, my wife landed on several unowned properties the first few times around the board and, being the ruthless businesswoman that she is, she purchased all of them and built hotels on her first complete set—the grey properties. Around the time that my wife started building her hotels, I managed to trade and start building houses on the green properties, my only properties. Things were looking rather grim for the protagonist (that's me!) until my father-in-law, my mother-in-law, and then my wife landed on my green properties during the same lap around the board without anyone landing on my wife's grey properties.

Oh goodie!

This fortuitous turn of events inspired me to investigate this further. How does the “Monopoly hat” move around the board? Does the movement of the Monopoly hat make some property locations more valuable than others?

Fig. 1: Simulation of random 1000 Monopoly hats (dots) wandering around a Monopoly board. Colored arcs represent their respective properties in Monopoly. This simulation accounts for Chance and Community Chest cards that randomly move players to different parts of the Monopoly board.
Probabilities of occupation in monopoly
Fig. 2: Location, location, location! Ignoring rental revenue, the various property colors are valued by how likely they are to be occupied during a particular roll of the game. Up to around 30 rolls, some property colors are more lucrative than others at particular times of the game.

With these questions in mind, I sought to first visualize how a player travels around the Monopoly board by making a movie of 1000 random Monopoly hats over 100 rolls of a Monopoly game (Fig. 1). Notice how the Monopoly hats start out as a traveling wave that persists for almost 30 rolls. I am not sure how long a typical game of Monopoly lasts, but I would guess that 30 rolls constitutes a significant portion of most Monopoly games, suggesting that some property colors have more valuable locations on the board than others, depending on the time (measured in rolls) during game play. We can quantify this effect by measuring the probability that any particular property set is occupied after a certain number of Monopoly rolls (Fig. 2).

In some sense, the old adage “Location, location, location!” rings true with these simulations. If no one lands on your properties, it does not matter how much rental income your properties could provide. These results therefore illustrate an interesting property of the Monopoly game that we observed during our game last night (and you have probably noticed yourself); temporal fluctuations in property occupation can dramatically alter relative property values. Although this does not entirely explain the sequence of events that turned my wife's bad fortune into my fortune, it is interesting to consider how the dynamics of game play can ultimately influence performance and outcomes.

Of course, if you have some other ideas about what all of these means, please share! Perhaps I will explore other ideas in subsequent blogs...

contributors to this post

headshot of Dean Malmgren
headshot of Mike Stringer