Monopoly

I'm guessing that most of us have played, or at least heard of, the game of Monopoly. I'm also gonna take a guess that not many of us have ever finished a game of Monopoly, but that's beside the point. For any of you who may not no the rules, the main concept is to move around the board by rolling two die, and buy properties as you land on them so that you can make money when your opponents land on said properties. Here's a link to the complete rules of the game.

http://www.avcschool.com/wp-content/uploads/2013/02/HDMC-Basic-Rules.pdf

In this blog I'll explain the best strategy to Monopoly by finding out what spaces are expected to be visited most, as well as which properties get the most bang for their buck based on how many people are playing. 

Not All Spaces are Created Equal

Those of you who have played Monopoly before, you've probably noticed that spaces like jail are more frequently visited than other spaces. Why is this? If the only way to move around the board was by rolling the two die, then the probability of spaces being visited would be distributed equally. In other words, at any point in the game the chances of your pieces being on any given space would be the same as any other space. What's different about Monopoly is that there are spaces such as "CHANCE" and "GO TO JAIL" that can send you to different spaces around the board. One can approach finding the probability distribution of spaces visited via simulation, or theory (both yield similar results). 

One method that has been used to figure out the distribution is by creating a computer program that simulates hundreds of thousand of games and simply counts how many times each space is visited. The results that come from the simulations usually are almost identical to the method we will begin to dive into. 

The second method uses a model called Markov Chains. Lets first define Markov Chains.  

Definition: A stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. 

Mathematical Definition: a sequence x0 , x1 , x2, … of probability vectors, along with a stochastic matrix A so that x1 = Ax0 , x2 = Ax1 , and, in general, xn+1 = Axn. 

In the context of Monopoly, Markov Chains essentially allows us to find the probability of moving from another given space to any other. Lets think about how this precess would work. We'll start out by numbering each space 0-39, "GO" being 0 and "BOARDWALK" being 39. If we start at space 0, we know that when you roll two die you can move ahead anywhere from 2-12 spaces. Given two six-sided die, the only possible way to roll a 2 would be to roll snake eyes. The probability of this is 1/36=2.8%. So there is a a 2.8% chance of moving from space 0 to space 2. We can calculate the probability of moving to any given space from space 0. Note that because we have a "CHANCE" at space 7, it is possible to move to more spaces that 2-12. In fact, it turns out that you have a 1.2% chance of going to jail on your first turn (bummer summer dude). We can repeat this process with every space to find a final distribution of how often each space is visited. You can imagine this would take a while, so instead of wasting your time I'll draw a diagram in my presentation that helps explain how this process would work. Here's a graph of the final distribution. 

Lets Talk Money

It's not enough to know how often each space is expected to be visited, we now must look into the payoff we can expect to receive by investing in different properties. We can find the expected earnings of each individual space by pairing the number of times that space is expected to be visited with how much money you will gain per visit. So basically what we would be doing is finding the expected earnings per opponent role. Looking at individual properties, we can create a graph that shows how one would expect their profits to grow as the game goes on. This graph is shown below. 

Looking at this graph alone, Mayfair is the best space to own based on the expected amount of roles it would to take to break even, and potential long term earnings. However, in Monopoly you must buy properties in sets. For example if you own Baltic Avenue, you also own Mediterranean Avenue because thy are both in the brown set. Because of this, we must adjust our graph to show expected earnings when owning different colors, rather than individual spaces. Our new graph is displayed below. 

    

 Now we can begin to see what spaces we should consider purchasing. Clearly Train Stations aren't worth considering, while the orange and yellow spaces catch our interest.

Considering the Number of Players

One question you're probably asking is how many opponent rolls are expected in an average game. This is where the number of opponents comes into play. When we have fewer players, we should expect fewer rolls. When we expect fewer rolls, we should consider purchasing colors that break even early on in the game. On the flip side, if we have many players we should expect more rolls, in which case you would want to purchase colors that have greater long-term earnings. 

Conclusion

Knowing the expected earnings of each property, and considering the number of opponents, we can keep the following strategy in mind when playing Monopoly. 

-When facing one opponent, the Light Blue and Orange spaces are best

-When facing 2-3 opponents, the Orange and Red spaces are the best

-When facing 4+ opponents, the Green spaces are best

-Don't ever bother with the Navy Blue, Yellow, or Train Station spaces

Feel free to comment or ask any questions, and don't be afraid to be that weirdo who pulls up a math blog at your next family gathering to kick some ass in Monopoly!

Bibliography

Standupmaths. "The Mathematics of Winning Monopoly." YouTube. YouTube, 08 Dec. 2016. Web. 26 Mar. 2017. 

"Probabilities in the Game of Monopoly®." Probabilities in the Game of Monopoly®. N.p., n.d. Web. 26 Mar. 2017. 

"Markov Chain." Wikipedia. Wikimedia Foundation, 25 Mar. 2017. Web. 26 Mar. 2017.

"THE MONOPOLY GAME RULES." THE MONOPOLY GAME RULES: STANDARD OR LONG RULES (n.d.): n. pag. Avcschool.com. Web. 

Hoehn, Stacy. "Monopoly and Mathematics: Linear Algebra in Action." The Mathematics Teacher 108.4 (2014): 280-86. Web.