Simulating Monopoly: Finding the Best Properties Using MATLAB | by Jake Mitchell | Towards Data Science

With 40 spaces, and 28 different properties, Monopoly may seem completely random at first, but there are certain properties that can be more beneficial to you over the course of a game. I wanted to find which properties these were using a MATLAB simulation of 10,000,000 dice rolls.
I simulate two dice being rolled using a random number generator and then add the two values together. I then add that amount on to the current position of the player to get their new position. I also have a variable to track the amount of doubles in a row. If the player rolls 3 doubles in a row, they are sent to jail for speeding.
die1 = randi(6); % roll first diedie2 = randi(6); % roll second dieif die1 == die2 % if same valuedoubles = doubles + 1; % increase number of doubles in a rowelsedoubles = 0; % set doubles to 0 if different dice valuesend
What follows in the code is all of the special case situations. For example, if the position adds up to be greater than 40, it reverts it back to the starting spot as a reference since the board is a loop. A position value of 43 would be changed to 3. Also, if the player lands on position 31 (“go to jail” space) they are sent to position 11 (jail).
if doubles < 3 % if less than 3 doubles in a rowposition = position + die1 + die2; % finds new positionif position > 40 % if above 40position = position - 40; % reverts position back to starting position as a referencearound = around + 1; % indicates the player went around the boardendif position == 31 % if player lands on the "go to jail" spacelands(position) = lands(position) + 1; % indicates player landed on position 31position = 11; % sends player to jailend
Other special case examples are community chest spaces and chance spaces. Of the 16 community chest cards, only 1 changes the location of the player (go to jail). Of the 16 chance cards, 9 change the player’s location:
Go to Jail
Advance to Boardwalk
Advance to Illinois Ave
Advance to St. Charles Place
Advance to Reading Railroad
Advance to Nearest Railroad
Advance to Nearest Utility
Advance to GO
Go Back 3 Spaces
After every turn is over, the space that the player ended up on is tallied in an array of all 40 spots, and then each spot is divided by the total amount of simulations (10 million). This gives us a percent likelihood of all 40 spaces being landed on.
After 10,000,000 dice rolls, the likelihood of each space being landed on is shown below:
Shown below is the top 20 board spaces from the bar graph above:
While the amount of times a property is landed on does affect the benefit it will bring to you as a player, we also need to consider how much it costs to buy the property, and how much players will pay you every time they land on it.
The probabilities shown above can also be described as the chances of a player landing on that property after every roll. We can use that, along with the rent for every property, to find the expected amount of money you will make on every player’s dice roll if you own said property.
For example, Ventnor Ave (shown above) is at a 2.691% chance of being landed on every roll. The rent with no houses for Ventnor Ave is $22, which means that every roll results in an expected $0.59202 for the owner.
The revenue per roll is not the best thing to compare between properties because a property that costs more than Ventnor should yield more revenue, but this is not always the case. If we divide the initial cost of buying the property by the revenue per roll, we can find how many rolls it will take to break-even on a property. For example Ventnor costs $260 to buy, so the owner should expect to make all of that money back after 439.18 rolls. Properties that take less rolls to break-even would be considered better investments because you could start turning a profit before other players.
We also need to consider how building houses and hotels on properties affects the rolls per break-even. Below is a breakdown of each property on a Monopoly board, given it’s situation regarding houses and hotels (excluding railroads and utilities). The value in each cell represents to amount of rolls it will take on average to make back the money spent on the property.
As you can see, some properties are almost guaranteed to lose you money in the long run. Buying Mediterranean Ave and not putting any houses on it will certainly cost you money over the course of a game. It won’t be a significant amount (only $60 to buy), but it is still not the best purchase.
All properties without houses take an extraordinary amount of time to break-even, which is why buying houses and hotels is so important.
The plot above shows how buying at least 3 houses for every monopoly you own can greatly reduce the amount of rolls it takes to make your money back. Even the brown-colored properties (brown line above), which are a horrible investment at the start, can start to turn a profit over the span of a game with 3 or more houses on it.
Something else interesting to note is that as the properties get more expensive, the less change is seen between 3 houses, 4 houses, and a hotel. Take the light-blue and yellow properties for example:
Light-blue properties see a considerable drop between 3 and 4 houses, and between 4 houses and a hotel. The yellow properties show almost no change between the 3 rightmost columns. This means that it will take the same amount of time to make back the money spent on 3 houses as it does for 4, but the revenue after breaking-even will be bigger (for the more expensive properties).
But what about the railroads and utilities?
The railroads are shown below, with all the different permutations for which ones you own. The value in each cell represents to amount of rolls it will take on average to make back the money spent on the property or properties.
For ownership of just 1 railroad, it is best to own B&O, just because it gets landed on more. If you want to own 2, go for the Reading and B&O combination. For 3, look for Reading, Pennsylvania, and B&O. Overall, the best investment for the railroads is either getting 3 or all 4, just because as people start to build houses on their properties, your 1 or 2 railroads won’t be making nearly as much compared to earlier in the game.
As for utilities:
Owning the utilities early in the game is probably one of the best moves. Both have the lowest rolls per break-even compared to any other property without houses. However, just like the railroads, you can’t build houses on these, so as the game goes on, these become less and less useful. Even owning both utilities still puts them at the bottom, considering 21 of the 22 other properties have lower rolls per break-even with 3 houses.
The orange properties are the best spots in the game. The oranges are all within the best 5 properties with 3 houses, and are the top 3 locations for hotels. The orange properties New York Ave and Tennessee Ave place 3rd and 5th amongst the spots most landed on.
Boardwalk and Park Place (dark blues) are tricky ones. Boardwalk consistently places high on the list in terms of rolls per break-even, while Park Place is always near the bottom. This monopoly is a high-risk, high-reward option compared to the oranges.
The absolute worst group to put hotels on are definitely the greens. All 3 place in the bottom 5 in terms of rolls per break-even with hotels. They are also below-average in terms of likelihood of being landed on.
Railroads are only beneficial near the end of the game if you have 3 or all 4. If you can get them early and start breaking-even, then that is a great move. Utilities are the best properties until people start building houses. You can’t win with just utilities or railroads, so do not put all of your effort and money into them.
Overall, this simulation showed that there is some strategy that can be implemented into your next Monopoly game. If this type of content interests you, I recommend reading my other board game articles, I Taught a Machine How to Play Connect 4, and The Best and Worst Catan Board Setups.