Monopoly Simulations And Your Winning Strategy

All you need to know to win a Monopoly.

The first Monopoly post (New Year’s Eve + Monopoly Rules Analysis—Prep For Simulation) provided background and refreshed the game rules. Now, we are diving into the simulation and maybe what you care about the most: how to win the game!

Here is some vital information for you to become the Invincible One when it comes to Monopoly among your family and friends. Feel free to jump to the bottom of this article for the conclusion.

Since you stayed, there are three questions we asked ourselves, and the answers paved the path to the winning strategy:

1. How frequently do your opponents land on your property?
2. What’s the Return On Investment (ROI) of the square you bought? How many times do your opponents have to visit you before this property will break even?
3. Which offer should you accept or propose when negotiating a trading deal, i.e., Expected Value (EV)?

3000 Monopoly Games And Raw Data Explained

We ran three rounds of the simulations and 1,000 games each round to collect some data for further analysis.

Recap the game board if you don’t remember what one looks like. The whole game starts at GO, ie. square 0, and players move clockwise from 0–39 (40 squares in total). The square 10 is jail, square 30 is “go to jail”, etc. Players move around this loop and follow the rest of the movement rules until there’s a winner.

a Monopoly board I drew some time ago.

Raw Data Explained

A quick walk-through of the raw data is in the screenshot below.

Column A is the number that represents each square: 0–39; essentially, it is a projection of the illustration board above. Each column after Column A is the individual result of a 75-turn game. The cells of each column corresponding to each square corresponding to the counts of a virtual player landed on each square within a 75-turn game.

For example, Column B represents one of the 3000 simulations, and the number you see in each Cell: B1, 3; B2, 1; B3, 3… are the counts that the virtual players landed on each square. You will get 75 if you sum up column B since it’s a 75-turn game in total.

Then, we will average each row for the probability of landing on each square.

We did this simulation several years ago. While reviewing this, I wondered why we ran many simulations for the analysis. Thinking back, 10% of the data is more than enough to conclude :)

Probability. How Frequently Will You Have Visitors?

Let’s take a look at the screenshot below. The property numbers are marked with their color or the name of a special square so they are easier to follow. The Round 1, 2, and 3 columns each represent a round of the average counts of 1,000 games.

We can see the average counts on each square from these three rounds are relatively close. Then we divided the counts of landing on each square to 75 (the total turns of a game) for the probability of landing on each square.

The screenshot below is the result of the calculation that excluded all the special squares, properties only:

• In the “Round 1 probability column”, the cells filled with green are the ones with higher landing probability. Those are the Orange and Red properties on the game board. Where you and your opponents are likely to end up on these squares more than once throughout the whole game (this again is statistically speaking. * A disclaimer if none of your opponents went visit your properties)
• Red, the ones with more negligible landing probability, are the dark brown properties that you probably won’t even step on once in a game
• White, whether you buy the property depends on your strategy. But I always like to buy one in each color set while I can, just to avoid my opponents owning a set and/or give me better negotiating leverage

Now, below is the probability histogram that includes all squares (both properties and the special squares).

• That peak is the first thing that caught your eye about this screenshot. That’s the Jail Square.
  Simply, there are just so many ways in Monopoly that a player can end their turn in Jail. So make the Jail square the MOST frequently visited square throughout the game.

On a positive note, orange is the new black !!

from the Orange Is The New Black drama

• You should also see an empty space in the histogram at square 30, the “Go To Jail.” That’s because you are immediately sent to Jail whenever you land on it. So, no one ever ends their turn here.
• The Chance card squares, 7, 22, and 36, also have relatively lower chances for a player to end their turn there. The reason because, 9 of the total 15 Chance cards direct a player to another square:
  - Advance to 0 (Go)
  - Advance to 5 (Deeprun Tram)
  - Advance to 11 (Stranglethorn Vale)
  - Advance to 24 (Shadowmoon Valley)
  - Advance to 39 (Orgrimmar)
  - Advance to 10 (Go To Jail, go directly to the jail do not pass Go)
  - Advance to the nearest transport, 5, 15, 25, or 35
  - Advance to the nearest Electric or Water Utility, 12, or 28
  - Go Back 3 spaces

Since Monopoly is a 2 dice game, the chance of rolling 7 is the highest, followed by the chance of rolling 6 or 8. Since a lot of the movement rules direct a player to the jail block, naturally, a newly released prisoner is more likely to end up on the orange properties. For the same reason, you can easily see why the red set is also frequently visited.

Property ROI and Break-Even point

Before we dive in and talk about ROI and breakeven, let’s have a look at the screenshot below,

• Column A: properties on the board
• Columns R to V:
  - R, the cost to your opponent per visit
  - S, the costs of building a house on the property
  - T = R-S
  - U, ROI (R/S)
  - V, Investment Payback Period (IPP) -> how many visits before your investment is paid back
• group AC includes the same info but for 2 houses
• group AI includes the same info, but for 3 houses
• Column AJ to AN, is the numbers for 4 house

* Please be aware that this chart only considers each property as an individual. Although we calculated the cost as a set, we didn’t look at the probability of a visit as a set. For example, if your opponent visited your square 16 and then later on your square 18, you should see a much better ROI and IPP than what’s in this chart. It’s simply because the cost has gone down when it’s spread over all the property as a set (of 2 or 3 properties).

ROI allows us to compare the value of all Monopoly property through an equitable method. Here’s the formula we use to generate the ROI, ROI = Net Profit / Total Cost, which is column U, and AM in the screenshot below marked in conditional formatting.

The ROI of 1 house vs 4 houses

We now know that the red and orange sets are the most visited sets on the board. However, does it mean that both sets will give you a relatively high return compared to other property sets on the board?

We will see a different result by taking a look at ROI instead of probability.

The conditional formatting is to emphasize the ROI columns of the spreadsheet so we can focus on discussing the 1 house and the 4 houses ROI. The cells marked Green of both columns, means relatively higher ROI compared to the rest of the properties of the same column. The cells marked Red, means the relatively lower ROIs.

We can give three small summaries from the Column U and the Column AM (screenshot below),

• The red properties aren’t as profitable as we thought! Although we do have a slightly higher chance to land on the red properties, the ROI is much smaller than the other sets, such as the light blue (6, 8, 9) and the dark blue (37, 39) properties.
• Even just in the 1 house column the dark blue properties seem like a lot better investment! The square 39 has the highest ROI than others, it gets even better once you have collected 4 houses, the -20.93% ROI, means you will break even within 2 visits!
• Once a better deal, like squares 19 and 29, the ROI was high when you only have a house. But its relative value dropped as the house’s number increased. On the other hand, once it seems like a shit deal, such as Square 16 and 18, they can suddenly become one of your top earners.

Expected Value. Location Location Location! (And Time Value)

Finally, the most important factor, the Expected Value (EV) of each property, is the number you use to evaluate a property.

The difference between EV and ROI is that EV also takes Time into consideration. In other words, how many visits can you expect to see in one round of the game? By calculating the EV of each property, you will have a better idea when it comes to trading property. Once you understand this, you’ll be sitting comfortably and flooding yourself with cash by the end of the game!! (Again, all information found here, including any ideas, opinions, views, predictions, forecasts, commentaries, suggestions, or stock picks, expressed or implied herein, are for informational, entertainment, or educational purposes only and should not be construed as personal investment advice.)

Three factors we need to calculate an EV,

• Cost, the price listed for a bare land plus the houses/hotel price
• Probability, the possibility of stepping on each square
• Price (Gross profit): how much you can charge when your property is visited

EV = (Price * Landing Probability in x turns)- Cost. Please note, the cost of houses for each square is the sum of the house cost + sum of the properties cost of the same color region.

Take the chart of 150 rolls, for example (screenshot below); the first columns are the number and the color of the property that matches what’s on the game board. Column E to J is the EV with a different number of housing, E is a bare site with no house, F is one house on each property, and so on.

The numbers in EV of each property are the money that you will earn (or lose) in this 150-roll game. We used conditional formatting to visualize the chart. The deeper the green, the more valuable the site is in comparison to the rest of the sites in the same construction level (same column comparison).

* Welcome to reach out and ask for the monopoly EV spreadsheet. Go to the last tab and change the turns on as you like. For a two player game it takes averagely 150–180 turns ; and with every one player added into the game, the turns will go up from 75–90.

Some examples might help you understand the spreadsheet better,

• The brown properties (especially square 1), the value is always negative before the 4th house is built. It means that the chance of square 1 will still cost you -89$ when you update to 3 houses in a 150-turn game. And you could only expect to earn 212$ after you updated square 1 to a city in a 150-turn game. So this is something that is not worth your money compared to other properties. Attractive as the property might seem because of the low entry value needed, it really isn’t the best way to allocate your limited resources.
• The dark blue set (squares 37 and 39), on the other hand. Although it is going to cost you a fortune, to begin with, you see a steep increase in payback in 3 houses, and you can expect to earn 5,196$ in a 150-turn game if you upgrade square 39 into city/4 houses! So if your opponents ever ask if you’d like to sell square 39, just enter how many turns you expected this game would take on the spreadsheet, and think about which upgrade the opponent might possibly achieve. If you knew the opponent was looking to build a city, then quote $5196

Resource Limitation. Sounds Familiar? Just Like In The Real World!

If I were to pick a reason for liking this game, then it’d be that the housing resource is limited:

• 32 town tokens
• 12 city tokens

It is exactly what makes the calculation above even more valuable. In a two player condition, it’s unlikely to reach the point that all 32 house tokens run out, and one of you still isn’t bankrupt. However, everything would be different in a three or more-player game.

Let’s have a look at the following scenario: Player A and B decided to work together against Player C. Assume the following,

• Player C owns both square 37 and 39.
• Where player A owns the brown set (square 1 and 3) and the light blue set (square 6, 8, 9), and
• Play B owns the pink set (square 11, 13, 14).

It seems like Player A and B are deemed to lose if you only look at the chart above. However, because of the limitation of the housing resources, if both A and B had the chance to acquire all 32 houses, player C had no way to upgrade their property, which led to bankruptcy. However, because the land price of squares 37 and 39 is so expensive, it’s quite likely that whoever bought these two properties has little spare money to put down houses immediately after purchasing both lands.

Luck. Again, Just Like Real World

All the simulations above were based on machine randomization, so we can expect the dice-rolling results to look more like a bell curve in a limited amount of turns. We all know though, real world dice rolls can be a bitch, you could very likely roll a couple of 12s in a few rows until you got back to “normal”.

In the end, luck is still a significant factor in this type of limited turns game. Good luck!