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A Data-Driven Tactics Simulation for Monopoly

A visual guide on dominating your next Monopoly game using thousands of MATLAB simulations.

Photo by Joshua Hoehne on Unsplash
Photo by Joshua Hoehne on Unsplash

Introduction:

In this sequel to a previous article of mine, Simulating Monopoly, I will be looking into the buying strategies of players – and how those strategies affect game outcomes. I will be doing this by modeling the game in Matlab, and then simulating thousands of games to find patterns within the player choices.

Coding the Game:

To model monopoly, you just need to simulate the rolling of dice and move players based on that roll value. Then apply a list of conditional statements based on what kind of tile they land on. If they land on a property, either decide to buy the property or pay rent to the owner. If they land on "Go to Jail", change the location of the player to "Jail". That aspect of the game is pretty straightforward in the code, and is outlined in more detail in my previous article linked above.

The code that is most exciting to me is the decision-making when it comes to buying property, buying houses/hotels, and trading. When a player is deciding to buy a property, two factors are taken into account:

  • The Price Ratio – the player’s money divided by the price of the property. The smaller the ratio, the bigger the impact on the player’s financial health.
  • The Purchase Coefficient – a random value assigned at the start of the game. The lower the purchase coefficient, the more risky a player is with their buying habits.

These two values are fed into the Decision Factor Formula. This formula is a modified sigmoid function that yields a result between 0 and 1. Results closer to 0 mean that the player would never buy this property in this situation, and results closer to 1 would be a great deal for the player. Then a random number generator produces a number between 0 and 1. If that number if less than the decision factor, the player buys the property (see code below).

price_ratio = player_data(turn,1)/property_data(i,2); % calculates price ratio
decision_factor = 1/(1 + exp(-1 * price_ratio + player_data(turn,3))); % calculates decision factor
if change == 1 % if buying property would result in a monopoly
  player_data(turn,3) = player_data(turn,3) + mon_weight; % increases chance of getting property
end
if rand(1) < decision_factor % if player decided to buy property
  property_data(i,11) = turn; % assignes property to new owner
  player_data(turn,1) = player_data(turn,1) - property_data(i,2); % subtracts price from owner's money
end

These two values and associated function are used for all players at every decision point in the game. There is even an extra incentive added to properties that would create a monopoly for that player.

The game is simulated until all but 1 player is bankrupt, at which the game is over and the statistics are recorded. Then everything is reset, and a new game starts. Repeat 1,000 times.

The Results:

The following results and analysis were performed with the assumption that 6 people were playing the game.

Should you be more risky, or more conservative with your money?

The plot below shows how your buying strategy affects your average wealth over the course of a game. As you move from left to right along the x-axis, the buying strategy goes from risky to conservative.

Image provided by the author.
Image provided by the author.

You could argue that there is a slight advantage towards being more risky based on the darker cluster of points in the bottom-left corner, but it’s not much. This plot unfortunately shows that there is no correlation between buying strategy and success rates, which was really disappointing to see. To prove it is not dependent on player amount, here are the same results with 4 players and 8 players:

4 players (left) and 8 players (right). Images provided by the author.
4 players (left) and 8 players (right). Images provided by the author.

Once again, there is a little evidence to show that being more risky can increase your average wealth based on these plots alone. Also, the accumulation of points near 0 is because in a game where almost everyone ends with 0$, you would expect the average to be skewed in that direction.

So there is no ideal buying strategy that I could use?

As it turns out, there is a buying strategy that could benefit a player, but it actually has to do more with how your opponents are playing.

The plot below is nearly identical to the previous plots, but now the x-axis is the Purchase Coefficient Ratio, or the ratio of your coefficient to the average of your opponents. Values lower than 1 mean you are being more risky than your opponents. Values higher than 1 mean you are being more conservative.

Image provided by the author.
Image provided by the author.

Based in this plot, we could say with more certainty that being more risky with your buying strategy – compared to your opponents – increases your expected average wealth. This is because there is a significant increase in points located higher up the y-axis between the x-axis values of 0.5 and 1.

We could also say that being more conservative – compared to your opponents – decreases you expected average wealth. The right side of the chart shows that players who were more conservative than their opponents almost had no cases of ending with a significant amount of money.

So if I should be more risky with my property buying compared to my opponents, which properties should I be focusing on?

Over the course of the 1,000 simulated games, I looked at which properties the eventual winners had at the point of their first opponent going bankrupt:

Images provided by the author.
Images provided by the author.

The x-axis is simply the tile location on the board. It’s like if you broke apart the 4 sides of the board and lined them up in a row. The colors match the properties on a standard monopoly board, with the exception of railroads being black points and utilities being gray points.

When looking at this plot, it is clear that the winning players targeted 3 specific groups of properties. They targeted the light blue, pink, and orange property groups (see red circles on board above). **** All of these properties had ownership rates well over 35%, and some came close to 45% in the case of the light blues.

Why do players who win often own these properties?

The ideal property to own is one that has an excellent return on investment. You do not want to own properties that will take an expected 80 turns for you to make your money back. The plot below describes it perfectly:

Image provided by the author.
Image provided by the author.

The rounds to breakeven is calculated using the average money spent on the property, including the initial purchase, and any subsequent housing added. This is divided by the average amount of rent earned per round to find the average rounds it takes to get your money back.

The properties that were most likely to be owned by winners are near the bottom, meaning that it takes less time to earn your money back, hence more time to make money from it. The plot below shows how these winner really valued properties with excellent return on investment rates.

Image provided by the author.
Image provided by the author.

The light blue, pink, and orange properties are clearly the best on the board. This is important because not only should you be more risky than your opponents, you should also focus that risk on these 3 property sets.

What is a quantitative value that can be assigned to this level of risk?

I tracked the wealth of all 1,000 winners from the simulated games and then averaged the amount of money at each dice roll to produce this visual:

Image provided by the author.
Image provided by the author.

The average winner for a 6-player game spent money on properties until they reached between 750$ and 800$ left in their hand. This should be achieved over the course of roughly 60–70 dice rolls, or around 10 complete cycles amongst all the players.

Once the average winner reached nearly 3 times the starting amount of money, they began reinvesting in properties that become available as players go bankrupt. They also start to focus on buying houses and hotels at this point.

Conclusions:

  1. It is not enough to just play in a risky manner, you should also try to be more risky than your opponents. At the very least, match your opponent’s playing style.
  2. Your buying strategy should revolve around trying to acquire the light blue, pink, and orange properties.
  3. Aim to spend around half of your starting amount on properties early in the game.

Thank you for taking the time to read my article! If this type of content interests you, I recommend reading some of my other board game simulations including Teaching a Machine How to Play Connect 4 or Simulating Mancala.


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