How To Win A Monopoly Game-The Ultimate Game Analysis

So you’ll be crowned the Monopoly king/queen among your family!

This is to remember my recently passed-away grandmother (PóPo) as someone who raised me and loved me dearly. I didn’t have a great childhood (tons of family drama), but one of my few good memories was playing Monopoly and poker with my cousins and PóPo on New Year’s Eve.

New Year’s Eve + Monopoly- My Childhood Memory

Every year, we had a hot pot for New Year’s Eve dinner. Then we cleared the table, sat down around it again, and the game started with one person playing as “the bank.” We used to bet the Red Envelopes money we got from PóPo. Whoever won the games went home for a good night’s sleep plus a nice cash bonus.

As children, we always thought there was some kind of a mystery power working behind the game that determines win or lose. For someone who kept losing, they would want to change their seat or eat a different type of snack to refresh their luck or get closer to the excellent spirit. Everyone has their secret little ritual to make the win possible. Mine is, be the bank.

Here’s a disclaimer: I take no responsibility for your losses. :D

If you are not interested in the analysis part, go to this article for the result and game strategy.

A Classic But Poorly Designed Game

Klaas bought me this World of Warcraft Edition a few years back. You’d know what I mean if you played WoW.

Well… when I say poorly designed, it is compared to all the other strategic games we have acquired throughout the years. Monopoly is one with a simple set of rules, straightforward, repetitive, and rather unbalanced.

The result mainly depends on two factors:

1. odds (you are rolling dice)
2. Your strategy for buying/upgrading a property

Game Started

It was a 2-player game: me vs Klaas.

I have a simple strategy on top of the [official rule](https://en.wikipedia.org/wiki/Monopoly_(game)#Official_rules)s. I bought properties whenever I could, either paid in cash or by mortgaging some of the properties that I already held. After a while, Klaas collected the set of properties which is six to eight steps right after the jail block (the orange set). I ended up on these properties very often, I realized I was in a vicious loop:

• I was sent away to jail frequently
• I rolled, and there came a lot of 6, 7, and 8 most of the time after came out of jail (standard odds with 2 dice)
• So I ended up on the orange set of properties frequently

If I didn’t know better, I’d figure: Either Klaas cheated, the dice are loaded or the game has flaws!!

I am competitive, just like you, otherwise you wouldn’t be still reading this! Like most people, we **love to win and hate to lose.** Because I did not want to simply accept the results and move on, so we decided to look into the game from a different perspective.

Klaas and I no longer felt the excitement when we played as we were little. We still went for a few rounds and thought it’d be more fun to simulate the play and write down the best play strategy.

That’s It For Sentiment, Let’s Talk Analysis

Since Klaas would never cheat and I trust that the dice with this brand new game weren’t loaded…only one thing left: find the flaws in the game (and exploit them next time).

We started by calculating each block’s landing probability and then its expected value through 1000 simulations. Then the result can be used to instruct the players what and when to buy so that eventually it helps to win this game without using much of your voodoo potions. So you can confidently say yes, when next time someone suggests playing Monopoly at your family reunion or New Year’s Eve.

Probability & Expected Value

Objective: an instruction that can tell us which property to buy to achieve the highest possible Expected Value (EV).

For this, we need:

1. the probability of landing on each block (Pi, the probability of landing on the property i)
2. the payment to acquire each property and the cost to upgrade
   – CL, cost to buy bare land
   – C1, cost of adding 1 house to the property. Note the rule is we can only start building houses when we own all the properties with the same color
   – C2, C3, C4, … cost of adding the house 2/3/4/Hotel to our property
3. the income we earn when a player steps on a property we own
   – IL, earning from a bare land
   – I1, income from a color of land with 1 house
   – I2, I3, … income from a color of lands with 2/3/4/Hotel

Combine the numbers above, we then get a simple equation: EV of a property = probability* Ix — CL(if x >= 1, then CL has to be the sum of all the property costs of the same color bar lands)

So How Do We Get These Numbers To Calculate Ev?

Costs and income per property are easy to log as they are part of the game setup.

How Do We Know The Probability of landing on a square?

We then wrote a little program to simulate this game, ran the simulation 1000 times, and finally analyzed the gathered data in Excel. Let’s dive in.

Monopoly- Rules And Game Logic Breakdown

We’ll start this game analysis with a quick recap of the rules that govern the movements in a standard monopoly game. Then, as part of the simulation, we will examine the impact of game length and number of players.

The Rules That Affect The Probability

Our goal is to ensure you are the Champion of Monopoly among your friends, though they might not be your friends for much longer.

Hopefully, at the end of this Monopoly series, you will have a clear idea of which property to buy and how much you should trade your property for. With this purpose in mind, we need to first review and clarify the game rules for us to figure out the probability of landing on any given square.

I am sure most of you have played this game, so feel free to skip this step. And for those who have never played Monopoly (anyone?), the board of this game looks like:

There are 40 small squares surrounding the big square in the middle, starting from 0, i.e. “Go/Collect $200 salary as you pass”). And back to square 0 after square 39, i.e. Boardwalk (the UK edition) /Orgrimar (the WOW edition).

The Movement Rules That Dictate A Player’s Position At The End Of Each Turn

At the beginning of this game, all players start at GO (square 0). And where a player lands before they finished his turn depends on two factors,

1. The player moves according to the result from rolling 2 dice (2 regular d6)
2. Following the rules of the square could result in the player’s final position being changed. See the list below:

Go To Jail, square 30. Any player who lands on this square is sent straight to square 10 (Jail)

1. Treasure Chest, squares 2, 17 and 33. There are 16 cards in the Treasure Chest deck in total. When you land on a Treasure Chest square, you will have to draw a Treasure Chest card, and follow the instruction on the card which possibly you will advance to one of the following locations,
   – Advance to 0 (Go)
   – Advance to 10 (Go To Jail, go directly to the jail, do not pass “GO”)
2. Chance, squares 7, 22 and 36. Similar to Treasure Chest, when you land on a Chance square, draw a Chance card from a deck of 16. The following cards in the Chance deck have the potential to move you around,
   – Advance to 0 (Go)
   – Advance to 5
   – Advance to 11
   – Advance to 24
   – Advance to 39
   – Advance to 10 (Go To Jail, go directly to the jail do not pass Go)
   – Advance to the nearest transport, 5, 15, 25, 35, depending on your current location
   – Advance to the nearest Electric or Water Utility, 12, 28, depending on your current location
   – Go Back 3 spaces from where you are
3. Rolled a double three times in a row! You are clearly a criminal if you get double this often and deserve to be locked away in jail immediately!!

Does The Number Of Turns In A Game Affect The Result?

The good thing about simulating the game with a program is that we can repeat the game as many times as we like while making small changes, and do it quickly to improve the Signal/Noise ratio.

Our assumption is that the change in the length of a game will not impact the probability of landing on a given square. Well… as soon as the game has gone beyond the point that there is no more property with no owners. Obviously, the expected value of a property will be affected, given that the expected value is governed by how many times a player lands on a square.

Does The Number Of Players Make A Difference?

The number of players is something we didn’t take into account in our simulation. You could argue that it’s a lot easier to collect a set of squares with the same color in a two-player game than in a four-player game. You are absolutely right. We also didn’t cover trading properties in our simulation.

All we care about in our simulation is calculating the EV (Expected Value) of each property. The EV will be the benchmark of each property's net worth, depending on your game's progress. With the sheet (which you can download below), you can tell when you got a sweet deal or when you really should tell your friend to stop asking for impossible prices for a property.

Old School Monopoly Simulation

It's old school in the sense that we are fully aware of some more advanced monopoly simulations out there. Feel free to skip this part. Or not, if you’re interested in how we got from the game logic to actual code.

How Does “Turn” Work In Monopoly?

To calculate the probability of landing on each square, we need first to define “Turn.” A turn should be exactly the same as when you (any of us) play: you rolled, you moved accordingly, you bought a property, and your turn ended. Here are the six steps broken down in code, from a turn starts to its end:

1. At the beginning of a game, all players start from square 0 (ie. GO)
2. A player’s turn starts with rolling two dice (regular d6), and
3. The player moves accordingly
4. Follow the instructions of the square the player landed on
5. Record the final position of the player in the reporting object
6. Pass the turn to the next player and repeat step 1

How does the “withdraw a card action” work? (Treasure Chest & Chance)

We shuffle the cards before the game starts, we draw a card from the top of the deck, follow the instruction on the card, and then put it back to the bottom of the deck. So we do the same in our simulation. Here’s the list of the rules that dictate the draw card action. Both treasure chests and chance operate the same way,

• Each deck has 16 cards
• We only shuffle the cards when setting up the game. Since all we care about are the cards that move a player, we randomly distribute this in a list of 16 items and set a variable of where in the deck we are (initially we are at card 0)
• When a player lands on Treasure Chest or Chance, look at the item where the variable currently points to and increment the pointer by 1.
• Move the player according to the result of the card that was drawn
• Restart from 0 after reaching card 15th, i.e. no reshuffle of the deck throughout the game

Result Of The Simulation

Now we have the code that simulates 1,000 games for us. Below is a screenshot of the numbers in Excel:

The numbers don’t mean much unless we start analyzing and interpreting them. In our next post, we will discuss how the data is converted into a set of numbers, which we call game strategies. And now, we are one step closer to winning!!