It’s just after Christmas: and many of you will be sitting around with your family playing board games. Contrary to popular belief, these games are not about family spirit, instead they are about bringing out your competitive side and asserting your supremacy over the family, so let’s have a look at the statistics of Monopoly, and how to play these to your advantage!
So we all know the classic family board game, Monopoly, where players roll dice to move round the board, and purchase and improve their properties. If you land on an opponent’s property, you have to pay them rent. This rent is higher if they build houses or hotels on these properties, but you can only build on a property if you own all the properties of that colour group. If you land on a property that you cannot pay the rent for, you are bankrupt and out of the game. The winner is the last remaining player after everyone has been bankrupted.

The strategy is clearly to get a monopoly on one specific colour group (or more if you can!) rather than get several individual properties, but not all colour groups were created equal! As you go round the board they get more expensive, the cheapest being Old Kent Road at £60, to the most expensive, Mayfair, at £400.

To best decide what colour groups to go for, we need to put together some tables about the total cost of the colour group, and the income that would be received from landing on each of the properties.

In the tables, I have considered the income from each property being landed on once. Although there are only two properties in the Brown and Dark Blue groups, this is accounted for in the fact that the costs for houses and the properties themselves are lower. I then took the income that the number of properties or houses would get and divided them by the costs of the properties and improvements. Generally, the value gets larger (and better) the greater the number of houses are placed: the game is trying to provide an incentive. However, what happens to this ratio as we move round the board?

For the properties without improvements, it seems that the more expensive ones get better. But the more houses you add, the less true this becomes. Dark blue is still the best value for 1, 2 and 3 houses, but with 4 houses and hotels it is edged out by the much cheaper Light blue and Orange properties. Whilst the Browns are very clearly the worst investment when looking at the graphs of small developments, they actually overtake the Reds, Yellows and Greens in value when hotels are placed on them.

In the later graphs, a “wavy” pattern becomes clear. This arises because two adjacent colour groups have the same prices to add a house or hotel, for example, the Brown and Light blues will both cost £50 to put a house on. The Pinks and Oranges both cost £100, the Red and Yellows £15

According to these graphs, by far the best investment in the game would be building hotels on the Light Blues: at a total cost of just £1070, each of them being landed on once would return you £1700.0 and the Green and Dark blues £200. However, each successive property group has higher rent, meaning that the more expensive of the two property groups that share a house development price will give better value as more of them have been built, so this leaves the better half of the colour groups as being Light blue, Orange, Yellow and Dark blue.

There are two sets of properties that we haven’t yet considered: the Stations and the Utilities. You cannot build on either of these properties, so the options remain quite limited, but owning 3 or 4 railway stations is an excellent investment – the main advantage is that the profitability is extremely high for such a small input of capital, only £200 per station and no building is required.

The Utilities are on average a mediocre investment, although your income from them will depend much more on luck as the rent is directly proportional to the roll of the dice that brings the player to that square.

However, we haven’t considered whether people will ever land on your properties at all! This is not quite as random as you may think, as both Jail and the Chance/Community Chest cards have a massive impact on which squares are landed on. In order for a property to be a good investment, it must both be landed on frequently and have a high level of rent compared to its price of purchase and development.A long term computer simulation has been run on the American version of the game (only the place names are different) to rank the frequency of the various squares, and that is listed here.

Jail is, by far, the most common square to be on, and this is because there are several ways to get there: two Chance/Community Chest cards, rolling three consecutive doubles, or landing on the “Go to Jail” square. The railroads are also very popular, with 3 of them in the top 10, the exception being Liverpool St Station, as that loses 11.11% of its incoming traffic to the “Go to Jail” squares. The oranges are also very common as they fall 6, 8 and 9 squares after jail, giving you a 39% chance of landing on an orange after going to jail.

The Dark blues have an interesting contrast: Park Lane is very weak as it loses more than 16% of its traffic to “Go to Jail,” whilst Mayfair is very common as there is a Chance card that sends you directly there.

The first three colour groups are all heavily affected by Jail as well. The Browns and the Light blues are both much weaker than one would expect as whenever someone is sent to jail, they go back over all properties further on but to not go over those properties again! Even though the Pink properties are after jail they still suffer, as there is only a 13% chance of hitting a Pink after jail.

The Reds are reasonable frequently visited, although they are extremely poor for developing properties on, so they should not normally be a target.

Considering all this, there is only one property set that truly stands out well beyond all others: the oranges. They are among the best value building properties, and are also the most frequently visited colour group.

There remains only one more question to be asked: you have acquired a monopoly, but how many houses should you build on it?

This gives somewhat strange results, and does not seem to follow a particularly logical pattern. For the Browns and the Light Blues, there is an upward trend in profitability when increasing the houses, so hotels are the best options, but strangely, in Reds, Greens, Yellows and Dark blue, the profitability actually decreases after the third house! So if you have three houses on there, resist the temptation to continue investing; the only situation you’d want to do this is if you were attempting to completely eliminate an opponent rather than increase your own value. However, on the Oranges and the Pinks, it still makes sense to keep building until you get hotels.

So in conclusion, my overall recommendations for the game would be the following: get hotels on the Oranges, get 3 houses on the Dark Blues, try and get all the railroads, and, if all else fails, sneak in a couple of extra notes from under the table!

We hope you’ve enjoyed this post! If you did then please check out our last two posts:

Santa Claus is coming to town and it is going to cost him £40m this year! – Though christmas is past, I’m sure you would like to see how we calculated the Santa’s schedule and why he needs 540 million litres of oats!

Nuclear Power: What is it and why is it important. (Fusion + Fission) – We take a look at the mechanics behind the power source of 18% of the world’s energy.