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Roll the Dice

Monday, 1 September, 2014

Monopoly: it’s all fun and games until somebody puts a hotel on Mayfair. Thanks to this Parker Brothers family favourite, we’ve all experienced the dizzying highs and soul-crushing lows of trying to become a capitalist land baron. We’ve all got our favourite properties, double-dipped into the Community Chest when times were tough, or won second prize in a beauty contest.

But what if we told you that Monopoly—much like life—is not just a game of chance? What if, by harnessing the power of maths, you could become an unstoppable Monopoly magnate? During secondary school, I bet you asked your maths teacher whether anything you were learning would ever be useful in real life. But here, we will demonstrate to you how a bit of probability and economics may help you win your next family games night.

What is the best property set to buy?

Some say it’s the red Trafalgar trio, others swear by the Pall Mall purples, and only mad dogs rely on the utilities. We consulted the Markov Chain, a giant matrix that houses the probabilities of moving from one square to another. We then threw in some economics, and came up with this equation: rent income (ie. the amount some sucker has to pay when they land on your property) divided by the purchase price, multiplied by the probability of landing on the square. To find the value of the entire set, we simply added the numbers together. Immediately, we can see that the light browns (Old Kent and Whitechapel) and the dark blues (Park Lane and Mayfair) are at a big disadvantage, as there’s only two properties in each set, as opposed to three. Here, we also notice the effects of the big Monopoly misprint, which went un-amended for almost two decades. The original rental income for Piccadilly was $22, when it should have been $24! This unfortunate mistake banishes the yellow set to fourth on our list.

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How does proximity to the big house effect property desirability?

In the real world, properties adjacent to prisons are not known for their investment value. However, in Monopoly, the fact that it is possible to go directly to jail from four squares (three Chance squares and the scary Go To Jail copper) means hardened crooks coming out of their stay in the cooler are likely to make a stop at the red and orange sets. This boosts the probability of landing on these squares, and thus the benefit of buying those properties to trap ex-cons into coughing up rent.

This is further confirmed by the uneven probability of the numbers that you’ll roll each turn.  When two dice are involved, seven becomes the most likely number to roll—followed by six and eight, then five and nine, and so on. So which property is seven squares from jail? It’s actually a Community Chest square, but it’s right in the middle of the orange set of property.

Is the secret to success really buying up all the stations?

To confirm or deny this old wives’ tale, we created a table that lists the expected return per lap of owning different numbers of stations. Owning all stations will give you an expected return of just under 10 per cent, which corresponds to an expected amount per player of $21 each lap. This means that if you have four people playing, you can expect to cash in $63 on average every time the players pass Go. This means the stations will pay for themselves in around 12 laps. They may not look like the most threatening properties, but they’ll offer a steady income stream.

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Will this new wisdom make me win every future game of Monopoly?

No. Monopoly still owes a lot to random chance, and the role of mathematics here is to try and bring some order back to the chaos—a way of wading through uncertainty. With that said, if you play a million times, you’re pretty much guaranteed to win at least a couple more times that you otherwise would have.