If you’ve played a bit of monopoly, you start to realize that the orange properties seem to be the best, while Boardwalk (the last dark blue one) is rarely ever worth it (until you crush your brother with the $2000 rent that comes with a hotel). However, how about the other colors? Is it worth buying one railroad property? And is staying in Jail so bad after all?
The role of the dice (no pun intended):
Of course, the dice is the most important part of the game. It will determine where each player lands, pays rent, and goes bankrupt. When rolling two dice, if we consider the possible combinations of the sum (2 to 12), we see that a sum of 7 appears the most frequently (16.7% of the time), while of course rolling a 2 or 12 only occurs 1 ⁄ 36 or 2.8% of the time. Thus, from the start one is most likely to land on the chance space, but also Vermont and Oriental avenue. Thus, as a player, if you land on an unowned space and other players are 6 – 8 spaces behind you, you should highly consider buying the space – and of course, a house if possible. With the rule that rolling doubles allows another roll (up to 3), landing on properties after are possible, but the chance is so low that don’t consider this possibility.
After many turns, with just this rule the probabilities would even out. So, we must consider another important aspect of the game – cards, and the existence of jail.
Community Chest and Chance cards:
While many of these cards give out money or take money away from you, some move you to different spaces, including Jail. Out of the 16 chance cards, 9 send you somewhere else, while in community chest 2 of the cards will change your position. The chance cards mean that the probability on landing on the 4 railroads, New York avenue (Red), St. Charles Place, Illinois Avenue and Boardwalk (Dark blue) are higher than they would normally be. For example, without drawing a chance card the probability of landing on boardwalk on the first turn would be 0 (you can’t go that far without ‘speeding’ (rolling 3 doubles) and thus being sent to jail). However, because of the chance card the probability of landing on boardwalk is 1 ⁄ 6 (probability of landing on Chance) x 1 ⁄ 16 (getting a ‘Go to Boardwalk’ card) = 1 ⁄ 96 = 1.04% – still low but possible, and higher than the probability of landing on free parking!
As there are 4 different ways of going to jail (go to jail spot, speeding, chance card, community chest card), it is perhaps the most important spot in monopoly. Jail is the reason why the orange properties are the best – looking back to the first point, everyone coming out of jail is most likely to land on Community chest, St. James (Orange) and Tennessee (Orange). Landing on the jail spot has a similar probability to surrounding properties, while speeding isn’t actually that rare. Although the probability of speeding on any given turn is (1 ⁄ 6)3 = 1 ⁄ 216 , with 4 players after each person plays 11 rolls, there is a 20% chance of one of them speeding. With this knowledge, we can now calculate the probability of landing on any space from one space. This is done using a Markov chain system, where the probabilities in a matrix eventually stop changing after many iterations. The actual Markov chain system in this case is too complicated, but there is a pretty good paper on it:
J. Laurie Snell Finite Markov Chains and their Applications, The American Mathematical Monthly (1959), 66 (2), 99-104.
If you want to know the basics, we essentially take Matrix A, which stores the probability of moving from one space to another (note it will not be symmetrical because the probability of moving from GO to Boardwalk is vastly different to the probability of moving from Boardwalk to GO). We multiply this by Matrix B which stores the number of players in each space.
The resulting matrix is multiplied by Matrix A, and this repeated until we reach a steady state probability matrix. In monopoly, this can be graphed:
Considering the prices to buy each space, we can clearly see that Orange and Red are the best ones!
We are almost done, but there is just one thing left to consider – What about the houses?
Investing in houses and hotels:
This is a really short point. By taking the steady state probability matrix into account but also the costs of building houses (or a hotel if you’re that rich), one can come up with a breakeven time (ie how many rolls before the amount you paid is recuperated). Of course, anything after that is a profit:
As you can see, buying hotels aren’t always the best idea in terms of profits. Alternatively, if you’re playing a short game, you might not want to invest in houses at all!
Buy the Red and Oranges if possible. Also, invest in B&O railroad especially if you can get more of them. Water Works isn’t such a bad idea. Try to buy 3 or 4 houses. Lastly, don’t worry if you get sent to jail – it might be a blessing!
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