Table of Contents

• Long term probabilities for each square
• Expected income per roll for each property
• Expected income per roll for other squares
• Payback times for purchases
• Mortgage payback and loss times
• Conclusions from all of this
• The Markov matrix

I was intrigued enough with this problem that I started working on trying to find the probabilities for landing on the different squares with all of the rules taken into account. I ran into some interesting problems but finally came up with the right answers, which you will find here along with some other useful derived data. Incidentally, I'm not much of a Monopoly® player myself, but I've always enjoyed interesting problems involving probability and statistics, of which this was one.

I first wrote a C program that simulates a single person rolling the dice and moving around the board a great number of times. It included all of the rules for going to jail and the Chance and Community Chest cards. Although this gave good aproximate answers, I decided that I wanted to write another program that would find the exact probabilities using a Markov matrix, which was the method described in a simplified form in the Scientific American article. I used an extended version of this program to generate this web page.

I discovered that it is really necessary to model two different strategies. When you are in jail, it is possible to get out immediately by paying $50 or by using a Get out of Jail Free card, but you can also wait until you roll doubles or pay on your third turn in jail. Early in the game it is often best to get out as soon as possible so as to have more opportunities to buy property. Later in the game, it is best to stay in jail as long as possible to avoid landing on an opponent's property. The choice of strategy changes the probabilities for all of the squares of the board because the more time you spend in jail, the less you will land on other squares. I computed probabilities for both cases and have included them here. I have compared the results from my simulation program and the Markov one and the results are very close, which leads me to believe that these results are reliable.

In the process of figuring all of this out I ran into an interesting difficulty. When trying to calculate the probabilities exactly using the Markov matrix, it is necessary to estimate the probability--for each square--that the last two rolls of the dice are doubles (since three doubles in a row sends you to jail). First I used an estimate of 1/36, but in practice it's different for each square and it's not that high for any square. I used my simulation program to find the empirical probability for each square and then used these values in my Markov matrix program. I simulated 32 billion rolls to make these estimates, so I believe they are reliable and any deviation from their exact values is extremely small. Interestingly, the probabilities of two previous rolls being doubles is slightly different on certain squares for the two jail strategies. Additionally, the average roll when landing on a utility is a bit lower or higher than 7 depending on the utility and the jail strategy, which affects the rent value.

Just as I finished putting together this web page, I realized that there is a more efficient way of making the probability calculations. It would have been better if I had created a larger Markov matrix with three entries for each square. The first would be for being on that square not having rolled doubles in the last roll. The second would be for having rolled doubles on the last roll but not the one before, and the third for having rolled doubles the last two times. This much larger matrix would insure that the exact probabilities would be generated and there would be no need for doing simulation at all. If I ever get the time and inclination I may redo this in that way. I don't expect it would change any of the probabilities by more than one or two in the third decimal place, but it would still be a better way of doing it.

Below are the two tables of probabilities, expressed as percentages, for landing on any of the squares of the game. I have separated the probabilities for visiting jail and being in jail. The number given for a particular square is the percent chance that a player will land on it after a single roll, in the long run. For example, on average, about 3.18% of the rolls will result in the player landing on the most probable non-jail square, Illinois Avenue. Of course, for a single roll starting on a particular square, the chances will be quite different, but in the long run it will average out. At the bottom of this page I have included a matrix containing the probabilities for landing on any square after one roll starting from each square on the board. It's very big and cumbersome but it's there if you want it.

The first column indicates the average probability for landing on each square when the player's strategy is to get out of jail immediately when he gets in. The third column is for a player who stays in jail as long as he can once there. The second and fourth columns hold the rank of the squares from most often to least often visited. Note the higher rank of the squares 2, 4, 6, 8, 10, and 12 in front of Jail caused by the increased likelihood of getting out of jail with doubles. In this table, and each of those following, I have split the jail square into two, "visiting jail" and "in jail", the latter of which is listed at the bottom of the tables.

Note that these numbers indicate the percent chance that you will "end up" on each square after a roll--not just land on. This takes into account being sent to another square because of a Chance card or for some other reason. For example, there is no possibility that you will end up on the Go To Jail square, because if you land on it, you immediately go to the Jail square and that one is counted. This is the reason the Chance and Community Chest probabilities are low.

Long Term Probabilities for Ending Up on Each of the Squares in Monopoly®

I've used the above information to draw other conclusions about the game such as how much money you can expect to earn on average from each property, and more importantly, how many opponent's rolls it takes on average to earn back the money invested to buy property, houses, or hotels.

First, I have created two sets of tables containing the average income per opponent roll for each property, including railroads, and utilities. One set assumes preferred short jail stay and the other preferred long jail stay. The values for the railroads include the extra income from the two Chance cards that result in double railroad rent.

Expected Income Per Opponent Roll on all Properties Assuming Preferred Short Jail Stay

Expected Income Per Opponent Roll on all Properties Assuming Preferred Long Jail Stay

The following table details the expected income and expense from the non-property squares on the board. The value for passing or landing on Go was arrived at through simulation, whereas the others were computed exactly. The income associated with the Chance and Community Chest squares is somewhat high because I was unable to include the cards for property repairs since the cost is based on the number of buildings you own. I also left out the Community Chest card where you collect $50 from each player and the Chance card where you pay each player $50. These last two should even out in the long run. I have assumed that landing on the Income Tax square always results in a $200 payment.

Average Income per Roll from other Squares

The following two sets of tables give the average number of opponent rolls needed to regain the cost of the last improvement to the property or the cost of the property itself. These figures are quite useful for determining what improvements to make first. Two of the groups of figures in this table are a bit fuzzy. I have assumed that when one lands on a utility, the roll had the usual distribution probabilities, which isn't necessarily true, but it should be very close. Also, when I give the payback number of rolls for the railroads, I average the payoff of the other railroads that might be owned. For example, if you buy Pennsylvania Railroad and you already own another railroad, your payback is faster if that other railroad is B. and O. rather than Short Line. So, I have averaged out the contributions from the other owned railroad.

These numbers were calculated by taking the cost of whatever was bought and dividing it by the increased average rent from the table above. Note that the figures in this table are rounded to four decimal places, but the ones used to make the calculations for the following table are to full precision. In other words, if you try to calculate these numbers by using those in the table above, you won't get quite the same answer, and it will not be as accurate as what is here. Once again, I have included a set of tables for a preferred short jail stay and for a long one.

Expected Number of Opponent Rolls to Recoup Incremental Cost (Short Jail Stay)

Expected Number of Opponent Rolls to Recoup Incremental Cost (Long Jail Stay)

It's also useful to know what the payback times are involving mortgages. For example, if you own all of the red properties and have no houses on them, you may want to know which one to mortgage so that you lose the least income while it's mortgaged. Below there are two sets of tables -- for both short and long jail stay strategies -- containing the expected number of opponent rolls it takes on average to lose the mortgage value of each property due to lost rent and the expected number of opponent rolls it takes to pay back the cost of unmortgaging each property. I have included values for all of the different possible ownership situations. The calculations were performed in the same way as in the tables above.

Expected Number of Opponent Rolls to Lose or Recoup Mortgages (Short Jail Stay)

Expected Number of Opponent Rolls to Lose or Recoup Mortgages (Long Jail Stay)

So, now that we have all of this data, how can it be used to play better Monopoly®? It should be possible to add up all of the expected income values for all of the properties in a partially completed game and get an estimate of who is likely to win in the end. Of course, this ignores lots of factors in a particular game such as cash on hand, bargaining, and luck. The capricious nature of short-term probability insures that you never know what will happen in the next few rounds.

It's also possible to make some general observations about the properties. The railroads are excellent investments, particularly when owned together, although in absolute income terms they don't keep up with heavily built on properties later in the game. The best return on investment to be found is from putting a third house on New York Avenue. In fact, the third house has the fastest payoff of any building on almost all of the properties. The square most landed on other than Jail is Illinois Avenue, and in fact a hotel there will bring the most income other than a hotel on Boardwalk. By far the worst individual investment is to buy Medeterranean Avenue without first owning Baltic. That's not to say that you shouldn't buy it, but it's not going to make you much money without quite a bit of construction. The properties between the Jail square and the Go To Jail square are landed on the most, because of the jump caused by landing on Go To Jail. The Orange ones have the biggest bang for the buck as far as building goes.

One more thing is worth pointing out if it's not obvious. The numbers given here are all long term averages. They are useful as guidelines, but particular circumstances must be considered in practice. For example, if you have the opportunity to put a house on either Illinois Avenue or Marvin Gardens, in general you should put it on Illinois Avenue. If one of your opponent's tokens is currently on Indiana Avenue, however, then he's got about a 14% chance of landing on Marvin Gardens on the next roll but no chance of landing on Illinois Avenue. Considering this, it's better to put the new house on Marvin Gardens in this particular situation.

Here are the two Markov matrices, which are quite large. The point of these tables is to show how likely you are to land on any square of the board starting from any other square. For example, the figures in the Baltic Avenue row are the probabilities that you will land on each one of those squares with one roll of the dice starting on Baltic Avenue. Similarly, the column associated with Baltic Avenue contains the probabilities that you will land on Baltic in one roll starting from any other square (those appearing in the left-most column). Note that these numbers take into account the probability that the two previous rolls before this one were doubles. Because this table is so large, I've duplicated the row and column headings a couple of times for readability.

Individual Square Probabilities for Preferred Short Term Jail Stay (as Percentages)

Individual Square Probabilities for Preferred Long Term Jail Stay (as Percentages)

I'm interested in comments on this page, particularly concerning any mistakes you may have noticed and suggestions on readability.

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