Probabilities in the Game of Monopoly&reg (2023)

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 startedworking on trying to find the probabilities for landing onthe different squares with all of the rules taken intoaccount. I ran into some interesting problems but finallycame up with the right answers, which you will find herealong with some other useful derived data. Incidentally,I'm not much of a Monopoly&reg player myself, but I'vealways enjoyed interesting problems involving probabilityand statistics, of which this was one.

I first wrote a C program thatsimulates a single person rolling the dice and moving aroundthe board a great number of times. It included all of therules for going to jail and the Chance and Community Chestcards. Although this gave good aproximate answers, Idecided that I wanted to write anotherprogram that would find the exactprobabilities using a Markov matrix, which was the methoddescribed in a simplified form in the Scientific Americanarticle. I used an extended version of this program togenerate this web page.

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

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

Just as I finished putting together this web page, Irealized that there is a more efficient way of making theprobability calculations. It would have been better if Ihad created a larger Markov matrix with three entries foreach square. The first would be for being on that squarenot having rolled doubles in the last roll. The secondwould be for having rolled doubles on the last roll but notthe one before, and the third for having rolled doubles thelast two times. This much larger matrix would insure thatthe exact probabilities would be generated and there wouldbe no need for doing simulation at all. If I ever get thetime and inclination I may redo this in that way. I don'texpect it would change any of the probabilities by more thanone or two in the third decimal place, but it would still bea better way of doing it.

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

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

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

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

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

First, I have created two sets of tables containing theaverage income per opponent roll for each property,including railroads, and utilities. One set assumespreferred short jail stay and the other preferred long jailstay. The values for the railroads include the extra incomefrom 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 fromthe non-property squares on the board. The value for passing orlanding on Go was arrived at through simulation, whereas the otherswere computed exactly. The income associated with the Chance andCommunity Chest squares is somewhat high because I was unable toinclude the cards for property repairs since the cost is based onthe number of buildings you own. I also left out the CommunityChest card where you collect $50 from each player and the Chancecard where you pay each player $50. These last two should evenout in the long run. I have assumed that landing on the IncomeTax square always results in a $200 payment.

Average Income per Roll from other Squares

The following two sets of tables give the average number ofopponent rolls needed to regain the cost of the lastimprovement to the property or the cost of the propertyitself. These figures are quite useful for determining whatimprovements to make first. Two of the groups of figures inthis table are a bit fuzzy. I have assumed that when onelands on a utility, the roll had the usual distributionprobabilities, which isn't necessarily true, but it shouldbe very close. Also, when I give the payback number ofrolls for the railroads, I average the payoff of the otherrailroads that might be owned. For example, if you buyPennsylvania Railroad and you already own another railroad,your payback is faster if that other railroad is B. andO. rather than Short Line. So, I have averaged out thecontributions from the other owned railroad.

These numbers were calculated by taking the cost of whateverwas bought and dividing it by the increased average rentfrom the table above. Note that the figures in this table arerounded to four decimal places, but the ones used to makethe calculations for the following table are to fullprecision. In other words, if you try to calculate thesenumbers by using those in the table above, you won't getquite the same answer, and it will not be as accurate aswhat is here. Once again, I have included a set of tablesfor 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 areinvolving mortgages. For example, if you own allof the red properties and have no houses on them,you may want to know which one to mortgage so thatyou lose the least income while it's mortgaged.Below there are two sets of tables -- for bothshort and long jail stay strategies -- containingthe expected number of opponent rolls it takes onaverage to lose the mortgage value of each propertydue to lost rent and the expected number of opponentrolls it takes to pay back the cost of unmortgagingeach property. I have included values for all ofthe different possible ownership situations. Thecalculations were performed in the same way as inthe 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 toplay better Monopoly&reg? It should be possible to add upall of the expected income values for all of the propertiesin a partially completed game and get an estimate of who islikely to win in the end. Of course, this ignores lots offactors in a particular game such as cash on hand, bargaining,and luck. The capricious nature of short-term probabilityinsures that you never know what will happen in thenext few rounds.

It's also possible to make somegeneral observations about the properties. The railroadsare excellent investments, particularly when owned together,although in absolute income terms they don't keep up withheavily built on properties later in the game.The best return on investmentto be found is from putting a third house on New York Avenue.In fact, the third house has the fastest payoff of anybuilding on almost all of the properties.The square most landed on other than Jail is IllinoisAvenue, and in fact a hotel there will bring the most incomeother than a hotel on Boardwalk. By far the worstindividual investment is to buy Medeterranean Avenue withoutfirst owning Baltic. That's not to say that you shouldn'tbuy it, but it's not going to make you much money withoutquite a bit of construction. The properties between the Jailsquare and the Go To Jail square are landed on the most,because of the jump caused by landing on Go To Jail. TheOrange ones have the biggest bang for the buck as far asbuilding goes.

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

Here are the two Markov matrices, which are quite large.The point of these tables is to show how likely you are toland on any square of the board starting from any othersquare. For example, the figures in the Baltic Avenue roware the probabilities that you will land on each one ofthose squares with one roll of the dice starting on BalticAvenue. Similarly, the column associated with Baltic Avenuecontains the probabilities that you will land on Baltic inone roll starting from any other square (those appearing inthe left-most column). Note that these numbers take intoaccount the probability that the two previous rolls beforethis one were doubles. Because this table is so large, I'veduplicated the row and column headings a couple of times forreadability.

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, particularlyconcerning any mistakes you may have noticed and suggestionson readability.

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