Board Games and probabilistic analysis – has there been a more iconic duo? I thought my idea was so novel, to combine these two interests into a little study, but a simple web search of "Markov Monopoly" shows the wide extent to which the topic has already been covered. Nevertheless, there are various different approaches out there that all produce different results, and that’s part of the beauty. This one leans more theoretical than practical, but perhaps someday I will seek out a deeper scope somewhere further down the rabbit hole.
Monopoly has been one of the world’s most widely played board games for the greater part of the last century, and to be a successful player requires deliberated strategy in negotiation and resource management. While such human-controlled skills are crucial, there is a heavy reliance on chance as gameplay is governed by rolling dice and often drawing cards. Monopoly is infamous for stirring up cutthroat in-game competitiveness, so knowledge about the likeliest landing areas on the board equips the player to make a more informed decision on how to achieve the coveted end-goal of driving their friends and family to bankruptcy.
Finance is the governing indicator of player progress in Monopoly, where every decision made by a player should ultimately improve their relative net worth. Having said that, analyzing the financial aspects of the game is beyond the scope of this investigation, which focuses solely on the player’s movement around the board. The objective here is to determine the probabilities of landing on any of the 40 spaces on the board when the number of rolls approaches infinity. These are known as the steady-state probabilities, which will be computed by employing the law of total probability and performing a Markov chain.
Overview of gameplay and assumptions for analysis
Hundreds of spinoffs from the original Monopoly exist today, each with their own set of themes and variations to gameplay. This investigation is conducted in reference to the original, standard rules version of the game, nowadays produced by Hasbro Inc.
Board of play
The Monopoly board is square in shape, with its edges lined by 40 positions of play, which we’ll refer to as spaces. The game begins at the space on the bottom-right corner of the board, known as Go. Most spaces represent a property that a player can choose to buy, though some serve other functions, discussed shortly. As mentioned earlier, this investigation is only concerned with the movement of the token around the board, so any details regarding monetary implications of the game are disregarded. Figure 1 above shows an outline of the game board with its spaces labeled numerically from #0 to #39. This numbering convention will be used throughout the investigation.
Drawing cards
Three spaces are labeled "Community Chest" (#2, #17, #33, referred to from here as CC1, CC2 and CC3 respectively) and three are labeled "Chance" (#7, #22, #36, or CH1, CH2 and CH3 respectively). Landing on any of these will prompt the player to take a draw card from its corresponding deck at the center of the board, and follow its instructions to perform either a monetary transaction, or a movement of the token to some other space; as mentioned earlier, this investigation is only concerned with the latter. The official rules call for a drawn card to be returned to the bottom of its deck (i.e. the sequence of drawn cards is constant throughout a game), but for this investigation, we will reshuffle drawn cards back into the deck to ensure that all draw events are random and statistically independent.
Jail
The Jail space (#10) serves two functions. If a player lands in jail solely by advancing the number indicated by the dice, they are said to be visiting. All other means of landing in jail warrant arrest, such as being instructed by a drawn card, or landing on the Go to Jail (#30) space (which we’ll denote as GTJ). To be released from jail, the player must either pay a fine, use a "get out of jail free" card, or roll doubles on their turn, else they remain under arrest. As mentioned, we are disregarding the aspect of the game that concerns rolling doubles, so we can assume that upon arrest the player immediately pays a fine and leaves on their next turn; for the purposes of this investigation there is no difference between being in jail as a visitor or a detainee.
Analysis
Using the conventions established above, we can begin the procedure for calculating the steady-state probabilities of occupying each board space.
Dice rolling events
Let m and n denote the values shown on each of the two dice rolled, where [m, n __ ∈ 1,6]. Since the dice are fair and their results are statistically independent, the probability of rolling each combination of m and n is 1/6 ∙ 1/6 = 1/36. Let r denote the total value of a roll, i.e. r = m + n (thus [r ∈ 2,12]), and q the number of combinations of m and n that produce r. For example, r = 12__ is produced by just q = 1 combination (where m = 6 and n = 6), and r = 3 is produced by q = 2 combinations (where m = 2 and n = 1, or m = 1 and n = 2). The probability distribution of P(R) is shown in Figure 2 below.
Active and inactive spaces
Movement is occasionally ordered by a card drawn upon landing on any of the six CH or CC spaces (#2, #7, #17, #22, #33, #36), or by GTJ (#30). These seven spaces are the only ones on the board that, if landed on, may send the player to some other space as part of the same roll, so we’ll call them active spaces. The remaining 33 spaces are with complete certainty the final destination in any roll that involves them, and are hence dubbed inactive spaces.
A noteworthy phenomenon occurs when a player landing on CH3 (#36) draws the card instructing them to "go back 3 spaces" and thus land on CC3 (#33). This creates a nested probability distribution within CH3 for destinations that may be directed by CC3. I have incorporated these into the total probability distributions presented in Table 1 below, which show the individual probabilities of being of being moved to space j by each active space.
Notice that Go to Jail (#30) has a very simple distribution: a probability of 1 of being moved to Jail (#10) and 0 anywhere else. Landing here, you are certain to always be sent to jail.
Probability of moving between any two spaces
Let M_{i,j} denote the event that the token will move from space i to space j. To compute its probability, every potential contributor to movement must be considered, which consists of the probability of being moved solely by the roll of the dice in addition to being moved by each of the seven active spaces. These eight events in total are mutually exclusive and collectively exhaustive, as expressed in the equation below per the law of total probability, where [a __ ∈ CC1 CC2 CC3 CH1 CH2 CH3 GTJ]. Substituting these active space descriptors with their corresponding index numbers gives [a ∈ 2 7 17 22 30 33 36].
This is good start, but it has a few holes; we need to make some adjustments to account for the behavior of the active spaces. If j is an active space, the first term in the equation must be excluded from the calculation to avoid double-counting probabilities. When this is the case, P(M) becomes the sum of seven probabilities rather than eight.
Otherwise, when j is an inactive space, a dice rolling event is certainly the sole instigator of motion. Mathematically this is described as "the probability of moving from i to j, given that the dice roll was j-i, is 100%", or P(M|R) = 1.
After taking these into consideration, we are able to update the initial formula and split it into cases dependent on the status of j as an active space.
This updated equation, which fully describes the probabilities of moving between any two spaces on the board, is fairly easy to solve, since all terms for P(R) and P(M|R) have been determined earlier (shown in Figure 2 and Table 1).
Transition matrix
For any given space i, there are a total of 40 different destinations j where a roll could theoretically end. Since there are also 40 values that i can take, a total of 1600 different combinations of i and j exist, thus 1600 values of P(M) that fully describe the probabilities of moving between any two spaces. With a little bit of cell reference magic, I enumerated these into a 40×40 array in an Excel spreadsheet. This is known as the transition matrix, as it represents the probabilities of transitioning across pairs of states. Then, with a little bit of conditional formatting magic, I visualized them into Figure 3 below, where P(M) for each combination of i and j is shaded based on a pale-to-dark linear gradient between the lowest nonzero probability (0.001532) and the greatest (0.173611); cases of zero probability are in white.
Note the transition matrix describes the probability of landing on j with the assumption that the roll began at i , regardless of how it got to i in the first place. This is why row #30 (which describes a turn starting at GTJ) contains probabilities at all, even though we know that a player could have never actually arrived at that space in the first place without being immediately sent to jail.
Steady state probabilities
Let T denote the 40×40 transition matrix computed above. Since the sum of all probabilities P(M) for a given i is 1, every row of T sums to 1.
Let O_j denote the event that space j is occupied, and let U_i denote the 1×40 initial state vector which contains the probabilities of occupying each space on the board at the beginning of the game (i.e. prior the first dice roll). The rules state that every player starts their game at Go, therefore the first entry (i.e. index #0) of U_i is 1 and all others are 0.
As the number of dice rolls approaches infinity, the probability distribution of O_j should converge to a constant set of 40 values that comprise the steady-state probabilities of occupying each space on the board.
All of this is tied together by applying the theory of Markov Chains. We begin by setting the initial state vector as the current-state vector U_c, and then multiply it by the transition matrix T to obtain an updated current-state vector U_cu . This process continues until the current-state vector ceases to change values, at which point it becomes known as the steady-state vector. Each iteration of multiplying the current-state vector by transition matrix simulates a dice roll event. The Python script below follows this process to compute the steady-state vector by extracting probabilities from the transition matrix that I stored in "Monopoly.xlsx".
The resulting vector is the steady-state probability distribution of O_j. According to the program, a total of 284 iterations of the Markov chain (i.e. 284 theoretical rolls of the dice) were performed for the solution to converge.
Results
Let’s plot our shiny new steady-state distribution, sorted by probability from highest to lowest.
Jail (#10) is by a large margin the most commonly landed-on space, with a steady-state probability of 0.0587. This means that as the number of rolls approaches infinity – or, more specifically, the iteration count of 284 – the player has a 5.87% chance of being there. Such a high relative probability is not surprising, considering the combined influence of all active spaces to send a token to jail.
As a result of this phenomenon, Jail acts as somewhat of a magnet for tokens, consequently increasing their probability of landing in the spaces closely following. We see this in the fact that seven of the twelve most commonly landed-spaces after Jail are within one dice-roll away.
The highest-probability property is Illinois Ave. (#24), which is 14 spaces ahead of Jail and thus equivalent to twice rolling a 7, which is of course the most likely value shown by the dice (per Figure 2). This fact, in addition to the potential for CH2 (#22) to give the instruction to "advance to Illinois Ave.", gives it a higher steady-state probability than even Go, which is influenced by 6 of the 7 active spaces.
Implications of steady-state probabilities
So, we know about where we’re likely to end up on the board, but how much can this help us improve our strategy? It’s hard to say without raising some new questions. For starters, what Monopoly game lasts 284 rounds?! I mean, are there even enough tables in the world to flip to make it that far in a game?
Since a typical game lasts only about 30 rounds per player, it’s important to know how reliable our steady-state distribution is after fewer iterations. The quantity 284 itself is somewhat arbitrary because it is dependent on the tolerance used by the program to determine equality between floating point values. Let’s explore this a little further by adding a function to the Python code from Script 1 that takes a set number of iterations i_max as an argument.
It’s quite a mouthful to constantly refer to "the greatest difference in Probability observed between corresponding elements in the steady-state vector and the current-state vector", so let the word tolerance define it here. The function above computes the tolerance for each number of iterations between 1 and i_max = 30. Still a mouthful? Then feast your eyes on this:
It’s clear to see from the plot that the tolerance drops below 1% after about 11 rolls, and is hardly detectable at all after about 25 rolls. We can be therefore be confident that there is no need to wait an absurdly large number of rolls to have faith in the steady-state probabilities, but should also keep in mind that they don’t really start to take shape until a few rolls into the game.
Optimizing property ownership with groups
Though the financial aspects of the game are outside the scope of this investigation, it is worth taking a glance at how our steady-state results can be meaningfully clustered. Monopoly rewards players that own and develop upon the board’s 28 property spaces in their predefined categories, whether by railroads, utilities, or real estate in eight color groups. These are shown with their combined probabilities below.
At a high-level, the railroads are the winners. This is intuitive because their group has greater membership than any other – four railroad spaces out of the forty on the board equals 10%, plus some change to account for influence from active spaces. The runners-up are the orange/red groups, which is also consistent with our earlier observation of the spaces that are a couple of high-probability dice rolls away from Jail.
Where do we go now?
We can continue to pick out observations like this all day, but these are not particularly useful, since they neglect to consider that not all properties are created alike.
Without taking into account the properties’ costs and expected rates of return, only part of the picture is painted by their steady-state probabilities. The objective of the game is, after all, a monetary one, so we cannot draw a robust plan for strategy until we incorporate money into our analysis.
These financial implications have also been widely explored by others, and though I’m a little burned out from Monopoly at the moment, I fully expect to take my own stab at it sometime in the future. For now, though, I am content with my main takeaway from the investigation so far: this game is way more fun to analyze than to actually play.
