This article was written by Paul du Pontavice, an IB student from Bromsgrove, UK.
Introduction
Monopoly, the classic board game, is not just a test of chance and negotiation but also a dynamic system governed by probabilities. Players aim to accumulate wealth by purchasing, trading, and developing properties while avoiding bankruptcy. To optimise decisions and improve chances of winning, we can employ Markov Chains, a mathematical model for analysing systems that transition from one state to another. The key factor to consider is that not all squares are created equal. Therefore, the probability of a player ending up at certain locations, as opposed to others, is subject to vary.
Understanding Monopoly & Markov Chains
A Markov Chain is a sequence of events (or states), where the probability of moving to the next state solely depends on the current state. In Monopoly, each square on the board is a “state”; rolling the dice determines the transition probabilities between states. Certain rules, such as the “Go to Jail” square, the “Chance” and “Community Chest” cards and the “Doubles system” add layers of complexity to these probabilities.
Monopoly’s 40 spaces can be mapped into a Markov Chain’s state space. Each board square represents a state (e.g., “Go”, “Whitechapel Road”, “Jail” etc.). Special states like “Go to Jail” and “Chance” create transition probabilities outside normal dice rolls.
Modelling Monopoly as a Markov Chain
Transition probabilities depend on dice rolls (2–12) and the game rules. The former is more significant, as each roll outcome has specific probabilities. For example, a 7 is the most common roll with 1/6 probability; special squares such as “Go to Jail”, which always transitions to “Jail” or “Chance” and “Community Chest”, redirect to other spaces probabilistically. “Double rolls” also follow this principle as rolling three doubles in a row means you are “caught speeding”, landing you in “Jail”.
As there are forty possible locations on which one can end up, modelling the game requires creating a 40x40 matrix P, where each element represents the probability of moving from state i to state j. Rows sum to 1, as they represent the probability of all possible transitions from a given state.
Calculate the steady-state probabilities by solving for the stationary distribution, π, where:
πP = π
This gives the long-term probabilities of landing on each square.
Figuring out the winning strategies
The steady-state probabilities reveal which squares players are most likely to land on. Historically, these include orange properties because high traffic from “Jail” guarantees these spaces (e.g., “Bow Street” etc.) are valuable investments. It also includes red properties, as their positioning after the “Chance” square means that they also attract frequent visits.
The steady-state probabilities can also be used to prioritise houses and hotels in high-probability spaces to maximise rent collection and avoid low-probability investments. Notably, the focus should be on securing these monopolies (e.g. orange and red sets) first, as they yield consistent returns. Squares like “Mayfair” and “Park Lane”, have low landing probabilities, despite high rents. Therefore, securing a monopoly on this set of properties should be deprioritised.
Further, some cards redirect players to specific properties (e.g., “Trafalgar Square, “Pentoville Road”) increasing players’ probability of ending up on it. It is therefore rational to invest in these properties to capitalise on card-induced landings.
Taking into account other factors
While Markov Chains show the probability of landing on a square rather than another, it does not present other factors such as the cost-to-profit ratio or the number of players in the game. Thus, train stations such as “Marylebone Station” may seem attractive. In reality, however, the rent generated will remain modest because houses and hotels cannot be built on it.
Another factor that might be considered is the extent to which a player can build houses and hotels on their square. This would depend both on their liquidity and the stage at which the game is at. Since there can be only 32 houses at any moment in the game, one can buy four houses per square. Therefore, it might be rational to have four houses instead of one hotel to block another player from building houses on their properties.
Conclusion
Winning at Monopoly takes more than luck—it is a game of probabilities, and Markov Chains provide a mathematical edge. Players can outmanoeuvre opponents by modelling the board, analysing steady-state probabilities and aligning strategies with high-traffic properties. While negotiations and chance play their respective roles, understanding the game’s probabilistic underpinnings transforms Monopoly into a game of calculated dominance.
The views and opinions expressed in this article belong solely to the writer and do not necessarily reflect the views and opinions of the Warwick Economics Summit.
References: Another Voice In The Room. (2015). Everything You Never Needed To Know About Monopoly (with charts). [online] Available at: https://cjamesanderson.wordpress.com/2015/11/04/everything-you-never-needed-to-know-about-monopoly-with-charts/.
Fry, H. and Thomas Oléron Evans (2017). The Indisputable Existence of Santa Claus. Abrams.
mathsbyagirl. (2015). 7 Days of Christmas: Day 3. [online] Available at: https://mathsbyagirl.wordpress.com/2015/12/20/7-days-of-christmas-day-3/
THE GAME IN BRIEF. (n.d.). Available at: https://www.hasbro.com/common/instruct/Monopoly_Vintage.pdf.
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