Markov recommends Avenue Henri-Martin

Buying properties and building houses and hotels on them, forcing the other players to pay ever more exorbitant rents… You guessed it: Monopoly, invented in the United States a century ago (see box below).

The game of Monopoly ------------------

The game is played on a board with 40 spaces arranged around a square, with 10 spaces on each side. On each turn, a player rolls two dice and moves their token forward by the corresponding number of spaces, starting from the initial space ("Go"). Players may buy some of the spaces they land on (at a predetermined price for each one), allowing the owner of a space to charge a certain amount to any player whose token subsequently lands there. By paying a certain price, players can build houses or a hotel on their properties, increasing the amount other players must pay them when they land there. The spaces have different names depending on the country—or even, in recent years, the city. In the best-known French version, they are named after streets in Paris.

Some spaces cannot be bought, including those that require the player to draw a card containing an instruction to follow. Then there is the famous "Jail" space, as well as the space that sends players there and forces them to remain there.

Since tokens move according to the chance outcome of the dice, one might think that Monopoly has little to offer from a mathematical standpoint. In fact, the game provides a textbook example of a Markov process, an important category of stochastic processes—systems that evolve randomly and can be studied using probability theory. Today, these processes are a major concern in applied mathematics, to the point that this field, which emerged in the early 20^th century, is now among the most widely studied topics in research laboratories, across fields including telecommunications, epidemiology, and finance.

Markov processes and Markov chains -----------------------------------------

A Markov process is a stochastic process in which knowing the situation at time n is enough to describe the probabilities of the various possible situations at time n + 1. Monopoly is one such process. If a token is on the Go space, it makes no difference whether the game has just begun or the player has just completed a full circuit: the probability that the dice will take the token to Rue Lecourbe is the same in both cases. Such a process is said to be memoryless: the past is of no help in predicting the future; only the present situation matters (see box below).