Mathematical explorations of Monopoly—Internal Assessment
Introduction:
It was a late Saturday evening, when my friends and I took a study break and decided to play
Monopoly.
Monopoly, one of the most classical family dinner games, ends, according to the rules, when
all players go bankrupt except for one. For me, Monopoly ended that night with my friend
refusing to lose and in a fit of rage rampaging through the board, leaving a mess of miniature
hotels and fake money behind.
Every time I play the vicious game, momentarily turning into a money-grubbing monster
devoted to thrust my loved ones into poverty, I ask myself: if my brother previously landed
on Avenue Mozart, should I build there? Should I stay in Jail and wait my turn or pay the
fine? Are Rue de la Paix and Avenue des Champs Elysées worth purchasing if no one ever
lands there? All of these questions can be answered, to some extent, with probabilities. In
other words, if you are already mercilessly battling for money with your family and friends,
might as well do it smartly and increase your odds of winning.
Monopoly is a game of chance, and like every game of chance, the math behind it can crack
it whilst efficiently annihilating your opponents/loved ones. Extensive research led me to
realise that the math behind Monopoly in fact, has long been developed by other,
competitive and passionate players. However, I also realised that despite the clear
indications in the rule booklet to not by any means make up your own rules, many of us do
just that. It was hence thanks to my friends and family’s rebellious rule-breaking attitude
that I decided to look into the mathematics behind Monopoly and the implications making
our own rules have in said math.
Monopoly: the rules
Monopoly is a classic American board game “in which players engage in simulated property
and financial dealings using imitation money.”1
The very basics of the game are similar to that of most board games. To decide who goes
first every player throws the two dice and the highest scoring one begins. Whenever its
one’s turn, you roll the dice, and move along the board in the indicated direction according
to the number drawn from the dice— the square you fall into will determine your next
move. You can land in various types of squares:
Properties—
Can be bought from the bank and built on only after owning the rest of its equally
coloured neighbouring ones.
Vary in price for purchasing and building according to its location on the board.
Their possession includes a card with its details including worth, mortgage and rent
value
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, If already owned by someone else one has to pay a fee for passing through it which
varies according to its developments (whether it has houses, hotels or neither).
If the player who landed on it does not want to purchase it, the property will then be
auctioned and can be purchased by the other players.
Railways—
There are only four of these in the game, all of equal value (M200).
Their interest rate depends on the number of railway cards one owns (one: M 25,
two: M50, three: M100, four: M200)
Public service companies—
There are only two of these in the game, both of equal value (M150).
If one owns one, the player who lands on it must pay the dice value times four
If one owns both, the player who lands on one of them must pay the dice value times
ten
Chance and Community Chest—
When one falls into one of these two types of squares one has to pick a card located
over the corresponding stack, follow its instructions and put it back.
Income tax and Luxury tax—
If on falls into one of these one has to pay the indicated tax
Miscellaneous squares—
Includes those labelled: Go, Just Visiting/Jail, Free Parking and Go to jail
If one goes to jail, there are three ways to get out:
1. Paying a fee of M50 on the next round and starting off on the Jail square
2. Using a Get out of Jail card, can be obtained from the Chance and/or Community
chest stack or bought from another player. After being used one starts off on the Jail
square and places the card back on its corresponding deck.
3. Waiting three rounds whilst throwing the dices every round expecting to draw
doubles. If doubles are obtained one is freed if not one has to wait out their
‘sentence’ and pay a fine of M50 at the end.
Lastly, every player begins with M1500 and can only be declared winner when the rest have
gone bankrupt and own no property.
The mathematics behind monopoly:
First of all, rolling the dice. Every outcome in Monopoly is derived from the dice rolls.
Assuming the Monopoly game is a 2011 French version and that both dices are fair:
2
Introduction:
It was a late Saturday evening, when my friends and I took a study break and decided to play
Monopoly.
Monopoly, one of the most classical family dinner games, ends, according to the rules, when
all players go bankrupt except for one. For me, Monopoly ended that night with my friend
refusing to lose and in a fit of rage rampaging through the board, leaving a mess of miniature
hotels and fake money behind.
Every time I play the vicious game, momentarily turning into a money-grubbing monster
devoted to thrust my loved ones into poverty, I ask myself: if my brother previously landed
on Avenue Mozart, should I build there? Should I stay in Jail and wait my turn or pay the
fine? Are Rue de la Paix and Avenue des Champs Elysées worth purchasing if no one ever
lands there? All of these questions can be answered, to some extent, with probabilities. In
other words, if you are already mercilessly battling for money with your family and friends,
might as well do it smartly and increase your odds of winning.
Monopoly is a game of chance, and like every game of chance, the math behind it can crack
it whilst efficiently annihilating your opponents/loved ones. Extensive research led me to
realise that the math behind Monopoly in fact, has long been developed by other,
competitive and passionate players. However, I also realised that despite the clear
indications in the rule booklet to not by any means make up your own rules, many of us do
just that. It was hence thanks to my friends and family’s rebellious rule-breaking attitude
that I decided to look into the mathematics behind Monopoly and the implications making
our own rules have in said math.
Monopoly: the rules
Monopoly is a classic American board game “in which players engage in simulated property
and financial dealings using imitation money.”1
The very basics of the game are similar to that of most board games. To decide who goes
first every player throws the two dice and the highest scoring one begins. Whenever its
one’s turn, you roll the dice, and move along the board in the indicated direction according
to the number drawn from the dice— the square you fall into will determine your next
move. You can land in various types of squares:
Properties—
Can be bought from the bank and built on only after owning the rest of its equally
coloured neighbouring ones.
Vary in price for purchasing and building according to its location on the board.
Their possession includes a card with its details including worth, mortgage and rent
value
1
Cite
1
, If already owned by someone else one has to pay a fee for passing through it which
varies according to its developments (whether it has houses, hotels or neither).
If the player who landed on it does not want to purchase it, the property will then be
auctioned and can be purchased by the other players.
Railways—
There are only four of these in the game, all of equal value (M200).
Their interest rate depends on the number of railway cards one owns (one: M 25,
two: M50, three: M100, four: M200)
Public service companies—
There are only two of these in the game, both of equal value (M150).
If one owns one, the player who lands on it must pay the dice value times four
If one owns both, the player who lands on one of them must pay the dice value times
ten
Chance and Community Chest—
When one falls into one of these two types of squares one has to pick a card located
over the corresponding stack, follow its instructions and put it back.
Income tax and Luxury tax—
If on falls into one of these one has to pay the indicated tax
Miscellaneous squares—
Includes those labelled: Go, Just Visiting/Jail, Free Parking and Go to jail
If one goes to jail, there are three ways to get out:
1. Paying a fee of M50 on the next round and starting off on the Jail square
2. Using a Get out of Jail card, can be obtained from the Chance and/or Community
chest stack or bought from another player. After being used one starts off on the Jail
square and places the card back on its corresponding deck.
3. Waiting three rounds whilst throwing the dices every round expecting to draw
doubles. If doubles are obtained one is freed if not one has to wait out their
‘sentence’ and pay a fine of M50 at the end.
Lastly, every player begins with M1500 and can only be declared winner when the rest have
gone bankrupt and own no property.
The mathematics behind monopoly:
First of all, rolling the dice. Every outcome in Monopoly is derived from the dice rolls.
Assuming the Monopoly game is a 2011 French version and that both dices are fair:
2
