Monopoly
I was enthralled when I saw a research poster that a number of students had put together describing the optimal strategy for playing Monopoly. As it turns out, people have tried to figure this out before (here and here), but this one looked like a much clearer technique and presentation. I thought I’d share their results, after a blurb about their technique:
“
If the monopoly board had only 40 property spaces, the probability of landing on any given space would be a uniform 2.5%. However, because of chance cards, community chest cards, and the existence of jail, these otherwise uniform probabilities are skewed. To figure out the new board probabilities, we created a probability matrix that answered the scenario: given that we are on square X, what is the probability that we came from square Y? This matrix corresponds to a system of 40 linear equations with 40 variables, which we can then solve in MATLAB to determine the steady state probabilities of being on any given square. To make sure that our steady state solution is applicable to our problem, we also used Excel to dynamically generate the board probabilities for a given turn, given that we start on Go. Our analysis shows that we approach steady state by turn 30, well within the bounds of a typical monopoly game.
”
These are the probabilities of landing on any given square (boards derived from Wikipedia graphics):
| Free Parking 2.92% |
Kentucky Avenue
2.66% |
Chance 1.08% ?
|
Indiana Avenue
2.62% |
Illinois Avenue
3.07% |
B&O Railroad
2.96% |
Atlantic Avenue
2.60% |
Ventnor Avenue
2.58% |
Water Works
2.87% |
Marvin Gardens
2.51% |
Go To Jail 0% |
|
| New York Avenue
2.85% |
|
Pacific Avenue
2.85% |
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| Tennessee Avenue
2.91% |
North Carolina Avenue
2.91% |
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| Community Chest 2.32% |
Community Chest 2.32% |
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| St. James Place
2.79% |
Pennsylvania Avenue
2.79% |
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| Pennsylvania Railroad
2.44% |
Short Line
2.44% |
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| Virginia Avenue
2.57% |
Chance 0.87% ?
|
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| States Avenue
2.29% |
Park Place
2.29% |
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| Electric Company
2.67% |
Luxury Tax
2.67% |
||||||||||
| St. Charles Place
2.72% |
Boardwalk
2.72% |
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| In Jail/Just Visiting 3.49%/4.31% |
Connecticut Avenue
2.31% |
Vermont Avenue
2.33% |
Chance 0.87% ?
|
Oriental Avenue
2.27% |
Reading Railroad
2.98% |
Income Tax
2.34% |
Baltic Avenue
2.18% |
Community Chest 1.90% |
Mediterranean Avenue
2.14% |
Go
3.07%
|
|
or if (like me) you’re more used to the British version:
| Free Parking 2.92% |
Strand
2.66% |
Chance 1.08% ?
|
Fleet Street
2.62% |
Trafalgar Square
3.07% |
Fenchurch Street Station
2.96% |
Leicester Square
2.60% |
Coventry Street
2.58% |
Water Works
2.87% |
Piccadilly
2.51% |
Go To Jail 0% |
|
| Vine Street
2.85% |
|
Regent Street
2.85% |
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| Marlborough Street
2.91% |
Oxford Street
2.91% |
||||||||||
| Community Chest 2.32% |
Community Chest 2.32% |
||||||||||
| Bow Street
2.79% |
Bond Street
2.79% |
||||||||||
| Marylebone Station
2.44% |
Liverpool Street Station
2.44% |
||||||||||
| Northumberland Avenue
2.57% |
Chance 0.87% ?
|
||||||||||
| Whitehall
2.29% |
Park Lane
2.29% |
||||||||||
| Electric Company
2.67% |
Luxury Tax
2.67% |
||||||||||
| Pall Mall
2.72% |
Mayfair
2.72% |
||||||||||
| In Jail/Just Visiting 3.49%/4.31% |
Petonville Road
2.31% |
Euston Road
2.33% |
Chance 0.87% ?
|
the Angel Islington
2.27% |
King’s Cross Station
2.98% |
Income Tax
2.34% |
Whitechapel Road
2.18% |
Community Chest 1.90% |
Old Kent Road
2.14% |
Go
3.07%
|
|
and here’s a spreadsheet with more data on the properties by group:
It turns out they define the “best” monopoly in terms of the profit, which is a function of the length of the game and the number of players.