how to win at monopoly

How to Win at Monopoly: Markov Chains for Fun and Profit

Derek Bruff, PhDDirector, Center for TeachingSenior Lecturer, Mathematics

Which properties are landed on most

often?

Which properties are landed on most

frequently?

Which properties are most profitable to own?

Monopoly Rules

• 40 spaces (Go through Boardwalk)• Roll two six-sided dice to move.• “Go to Jail” sends you to Jail.• Rolling three doubles in a row sends you to Jail.• Get out of jail by…– Paying $50,– Using a “Get out of Jail, Free” card,– Rolling doubles, or– Spending three turns in Jail.

• Chance and Community Chest cards have various effects.

Monopoly Rules

• 40 4 spaces (Go through Boardwalk)• Roll two six-sided dice Flip a coin to move.• “Go to Jail” sends you to Jail.• Rolling three doubles in a row sends you to Jail.• Get out of jail by…– Paying $50,– Using a “Get out of Jail, Free” card,– Rolling doubles, or– Spending three turns in Jail.


Suppose we only have four spaces (A, B, C, and D) and that a move consists of flipping a coin.• Heads = Move two

spaces• Tails = Move one space

TERMINALLY BORING

Monopoly: Terminally Boring Edition

x0 = x1 =

What is x2?


x0 = x1 =

x2 =

If x2=Px1, then what is P?

If x2=Px1, then what is P?

=

x2 x1P


Model: xk+1=Pxk

P =


Model: xk+1=Pxk

x0 = (1, 0, 0, 0)


Model: xk+1=Pxk

x0 = (1, 0, 0, 0)

x1 = (0, .5, .5, 0)


Model: xk+1=Pxk

x0 = (1, 0, 0, 0)

x1 = (0, .5, .5, 0)

x2 = (.25, 0, .25, .5)


Model: xk+1=Pxk

x0 = (1, 0, 0, 0)

x1 = (0, .5, .5, 0)

x2 = (.25, 0, .25, .5)

x3 = (.375, .375, .125, .125)


Model: xk+1=Pxk


• 40 4 spaces (Go through Boardwalk)• Roll two six-sided dice Flip a coin to move.• “Go to Jail” sends you to Jail.• Rolling three doubles in a row sends you to Jail.• Get out of jail by…– Paying $50,– Using a “Get out of Jail, Free” card,– Rolling doubles, or– Spending three turns in Jail.


Monopoly: Simple Model



Rolling Two Six-Sided Dice

Spaces Moved Probability

2 1/363 2/364 3/365 4/366 5/367 6/368 5/369 4/36

10 3/3611 2/3612 1/36

Markov Chains

Definition: A vector with the property that the sum of its entries is 1 is called a probability vector.

Definition: A square matrix with the property that the sum of the entries in each of its columns is 1 is called a stochastic matrix.

Andrey Markov, 1856 – 1922

Markov Chains

Definition: A Markov chain is a dynamical system for which

• the probability vector xk describes the state of the system at time k and

• successive state vectors are related by the following equation, where P is a stochastic matrix called the transition matrix for the system.

xk+1=Pxk

Markov Chains

Theorem: If P is the transition matrix for a Markov chain (and P is regular), then…

• There is a unique probability vector q such that Pq=q.

• For any initial state vector x0, xk q as k

Finding q means solving the equation

Pq=q



Pq=q

Monopoly: Model #2



Monopoly: Model #2


Pq=q

Monopoly: Model #3



Monopoly: Model #3


Pq=q

Monopoly: Model #4



Monopoly: Model #4


Pq=q

What’s Left?

• Rolling three doubles in a row sends you to Jail.


You still have two underlying models—leave jail quickly or stay as long as you can.

Probabilities by Truman Collins

Short Jail Stay

Probabilities by Truman Collins

Long Jail Stay

OTHER APPLICATIONS OF MARKOV CHAINS

RISK—and other board games

Baseball, tennis, jai alai,…

Migration Models

Google’s PageRank Algorithm

Flickr Credits• “monopoly,” foreverdigital• “Black Dice,” Mariano Kamp• “Monopoly,” unloveablesteve• “last man standing,” Robert Terrell• “Racing for Home,” Scott Ableman• “Nomads (brog pa) crossing Lha chu at Kailash

Kora,” reurinkjan

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