how to win at monopoly
DESCRIPTION
An introduction to Markov chains via MonopolyTRANSCRIPT
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How to Win at Monopoly: Markov Chains for Fun and Profit
Derek Bruff, PhDDirector, Center for TeachingSenior Lecturer, Mathematics
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Which properties are landed on most
often?
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Which properties are landed on most
frequently?
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Which properties are most profitable to own?
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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.
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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.
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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.
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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.
![Page 10: How to Win at Monopoly](https://reader035.vdocument.in/reader035/viewer/2022062404/5539e1a24a79590a7f8b49e5/html5/thumbnails/10.jpg)
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.
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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.
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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.
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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.
• Chance and Community Chest cards have various effects.
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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
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Monopoly: Terminally Boring Edition
x0 = x1 =
What is x2?
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Monopoly: Terminally Boring Edition
x0 = x1 =
x2 =
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If x2=Px1, then what is P?
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If x2=Px1, then what is P?
=
x2 x1P
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Monopoly: Terminally Boring Edition
Model: xk+1=Pxk
P =
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Monopoly: Terminally Boring Edition
Model: xk+1=Pxk
x0 = (1, 0, 0, 0)
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Monopoly: Terminally Boring Edition
Model: xk+1=Pxk
x0 = (1, 0, 0, 0)
x1 = (0, .5, .5, 0)
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Monopoly: Terminally Boring Edition
Model: xk+1=Pxk
x0 = (1, 0, 0, 0)
x1 = (0, .5, .5, 0)
x2 = (.25, 0, .25, .5)
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Monopoly: Terminally Boring Edition
Model: xk+1=Pxk
x0 = (1, 0, 0, 0)
x1 = (0, .5, .5, 0)
x2 = (.25, 0, .25, .5)
x3 = (.375, .375, .125, .125)
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Monopoly: Terminally Boring Edition
Model: xk+1=Pxk
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Monopoly: Terminally Boring Edition
• 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.
• Chance and Community Chest cards have various effects.
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Monopoly: Simple Model
• 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.
![Page 28: How to Win at Monopoly](https://reader035.vdocument.in/reader035/viewer/2022062404/5539e1a24a79590a7f8b49e5/html5/thumbnails/28.jpg)
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
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P =
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Monopoly: Simple Model
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Monopoly: Simple Model
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Monopoly: Simple Model
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Monopoly: Simple Model
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Monopoly: Simple Model
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Monopoly: Simple Model
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Monopoly: Simple Model
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Monopoly: Simple Model
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Monopoly: Simple Model
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Monopoly: Simple Model
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Monopoly: Simple Model
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Monopoly: Simple Model
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Monopoly: Simple Model
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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
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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
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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
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Monopoly: Simple Model
Finding q means solving the equation
Pq=q
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Monopoly: Model #2
• 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.
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P =
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Monopoly: Model #2
Finding q means solving the equation
Pq=q
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Monopoly: Model #3
• 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.
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Monopoly: Model #3
Finding q means solving the equation
Pq=q
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Monopoly: Model #4
• 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.
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Monopoly: Model #4
Finding q means solving the equation
Pq=q
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What’s Left?
• Rolling three doubles in a row sends you to Jail.
• Chance and Community Chest cards have various effects.
You still have two underlying models—leave jail quickly or stay as long as you can.
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Probabilities by Truman Collins
Short Jail Stay
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Probabilities by Truman Collins
Short Jail Stay
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Probabilities by Truman Collins
Short Jail Stay
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Probabilities by Truman Collins
Short Jail Stay
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Probabilities by Truman Collins
Long Jail Stay
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Probabilities by Truman Collins
Long Jail Stay
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Probabilities by Truman Collins
Long Jail Stay
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Probabilities by Truman Collins
Long Jail Stay
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OTHER APPLICATIONS OF MARKOV CHAINS
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RISK—and other board games
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Baseball, tennis, jai alai,…
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Migration Models
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Google’s PageRank Algorithm
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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