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

Derek Bruff, PhD
Director, Center for Teaching
Senior Lecturer, ...
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...
Monopoly Rules
•
•
•
•
•

40 spaces (Go through Boardwalk)
Roll two six-sided dice to move.
“Go to Jail” sends you to Jail...
Monopoly Rules
•
•
•
•
•

40 spaces (Go through Boardwalk)
Roll two six-sided dice to move.
“Go to Jail” sends you to Jail...
Monopoly Rules
•
•
•
•
•

40 spaces (Go through Boardwalk)
Roll two six-sided dice to move.
“Go to Jail” sends you to Jail...
Monopoly Rules
•
•
•
•
•

40 spaces (Go through Boardwalk)
Roll two six-sided dice to move.
“Go to Jail” sends you to Jail...
Monopoly Rules
•
•
•
•
•

40 spaces (Go through Boardwalk)
Roll two six-sided dice to move.
“Go to Jail” sends you to Jail...
Monopoly Rules
•
•
•
•
•

40 spaces (Go through Boardwalk)
Roll two six-sided dice to move.
“Go to Jail” sends you to Jail...
Monopoly Rules
•
•
•
•
•

40 4 spaces (Go through Boardwalk)
Roll two six-sided dice Flip a coin to move.
“Go to Jail” sen...
Suppose we only have four
spaces (A, B, C, and D) and
that a move consists of
flipping a coin.
• Heads = Move two
spaces
•...
Monopoly: Terminally Boring Edition
x0 =

x1 =

What is x2?
Monopoly: Terminally Boring Edition
x0 =

x1 =

x2 =
If x2=Px1, then what is P?
If x2=Px1, then what is P?

=

x2

P

x1
Monopoly: Terminally Boring Edition
Model: xk+1=Pxk

P=
Monopoly: Terminally Boring Edition
Model: xk+1=Pxk

x0 = (1, 0, 0, 0)
Monopoly: Terminally Boring Edition
Model: xk+1=Pxk

x0 = (1, 0, 0, 0)
x1 = (0, .5, .5, 0)
Monopoly: Terminally Boring Edition
Model: xk+1=Pxk

x0 = (1, 0, 0, 0)
x1 = (0, .5, .5, 0)
x2 = (.25, 0, .25, .5)
Monopoly: Terminally Boring Edition
Model: xk+1=Pxk

x0 = (1, 0, 0, 0)
x1 = (0, .5, .5, 0)
x2 = (.25, 0, .25, .5)
x3 = (.3...
Monopoly: Terminally Boring Edition
Model: xk+1=Pxk
Monopoly: Terminally Boring Edition
•
•
•
•
•

40 4 spaces (Go through Boardwalk)
Roll two six-sided dice Flip a coin to m...
Monopoly: Simple Model
•
•
•
•
•

40 spaces (Go through Boardwalk)
Roll two six-sided dice to move.
“Go to Jail” sends you...
Rolling Two Six-Sided Dice
Spaces Moved

Probability

2

1/36

3

2/36

4

3/36

5

4/36

6

5/36

7

6/36

8

5/36

9

4/...
P=
Monopoly: Simple Model
Monopoly: Simple Model
Monopoly: Simple Model
Monopoly: Simple Model
Monopoly: Simple Model
Monopoly: Simple Model
Monopoly: Simple Model
Monopoly: Simple Model
Monopoly: Simple Model
Monopoly: Simple Model
Monopoly: Simple Model
Monopoly: Simple Model
Monopoly: Simple Model
Markov Chains
Definition: A vector with the
property that the sum of its
entries is 1 is called a probability
vector.
Defi...
Markov Chains
Definition: A Markov chain is a dynamical system for
which
• the probability vector xk describes the state o...
Markov Chains
Theorem: If P is the transition matrix for a Markov
chain (and P is regular), then…
• There is a unique prob...
Monopoly: Simple Model
Finding q means solving the equation

Pq=q
Monopoly: Model #2
•
•
•
•
•

40 spaces (Go through Boardwalk)
Roll two six-sided dice to move.
“Go to Jail” sends you to ...
P=
Monopoly: Model #2
Finding q means solving the equation

Pq=q
Monopoly: Model #3
•
•
•
•
•

40 spaces (Go through Boardwalk)
Roll two six-sided dice to move.
“Go to Jail” sends you to ...
Monopoly: Model #3
Finding q means solving the equation

Pq=q
Monopoly: Model #4
•
•
•
•
•

40 spaces (Go through Boardwalk)
Roll two six-sided dice to move.
“Go to Jail” sends you to ...
Monopoly: Model #4
Finding q means solving the equation

Pq=q
What’s Left?
• Rolling three doubles in a row sends you to
Jail.
• Chance and Community Chest cards have
various effects.
...
Short Jail Stay

Probabilities by Truman Collins
Short Jail Stay

Probabilities by Truman Collins
Short Jail Stay

Probabilities by Truman Collins
Short Jail Stay

Probabilities by Truman Collins
Long Jail Stay

Probabilities by Truman Collins
Long Jail Stay

Probabilities by Truman Collins
Long Jail Stay

Probabilities by Truman Collins
Long Jail Stay

Probabilities by Truman Collins
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 standin...
How to Win at Monopoly
How to Win at Monopoly
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An introduction to Markov chains via Monopoly

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  • “monopoly,” by Flickr user foreverdigital, http://www.flickr.com/photos/foreverdigital/4159039717/
  • “Black Dice,” by Flickr user Mariano Kamp, http://www.flickr.com/photos/mkamp/2478311790/
  • “Monopoly,” by Flickr user unloveablesteve, http://www.flickr.com/photos/unloveable/2400877902/
  • Strategic LearnersThese learners react well to competition and the chance to do better than anyone else.They often become strategic learners, making high grades but seldom grappling deeply enough to change their own perceptions.They are often “regurgitators”—learning material for the test and then quickly expunging the material to make room for something else.
  • “Racing for Home,” by Flickr user Scott Ableman, http://www.flickr.com/photos/ableman/183059002/
  • “Nomads (brog pa) crossing Lhachu at KailashKora,” by Flickr user reurinkjan, http://www.flickr.com/photos/reurinkjan/3127359433/
  • Mathematical PageRanks (out of 100) for a simple network (PageRanks reported by Google are rescaled logarithmically). Page C has a higher PageRank than Page E, even though it has fewer links to it; the link it has is of a much higher value. A web surfer who chooses a random link on every page (but with 15% likelihood jumps to a random page on the whole web) is going to be on Page E for 8.1% of the time. (The 15% likelihood of jumping to an arbitrary page corresponds to a damping factor of 85%.) Without damping, all web surfers would eventually end up on Pages A, B, or C, and all other pages would have PageRank zero. Page A is assumed to link to all pages in the web, because it has no outgoing links.http://en.wikipedia.org/wiki/PageRank
  • How to Win at Monopoly

    1. 1. How to Win at Monopoly: Markov Chains for Fun and Profit Derek Bruff, PhD Director, Center for Teaching Senior Lecturer, Mathematics
    2. 2. Which properties are landed on most often?
    3. 3. Which properties are landed on most frequently?
    4. 4. Which properties are most profitable to own?
    5. 5. 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.
    6. 6. 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.
    7. 7. 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.
    8. 8. 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.
    9. 9. 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.
    10. 10. 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.
    11. 11. 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.
    12. 12. 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.
    13. 13. 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
    14. 14. Monopoly: Terminally Boring Edition x0 = x1 = What is x2?
    15. 15. Monopoly: Terminally Boring Edition x0 = x1 = x2 =
    16. 16. If x2=Px1, then what is P?
    17. 17. If x2=Px1, then what is P? = x2 P x1
    18. 18. Monopoly: Terminally Boring Edition Model: xk+1=Pxk P=
    19. 19. Monopoly: Terminally Boring Edition Model: xk+1=Pxk x0 = (1, 0, 0, 0)
    20. 20. Monopoly: Terminally Boring Edition Model: xk+1=Pxk x0 = (1, 0, 0, 0) x1 = (0, .5, .5, 0)
    21. 21. Monopoly: Terminally Boring Edition Model: xk+1=Pxk x0 = (1, 0, 0, 0) x1 = (0, .5, .5, 0) x2 = (.25, 0, .25, .5)
    22. 22. 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)
    23. 23. Monopoly: Terminally Boring Edition Model: xk+1=Pxk
    24. 24. 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.
    25. 25. 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.
    26. 26. Rolling Two Six-Sided Dice Spaces Moved Probability 2 1/36 3 2/36 4 3/36 5 4/36 6 5/36 7 6/36 8 5/36 9 4/36 10 3/36 11 2/36 12 1/36
    27. 27. P=
    28. 28. Monopoly: Simple Model
    29. 29. Monopoly: Simple Model
    30. 30. Monopoly: Simple Model
    31. 31. Monopoly: Simple Model
    32. 32. Monopoly: Simple Model
    33. 33. Monopoly: Simple Model
    34. 34. Monopoly: Simple Model
    35. 35. Monopoly: Simple Model
    36. 36. Monopoly: Simple Model
    37. 37. Monopoly: Simple Model
    38. 38. Monopoly: Simple Model
    39. 39. Monopoly: Simple Model
    40. 40. Monopoly: Simple Model
    41. 41. 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
    42. 42. 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
    43. 43. 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
    44. 44. Monopoly: Simple Model Finding q means solving the equation Pq=q
    45. 45. 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.
    46. 46. P=
    47. 47. Monopoly: Model #2 Finding q means solving the equation Pq=q
    48. 48. 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.
    49. 49. Monopoly: Model #3 Finding q means solving the equation Pq=q
    50. 50. 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.
    51. 51. Monopoly: Model #4 Finding q means solving the equation Pq=q
    52. 52. 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.
    53. 53. Short Jail Stay Probabilities by Truman Collins
    54. 54. Short Jail Stay Probabilities by Truman Collins
    55. 55. Short Jail Stay Probabilities by Truman Collins
    56. 56. Short Jail Stay Probabilities by Truman Collins
    57. 57. Long Jail Stay Probabilities by Truman Collins
    58. 58. Long Jail Stay Probabilities by Truman Collins
    59. 59. Long Jail Stay Probabilities by Truman Collins
    60. 60. Long Jail Stay Probabilities by Truman Collins
    61. 61. OTHER APPLICATIONS OF MARKOV CHAINS
    62. 62. RISK—and other board games
    63. 63. Baseball, tennis, jai alai,…
    64. 64. Migration Models
    65. 65. Google’s PageRank Algorithm
    66. 66. 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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