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# How to Win at Monopoly

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An introduction to Markov chains via Monopoly

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### 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