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An introduction to Markov chains via Monopoly

An introduction to Markov chains via Monopoly

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- 1. How to Win at Monopoly: Markov Chains for Fun and Profit Derek Bruff, PhD Director, Center for Teaching Senior Lecturer, Mathematics
- 2. Which properties are landed on most often?
- 3. Which properties are landed on most frequently?
- 4. Which properties are most profitable to own?
- 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. 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. 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. 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. 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. 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. 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. 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. 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. Monopoly: Terminally Boring Edition x0 = x1 = What is x2?
- 15. Monopoly: Terminally Boring Edition x0 = x1 = x2 =
- 16. If x2=Px1, then what is P?
- 17. If x2=Px1, then what is P? = x2 P x1
- 18. Monopoly: Terminally Boring Edition Model: xk+1=Pxk P=
- 19. Monopoly: Terminally Boring Edition Model: xk+1=Pxk x0 = (1, 0, 0, 0)
- 20. Monopoly: Terminally Boring Edition Model: xk+1=Pxk x0 = (1, 0, 0, 0) x1 = (0, .5, .5, 0)
- 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. 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. Monopoly: Terminally Boring Edition Model: xk+1=Pxk
- 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. 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. 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. P=
- 28. Monopoly: Simple Model
- 29. Monopoly: Simple Model
- 30. Monopoly: Simple Model
- 31. Monopoly: Simple Model
- 32. Monopoly: Simple Model
- 33. Monopoly: Simple Model
- 34. Monopoly: Simple Model
- 35. Monopoly: Simple Model
- 36. Monopoly: Simple Model
- 37. Monopoly: Simple Model
- 38. Monopoly: Simple Model
- 39. Monopoly: Simple Model
- 40. Monopoly: Simple Model
- 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. 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. 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. Monopoly: Simple Model Finding q means solving the equation Pq=q
- 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. P=
- 47. Monopoly: Model #2 Finding q means solving the equation Pq=q
- 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. Monopoly: Model #3 Finding q means solving the equation Pq=q
- 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. Monopoly: Model #4 Finding q means solving the equation Pq=q
- 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. Short Jail Stay Probabilities by Truman Collins
- 54. Short Jail Stay Probabilities by Truman Collins
- 55. Short Jail Stay Probabilities by Truman Collins
- 56. Short Jail Stay Probabilities by Truman Collins
- 57. Long Jail Stay Probabilities by Truman Collins
- 58. Long Jail Stay Probabilities by Truman Collins
- 59. Long Jail Stay Probabilities by Truman Collins
- 60. Long Jail Stay Probabilities by Truman Collins
- 61. OTHER APPLICATIONS OF MARKOV CHAINS
- 62. RISK—and other board games
- 63. Baseball, tennis, jai alai,…
- 64. Migration Models
- 65. Google’s PageRank Algorithm
- 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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