How to Win at Monopoly

1. How to Win at Monopoly: Markov Chains for Fun and Profit Derek Bruff, PhD Director, Center for Teaching Senior Lecturer, Mathematics

3. Which properties are landed on most often?

4. Which properties are landed on most frequently?

5. Which properties are most profitable to own?

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.

13. 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.

14. 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

15. Monopoly: Terminally Boring Edition x0 = x1 = What is x2?

16. Monopoly: Terminally Boring Edition x0 = x1 = x2 =

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

19. If x2=Px1, then what is P? = x2 P x1

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

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

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

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

24. 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)

25. Monopoly: Terminally Boring Edition Model: xk+1=Pxk

26. 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.

27. 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.

28. 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

29. P=

30. Monopoly: Simple Model

43. 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

44. 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

45. 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

46. Monopoly: Simple Model Finding q means solving the equation Pq=q

47. 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.

48. P=

49. Monopoly: Model #2 Finding q means solving the equation Pq=q

54. 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.

55. Short Jail Stay Probabilities by Truman Collins

59. Long Jail Stay Probabilities by Truman Collins

63. OTHER APPLICATIONS OF MARKOV CHAINS

64. RISK—and other board games

65. Baseball, tennis, jai alai,…

66. Migration Models

67. Google’s PageRank Algorithm

68. 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

How to Win at Monopoly

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