How to Win at Monopoly
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The document discusses using Markov chains to model and analyze the game of Monopoly. It explains the basic rules of Monopoly and then presents increasingly complex Markov chain models of landing on spaces and properties. The models start simply with a few spaces and coin flips and become more detailed representations incorporating jail stays and dice rolls. The document also provides background on Markov chains and their properties, explaining how the transition matrix P and stationary distribution q can be used to analyze Monopoly and other systems.
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.
- 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 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=
- 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
- 50. 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.
- 51. Monopoly: Model #3 Finding q means solving the equation Pq=q
- 52. 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.
- 53. Monopoly: Model #4 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
- 56. Short Jail Stay Probabilities by Truman Collins
- 57. Short Jail Stay Probabilities by Truman Collins
- 58. Short Jail Stay Probabilities by Truman Collins
- 59. Long Jail Stay Probabilities by Truman Collins
- 60. Long Jail Stay Probabilities by Truman Collins
- 61. Long Jail Stay Probabilities by Truman Collins
- 62. Long Jail Stay Probabilities by Truman Collins
- 63. OTHER APPLICATIONS OF MARKOV CHAINS
- 64. RISK—and other board games
- 66. Migration Models
- 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
Editor's Notes
- #2: “monopoly,” by Flickr user foreverdigital, http://www.flickr.com/photos/foreverdigital/4159039717/
- #4: “Black Dice,” by Flickr user Mariano Kamp, http://www.flickr.com/photos/mkamp/2478311790/
- #6: “Monopoly,” by Flickr user unloveablesteve, http://www.flickr.com/photos/unloveable/2400877902/
- #65: 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.
- #66: “Racing for Home,” by Flickr user Scott Ableman, http://www.flickr.com/photos/ableman/183059002/
- #67: “Nomads (brog pa) crossing Lhachu at KailashKora,” by Flickr user reurinkjan, http://www.flickr.com/photos/reurinkjan/3127359433/
- #68: 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