1. A Game Of Strategy … Or Luck? Serene Li Hui Heng Xiaojun Jiang Cheewei Ng Li Xue Alison Then Team 5, MS&E220 Autumn 2008

2. What about Monopoly? Background One of the most popular board games 480 million players worldwide Dependent on both LUCK and STRATEGY Dominance prevails if one has a strategy to complement luck Project Purpose Discover the importance of strategy in Monopoly Determine successful strategies Learn how some decisions affect game dynamics E.g. Order of Starting

3. Models: Solving the Problem The 3 Models and their Assumptions: Model 1 ▫40 squares in the game ▫No jail, chance cards, community chest cards, etc. ▫Only one player Model 2 ▫Includes jail term ▫Going to jail equates to missing the next 3 turns ▫There is still only one player Model 3: ▫Four players ▫Four property colors were chosen and investigated - light blue, orange, yellow, dark blue ▫Each player owns one of the four property colors ▫Fixed rents ▫Cumulative expected earnings modeled

4. Model 1: No Jail We used the probabilities of rolling different numbers with two dice to find the probability of landing on a square on the certain throw. Here are the probabilities of throwing a certain number with two dice: 2 Dice:121110987654321 Prob:0.02780.05560.08330.11110.13890.16670.13890.11110.08330.05560.02780.0000 Here are the landing probabilities for the first 6 throws :

5. This shows the probability of landing on each square in each of the first 10 throws. A Simple Board

6. Model 1: Light Blue is Best Spreadsheet model allows us to monitor the probability of landing on each spot at least once, after n throws. From the graph, it suggests that the light blue properties (squares 7, 9 and 10) are the most optimal choice. Highest probability of being in light blue properties at least once Highest probability for light blue properties

7. Model 2: With Jail Probability of “Going to Jail” flows into 3 jail states over the next 3 turns This is due to the assumption that one misses 3 turns when he/she goes to jail Throw11 Jail 31 Go to JailJail 1Jail 2Jail 3Being in JailVisiting Jail 10.08330.000000000.0833 20.06170.000000000.0617 30.00270.009800000.0027 40.00000.07480.009800 0.00002 50.00110.04850.07477190.0097700.0845455150.0011 60.03450.00880.04846440.074770.009770.1330098830.0247 Being at jail state 3 means that the player is allowed to play again in the next throw, so the probability 0.00977 is added to the probability of merely “visiting jail” at square 11

8. Model 2: Orange is best Peaks at square 18 (Orange) i.e. P(being in square 18 at least once after n throws) is highest Suggests that Orange may be the best color set for a player As the number of throws increases, the probability of landing in jail converges to a probability of 0.075 Highest probability for orange properties

9. Model 3: Multi-Players There are four players, each one already designated with a different color set of properties – either Light blue, Orange, Yellow or Dark Blue Light Blue is chosen as Model 1 suggests that it is the best color Orange is chosen as Model 2 suggests it is the best color Dark Blue is chosen as it comprises of the most expensive properties Yellow is chosen as it is the next most expensive set of properties, after Green and Dark Blue. Green is not chosen as it is right after the “Go to jail” square. Each player collects rent from the other three, thus we can calculate the expected earnings per throw We assumed a fixed constant rent for each property What we are more interested in is the cumulative earnings, so that we see who does the best in the long run We also took into account the collecting of $200 as the player passes “Go” in each round

10. With the probabilities of landing on each square, and the rent of each colored property as determined by the rules of the game, we find that the yellow properties give the highest expected return to a player in the long term We had earlier expected Orange to be the best performer, but its actually the third best in this case because its rent is lower than both Yellow and Dark Blue Expected Earnings (I)

11. Taking away the worst performer in the previous case (Light Blue), and adding Green, we discovered that Green will outperform even Yellow This leads us to believe that the expected earnings are determined more by the rent the properties collect than the probability of landing on those squares Expected Earnings (II) Cumulative Expected Earnings (Case 2)

12. Sensitivity Analysis In our sensitivity analysis, we seek to: Find how expected net earnings can change as certain factors vary. In particular, we explore the effects of the following: Rent: Changes with the establishment of houses and hotels For 100 throws, the following rent distribution is assumed: 0 – 5 throws: no houses 5 – 30 throws: 1 houses 31 – 55 throws: 2 houses 56- - 80 throws: 3 houses 80 – 95 throws: 4 houses 95 – 100 throws: hotel Additional player: Affects the dynamics of the game e.g. how quickly will a player be bankrupt etc. A fifth player (owner of green property) is added

13. Sensitivity Results (I) From the results, we observe that the owner of the light blue properties starts of with more cash at hand, but quickly experiences a rapid cash outflow, which eventually leads to bankruptcy (i.e. cash at hand becomes negative) Owner of yellow properties ends up with the most cash at hand, given the various rent collected as the game progresses. This is in agreement with results of Model 3.

14. Sensitivity Analysis (II) Adding the 5 th player, who is owner of the green properties, we find that the expected cash at hand of the previous 4 players becomes more pronounced i.e. If expected cash of a player used to increase, it now increases at a greater rate and to greater amounts. Owner of the orange, dark blue and yellow properties all experience more expected cash as number of throws increases. Similarly, expected cash of a player decreases more drastically if it used to decrease when there are 4 players. This is seen for the owner of blue properties.

15. Conclusion & Recommendations Our Conclusions: Expected earnings of each player is influenced more by the amount of rent the property collects rather than the probabilities of landing on that property Strategy is useful for winning given that expected payoffs associated with different color sets can vary substantially Monopoly is a game of both STRATEGY and LUCK, given that there is a fair amount of uncertainty in the process. Our Recommendations: Focus on buying the yellow and green properties to maximize chances of winning In particular, one may want to avoid only owning light blue properties, since an owner of only light blue properties faces negative cash at hand over the long run i.e. becomes bankrupt Suggested Follow-up Actions: Study the effect of community and chest cards Study the effect of increased uncertainty from collusion among players