MONOPOLY

MONOPOLY

• 1.
  MONOPOLY: A MARKOVIAN APPROACH BY:SACHIN NOWAL (2014A3PS143G) AVIRAL JAIN (2014B4A8683G)
• 2.
  Introduction In monopoly pawnsare marched around a board with 40 positions according to the dictates of a pair of dice. The person in control of the pawn pays out or receives money to or from other players and the bank, depending on the position of the pawn and the state of development of the particular position. Thus, In order to assess the relative value of controlling the various properties one must have an idea of the frequencies with which it can be expected that pawns will occupy each position. These are called the limit frequencies of the positions (or probability of being on a particular square). The limit frequencies of the positions in the game of Monopoly are calculated on the basis of Markov chains. In order to make the process Markovian some minor modifications in the rules are necessary. Monopoly Board Game Monopoly is a board game that originated in the United States in 1903 as a way to demonstrate that an economy which rewards wealth creation is better than one in which monopolists work under few constraints. The game is named after the economic concept of monopoly—the domination of a market by a single entity. Players move around the game-board buying or trading properties, developing their properties with houses and hotels, and collecting rent from their opponents, with the goal being to drive them all into bankruptcy leaving one monopolist in control of the entire economy. The game has 40 squares and the moves are solely determined by the pair of dice. The long term probability for all squares would be 1/40 if all the squares behave similarly. Although this is not the case, there are some squares that can move a player in a manner different than the others. These squares are Chance, Community Chest and the "Go to the Jail". Assumptions and changes in the project: In this project we have modified the standard Monopoly game to "The BITS Monopoly”. The image given below depicts the new states and the new model. There are few features which make this game process a non-markovian process: the chance and community chest position-transfer cards. We can make the process for Monopoly Markovian by is distinguishing three, rather than one, jail states and changing the rules slightly so as to remove the choice of the player in leaving jail and the historical dependence on the order of the chance and community chest cards. We shall consider the jail position as determining three states in jail, having just arrived on the previous turn in jail, having stayed one turn and in jail, having stayed two turns. We name these states as j0, j1,j2. To remove player choice we assume probability of going from state j0 to state j1 as 2/3 and probability of going from state j1 to state j2 as 2/3. Also for the ? state or the chance state ,it has been assumed that the player will have four choices: “Go to B-Dome”, ”Disco/(Jail)”, ”Go to BORKAR(9)” / ”Go to Library(38)” and do nothing(i.e. remain on the chance square) with probabilities ¼ for all. And “Caught Smoking” is as same as “Go to Jail” square in the standard monopoly.
• 3.
  BITS MONOPOLY MODEL(Transitional states)
• 4.
  S.No. Name ofthe Sqaure Square No. 1 Bdome 1 2 AH-1 2 3 A-MESS 3 4 AH-2 4 5 AH-7 5 6 GAJA 6 7 AH-8 7 8 ATM 8 9 BORKAR 9 10 PERSIAN 10 11 MED-C 11 12 Laundry 12 13 Community Center 13 14 AH-3 14 15 AH-4 15 16 CHANCE 16 17 AH-5 17 18 AH-8 18 19 Caught Smoking 19 20 VGH 20 21 Gate 21 22 CH-6 22 23 CH-5 23 24 Caught Smoking 24 25 CH-4 25 26 C-MESS 26 27 CHANCE 27 28 CH-3 28 29 SWD 29 30 CH-2 30 31 DISCO(J0) 31 32 CH-1 32 33 CC 33 34 Food King 34 35 Auditorium 35 36 Community Center 36 37 IC 37 38 Library 38 39 Pragati 39 40 ICE 'n' SPICE 40 41 J1 41 42 J2 42
• 5.
  Methodology For the purposeof calculating the steady state probabilities we created a transitional matrix. And for the initial state vector we used B-Dome( square1) as the starting point. The states are the squares on the board and each turn corresponds to a step. The transition matrix is a 42x42(including the extra Disco states) matrix containing the probabilities of moving from each square to each other square. If the state vector X at time k is then the state vector at time k+n is the matrix product XPn , where P is the transition matrix of the chain. In particular, the element Pij (n) in row I and column j of Pn is the probability starting from state I of being in j in n steps. We use this simple fact to compute the matrix for the per turn chain in terms of the matrix for the per throw chain. Observations (Graph-1)The graph below shows the probability of occupying a particular square. Here it can be seen that the probability of being on caught (square 19 and square 24) is 0.This is because as soon as the player lands on this square he instantly goes to jail and stays there until next turn. In other words the player does not spend any time on these particular squares. The low probabilities of being on chance (square 16 and square 27) can also be explained by the fact that when a player lands on the chance square, he instantly acts as per the chance card he receives. The probability of being in jail = j0+j1+j2= .0732 + .0488 + .0325 = .1545
• 6.
  Note: X-Axis:Square no.(Refer thegiven table table) Y-Axis:Probablity of being on a particular square . (Graph-2)The graph below shows the probability of landing on a particular square.
• 7.
  Note: X-Axis: Square no.(Referthe given table table) Y-Axis: Probability of on a particular square. Conclusion  The square with maximum frequency is DISCO (jail) with 15.45% probability.  The square with minimum frequency in CH-1 with 1.55% probability.  The property with the maximum frequency in BORKAR with 3.30% probability.  The ‘light blue’ property cluster has the highest probability among all the property clusters.  To maximize the profits, the player must invest in the property whose limit frequency is the highest. (Considering equal rent from each property).