Matrices

Note: In the LJ model we have 118 states [no (Jail,1) or (Jail,2)], so all matrices are 118 × 118. In the RIJ model we have 120 states, so all matrices are 120 × 120.

Also, we index the rows by the current state and the columns by the next state, but Abbott and Richey reverse this convention. In other words, they consider the transposes of the (40 or 42 state versions of the) matrices given here.

• Roll Matrix R

  block matrix with nine blocks, four of which are blocks of zeroes
  
  

• Jail Matrix J

  Abbott and Richey use a matrix to simulate the Go to Jail space and to approximate the rule about three doubles sending a player to Jail, but our choice of state space avoids this approximation.
  
  For consistency, we will take J to be the identity matrix.
  
  

• Community Chest Matrix Cc

  differs from the identity matrix in exactly nine rows
  (since there are three Community Chest spaces and three values for d)
  
  

• Chance Matrix Ch

  differs from the identity matrix in exactly nine rows
  (since there are three Chance spaces and three values for d)
  
  

• Final Transition Probability Matrix
  P = R · J · Ch · Cc

  Note that the order of multiplication is important. In particular, see spaces 33 (CommunityChest3) and 36 (Chance3).

Main Previous Next