# State Space

• Consists of 120 ordered pairs (s,d), where s = 0, 1, ..., 39 and d = 0, 1, 2. Here, s represents the space on the board (0 means Go, 39 means Boardwalk) and d is the number of doubles rolled so far in this turn. By the rules of the game, a player who rolls three doubles in one turn must go directly to Jail.

• Since a player never remains on the "Go to Jail" space, and since we need separate states for Jail and Just Visiting, it is convenient to assign Just Visiting to s = 10 and Jail to s = 30. We will need to be careful in setting up the roll matrix to account for this convention.

• In the LJ model, we have no use for three separate Jail states, so we use only (30,0) and discard (30,1) and (30,2), leaving only 118 states. However, in the RIJ model, we use state (30,1) to represent the situation when a player has failed at doubles exactly once and state (30,2) to represent the situation when a player has failed at doubles exactly twice.

• Note: Abbott and Richey use only 40 states (42 for RIJ), but at the expense of less accurate results.