Question 11.5.21: A highly simplified game of “Monopoly” is played on a board w......
A highly simplified game of “Monopoly” is played on a board with four squares as shown in Figure 11.8. You start at GO. You roll a die and move clockwise around the board a number of squares equal to the number that turns up on the die. You collect or pay an amount indicated on the square on which you land. You then roll the die again and move around the board in the same manner from your last position. Using the result of Exercise 20, estimate the amount you should expect to win in the long run playing this version of Monopoly.
The transition matrix is
P= \begin{matrix} GO \\ A \\ B \\ C \end{matrix} \overset{\begin{matrix} GO && A && B && C\end{matrix} }{\begin{pmatrix} 1/6 & 1/3 & 1/3 & 1/6 \\ 1/6 & 1/6 & 1/3 & 1/3 \\ 1/3 & 1/6 & 1/6 & 1/3 \\ 1/3 & 1/3 & 1/6 & 1/6 \end{pmatrix} } .
Since the column sums are 1, the fixed vector is
w = (1/4, 1/4, 1/4, 1/4) .
From this we see that wf = 0. From the result of Exercise 20 we see that your expected winning starting in GO is the first component of the vector Zf where
\text{f}= \begin{pmatrix} 15 \\ -30 \\ -5 \\ 20 \end{pmatrix}.
Using the program ergodic we find that the long run expected winning starting in GO is 10.4.