## Math 160: Study Guide - Chapter 8 & 9

1. Solve a strictly determined game. Show all work.
2. Solve a game. Show all work. Know the formulas from section 8.2.
3. Show that a the square of a probability matrix is also a probability matrix. To do this, you must find the square, and then show that the rows add to one.
4. Show the strategies and find the value of the game if both players randomly select a strategy. Show the strategies and find the value of the game if one player randomly chooses a strategy, but the other player uses their optimal strategy (use the calculator to find the optimal strategy).
5. Setup a game problem and then use the calculator to find the solution.
6. Markov chain problem. Fill in the missing probabilities in the transition diagram, find the transition matrix, find the limiting and steady-state matrices.
7. Markov chain application problem. Write the transition matrix, initial state matrix. Find the second state matrix and the third state matrix. Look at problems 9.1.43 - 53*.
8. Solve a game using the calculator. The column player lets the row player know what her strategy will be, find the best a priori strategy for the row player (this is the best strategy for the row player if he knows what the column player will do) and the value of the game under this strategy. Find the value of the game if the row player uses his a priori strategy, but the column player uses her optimal strategy instead of the one she said she was going to use. Find the best a priori strategy for the column player (what should she play if she knows what the row player will do because he thinks he knows what she is going to do).
9. Write the linear programming problems necessary to solve a game. Then, find the solution using the geometric approach to linear programming.
10. Find the solution to a game using the calculator. Find the value if both players play randomly (all choices are not equally likely - you need to have a concept of what a relative frequency is). Find the payoffs for the row player under the expected value criterion, maximax criterion, and maximin criterion.
11. Absorbing Markov chain application problem. Draw a transition diagram, write the transition matrix, find the fundamental matrix F, the limiting matrix. Find various long-term probabilities and the average duration spent in the matrix. Look at problems 9.3.39 - 9.3.45*.
12. Absorbing Markov chain problem. Create the transition matrix. Find the number of moves before exiting the system. Find the probability of ending up in a particular absorbing state when starting in a transient state.

### Notes

• The solution to a game consists of the optimal row strategy P*, the optimal column strategy Q*, and the value of the game v.
• When solving a game, first check for strictly determined games. Then check for recessive rows or columns.
• The calculator can be used to solve all the problems (except 3), but show work on the first three problems.
• Problems 1 - 7 must be worked individually and turned in before picking up the second half of the test.
• Problems 8 - 12 may be worked in groups of up to three people or alone using notecards. The notecards may contain formulas for absorbing Markov chains and the geometric approach to linear programming.
• Move swiftly through the first portion of the test - the group problems will be lengthy.
 # 1 2 3 4 5 6 7 8 9 10 11 12 Total Pts 4 4 4 8 8 9 9 10 8 14 14 8 100