# Math 160: Study Guide - Chapters 9-10

1. Solve a strictly determined game. Show all work.
2. Eliminate all recessive rows and columns. Do NOT solve the game.
3. Create a transition matrix from the transition diagram.
4. Solve a 2×2 non-strictly determined game. Show all work. Know the formulas from section 10.2.
5. Take an Absorbing Markov Chain story problem and create a transition diagram and the transition matrix. Be sure to put the matrix into standard form.
6. Take a Markov chain story problem and create a transition matrix and initial state matrix.
7. Markov chain problem. Some state matrices are given, provide an interpretation in the context of the story.
8. Know how to adjust a non-strictly determined game before forming linear programming problems. The graph of a feasible region, with the coordinates of the corner points, is shown. Identify whether the feasible region is for the row player or column player and determine where the optimal solution occurs.
9. Setup a game problem and then use the calculator to find the solution. Look at problem 10.4.9-12.
10. A final tableau from a game problem is given. Give the optimal row and column strategies and the value of the game.
11. Find the solution to a game using the calculator. Find the payoffs for the row player under the expected value, maximax, maximin, and minimax criteria. 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).
12. Absorbing Markov chain. Find probabilities of a certain transition, how many transitions should be expected before reaching an absorbing state and the long term probability of reaching an absorbing state. Use the information on the take home portion of the exam to answer this question.
13. Absorbing Markov chain. The fundamental matrix F and the long term probabilities FR are given. Use them to answer questions. Similar to question 12 except that I've given you the matrices and on question 12, you had to find them yourself on the take-home exam.

## Notes

• Problems 1-8 must be answered without a calculator. The game program on the calculator can be used for the rest of the test whenever you need to find an optimal strategy.
• Problems 9-13 should make full use of the calculator programs and other capabilities.
• The solution to a game consists of the optimal row strategy P*, the optimal column strategy Q*, and the value of the game v. Make sure you give all three parts, not just the value of the game.
• Move swiftly through the first portion of the test.
• Question 12 is 14 points on the actual test (7 parts, 2 points each) and 8 points for the take home question.

## Points per problem

 # Pts 1 2 3 4 5 6 7 8 9 10 11 12 13 Total 6 4 6 10 10 8 6 8 6 10 20 14+8 9 125