# 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. Setup a game problem and then use the calculator to find the solution. Look at problem 10.4.9-12.
8. A final tableau from a game problem is given. Give the optimal row and column strategies and the value of the game.
9. Write the linear programming problems necessary to solve a game. Do not attempt to solve the game.
10. Markov chain problem. Fill in the missing probabilities in the transition matrix, give an initial state matrix and find a state matrix for future time period. Find the steady state matrix.
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 1-6 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 7-13 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 7 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 8 8 8 8 10 10 10 20 14+7 6 125