Math 160: Study Guide - Chapters 9-10

1. Solve a strictly determined game. Show all work.
2. Eliminate any recessive rows and/or columns. Do NOT solve the game.
3. Solve a 2×2 non-strictly determined game. Show all work. Know the formulas from section 9.2.
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. Absorbing Markov Chain problem. Given a transition matrix, find the fundamental matrix and the long term probabilities of moving from any transient state to an absorbing state.
6. Setup a game problem and then use the calculator to find the solution. Look at problem 9.4.10.
7. A final tableau from a game is given. Give the optimal row and column strategies and the value of the game.
8. Markov chain problem. Fill in the missing probabilities in the transition matrix, draw a transition diagram, give an initial state matrix and find a state matrix for future time period. Find the steady state matrix.
9. Absorbing Markov chain application problem. The fundamental matrix F, the probabilities of moving from a transient state to an absorbing state (matrix FR), and the sum of each row of the fundamental matrix are given. Use them to answer the questions.
10. Write the linear programming problems necessary to solve a game. Do not attempt to solve the game.
11. Absorbing Markov chain problem. Questions similar to #9 will be asked, but you'll need to find all the matrices yourself (this is on the take home portion).
12. 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).
13. 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.

Notes

• Problems 1, 2, and 3 must be solved 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 4-11 must be worked alone, but the game program on the calculator can be used.
• Problems 12-13 can be worked in groups of up to three people.
• 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, the later questions are harder (which is why you get to work in a group).
• Question 11 is composed of two parts. 8 of the points are on the in-class portion and 4 points are on the take home.

Points per problem

 # Pts 1 2 3 4 5 6 7 8 9 10 11 12 13 Total 3 3 3 6 6 6 8 11 12 8 8+4 10 12 100