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

1. Solve a strictly determined game. Show all work.
2. Solve a game. Show all work. Be sure to check for recessive rows and columns. Know the formulas from section 8.2.
3. 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).
4. Absorbing Markov Chain problem. Given a transition matrix, find the fundamental matrix and the expected number of trials from a non-absorbing state to an absorbing state.
5. Setup a game problem and then use the calculator to find the solution. Look at problem 8.4.10.
6. A final tableau from a game is given. Give the optimal row and column strategies and the value of the game.
7. 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.
8. Absorbing Markov chain problem. Finish the transition matrix. Find the fundamental 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. Find the expected value if each absorbing state is assigned a numerical value.
9. 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).
10. Absorbing Markov chain application problem. Finish the transition matrix and find the average number of trials before entering an absorbing state.
11. 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.
12. Write the linear programming problems necessary to solve a game.

### Notes

• Problems 1 and 2 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 8-12 can be worked with a partner. If you work alone, you may have a notecard with the formulas for absorbing Markov chains on it.
• The solution to a game consists of the optimal row strategy P*, the optimal column strategy Q*, and the value of the game v.
• 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 8 7 8 8 13 12 10 6 12 8 100