ReviewDay 6

Unit 2 - Matrices & Game Theory

Warm UpSuppose that Sol and Tina change their game so that the payoffs to Sol are

a. Use the row matrix to find Sol’s best strategy for this game.

b. Use the column matrix to find Tina’s best strategy for this game.

c. Find Sol’s expected payoff matrix equation for this game and solve.

1 3

2 4

1p p

1

q

q

Warm UpSuppose that Sol and Tina change their game so that the payoffs to Sol are

a. Use the row matrix to find Sol’s best strategy for this game.

b. Use the column matrix to find Tina’s best strategy for this game.

c. Find Sol’s expected payoff matrix equation for this game and solve.

1 3

2 4

1p p

1

q

q

Warm Up Answers

a. Use the row matrix to find Sol’s best strategy for this game.

b. Use the column matrix to find Tina’s best strategy for this game.

c. Find Sol’s expected payoff matrix equation for this game and solve.

1p p

1

q

q

1 3 .7

.6 .4 .22 4 .3

a. Sol should play heads 6 of the 10 times and tails 4 of the 10 times.

b. Tina should play heads 7 of the 10 times and tails 3 of the 10 times.

c. Sol should expect to win 2 cents every 10 times.

Matrices to Calculate Expected Payoff to Sol:

Homework Questions?!

Unit 4 Matrix Test Topics1. Strictly Determined Games: Find maximin, minimax and saddle

points - be able to interpret each based on the context

2. Non- Strictly Determined Games: Find payoff matrix, best strategy for row and column player, and expected value for row player

3. Markov Chains: Create transition matrix and initial-state matrix and interpret values after a certain number of cycles

4. Leslie Matrix: Create Leslie matrix to find population distribution, total population, long term growth rate, and time when a maximum population is reached

5. Matrix operations: Perform calculations and interpret properly, especially Matrix multiplication and Scalar multiplication

PracticeThere are 20 students in student council and every week they bring snacks to their meeting. This week 8 brought chips, 7 brought drinks and 5 brought dessert. 18% of those who brought chips to the first meeting brought chips again and 42% brought drinks. Of those that brought drinks, 35% brought drinks again and the rest brought dessert to the next meeting. And of those that brought dessert to the first meeting, 26% brought dessert again and 48% brought chips.

a. What is the initial matrix for the student council?

b. What is the transition matrix for the student council?

c. Approximately how many students will bring drinks to the 4th meeting??

Practice AnswersThere are 20 students in student council and every week they bring snacks to their meeting. This week 8 brought chips, 7 brought drinks and 5 brought dessert. 18% of those who brought chips to the first meeting brought chips again and 42% brought drinks. Of those that brought drinks, 35% brought drinks again and the rest brought dessert to the next meeting. And of those that brought dessert to the first meeting, 26% brought candy again and 48% brought chips.

a. What is the initial matrix for the student council?

b. What is the transition matrix for the student council?

c. Approximately how many students will bring desserts to the 4th meeting?

0 8 7 5D .18 .42 .40

0 .35 .65

.48 .26 .26

T

6.59 so approximately 6-7 students(Find D4, then find the dessert column.)

Practice

2. Use the concept of dominance to solve each of the following games. Give the best row and column strategies and the saddle point of each game.

d. e. f.

1. Each of the following matrices represents a payoff matrix for a game. Determine if the game is strictly determined or not. If it is, find the best strategies for the row and column players, and the saddle point of the game.

Practice

2. Use the concept of dominance to solve each of the following games. Give the best row and column strategies and the saddle point of each game.

d. e. f.

1. Each of the following matrices represents a payoff matrix for a game. Determine if the game is strictly determined or not. If it is, find the best strategies for the row and column players, and the saddle point of the game.

Practice Answers

Not a Strictly Determined Game because maximin does not equal minimax.

Saddle Point at 4.Best strategies are row 3 and column 3.

Saddle Point at 0.Best strategies are row 1 and column 1.

0-2

3 1 2

-6-44

6 8 4

0-3-4

0 3 2

1. Each of the following matrices represents a payoff matrix for a game. Determine if the game is strictly determined or not. If it is, find the best strategies for the row and column players, and the saddle point of the game.

2. Use the concept of dominance to solve each of the following games. Give the best row and column strategies and the saddle point of each game.

1031

4 3 7

-20-22

4 2 5

Saddle point at 2.Best strategy for the Row Player is option D, and for the Column Player is option F.

Saddle Point at 3.Best strategy for the Row Player is option C, and for the Column Player is option F.

Practice Answers

Packet p. 12 #5 and #6 Review

Packet p. 12 #5 Review5. The matrix to the right represents a payoff matrix for a game.

a. Calculate the best strategy for the row player.

b. Calculate the best strategy for the column player.

c. If both players play their best strategy what is the expected payoff for the row player?

4 4

5 5

Work on Test Review