Input – Output Models P = # of items produced D (demand) = # of items available for external sales So, the number produced – the number used internally = the number available for external sales P – (% used)P = D

Example: For every 100 batteries that are produced 5 (5% or .05) are used inside the company. P - .05P = D Suppose the company receives an order for 2000 batteries, how many need to be produced to fill this order?

1P – .05P = D 1P - .05P = 2000 (combine like terms) .95P = 2000 (divide by .95) 2105.2 = P 2106 batteries (cant make portion of a batteries so round up to next whole number) So, 2106 are produced, 106 batteries are used internally and 2000 are available for external sale.

Two Variable Economies Battery Division For the battery division to produce 100 batteries, it must use 5 (5%) of its own batteries. For every 100 batteries produced, 2 motors (2% of the number of batteries produced) are required from the motor division Motor Division For the motor division to produce 100 motors, it must use 3 (3%) of its own motors For every 100 motors produced, 8 batteries (8% of the number of motors produced) are required from the battery division.

Making a consumption matrix To Batteries Motors Batteries .05 .08 From Motors .02 .03 The consumption matrix is usually matrix [C] in the book. When making your consumption matrices, think of where the items are coming FROM and where are they going TO.

Making a Weighted Digraph B M B .05 .08 M .02 .03 .02 .03 M B .05 .08

Formulas you need for this section 1.) When production (P) is given and you need to find demand (D) or how many will be left to use (or sell), use D = P – CP 2.) When demand (D) is given and you need to find production (P) to know how many to make to fill the order, use P = (I – C)-1 D

1.) In a given day you produce 2000 batteries and 500 motors at the factory, how many will be available for sale after production?

2.) You have an order come in for 1000 batteries and 2000 motors, how may of each do you need to produce in fill this order?

Use the same consumption matrix for these two problems. 3.) On a given day you produce 500 batteries and 250 motors, how many will be available for external use (sale)? How many are consumed internally? 4.) An order comes in for 10,000 batteries and 3000 motors, how many will the factory have to produce to fill this order?

3.) Use

4.) Use

Given the following consumption matrix answer the following questions. A B C D A .11 .12 .23 .04 B .22 .13 .31 .21 C .06 .22 .04 0 D .20 .31 .22 .11 Which sector is Sector B most and least dependent on? If Sector A has an output of $5 million, what is the input from Sector B

Markov Chains There is 70% chance of rain today. It is known that tomorrow’s weather depends on today’s. If it rains today, then there is a 60% chance of rain tomorrow. If it does not rain today, then there is a 35% chance it will rain tomorrow. 1.) Make a tree diagram to show the possible outcomes and the chances for rain tomorrow.

Today Tomorrow rain=.7x.6 .6 = .42 rain no rain=.7x.4 .7 .4 =.28 .6 = .42 rain no rain=.7x.4 .7 .4 =.28 .35 rain = .3x.35 .3 = .105 no rain .65 no rain=.3x.65 =.195 Chance it rains tomorrow is .42+.105= .525

What do we do if it is more than one day? D1 = D0T The first day is called the initial distribution and is a row matrix. For our example it would be .70 .30 The percentages for the following day’s outcome make up what is called the transition matrix. For our example it would be .60 .40 .35 .65 Transition matrices must be square and the rows add up to 1.

Game Theory Using matrices to determine a player’s best strategy. The matrix is called a payoff matrix. The first types of game are very simple and easy to see the best strategy, we will get the harder game in the next section. A game in which the best strategy for both players is to pursue the same strategy every time is called strictly determined. We will start with strictly determined games.

Jill Heads Tails Jack Heads 4 -1 Tails -1 -3 Two players are playing a simple coin-matching game. Each player conceals a coin in their hand and show the other player at the same time. Our two players are Jack and Jill. Here are the payouts: Jack will win 4 pennies from Jill if both are heads. Jill will win 3 pennies from Jack if they are both tails and one penny if they do not match. Payout matrix Jill Heads Tails Jack Heads 4 -1 Tails -1 -3

Jill Heads Tails Jack Heads 4 -1 Tails -1 -3 The payout matrix is always written from the row players point of view. If we want to think about this from Jill point of view all of the values are the opposite of the current payout matrix. Jill’s POV Jill Jack Heads -4 1 Tails 1 3

To find the Row players (Jack’s best strategy) find the minimum value in each row and then circle the largest of those minimums (maximin). To find the column players (Jill’s best strategy) find the maximum value in each column and circle the smallest of those maximums (minimax). Jill (Row Heads Tails min) Jack Heads 4 -1 -1 Tails -1 -3 -3 (Column max) 4 -1 Since the minimax and the maximin are the same number the game is strictly determined and this value is called the saddle point

Eliminating Rows and Columns It is sometimes beneficial to eliminate certain strategies if other strategies have them dominated (are clearly a better choice for all possibilities). You will need to use this in the next section! You can cross off any row where all of the values are lower then the corresponding values of another row. You can cross off any column where all of the values are higher then the corresponding values of another column.

E F G A 3 1 7 B 0 1 3 C 4 3 4 D 1 2 6 B 0 1 3 Compare C 4 3 4 row B and row C

Changing the Game Our two players are Jack and Jill. Here are the payouts: Jack will win 4 pennies from Jill if both are heads and will win 2 pennies from Jill if they are both tails. Jill will win three penny if he has heads and she has tails and 2 pennies if he has tails and she has heads. Make the payout matrix for this game. Find the minimax and the maximin.

Jill H T Jack H 4 -3 -3 T -2 2 -2 4 2 The game is not strictly determined, therefore has no saddle point. The best strategy is to play both options at random to keep your opponent from guessing your choice. But at what percentage should we play each of the choice to optimize our payout.

Let’s first figure what would happen if each player played each of the strategies 50% of the time. HH = .25 HT = .25 TH = .25 TT = .25 Jack Jill H T .5

Use the percentages found in the tree diagram to find the expectations for this game, if both players play 50% heads and 50% tails. Outcome HH HT TH TT Probability .25 .25 .25 .25 Amount 4 -3 -2 2 Expectations = .25(4) + .25(-3) + .25(-2) + .25(2) = .25 The row player’s advantage is .25 of a cent per play

Finding your best strategy. Row Player p 1-p 4 -3 = -2 2 Set equal and solve:

Finding your best strategy. Column Player 4 -3 q = -2 2 1-q Set equal and solve:

Finding the Expectations 5/11 H H 4/11 Jill 6/11 T Jack 5/11 H 7/11 T Jill 6/11 T

Finding the Expectations = The interpretation of this answer is that if the row player plays their best strategy and the column plays their best strategy, then the row player will win .18 cents per play. A positive number is in the row player’s advantage A negative number is in the column player’s advantage.

For the following matrices find the row and column player’s best strategies and the expectations for the games. 2 1 -4 3 2 -4 5 -2 5 6 -2 -3 1