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Finite Math B Chapter 11: Game Theory

Game Theory: A branch of mathematics that provides a systematic way to attack problems of decision making when some alternatives are unclear or ambiguous. Examples:

A football team tries to decide which play to run. The success of the play depends on the reaction of the opposing team and how well they “read” the play.

A hospital administrator must choose nursing staffing levels even though the demand is unpredictable.

A concert venue must make plans to set up inside (which holds less seats) or outside (which holds more seats) even though an outside concert may have to be cancelled due to weather.

Example: (from text, pg 599) A small manufacturer of Christmas cards must decide in February what type of cards to emphasize in her fall line. She has three possible strategies: emphasize modern cards, emphasize old-fashioned cards, or emphasize a mixture of the two. Her success depends on the state of the economy in December. If the economy is strong, she will do well with her modern cards. In a weak economy, people long for the “old days” and buy old-fashioned cards. In an in-between economy, a mixture of cards do the best. Define the problem: PAYOFF MATRIX: States of Nature: alternatives over which you have ___________________________ Strategies: Payoffs: choices you can make consequences of each strategy for each state of nature

What would an optimist do? try for ____________________ payoffs, produce ____________________ cards

What would pessimist do? “What’s the worst that could happen?” try to avoid ____________ payoffs, produce ______________________ cards

The manufacturer reads in a business magazine that leading experts think there is a 50% chance of a weak economy at Christmas, a 20% chance of an in-between economy, and a 30% chance of a strong economy. Now what should her strategy be?

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History Lesson for the Day Game Theory - developed in 1940’s by John von Neumann to

analyze aggressive/competitive situations in business, warfare, and social interactions.

Neumann and pal Oskar Morganstern publish book in 1944 called

“Theory of Games and Economic Behavior”

Movie “A Beautiful Mind” based on work of John F. Nash, Jr. who won a Nobel Prize in Economics for work in Game Theory in 1994.

R.J. Aumann & T.C. Schelling won Nobel Prize in Economics for

Game Theory work in 2005.

Game Theory mentioned in many episodes of “Numb3rs” TV Show as FBI attempts to predict outcome of interaction with criminals.

----------------------------------------------------------------------------------------------------------------------------------------------------- Two Person Games Payoff Matrix: A picks row. B picks column. A Player’s Choice is a strategy. Positive Value = payout to player A. Negative Value = payout to player B. Who gets what if A chooses strategy 2 and B chooses strategy 1? ______________________________________ Example:

a)Determine the payoff when strategy (3, 2) is used. b) Determine the payoff when strategy (1,3) is used.

Any game where no money enters the game from the outside is called a zero-sum game. This means that someone LOSES and someone WINS each time. The stock market is not a zero-sum game. It is possible for outside forces to affect game and for all players to win or to lose.

John von Neumann 1903-1957

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ACTIVITY: Game Theory GAME STRATEGIES Yellow Game Pink Game

B1 B2

A1 -4 8

A2 5 -1 Green Game

B1 B2

A1 -2 -4

A2 -3 7 Light Blue Game

B1 B2

A1 2 3

A2 -3 -1

B1 B2

A1 2 -1

A2 -3 4

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GAME STRATEGIES Violet Game

B1 B2 B3

A1 -2 -1 9

A2 4 5 -5 Brown Game

B1 B2

A1 2 -6

A2 -4 5

A3 3 -1 Orange Game

B1 B2 B3

A1 -2 -1 6

A2 -3 5 -6 Dark Blue Game

B1 B2

A1 1 4

A2 -2 -5

A3 -5 7

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DOMINATED STRATEGIES In a zero sum game, not all strategies are equal. Some strategies are much better than others. If one strategy “dominates” another, it can be removed from consideration. Example: Assume you are player B

Example: Find any dominated strategies in the games with the given payoff matrices. Then remove any dominated strategies from the game. a) b) c) d)

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Strictly Determined Games What strategies should the players use in the following game? Goal: Optimum strategies – most profitable payoff to respective players. Value of the game - the payoff when each player chooses the optimum strategy. Pure strategy: Always choosing the same row (discussed in this section) Mixed strategy: Random choice of rows (discussed in the next section) Assume players are going to both play by PURE STRATEGY: A: Chooses R1 B: Chooses C1 A: Chooses R2 B: Chooses C2 A: Chooses R3 B: Chooses C3 The optimum strategy here is for A to play ______ and B to play ______. The value of the game is: ______ The position _______ is called the saddle point. A game with a saddle point is called a strictly determined game. The value of the saddle point = the value of the game = the payout that will happen if each player uses pure strategies. A game with a value of 0 is a “fair game”. A game with a positive value benefits player _____. A game with a negative value benefits player ______.

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Examples: Find the saddle points in the following games. (Hint: Underline the smallest number in each row. Circle the largest number in each column. Any the same?) a) b) c) d) e) f)

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Example: A company has developed a new product. If the new product is better than the old product and the company decides to market it, sales should increase by $70,000. If it is not better and they market the new product, they will lost $35,000. If they decide not to market the new product, they will lost a total of $35,000 if it is better, and just research costs of $15,000 if it is not. They believe there is 40% chance the new product is better. a) Prepare a playoff matrix b) What strategy should the company take? What would the predicted outcome on sales be? c) Are there any dominated strategies? Is there a saddle point? Example: College kids are playing capture the flag on campus. One team’s “home base” is located somewhere between the student union and the library. A person playing figures he has two options. If he tries to sneak in behind the student union, his chance of success is 80% if extra guards are sent to the library and 40% if extra guards are sent to the student union. If he tries to sneak in at a library, his chance of success is 20% if extra guards are sent to the student union and 30% if extra guards are sent to the library. Let the payoffs be the probabilities of success. Find the strategies producing the saddle point and find the value of the game. What is the best “sneaking” strategy for the person?

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11.2 Mixed Strategies Scenario: Zero Sum Game Optimum Strategy: Pure Strategy Two Players A picks Row with Saddle point Saddle Point Exists B picks Column with Saddle point (Strictly Determined Game) Game Value (payout) = Saddle Point If there is no saddle point, it is necessary for both players to _______ their strategies! Example 1: Suppose you have the payoff matrix.

What if Player A decides to flip a coin to decide which row to play. Over time he will play row 1 50% of the time and row 2 50% of the time. Player B decides to play column 1 75% of the time and column 2 25% of the time. What is the probability of each entry being selected? P(row 1, col 1) = P(row 1) x P(col 1) = _______ x _______ = _______ P(row 1, col 2) = P(row 1) x P(col 2) = _______ x _______ = _______ P(row 2, col 1) = P(row 2) x P(col 1) = _______ x _______ = ________ P(row 2, col 2) = P(row2 ) x P(col 2) = _______ x _______= ________ What is the expected value of the game using these strategies? P(row1,col1) x a1,1 + P(row1,col2) x a1,2 + P(row2,col1) x a2,1 + P(row2,col2) x a2,2 = Expected Value

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If A and B vary their strategies even a little bit, the outcome of the game can be VERY different. So how do we decide the BEST Mixed strategy? In other words, how often should we play each row or play each column? One way to do this in a two player game is to use the following formulas:

Example 2:

Non-strictly Determined Game:

Payoff Matrix: 11 12

21 22

a a

a a

Optimum Strategies

Divisor: Row Player 1 2p p

d = 11 21 12 22a a a a p1 = play row 1 22 211

a ap

d

p2 = play row 2 2 11p p

Value of the game:

g = 11 22 12 21a a a a

d

Column Player

1

2

q

q

q1 = play column 1 22 121

a aq

d

q2 = play column 2 2 11q q

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Example 3: Write the row and column matrices that represent the optimal strategies, and find the value of the game. Be sure to look for a saddle point first! a) b)

c) 2 3

4 7

d) (Hint: remove dominated strategies!)

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11.3 Game Theory and Linear Programming Scenario: No saddle point (not strictly determined) – no “pure strategy” More than 2 choices for a player and matrix cannot be reduced by dominance (cannot use yesterday’s formulas)

Solving a Zero Sum Game 1. Set up the payoff matrix. 2. Remove any dominated rows or columns. (Section 11.1, Day 1) 3. Check for a saddle point. (Section 11.1, Day 2) If the saddle point exists, value of the game = saddle point and optimum strategies = (row, column) of saddle point. If no saddle point exists, but payoff matrix = 2x2, used mixed strategy formulas (Section 11.2) If no saddle point exists, and payoff matrix isn’t 2x2,proceed to step 4. 4. If payoff matrix contains negative numbers, add “K” a positive number to all entries so that all entries are non negative. 5. Set up a linear programming problem in the following format:

1 2 3

11 1 1

1 1

1 2

Maximize

subject to 1

1

with 0, 0, 0

n

n n

m mn n

n

z x x x x

a x a x

a x a x

x x x

6. Solve.

Value of game: 1

gz

Optimum strategy Player B:

1 1

2 2

...

n n

q x g

q x g

q x g

Optimum strategy Player A:

1 1

2 2

...

n n

p y g

p y g

p y g

7. If you added a constant “K” in step 4, subtract from “g” to find the actual value of the game.

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Example 1: Convert the following playoff matrix to a linear programming problem. Then convert to a simplex tableau.

Example 2: Given the following payoff matrix Set up the linear programming problem and write the corresponding simplex tableau.

After all necessary row operations, you obtain the following simplex tableau:

What are the optimum strategies? What is the value of the game?

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Example 3: Given the following payoff matrix Set up the linear programming problem and write the corresponding simplex tableau.

After all necessary row operations, you obtain the following matrix:

What are the optimum strategies? What is the value of the game?

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Example 4: Use simplex method to find the optimum strategies and the value of the game for: