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Analyzing Strategies in Game Theory and Estimating the Golden Ratio using Fibonacci Numbers

## Task 1: Analyzing the Game of Matching Pennies

1. For the game of matching pennies shown below, the payoff is for player 1. Determine the best strategy for each player, the expected value of the game, and which player (if either) has an advantage. Determine the expected outcome if the game is played 1000 times.

Player 1 Should _______________________________________________________

Player 2 Should _______________________________________________________

Expected Value of Game _______________________

Expected Outcome if Played 1000 Times ________________________________

Suppose Player 2 decides to "flip a coin" rather than play the best strategy. What should Player 1 do? How does this change the outcome of the game?

Player 2

H T

Player 1 H -7 11

T 5 -8

Player 1 Should _______________________________________________________

Player 2 Should _______________________________________________________

Expected Value of Game _______________________

Expected Outcome if Played 1000 Times ________________________________

Suppose Player 2 decides to "flip a coin" rather than play the best strategy. What should Player 1 do? How does this change the outcome of the game?

2. Suppose you have a calculator but no access to the internet.Explain how you could use Fibonacci Numbers to find an extremely accurate estimate the value of the Golden Ratio. Next, use the method you described to estimate the Golden Ratio to 9 decimal places. [Note: You need to describe your method and use it to justify your answer. Simply writing down an answer is NOT sufficient]

3. Determine the minimax and maximin strategies for the game shown below. PAYOFF IS FOR PLAYER 1.

Player 1

Col 1 Col 2 Col 3 Col 4 Col 5 Col 6

Row 1 1 -1 6 4 3 2

Player 2 Row 2 -1 0 -2 -1 -1 1

Row 3 -2 4 9 -2 -3 3

Row 4 5 2 1 -5 -4 7

Row 5 2 1 7 0 -1 2

Row 6 4 6 3 9 2 6

Circle one State Specific Strategy

Player 1 uses the maximin\minimax _____________________________

Player 2 uses the maximin\minimax _____________________________

The game does\does not have a saddle point

Should Both Players use their minimax or maximin strategy? _________________

Expected Value of the game (or a Range for EV) ___________________________

Value of the game if both players play their “pure” strategy __________________

If Player 1 uses his “Pure” Strategy, what should Player 2 do? ________________

4.Determine the minimax and maximin strategies for the game shown below. PAYOFF IS FOR PLAYER 2.

Player 1

Col 1 Col 2 Col 3 Col 4 Col 5 Col 6

Row 1 1 -1 6 4 3 2

Player 2 Row 2 -1 0 -2 -1 -1 1

Row 3 -2 4 9 -2 -3 3

Row 4 5 2 1 -5 -4 7

Row 5 2 1 7 0 -1 2

Row 6 4 6 3 9 2 6

Circle one State Specific Strategy

Player 1 uses the maximin\minimax _____________________________

Player 2 uses the maximin\minimax _____________________________

The game does\does not have a saddle point

Should Both Players use their minimax or maximin strategy? _________________

Expected Value of the game (or a Range for EV) ___________________________

Value of the game if both players play their “pure” strategy __________________

If Player 1 uses his “Pure” Strategy, what should Player 2 do?

5. Suppose in the game shown in below (payoff for player 1) Player 2 notices that Player 1 is playing heads three fourths of the time and playing tails one fourth of the time. What should player 2 do to take advantage of this? How does this change the value of the game? What is the expected outcome if the game is played 1000 times under these new conditions?

Player 1

H T

Player 2 H 7 -15

(payoff) T -5 11

Player 2 Should __________________________________________

The new Value of the game is ________________________________________

The expected outcome is played 1000 times

6. The Numbers:

(a) The numbers 24157817 and 63245986 are both Fibonacci numbers. There is also one Fibonacci Number between these two. What is it?