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# Quantitative Analysis and Game Theory

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Quantitative Analysis (Show All Work)

#1

Solve the following Game:

Y1 Y2

X1 -5 -10

X2 12 8

X3 4 12

X4 -40 -5

#2

For the following two-person, zero-sum game, are there any dominated strategies? If so, eliminate any dominated strategy and find the value of the game.

Y1 Y2 Y3
X1 4 5 10

X2 3 4 2

X3 8 6 9

#3 Probability

A silver dollar is flipped twice. Calculate the probability of each of the following occurring:

a. a head on the first flip
b. a tail on the second flip given that the first toss was a head
c. two tails
d. a tail on the first and a head on the second
e. a tail on the first and a head on the second or a head on the first and a tail on the second
f. at least one head on two flips

##### Solution Summary

The solution produces a game theory based on:

Quantitative Analysis (Show All Work)

#1

Solve the following Game:

Y1 Y2

X1 -5 -10

X2 12 8

X3 4 12

X4 -40 -5

#2

For the following two-person, zero-sum game, are there any dominated strategies? If so, eliminate any dominated strategy and find the value of the game.

Y1 Y2 Y3
X1 4 5 10

X2 3 4 2

X3 8 6 9

#3 Probability

A silver dollar is flipped twice. Calculate the probability of each of the following occurring:

a. a head on the first flip
b. a tail on the second flip given that the first toss was a head
c. two tails
d. a tail on the first and a head on the second
e. a tail on the first and a head on the second or a head on the first and a tail on the second
f. at least one head on two flips

##### Solution Preview

Quantitative Analysis (Show All Work)

#1

Solve the following Game:

Y1 Y2

X1 -5 -10

X2 12 8

X3 4 12

X4 -40 -5

#2

For the following two-person, zero-sum game, are there any dominated strategies? If so, eliminate any dominated strategy and find the value of the game.

Y1 Y2 Y3
X1 4 5 10

X2 3 4 2

X3 8 6 9

#3 Probability

A silver dollar is flipped twice. Calculate the probability of each of the following occurring:

a. a head on the first flip
b. a tail on the second flip given that the first toss was a head
c. two tails
d. a tail on the first and a head on the second
e. a tail on the first and a head on the second or a head on the first and a tail on the second
at least one head on two flips

EXPLANATION FOR #1 & #2
Game theory describes the situations involving conflict in which the payoff is affected by the actions and counter-actions of intelligent opponents. Two-person zero-sum games play a central role in the development of the theory of games. In order to develop the concepts, consider the following game in which player I has two choices from which to select, and player II has three alternatives for each choice of player I. The payoff matrix T is given below:
player II
j=1 j=2 j=3
player I i=1 4 1 3
i=2 2 3 4
______________________________________
The Payoff Matrix
In the payoff matrix, the two rows (i = 1, 2) represent the two possible strategies that player I can employ, and the three columns (j = 1, 2, 3) represent the three possible strategies that player II can employ. The payoff ...

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