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finish this assignment and draw Econ 0200 Game Theory Problem Set \#3 1. What is...

Sep 25, 2024

finish this assignment and draw

Solution

Expected Value Calculation: To find the expected value of the lottery ticket, we calculate the expected value (EV) using the formula:

EV = \sum (P_i \times V_i)

where

P_i

is the probability of each outcome and

V_i

is the value of that outcome. The probabilities and values from the prize table are: - Win 600 with probability $ \frac{1}{1000} $ - Win 50 with probability

\frac{1}{20}

- Win 4 with probability $ \frac{1}{5} $ - Win 1 with the remaining probability

1 - \left(\frac{1}{1000} + \frac{1}{20} + \frac{1}{5}\right)

Calculation of Probabilities: First, we calculate the remaining probability:

1 - \left(\frac{1}{1000} + \frac{1}{20} + \frac{1}{5}\right) = 1 - \left(0.001 + 0.05 + 0.2\right) = 1 - 0.251 = 0.749

Thus, the probability of winning $1 is $0.749$

Total Expected Value: Now we can calculate the expected value:

EV = \left(\frac{1}{1000} \times 600\right) + \left(\frac{1}{20} \times 50\right) + \left(\frac{1}{5} \times 4\right) + \left(0.749 \times 1\right)

Calculating each term: -

\frac{1}{1000} \times 600 = 0.6

\frac{1}{20} \times 50 = 2.5

\frac{1}{5} \times 4 = 0.8

0.749 \times 1 = 0.749

Adding these together gives:

EV = 0.6 + 2.5 + 0.8 + 0.749 = 4.649

Rounding to the nearest penny, the expected value is $4.65

Finding Mixed Strategy Nash Equilibria: For the mixed strategy Nash equilibria, we need to set the expected values equal for both players. For player 2, we set:

b(x) + f(1-x) = d(x) + h(1-x)

and solve for

x

. Similarly, for player 1, we set:

a(1-p) + c(p) = e(1-p) + g(p)

and solve for

p

. This process will be repeated for each game scenario from 2 to 9

Answer

The expected value of the lottery ticket is $4.65

Key Concept

Expected Value in Game Theory

Explanation

The expected value is a crucial concept in game theory, representing the average outcome of a probabilistic event, which helps in decision-making under uncertainty.