tentamen techieken

Question 1

What is the definition of Pareto efficiency?

Answer 1

Question 2

How to verify the definition of Pareto efficiency?

Answer 2

Show that by the definition no Pareto improvement is available

Question 3

How to calculate Pareto efficient allocations by weighed utility?

Answer 3

1. Create weighted utility function
2. Minimize (by derivation, or just spot it)

Question 4

How to put a game in normal form, how to find Nash equilibrium using this data?

Answer 4

Write N = {1, 2, .. (number of players)}, A₁ = [range of avaiblable investment] (do this for all), 𝜋_i (I_i, I_-i) = … profit function.

Just derive the profit functions (with regard to I₁, I₂, …) and equate to 0.

Question 5

How to solve subgame with imperfect information?

Answer 5

You can just simply rewrite it as a Nash-table and solve it like that.

Question 6

How to calculate probability of choosing something in a game?

Answer 6

Using the logit function where e^𝜆(value)/(e^𝜆(value)+e^{𝜆(other value)})

Question 7

How to find a players best response?

Answer 7

Just take the derivative of the utility function and equate to zero

Question 8

How to find Pure Nash equilibrium (with formulas)?

Answer 8

Derive and equate to zero (you get a matrix, and solve that).

Question 9

How to find weighted utility for a game in a table?

Answer 9

Question 10

How to compute Mixed Nash equilibrium?

Answer 10

1. Set (q, 1 - q) to the first two options. Thus crate U(action, q) and U(other action, q), where the action is from the other player.
2. Equate the utilities
3. Do the same for the other player.

Question 11

How to calculate the contract curve?

Answer 11

1. Set weights (𝛾, 1 - 𝛾)
2. Rewrite utility of second to s.t. x^B = (a, b) - x^A
3. Create 𝛾 u(x^A) + (1 - 𝛾) u(x^B)
4. Derive by x^A₁ , x^A₂ and equate to 0.
5. Write as x(𝛾) = (x^A(𝛾), x^B​(𝛾))

Question 12

How to find the barter equilibrium?

Answer 12

Calculate u_A(x^A(𝛾)) ≥ a, and u_B(x^B(𝛾)) ≥ b