Derive the nash equilibrium prices in the game played

Question

Assignment

Question 1

There is a collection of 5 NGOs operating in the health sector of a country. Each NGO can choose either to invest in a large-scale and collaborative project with other NGOs, or invest in an independent, small-scale, project where the NGO does not have to collaborate with other NGOs. A small-scale project is always successful and gives the NGO a payoff of S. For the large-scale project to succeed, at least 3 NGOs have to invest in it. If the large-scale project is successful, the value that it generates is L, which is evenly divided between the collaborating NGOs. Assume that L/5 > S > 0. If the large-scale project fails, each collaborating NGO gets 0.

a) Describe the normal form of this game
b) Find the BR function for an NGO.
c) Find all NE (in pure strategies).

Question 2- Complementary Monopoly

A firm sells vacation packages consisting of a plane trip and a hotel stay. Demand for complete vacations is given by q(p_v) = 100 - p_v, where p_v is the total price of the vacation. That is, p_v = p_h + p_a, where ph and pa are the prices of the hotel stay and plane trip, respectively. Let ch be the cost of each hotel stay and ca be the cost of each plane trip.

a) Suppose the firm chooses pv in order to maximize total profit. How much should it charge for each vacation? How many vacations are sold? What is its profit? (Note that profits π = p*q-c(q) : the total revenue from selling q minus the total cost)

b) Suppose that the hotel division and airline division are run by separate managers, each of whom cares only about the profit earned by his own division. Given that the hotel division charges p_h, what price would the airline division charge (i.e., what price maximizes profit earned by the airline division)? Note: your answer will give pa as a function of ph.

c) Derive the Nash equilibrium prices in the game played between the managers of the two divisions. What are the corresponding quantities and profits?

Question 3- Revisiting Hotelling's Model

Two political parties, Left wing and Right wing compete in electoral elections. There is a continuum of voters uniformly distributed on the interval [0, 1] where 0 represents the leftmost view (extreme left) and 1 the rightmost. Each party can declare that their view is x where 1 ≥ x ≥ 0 and each voter votes to the party that declare that their views are closest to his views. For example, if a voter's view is represented by 0.3, the left party chose 1/4 and the right party chose 1/2, this voter votes for the left party as 1/4 is closer to 0.3. If both parties choose the same spots they will split the votes between them.

Each party's objective is to have the highest percentage of voters voting for it.

i) Describe the normal form of the game (players, strategies, payoffs).
ii) Solve the game using (pure strategy) Nash equilibrium.