Total of n>2 companies considering entry into a new market. The cost of entry is 30. If only one company enters, then its gross profit is 200. If more than one company enters, then each entrant earns a gross profit of 40. The payoff to a company that enters is its gross profit minus its entry cost, while the payoff to a company that does not enter is 50. a) Show that if n=3, then there is a symmetric Nash equilibrium in mixed strategies in which each firm enters with probability 0.5 b) Find a symmetric Nash equilibrium in mixed strategies for arbitrary number of firms n (hint: the probability of each firm enters the market will depend on n)

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Q. There are two firms in the breakfast cereal industry of food, Kellogg and General Mills. Each firm has three strategies to follow to get the maximum payoff. In the below matrix find if there are any dominated strategies and a Nash equilibrium strategy for each firm. If so, what are they? What is the payoff (market share) for Kellogg and General Mills respectively?

All Payoffs are for Kellogg

Kellogg/GM	B1	B2	B3
A1	0.2	0.3	0.5
A2	0.6	0.5	0.3
A3	0.5	0.4	0.6

Question

Consider a market with 100 consumers. Each consumer would like to buy at most one unit and is willing to pay up to 10$. There is an incumbent firm that already operates in the market and a potential entrant firm that considers the possibility of entering the market. Both firms do not have any production costs but the entrant has a fixed entry costs of F = 280, that the entrant pays only if he enters the market. Out of the 100 consumers, 60 consumers are loyal to the incumbent and will not buy from any other firm. The remaining 40 consumers are non-loyal and will buy from the firm that charges the lowest price. At equal prices, non-loyal consumers will buy from the entrant. The timing is the following. In the first stage, the incumbent chooses a price, pI. In the second stage, the entrant observes pI and chooses whether to enter the market. If the entrant enters, the entrant pays F = 280 and then chooses a price, pE. If the entrant stays out, the entrant earns 0. Prices are continuous numbers. In the third stage, consumers decide from which firm to buy.

(a) What is the market outcome (entrantâs entry decision, prices and profits of the two firms)?

(b) Solve (a) for the case where F = 200 (instead of F = 280). Explain in words the differences between the two sections.

(c) Suppose again that F = 280. Suppose that the in first stage, the incumbent chooses a price, pI, and cannot change it later on. In the second stage, the entrant chooses whether to enter or not, and if the entrant enters, the entrant chooses a price, pE. However, the entrant cannot observe pI. The entrant knows that the incumbent charged a certain pI in the first stage and cannot change it, but the entrant cannot observe this pI. What are the prices and profits of the incumbent and the entrant? Explain your answer.

(d) Suppose now that there are two markets, A and B. Each market has 100 consumers, out of which 60 consumers are loyal. Consumers in market A are willing to pay up to 10$, while consumers in market B are willing to pay up to 15$. In each market there is an incumbent: incumbent A in market A and incumbent B in market B. There is one entrant that contemplates whether to enter into market A, market B, or non. The entrant does not have any entry costs (F = 0) and can enter only one of the markets. In the first stage, incumbent A chooses a price, pA. In the second stage, incumbent B observes the price of incumbent A and then chooses a price, pB. In the third stage, the entrant observes the two prices and chooses whether to enter one of the markets and if so, the entrant chooses a price. What is the market outcome (entrantâs entry decision, prices and profits of the three firms)?

Two firms are considering introducing new Microeconomics textbooks. The market can support only one. The payoff matrix is as follows. (Firm 1's payoff is listed first in

each cell). *For part c, ignore "mixed" or randomized strategies.

	Firm 2 Introduce	Firm 2 Don't Introduce
Firm 1 Introduce	-10, -10	50, 0
Firm 1 Dont Introduce	0, 50	0, 0

1. What are the cooperative solutions (if any)? Explain.
(Hint: In a "non-cooperative" solution, each party maximizes its own rewards regardless of the results for others. In a "cooperative" solution, the strategies of the participants are coordinated so as to attain the best result for the whole group.)

2. What are the dominant strategies (if any)? Explain.

3. What are the (pure) Nash Equilibria (if any)? Explain.

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