Problem Set 6

1. The following is an Entry Game with incomplete information. An entrant firm, Firm 1 decides whether or not to enter a market that already has an incumbent firm, Firm 2. If Firm 1 enters, Firm 2 can either fight the entrant or accommodate it.

Firm 1 can be of two types – Aggressive or Passive. The payoffs for each type of entrant are given in the game tree below.

Suppose Firm 1’s type is known to both players, so that there is complete information.

1. Using the extensive form representation provided above, fill in the payoffs for each type of Firm 1 in the payoff matrices below.

Aggressive

Passive

2. Find the Nash Equilibrium for each type of Firm 1 by using the two separate payoff matrices.

Aggressive

Passive

2. Suppose now that Firm 2 does not know Firm 1’s type. The game of incomplete information is represented below. The prior belief about Firm 1’s type is given by

PrAggressive=p,  PrPassive=1-p,

1. List the strategies available to each player. Remember that Firm 1’s strategy will be contingent on its type, so it has 4 possible strategies. Firm 2’s strategies do not depend on Firm 1’s type, so it has 2 strategies.

2. Construct a new payoff matrix with Firm 1’s strategies listed along rows and Firm 2’s strategies along columns and fill in the expected payoffs.

3. Find all the Bayesian Nash Equilibria (BNE). Specify the condition on prior beliefs, p

   

    where necessary. The BNE will be different for p≤12

   

    and p≥12.

   

    

4. Check if each of the BNE found in part e) are sequentially rational.

3. In the Entry Game described above, let us check if there are separating equilibria that qualify as a Perfect Bayesian Equilibria (PBE).

First consider Firm 1’s strategy EN.

1. What is the updated belief, q

   

   , that Firm 2 must hold about Firm 1’s type if Firm 1 follows the given separating strategy?

q=Pr?[Aggressive|E]

2. What is Firm 2’s best response if Firm 1 enters given its updated belief q

   

   ?

3. Is Firm 1’s chosen separating strategy rational for each of its type given Firm 2’s strategy chosen in part b)?

4. Is there a PBE with the given separating strategy for Firm 1? If so, specify the strategy profile and associated beliefs.

Next consider Firm 1’s strategy NE.

5. Redo parts a) to d) above to test if a PBE exist with this separating strategy.

4. In the Entry Game described above, let us check if there are pooling equilibria that qualify as a Perfect Bayesian Equilibria (PBE).

First consider Firm 1’s strategy EE.

1. What is the updated belief, q

   

   , that Firm 2 must hold about Firm 1’s type if Firm 1 follows the given the pooling strategy?

q=Pr?[Aggressive|E]

2. What is Firm 2’s best response if Firm 1 enters given its updated belief q

   

   ?

3. Is Firm 1’s chosen pooling strategy rational for each of its type given Firm 2’s strategy chosen in part b)?

4. Is there a PBE with the given pooling strategy for Firm 1? If so, specify the strategy profile and associated beliefs.

Next consider Firm 1’s strategy NN.

5. Redo parts a) to d) above to test if a PBE exist with this pooling strategy.