Question

3. Find the Bayesian-Nash Equilibrium for the following Entry game." Two firms in same product market Incumbent chooses Build (B) or Don't Build (D) capacity Entrant chooses Enter E or Stay Out S Incumbent has two types, which affect cost of building capacity a = high cost type, and θι low cost type-o, has a higher capacity cost than θ Prior probability of 0h is p. Idea is incumbent earns a higher profit if entrant stays out. So may want to build capacity to threaten a price war and deter entry Payoffs are as follows: Entrant Entrant Incumbent B 0,-1 2,0 D 2,1 3,0 Payoffs if O В 3-1 5,0 D 2,1 3,0 Payoffs if θι a) What is the Bayesian-Nash equilibrium? Describe it based on the value of pi which denotes the common prior probability that Entrant's cost is high (0) (NOTE: A Bayesian Nash equilibrium is described by a strategy for the 0, type, a strategy for the 0 type, and a strategy for the Incumbent.)

Answer 1

In case of high type, if entrant chooses to enter or stay out ,incumbent Won't build as it follows a dominant strategy in not bulding. And if the incumbent doesn't build, entrant enters

In case of low cost type, the incumber builds irrespective of the entrant enterning or staying out as building is the dominant strategy. And if the incumbent builds, entrant stays out

If p is the probability of high type, then for the incumbent, pay off is

2p+(1-p)*5=2p+5-5p=5-3p and the maximum is achived When p=0, i.e. in the case of low cost type where the incumbent chooses to build

Hence the Bayesian Nash equilibrium is (enters, stays out, builds)

Find the Bayesian-Nash Equilibrium for the following "Entry game?",

Two firms in same product market.

Incumbent chooses Build (B) or Don't Build(D) capacity.

θh=high cost type,and θl low cost type.  θh has a higher capacity cost than  θl.

Prior probability of  θh is p.

Idea is incument earns a higher profit if entrant stays out. So may want to build capacity to threaten a price war and deter entry.

Payoffs are as fallows:

Entrant Entrant

E S E

S

Incumbent B 0,-1 2,0 B 3,-1 5,0

D 2,1 3,0 D 2,1 3,0

Payoffs if  θh Payoffs if

θl

a) What is the Bayesian-Nash equlibrium? Describe it based on the value of p1, Which denotes the common prior probability that Entrant's cost is high ( θ).

(NOTE: A Bayesian Nash equilibrium is described by a strategy for the  θh type, a strategy for  θl type, and strategy for the Incumbent.)