Bayesian-Nash equilibrium paper

Q1. Consider a simultaneous-move game between an entrant and a monopolist. The monopolist
can accommodate entry, A, or fight entry, F. The entrant can enter, E, or stay out, S. The payoffs
depend on whether the monopolist is low-cost or high-cost. The monopolist knows whether it is
low-cost or high-cost. The entrant on the other hand believes that the monopolist could be low-cost
or high-cost with probabilities 2 and 3 respectively. These facts and the payoffs below are common

knowledge among the entrant and monopolist.

Low-cost Monopolist High-cost Monopolist
A F A F
E (1,1) (-1,2) E (1,1) (-1,-1)
S (0,3) (0,4) S (0,3) (0,0)

Determine the Bayesian-Nash equilibria of this game
The strategies for the entrant are EF and S. The monopolist is of two possible
types — low cost or high cost. The strategies for the monopolist specify an
action for each type. Therefore the monopolist has 4 possible strategies:

Monopolist’s Strategies Denoted as:
{Low-cost} — A; {High-cost} — A | (4, A)
{Low-cost} — A; {High-cost} — F | (A, F)
{Low-cost} — F; {High-cost} — A | (F, A)
{Low-cost} — F; {High-cost} — F | (FF)

The entrant’s expected payoffs from FE and S against each of the strategies
of the monopolist are:

Entrant’s Payofls
E S
Against (A, A) ZMH +in=1 200) +1(0)=0
Against (AF) | ZT 1(-D=1 [20 +10)=0
Against (FLA) | 2-1) +1) =-1 [200 +1(0)=0
Against (F,F) [ 2(-1D) + 1-1) = 1] 30) + 1(0)=0

(AA) | (AF) (F, A) (FF)
E 1,(1,1) 5 (1,1) — (2, 1) -1,(2,-1)
S510,(3,3)]0,(3,0) 0, (4,3) 0, (4,0)

We can then “box” the best responses for each player. Note that for the
monopolist we box the best response to E and S for each type.
Answer key:

AA) [AF (FA) FED
elmam |B -1|-ten|-1.a-1)
510,380,300 |0@&3 |0®@0)

The only cell in which all payoffs are “boxed” is (5, (F, A)). Therefore this
is the unique Bayesian-Nash equilibrium of the game.