Question

Eric is an owner of a big firm and James applied for that firm. If James is smart, he will make $10 for Eric and Eric will pay him $7 (so that Eric’s profit from James will be $3). If James is not-smart, he will make $4 for Eric and Eric will pay him $3 (so that Eric’s profit from James will be $1). The probability that James is smart is 1/2. Actually, Eric does not know if James is smart or not and he will pay the average wage ($7+$3)/2=$5 and the expected profit for Eric would be $7-$5=$2. Then, professor Goo opened a “Useless University (=ULU)” and he gives a degree only to the students who pass his difficult test. In order to pass the test, the students should take professor Goo’s class and the tuition is $1 per month. If the student is smart, it takes only 1 month for him to pass the test. On the other hand, if the student is not-smart, it takes 5 months to pass the test. We will assume that Eric can check if James passes the test or not, but Eric cannot check how many months James took professor Goo's class. Find out two Perfect Bayesian equilibria of this game, one separating and one pooling.

Answer 1

There doesn’t exist any separating Perfect Bayesian equilibrium.
Separating:
AB’: q=0 → Player 2 prefer to play Y rather than x and player 1 prefer not to play B’ → No PBE
A’B: q=1 → Player 2 prefer to play X rather t than Y and player 1 prefer not to play A’ → No PBE

There exists one pooling perfect Bayesian equilibrium.
Pooling:
AA’ : No restriction on q and player 2 will choose to play Y if q<3/5, knowing player 2 would like to
play Y, player 1 doesn’t have any incentive to deviate from AA’ →PBE: (AA’, Y) with q<3/5
BB’ : q=1/2, and player 2 prefer to play Y rather than X, knowing this, player 1 would like to play AA’
rather than BB’ → No PBE