You are the Dean of the Faculty at St. Anford University.

You are the Dean of the Faculty at St. Anford University. You hire Assistant
Professors for a probationary period of 7 years, after which they come up
for tenure and are either promoted and gain a job for life or turned down,
in which case they must find another job elsewhere.
Your Assistant Professors come in two types, Good and Brilliant.
Any types worse than Good have already been weeded out in the hiring
process, but you cannot directly distinguish between Good and Brilliant
types. Each individual Assistant Professor knows whether he or she is
Brilliant or merely Good. You would like to tenure only the Brilliant types.
The payoff from a tenured career at St. Anford is $2 million; think
of this as the expected discounted present value of salaries, consulting
fees, and book royalties, plus the monetary equivalent of the pride and
joy that the faculty member and his or her family would get from being
tenured at St. Anford. Anyone denied tenure at St. Anford will get a fac-
ulty position at Boondocks College, and the present value of that career
is $0.5 million.
Your faculty can do research and publish the findings. But each
publication requires effort and time and causes strain on the family;
all these are costly to the faculty member. The monetary equivalent of
this cost is $30,000 per publication for a Brilliant Assistant Professor and
$60,000 per publication for a Good one. You can set a minimum number,
N, of publications that an Assistant Professor must produce in order to
achieve tenure.
(a) Without doing any math, describe, as completely as you can, what
would happen in a separating equilibrium to this game.
(b) There are two potential types of pooling outcomes to this game.
Without doing any math, describe what they would look like, as
completely as you can.
(c) Now please go ahead and do some math. What is the set of possible
N that will accomplish your goal of screening the Brilliant profes-
sors out from the merely Good ones?