A game has incomplete information if not all players know all payoffs of the other players.
John Harsanyi ...
A version of Spence's Education Game:
Assume there are two kind of people: Clever people, able to learn fast and
with little "cost", and the normal kind. Furthermore assume that the clever
people are much more valuable for some high-level company. People decide
whether they want to get a (College) education, but note that whether they get it or not does
notaffect their value for the company. After having graduated (or not),
the company decides whether it wants to hire that particular person, but
the type of the person (clever or normal) is known only to the person but
not to the company. What happens? (compare [Spence 1973])
More precisely, we assume that p of the population is clever, that getting a degree costs
x units for the clever, and y units otherwise, with x < y. Getting the job at that high-level
company is worth z units of money. Of course, z must be larger than x,
othertwise nobody would bother to get an education. Furthermore, hiring a clever guy is worth v units
for the company, but hiring a normal person is worth w, usually w is negative).
In the first, nobody gets an education, and the company hires 2/3 of those with education but nobody without. Since nobody gets an education, this means that nobody is hired. The payoffs for emplyees and the company are both 0.
In the second equilibrium, nobody gets an education, and nobody is ever hired. The payoffs are 0 and 0 again.
The third equilibrium is certainly the most interesting. Here clever people get an education, but normal people don't. The company just hires educated people. The average payoff for a person is 1/10, and the average payoff for the company (per person) is 2/5. Since the company would want to hire somebody, this is what may happen.
The first two Nash equilibria are called "pooling equilibria", the third one is a "separating" equilibrium. "Pooling" means that clever and normal people act the same way (send the same signals), contrary to "separating".
See [Choo/Kreps 1987] for a different version of Spence's game with two companies bidding for employees. This is also described in [Noldeke/van Damme 1990].