Game 212.339 theory and Applications 2008 Autumn
In Ho Lee
Problem Set 4
(Due date : October 18th)
1. We have two partners who simultaneously invest in a project, where the level of
investment can be any non-negative real number. If partner i invests x i and the other
partner j invests x j , then the payoff of partner i will be
ixixj x3
i
Here, i is privately known by partner i, and the other partner believes that i is
uniformly distributed on [0, 1]. All these facts are common knowledge.
Find a symmetric Bayesian equilibrium (i.e., a Bayesian equilibrium in which both
partners play the same strategy) in which the investment of partner i is in the form
of x i i a b i
2. Yunjoon and Hyungjoon will go out for a dinner on coming saturday. They decide
the place for the dinner through the secret votes of the two members. They have two
options; Sol-bot cafeteria and Kitchen. They agree that the Sol-bot cafeteria is the
best if it is clear, and the Kitchen is the best if it is rainy. Their preference assigns
payoff of 4 if the best place for the weather condition of the day is chosen (obtains
more votes than the other place), payoff of 0 if the other place is chosen, and payoff
of 2 if the places tie. Yunjoon is informed of the coming saturdays weather because
he reads the newspaper daily, whereas Hyungjoon believes it will be clear with
probability 0.8 and rainy with probability 0.2. Each of them may either vote for the
zoo, vote for the aquarium, or not vote. Draw the strategic form of the game and
find a Bayesian Nash equilibrium (or equilibria). Why doesnt Hyungjoon vote in
some equilibrium?
3. A risk-neutral entrepreneur has a project that requires $100,000 as an investment,
and will yield $300,000 with probability 1/2, $0 with probability 1/2 after a year.
There are two types of entrepreneurs: rich has who a wealth of $1,000,000, and poor
who has $0. For some reason, the wealthy entrepreneur cannot use his wealth as an
investment towards this project. There is also a bank that can lend money with
interest rate . That is, if the entrepreneur borrows $100,000 to invest, after the
project is completed he will pay back $100,000(1)-if he has that much money. If
his wealth is less than this amount at the end of the project, he will pay all he has.
The order of the events is as follows.
- First, bank posts .
- Then, entrepreneur decides whether to borrow ($100,000) and invest.
- Then, uncertainty is resolved.
a. Compute the subgame perfect equilibrium for the case when the wealth is
common knowledge.
b. Now assume that the bank does not know the wealth of the entrepreneur. The
probability that the entrepreneur is rich is 1/4. Compute the equilibrium in this
case.
4. Consider a worker and a firm. Worker can be two types, High or Low. The worker
knows his type, while the firm believes that each type is equally likely. Regardless
of his type, a worker is worth 10 for the firm. The workers reservation wage (the
minimum wage that he is willing to accept) depends on his type. If he is of high
type his reservation wage is 5 and if he is of low type his reservation wage is 0.
First the worker demands a wage w 0 ; if the firm accepts it, then he is hired with
wage w 0 , when the payoffs of the firm and the worker are 10-w 0 and w 0 ,
respectively. If the firm rejects it, in the next day, the firm offers a new wage w 1 . If
the worker accepts the offer, he is hired with that wage, when the payoffs of the
firm and the worker are again 10-w 1 and w 1 , respectively. If the worker rejects the
offer, the game ends, when the worker gets his reservation wage and the firm gets 0.
Find a Bayesian equilibrium of this game.