Standard deviation: 12.44 Comments:
The average performance is very good. I think most of you fully understand the basic concepts in the lecture. Those of you receive 40 or lower might better study much harder. The last handout “game theory chapter (from Nicholson and Snyder)” should be helpful to complement the materials in the lecture.
(b) Now suppose there are n(> 2) individuals. Then, can we find a competitive equilibrium? (How) Does your answer depend on n?
4. Question 4 (8 points)
Consider a production economy with two individuals, Ann (A) and Bob (B), and two goods, leisure x 1 and a consumption good x 2 . Ann and Bob have equal en-
where α > 0 and 0 < β < 1. Let w 1 , w 2 > 0 be the prices for inputs x 1 and x 2
respectively. Then, answer the following questions. (a) Sketch the isoquant for this technology.
Hint: Isoquant is the combination of inputs that achieves a given level of output y. (similar to “indifference curve” in consumer theory.)
C) Now suppose that the rule of the game is modified as follows. If exchange occurs, each individual receives 3 times as much amount as the bill she will have. For example, if individual 1 receives $5 and 2 receives $10 initially and both wish to exchange, then 1 will receive $30 (= $10 x 3) and 2 will receive $15 (= $5 x 3). Nothing happens if they do not exchange. Then, does trade occur in a Bayesian Nash equilibrium? Explain.
(a) Show that there is no pure-strategy equilibrium in this game.
(b) Is there any strictly dominated strategy? If yes, describe which strategy is dominated by which strategy. If no, briefly explain the reason.
(c) Derive the mixed-strategy Nash equilibrium.
since there is no future play . The only possible outcome is a price war irrespective of the past history of the play.
In the second to the last period ( t = T-1 ), no firm has an
incentive to collude since the future play will be a price war no matter how each firm plays in period T-1 .
General Formulation of PD
The larger the payoff, the better the corresponding result.
Desirability of outcomes for each player:
g > c > d > l, that is, ( D , C ) > ( C , C ) > ( D , D ) > ( C , D )
Choose a subway station in Tokyo and write down its name.
You will win if you can choose the most popular answer.
Most of the students are expected to write “xxx”.
Like this experiment, there may exist a Nash equilibrium which stands out from the other equilibria by some
2. Simple 2-2 Games (18 points, take your time)
For the 2-2 games X, Y, and Z below, answer the following questions: i. Explain whether there exists a dominant strategy.
ii. Find (all) pure-strategy Nash equilibrium if it exists. iii. Find (all) mixed-strategy Nash equilibrium if it exists.
B) Consider the two-period repeated game in which the above stage game will be played twice. Suppose that the payoff for each player is simply the sum of the payoffs in the stage games. Then, can (U, L) be sustained as a subgame perfect Nash equilibrium? If yes, derive the equilibrium. If not, explain why.
a) Find all pure strategy Nash equilibria.
b) Find the mixed strategy Nash equilibrium in which each player randomizes over just the first two actions, i.e., A, B for P1 and D, E for P2, respectively.
c) Is there a mixed strategy Nash equilibrium in which both players randomize over all three strategies? If yes, derive the equilibrium. If not, explain why.
4. Incomplete Information (16 points, think carefully)
There are four different bills, $1, $5, $10, and $20. Two individuals randomly receive one bill each. The (ex ante) probability of an individual receiving each bill is therefore 1/4. An individual knows only her own bill, and is simultaneously given the option of exchanging her bill for the other individual’s bill. The bills will be exchanged if and only if both individuals wish to do so; otherwise no exchange occurs. That is, each individuals can choose either exchange (E) or not (N), and exchange occurs only when both choose E. We assume that individuals’ objective is to maximize their expected monetary payoff ($).
Strategy and Outcome
Strategy in dynamic game = Complete plan of actions
What each player will do in every possible chance of move.
Even if some actions will not be taken in the actual play, players specify all contingent action plan.
(a) The intersection of any pair of open sets is an open set.
(b) The union of any (possibly infinite) collection of open sets is open.
(c) The intersection of any (possibly infinite) collection of closed sets is closed. (You can use (b) and De Morgan’s Law without proofs.)
Problem Set 2: Posted on November 4
Advanced Microeconomics I (Fall, 1st, 2014)
1. Question 1 (7 points)
A real-valued function f (x) is called homothetic if f (x) = g(h(x)) where g : R → R is a strictly increasing function and h is a real-valued function which is homo- geneous of degree 1. Suppose that preferences can be represented by a homothetic utility function. Then, prove the following statements.