Question 1. Exercises S4 and U4 demonstrate that in zero-sum games, such as the Evert-Navratilova tennis rivalry, changes in a player's payoffs can sometimes lead to unexpected or unintuitive changes to her equilibrium mixture. But what happens to the expected value of the game? Consider the following general form of a two-player zero-sum game:
|
|
COLIN |
|
|
|
L |
R |
| ROWENA |
U |
a, -a |
b, -b |
|
D |
c, -c |
d, -d |
Assume that there is no Nash equilibrium in pure strategies, and assume that a, b, c, and d are all greater than or equal to 0. Can an increase in any one of a, b, c, or d lead to a lower expected value of the game for Rowenal if not, prove why not. If so, provide an example.
Question 2. Consider Spence's job-market signaling model with the following specifications. There are two types of workers, 1 and 2. The productivities of the two types, as functions of the level of education E, are
W1(E) = E and W2(E) = 1.5E.
The costs of education for the two types, as functions of the level of education, are
C1(E) = E2/2 and C2(E) = E2/3.
Each worker's utility, equals his or her income minus the cost of education. Companies that seek to hire these workers are perfectly competitive in the labor market.
(a) - If types are public information (observable and verifiable), find expressions for the levels of education, incomes, and utilities of the two types of workers.
Now suppose each worker's type is his or her private information.
(b) Verify that if the contracts of part (a) are attempted in this situation of information symmetry, then type 2 does not want to take up the contract intended for type 1, but type 1 does want to take up the contract intended for type 2, so "natural" separation cannot prevail.
(c) If we leave the contract for type 1 as in part (a), what is the range of contracts (education-wage pairs) for type 2 that can achieve separation?
(d) Of the possible separating contracts, which one do you expect to prevail? Give a verbal but not a formal explanation for your answer.
(e) Who gains or loses from the information asymmetry? How much?
There are Unsolved Problems from Games of Strategy by Dixit. 4th edition