Game Theory in the Social Sciences Quiz
Each numbered question has equal weight toward that total. Each lettered subpart has equal weight in the point value of its question.
1. Consider the in?nitely repeated version of the stage game. Assume that each player's payo? is the present value of her payo? stream and the discount factor is δ.
|
Player 1 |
|
|
|
|
|
a |
b |
c |
d |
|
a |
4,4 |
5,3 |
6,2 |
1,1 |
| player 2 |
b |
3,5 |
6,6 |
7,10 |
2,7 |
|
c |
2,6 |
10,7 |
8,8 |
2,11 |
|
d |
1,1 |
7,2 |
11,2 |
3,3 |
(a) Find a strategy pro?le that results in an outcome path in which both players choose b in every period and the strategy pro?le is a SPNE.
(b) Find a strategy pro?le that results in an outcome path in which both players choose b in every odd period and c in every even period and the strategy pro?le is a SPNE.
2o.Consider the following stage game of an in?nitely repeated game.
|
L |
M |
R |
| U |
4, 6 |
2, 2 |
2, 3 |
| M |
6, 3 |
3, 4 |
0, 3 |
| D |
2, 1 |
1, 0 |
0, 1 |
(a) What are the Nash equilibria of the stage game?
(b) Find a subgame perfect Nash equilibrium of the in?nitely repeated game where the players use a grim trigger strategy, and where the equilibrium outcome is ((U, L), (U, L), (U, L), ...). Be sure to specify the minimum δ that supports such an SPNE.
3. There are 3 students who are assigned to a group project. In each period, each student simultaneously decides how hard to work. Let et denote the e?ort chosen by student i in period t, and assume that et can take 1 of 11 levels: 0, 1, 2, ..., 10. The quality of the ?nal project is higher when the students work harder. More speci?cally, the quality (or, grade) of the project equals twice the amount of total e?ort, i.e.
t t t
T = 2(e1 + e2 + e3).
A student's utility is an equal share of T less the personal cost of e?ort, which is simply the amount of e?ort. This stage game is in?nitely repeated, where each student's payo? is the present value of the payo? stream and student i's discount factor is δ.
(a) What is each player's choice of ei in the Nash equilibrium of the stage game? What choice of ei for each player maximizes the sum of all players' payo?s in the stage game?
(b) In the in?nitely repeated game, assume that the history of the game is common knowledge. That is, in period t, the past choices of e?ort for all students over period 1, ..., t-1 are observed. Derive an SPNE in which each player chooses e?ort e∗ > 0 in each period.
(c) Assume that the history of the game is not common knowledge. In period t, only the total e?ort, (et-1 + et-1 + et-1), in period t - 1 is observed by all players. (A player knows his own past e?ort, but you can ignore that information.) Explain intuitively how the players could use strategies that support e?ort e∗ > 0 in each period of the in?nitely repeated game in SPNE.
4. As Elizabeth Arden said, "Repetition makes reputation and reputation makes customers." Let's see how that might work. In each period, a manufacture can choose to produce a low-quality product at a cost of 10 per unit or a high-quality product at a cost of 20 per unit. A consumer is willing to pay up to 15 for a low-quality product and up to 50 for a high-quality product.