1. Consider the infinitely repeated version of the stage game. Assume that each player's payoff is the present value of her payoff stream and the discount factor is δ.

(a) Find a strategy profile that results in an outcome path in which both players choose b in every period and the strategy profile is a SPNE.

(b) Find a strategy profile 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 profile is a SPNE.

2. Consider the following stage game of an infinitely repeated game.

(a) What are the Nash equilibria of the stage game?

(b) Find a subgame perfect Nash equilibrium of the infinitely 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 e_i^t denote the effort chosen by student i in period t, and assume that e_i^t can take 1 of 11 levels: 0, 1, 2, ..., 10. The quality of the final project is higher when the students work harder. More specifically, the quality (or, grade) of the project equals twice the amount of total effort, i.e.

T = 2(e^₁t + e₂^t + e₃^t).

A student's utility is an equal share of T less the personal cost of effort, which is simply the amount of effort. This stage game is infinitely repeated, where each student's payoff is the present value of the payoff 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' payoffs in the stage game?

(b) In the infinitely repeated game, assume that the history of the game is common knowledge. That is, in period t, the past choices of effort for all students over period 1, ...,t-1 are observed. Derive an SPNE in which each player chooses effort e∗ > 0 in each period.

(c) Assume that the history of the game is not common knowledge.

In period t, only the total effort, (e₁^t-1 + e₂^t-1 + e^₃t-1 ), in period t - 1 is observed by all players. (A player knows his own past effort, but you can ignore that information.) Explain intuitively how the players could use strategies that support effort e∗ > 0 in each period of the infinitely 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.

The problem is that a consumer cannot tell by looking at whether it is of low or high quality. However, once having purchased the product, a consumer learns its quality through usage. There is one consumer each period, and a consumer busy as long as the net surplus from the product is nonnegative. The net surplus is the expected value of the product (which is 15 if it is expected to be of low quality and 50 if of high quality), minus the price said. The manufacturer's payoff is the present value of its profit stream, where the profit in any period is zero if the consumer doesn't buy and is the price charged, less the cost of making the good, if the consumer does buy. The manufacture's discount factor is δ.

In each period, the manufacturer decides on the quality and price of the project. After observing the price (but not the quality), the consumer either buys or doesn't buy. In each period, all past actions, including all past prices, qualities, and purchasing decisions, are common knowledge. Find a strategy profile that results in the manufacturer's producing a high-quality product every period and charging a price of 50, and consumers buying the product. Derive conditions for this strategy profile to be a SPNE.