Game Theory Assignment: Pasture

There are five equally-weighted questions and you should answer all of them. All questions are based on the “Pasture” story below.

Explain concisely all steps you undertake to answer the questions. Some questions ask you to “explain”, state in words, or “discuss”. This should be possible with one or at most three short sentences; please type these answers. You may scan handwritten algebra and graphical illustrations. Please use at least 3 digits after the comma for interim calculations. In general, an answer like 5/3 should be preferred to an answer like “1.667”. This is an open book assignment, but you must always use your own words and calculations. No communication with others is allowed (except for clarification questions to the module lead via email when necessary). Keep in mind, with Turnitin, you need to upload a single pdf file. A submission is only complete with a confirmation number. See blackboard for detailed submission instructions and the deadline.

The goats of two villagers graze on a common pasture. The number of goats that villager i = 1, 2 keeps is xi, assumed to be real non-negative number.

The payoff of villager i depends directly on xi. It also depends on how much of the pasture is left unused for regeneration. Namely, if X = x1+x2 is the total number of goats, and K is some exogenous capacity of the pasture, then the payoff is increasing in the difference (K ? X). To be precise, the total payoff of agent i = 1, 2 is ui =

(If villagers select the number of goats so that K < X, then the pasture is destroyed, all goats die and the payoff drops to zero)

 

1. (20 points) Villagers simultaneously decide how many goats to keep. Find a Nash equilibrium of this game. Illustrate with a clearly annotated graph of best reply functions.

2. (20 points) Is this Nash equilibrium efficient (in the utilitarian sense)? Demonstrate and explain.

 

3. (20 points) Pigouvian tax. Suppose that we impose a tax of ? per goat. Namely, if villager i keeps xi goats on the pasture, she has to transfer ?xi goats to the other villager. The consumption of villager i consists of the remaining goats plus the transfer from the other villager, (1 ? ?) xi + ?xj . The utility, therefore, is ui =

What ? must be so that the total number of goats in equilibrium is efficient?

 

It seems that this tax just reshuffles the goats among the villagers, so it does not really change anything. How is it then that it can help the villagers to achieve the efficiency? Provide a crisp intuitive argument. 

 

4. (20 points) (Let’s go back to the no-tax case ? = 0) Suppose that villager 2 observes how many goats villager 1 buys, and then decides how many goats to buy. Specify the game formally, in particular explain the strategies of each player. What solution concept would you use? Find an equilibrium. Who has an advantage, the first mover or the second mover? Discuss.

5. (20 points) (Let ? = 0 again) Suppose that the simultaneous-move game in point 1 is repeated infinitely many times. Time periods are t = 0, 1, 2, .... Let xit be the number of goats that villager i chooses in period t, and let uit be the resulting utility of villager i in period t. The total payoff from the repeated game is equal to the discounted sum of stage payoffs uit with the discount factor 0 ? ? < 1. To be more precise, it is Ui =   

 

For concreteness, suppose that the pasture’s capacity is K = 600.

(a) Treating ? as a parameter, calculate the payoff that would emerge in the repeated game if both players used the following strategy: “choose the number of goats that is a Nash equilibrium in the single-period game in point 1, forever regardless of history”. Is this strategy profile a subgame perfect Nash equilibrium of the infinitely repeated game? Provide a concise argument.

(b) Assume that ? is sufficiently close to 1. Define a subgame perfect Nash equilibrium of the infinitely repeated game in which both villagers choose the efficient number of goats. Prove that the strategy you proposed is a subgame perfect Nash equilibrium.