Economy- Game theory problem set

1. (a) Fred and Sally are going to play a very simple game. At the start of the game Fred will choose one of three actions: a, b, or c. After seeing Fred’s choice Sally will choose one of two actions: x or y. How many strategies does Sally have, i.e., how many elements are in Sally’s strategy set?

(b) Now consider a more general version of this game. Suppose Fred has k possible actions at the start of the game and Sally has n possible actions in response to each of Fred’s actions (and so in the game we considered initially k = 3 and n = 2). How many strategies does Sally have?

2. 2. Jones TV and Smith TV are the only two stores in your town that sell flat panel TV sets. First, Jones will choose whether to charge high prices or low prices. Smith will see Jones’s decision and then choose high or low prices. If they both choose High, each earns $10,000. If they both choose Low, each earns$8,000. If one chooses High and the other chooses Low, the one that chose High earns $6,000 and the one that chose Low earns $14,000.

(a) Draw the game tree. Use backward induction to solve this game.

(b) Suppose Smith goes to Jones and promises to choose High if Jones chooses High. Is this a credible promise?

(c) Now suppose Jones starts a new policy that says they will always match or beat Smith’s price. They advertise their new policy heavily and so they must choose Low if Smith chooses Low. So the game now has the following structure. First, Jones chooses High or Low. Second, Smith chooses High or Low. Third, if Jones has chosen High and Smith has chosen Low, Jones matches Smith’s price and chooses Low. Draw the game tree. Use backward induction to solve this game.

3. 3. Consider the following version of the Stackelberg model. There are two firms, 1 and 2. 1 is the leader and chooses its quantity first.  2 is the follower and chooses its output after 1 chooses. They produce identical goods and so if q₁ = is 1’s output and q₂ = 2’s output then market output is q₁ + q₂.  Each produces output at a constant average and marginal cost of 3.  The market demand curve is p = 19 – 2(q₁ + q₂).

Find the subgame perfect Nash equilibrium in this game.

4. Consider a game of Mutually Assured Destruction played by two superpowers, Country 1 and Country 2. Country 2 has precipitated an international incident.  Country 1 can either ignore the incident or escalate the crisis.  If it ignores the incident the game ends and both countries receive a payoff of 0.  It if escalates, then Country 2 can either escalate further or back down.  If it backs down, then the game ends, Country 1 receives a payoff of 2, and Country 2 receives a payoff of ‑2.  If Country 2 escalates further then 1 and 2 play a game where they move simultaneously.  Each has two strategies in this subgame: Back Down or Doomsday.  The payoffs in the subgame are as follows:

Find all pure strategy subgame perfect Nash equilibria of this game.

5. Consider the following two player game.

(a) Find the Nash equilibrium (or equilibria) of this game.

(b) Find the subgame perfect Nash equilibrium of this game.