game theory 5

1. (15 points) The inverse demand function for a product may be P(Q) = 11 – Q or P(Q) = 7 – Q, with equal probability. Two firms sell the product in simultaneous Cournot competition. Firm 1 knows the demand function, Firm 2 does not. The marginal cost is 0 for both firms. What is the Bayesian Nash Equilibrium?

2. Consider the following game. Player 1 is either "Friendly" (with probability 2/3) or "Mean" (with probability 1/3), and decides to Smile or Not Smile. A Friendly type has no cost of smiling, but a Mean type has a cost of 4.

Player 2 decides to Ask or Not ask Player 1 for help. If Player 2 does not ask Player 1 for help, both players get 0 (minus any cost of smiling for 1). If Player 2 asks a Friendly Player 1 for help, both players get +3 units of utility (minus any cost of smiling for 1). If Player 2 asks a Mean Player 1 for help, Player 2 gets -3 units of utility while Player 1 gets +3 units of utility (minus any cost of smiling) because he gets a chance to be mean to the other player. Player 2 does not know if 1 is Friendly or Mean (but Player 1 does).

For each case below, (a) draw the extensive form of the game, and (b) solve the game using the appropriate solution concept using pure strategies.

I. (15 points) Player 2 does not see whether Player 1 is smiling when he’s deciding whether to ask for help.

II. (20 points) Player 2 sees whether Player 1 is smiling when he’s deciding whether to ask for help.