Consider the following simplification of poker (straight from the first lecture on dynamic games of incomplete information). There are two players, a sender and a receiver. First, each player first puts a dollar down (the ante). Notice however that there is no choice here, so you don't need to make this part of the extensive form of the game (the reason for describing the setup this way is that it is a natural way of generating the payoffs). Then the sender draws a single card from a deck consisting of an equal number of queens and kings. Only the sender draws a card. After observing the card, the sender may either "bet" or "fold". If the sender folds, the receiver wins the pot containing the ante (implying that the receiver wins 1 and the sender loses 1). If the sender "bets", he places another dollar in the pot. The receiver then has the option between "fold" and "call". If the receiver folds, then the sender takes the pot (implying a net transfer of 1 from the receiver to the sender). If the receiver "calls" he adds another dollar to the pot. In this case the sender wins if he has a king (in which case there is a net transfer of 2 from receiver to sender) and the receiver wins if the sender has a queen (in which case there is a net transfer of 2 from sender to receiver:

a. Draw the extensive form of the game.

b. Calculate all Nash equilibria in pure as well as mixed strategies.

c. Is the game "fair" in the sense that both players can expect to break even on average?

d. Change the game so that it costs x dollars for the sender to bet and x dollars for the receiver to call. Find the equilibrium.

e. Let it cost 1 dollar to bet and 1 dollar to call, but assume that the receiver has a third option. By paying 2 additional dollars player 2 can raise. If the receiver raises the sender either calls or folds. Payoffs are thus (2; 2) if the sender folds, (4; 4) if the sender with a king calls, and (4; 4) is the sender with a queen folds. Construct a weak perfect Bayesian equilibrium.

(I only need the answers for C, D, and E)