1. In the game below, find all subgame-perfect Nash equilibria (there are two). Identify the path through the tree that each one represents, and the utilities of each player. Does the game have any Nash equilibria that are not subgame perfect?

2. Compute the subgame perfect Nash equilibriua in the game below.

3. Alice and Bob own a dollar, which they need to share in order to consume. Alice makes an offer x ∈ X = {0.01, 0.02,..., 0.98, 0.99}; and observing the offer, Bob accepts it or rejects it. If Bob accepts the offer, Alice gets 1-x and Bob gets x. If he rejects, then each gets 0.

(a) Compute all the subgame-perfect equilibria in pure strategies.

(b) Now suppose that their cousin Carol sells a contract for $0.01. The contract requires that Bob is to pay 1 dollar to Carol if Bob accepts an offer x that is less than X', where x'∈ X is chosen by Bob at the time of purchase of the contract. In particular, consider the following time-line:

• Bob decides whether to buy a contract from Carol and determines x' if he chooses to buy;

• Alice observes Bob's decision (i.e. whether he buys the contract and x' if he buys);

• Then, they play the bargaining game above, where Bob pays Carol 1 dollar if he accepts an offer x < x'.

Find all the subgame-perfect equilibria in pure strategies.