1. Consider a negotiation between Player 1 and Player 2.  Player 1 makes an offer x  ∈ R and Player 2 observes the offer and chooses whether to accept.  If Player 2 rejects, the game ends and both players receive payoff zero.  If Player 2 accepts the offer, Player 1 receives payoff 1 − x and Player 2 receives payoff x.  Since Player 2 cannot make a counter offer and Player 1 will not make a second offer, the offer made by Player 1 is called a take-it- or-leave-it offer or an ultimatum. Find all the subgame perfect equilibrium of the game.

2. Consider the Stackelberg competition where each player’s payoff function is ui (qi , q −i ) = (A−qi − q −i )qi .

(a) Find all the subgame perfect equilibria of the game.

(b) Notice that Firm 1’s equilibrium output is the same as the monopoly output.  Should we try and

give an economic explanation of this particular result? If so, can you propose such an explanation?

3. There are two players, each drawn independently an integer uniformly between 1 and 100; the probability that any integer i between 1 and 100 is drawn with probability .01. Player 1 chooses whether to challenge Player 2.  If Player 1 challenges Player 2, Player 2 chooses to accept or concede.  If Player 1 does not challenge Player 2, each player receives payoff 0; if Player 1 challenges and Player 2 concedes, Player 1 receives payoff .5 and Player 2 receives payoff -.5. If Player 1 challenges and Player 2 accepts, the player who draws the bigger integer receives payoff 1 and the other player receives payoff -1; in case the two

players draw the same integer, each receives payoff 0.

(a) What is each player’s set of (pure) strategies?

(b) What is each player’s set of mixed strategies?

(c) What is each player’s set of behavior strategies?

(d) Verify that the following is a subgame perfect equilibrium: Player 1 challenges with probability one if his card is 67 or better; Player 1 challenges with probability 16/99 if his card is 66 or worse; Player 2 accepts a challenge with probability one if his card is 68 or better, accepts with probability 1/3 if his card is 67 and concedes with probability one if his card is 66 or worse.