1. Definitions-
Problem 1-(a) In a game, what does it mean for X to be common knowledge?
(b) What does it mean for a strategy si to weakly dominate another strategy si' ?
(c) What does it mean for a strategy si to be never a best response?
(d) What is an Nash equilibrium?
2. Application-
Problem 1- Recall Matching Pennies version 2, where player 2 observes what side of the coin player 1 has chosen before he chooses.
(a) Player 1, sensing this is quite unfair for her, demand that they revise the rule of the game. Player 2 agreed, but he demanded that he still be able to observe player 1's choice before he chooses. What can be changed about and/or added to the game so that the game is just as fair as when player 2 does not observe player 1's choice? Anything about the game can be changed except that 1) player 1 moves first then player 2 moves; 2) player 1 and 2 only may choose head or tail at each of his/her information sets; and 3) that player 2 observe player 1's choice. Represent the modified game in extensive form
(b) Represent the game in normal form. Is there a Nash equilibrium where both players play pure strategies?
Problem 2- Recall the Cournot competition discussed in class. Show that when firm 1 knows that firm 2 is rational and that firm 2 knows his profit function, 0 ≤ q1 < 2 is strictly dominated for firm 1 by q1 = 2 and also show that 2 < q1 ≤ 4 is not strictly dominated.
Problem 3- Represent the Hide-and-Seek game where there are only two cups and two players in normal form. If you hide a coin under a cup and seek where you hid, you do not get any payoff (unless your opponent happens to hide where you seek (and hide) as well). Is there any Nash equilibrium where both players play a pure strategy?
Problem 4- Recall the Bertrand competition, but now players can choose any price, not just the integer price. The market demand function is p = 8-q and only the firm that offers lowest price gets to earn revenue. When both firms offer the same price, say p, then they each sell the quantity 8-p/2 at the price of p (so of course firms will not want to charge a price p > 8) For simplicity, let the marginal cost be equal to zero for both firms. Show that
(a) both firms offering difference prices, both of which are greater than 0, is not a Nash equilibrium;
(b) both firms offering the same price higher than zero is not a Nash equilibrium; and
(c) both firms offering the price of zero is a Nash equilibrium.