Advanced Microeconomics - Game Theory Problem Set 1. Jehle/Reny, Exercise 7.3, 7.4, 7.5 (a), (c) 2. Consider a Bertrand game, in which …rms produce homogeneous goods, have identical and constant marginal costs, and no …xed costs. Show that the Nash equilibrium strategies are weakly dominated. 3. Derive the (mixed-strategy) Nash equilibria of the following strategic-form game between player 1 and player 2: Player 1#, Player 2 ! A B C A 0, 1 1, 0 0, 1 B 2, 0 0, 2 2, 0 C 3, 0 0, 3 3, 0 4. Consider a …rst-price sealed-bid auction with two bidders. Each bidder i’s valuation of the object is vi , which is known to both bidders. Each bidder submits a bid in a sealed envelope, with the bidder who has submitted the highest bid winning the object and paying the auctioneer the amount of his bid. If the bidders submit the same bid, each gets the object with probability 1 . Bids must be in euro multiples (assume that valuations are as well). 2 (a) Are any strategies strictly dominated? If yes, which ones? (b) Are any strategies weakly dominated? If yes, which ones? (c) Is there a Nash equilibrium? If yes, what is it and is it unique? 5. Consider a game between a Buyer and a Seller, in which the two players choose their strategies simultaneously. The Buyer must choose between Buy and Don’t Buy, and the Seller may choose High or Low quality for his product. There game comes in two variants, Game A and Game B. Buyer#, Seller ! high quality low quality 6, 10 -2, 4 Game A: buy don’t buy 0, 2 0, 0 Buyer#, Seller ! high quality low quality 6, 4 -2, 10 Game B: buy don’t buy 0, 2 0, 0 (a) Are there any dominant strategies for the seller and/or the buyer in game A? Are there any in game B? If yes, identify them. What is the Nash equilibrium of game A, what is the Nash equilibrium of game B?

(b) Now suppose that the Buyer does not know whether she is playing Game A or Game B. Her beliefs are that she plays Game A with probability 12 and Game B with probability 12 . Draw the game tree. (c) Derive the Bayesian Nash equilibrium of the incomplete-information game. 6. Jehle/Reny, exercises 7.16 and 7.17. 7. Consider the extensive-form game below. (a) Find the pure-strategy perfect Bayesian equilibria. (b) Which of these equilibria satisfy the intuitive criterion? Explain. (c) Write down the strategic form of the game.

8. Consider the extensive-form game below. (a) Derive the strategic form of this game and …nd all Nash equilibria in pure strategies. (b) Explain which of these equilibria are subgame perfect. (c) Now check if there are any equilibria in mixed strategies. To do this, consider …rst the subgame de…ned by player 2’s decision node. Find a (mixed!) joint behavioral strategy that constitutes a Nash equilibrium in this subgame. Then describe the subgame perfect joint behavioral strategy for the entire game that induces this (mixed) behavioral strategy in the subgame. (d) Find a joint mixed strategy that is equivalent to the joint behavioral strategy in (c) in the sense that it induces the same probability distribution over the end nodes.

9. Consider a public goods game, in which 2 agents must decide how much to contribute to a public good. Agent i = 1; 2 has utility function ai ln(g1 +g2 )+xi ; where gi 0 is the amount of the public good contributed by agent i and xi is the amount of private consumption by agent i. The budget constraint of agent i is simply xi + gi = wi , where wi is agent i’s income. The total amount of the public good is G = g1 + g2 . Assume that wi > ai for each agent. (a) Derive agent 2’s best response as a function of agent 1’s contribution. = y1 : Remember that gi 0 for i = 1; 2, and that dln(y) dy (b) Suppose that a1 = a2 : Describe the Nash equilibria. (c) Now suppose that a1 < a2 : Describe the Nash equilibria. (d) Now consider a Stackelberg game where agent 1 chooses his contribution …rst and agent 2 selects his contribution after observing 1’s contribution. If a1 = a2 , describe the subgame perfect (“Stackelberg”) equilibrium. (e) Describe the Stackelberg equilibrium when a1 < a2 . (f) Suppose that a1 > a2 . Describe the Stackelberg equilibrium. 10. Consider two …rms that compete in a market with demand p = 120 Q, where Q = q1 + q2 . The cost function of …rm i is Ci (qi ) = 30qi for i = 1; 2. (a) If the two …rms choose their quantities simultaneously (Cournot competition), determine the Nash equilibrium. (b) Suppose …rm 1 chooses output …rst and …rm 2 chooses output after observing 1’s output. Determine the subgame perfect equilibrium. Does this sequential game have other Nash equilibria that are not subgame perfect? If yes, state them and explain why this is the case. If not, explain why not. (c) Now suppose that the two …rms once again choose quantities simultaneously. Each …rm’s marginal cost can be either high cH i = 30 or low cLi = 10 , i = 1; 2; with equal probability. Each …rm knows the realization of its own marginal cost but not that of the other …rm. Derive the Bayesian Nash equilibrium.

11. The government wants to sell drilling rights to an oil …eld using a …rst-price sealed-bid auction. There are two bidders. The one with the highest bid wins and pays his bid. If both bidders make the same bid, each wins with probability 1 . The value of the drilling rights is either 2 or 0. Bidder 1 does not know this 2 value and assigns probability x to each possibility. Bidder 2 knows the value of the drilling rights. Assume that each bidder may choose between only two bids: 0 or 1. If both bid 0, the government does not sell the rights. (a) Write down the extensive form of this game (i.e., the game tree). (b) Determine the Bayesian Nash equilibrium of the game. How does it depend on x? 12. Suppose two …rms compete by simultaneously choosing prices. They may either choose a high price PH or a low price PL . Payo¤s are given in the form: (payo¤ of …rm 1, payo¤ of …rm 2). If both …rms price high, payo¤s are (50; 50). If both …rms price low, payo¤s are (36; 36). If …rm 1 prices high and …rm 2 prices low, payo¤s are (20; 60). If …rm 1 prices low and …rm 2 prices high, payo¤s are (60; 20). (a) Set up the strategic-form game and …nd all Nash equilibria. (b) Modify the game above by allowing each …rm to adopt a meeting competition clause (M CC). A …rm’s strategy can thus be PH , PL , or M CC. De…ne M CC as follows: if a …rm chooses an M CC, it sets a high price but promises to match any lower price announced by its rival. Set up the strategic-form game and …nd all Nash equilibria. Does one equilibrium seem more reasonable? (c) Modify the game in (b) by assuming that …rms …rst simultaneously choose whether or not to adopt M CC. After these decisions have been made and observed, any …rm that did not choose M CC gets to choose its price. In the event that neither …rm adopts M CC, both …rms choose their price simultaneously. Set up the extensive form game and solve for all subgame perfect equilibria. (d) Modify the game in (c) by assuming that if a …rm adopts MCC, the advertising costs of informing the public of this policy is 10. Solve for all subgame perfect equilibria. (e) Would it make sense for …rm 1 to take an action that would unilaterally increase its cost of informing the public were it to adopt M CC? Assume that the increased cost is observable to …rm 2. Why or why not? (f) Modify the game in (d) by assuming that …rm 2 gets to observe …rm 1’s M CC decision before making its own M CC decision. Set up the extensive form game and solve for all subgame perfect equilibria. Is there a …rst mover advantage for …rm 1?

13. Consider a market with one …rm and one consumer, in which the consumer wishes to buy either zero or one unit of a good. The …rm can choose to produce a high-quality (H) or a low-quality (L) unit, or no unit at all (N ). But any unit it sells must be sold at the exogenous price of P . The consumer, after observing the …rm’s choice, decides whether to buy (b) or not (n). A choice of N by the …rm gives a payo¤ of zero to both players. If the …rm produces a unit of quality i, its costs are Ci (i = H; L) whether or not the consumer buys, and the consumer gets a bene…t of Vi P (i = H; L) if she buys a unit of quality i, and zero if she does not buy. (a) Write down the game tree for this two-player, complete information game. (b) Assuming VH > VL > P > CH > CL > 0, …nd all the pure-strategy Nash equilibria. (c) Which, if any, of the equilibria listed in part (b) are subgame perfect? 14. From Roy Gardner: Players 1 and 2 are playing a special game of Poker, 1-card Stud Poker. The deck of cards consists of 50% aces and 50% kings. Prior to the deal, each player puts $1, called the ante, into the center of the table, called the pot. Each player is dealt one card face down, which neither the player nor the opponent sees. At this point, a player can bet $2 (also placed in the pot) or pass. The players make this decision simultaneously. Then the game ends. If one player bet and the other passed, the player who bet takes the pot. If both players bet or both players passed, both turn over their card (the showdown). The player with the highest card wins the pot, with an ace beating a king. If both players in a showdown have equal cards, they split the pot. (a) Write down a game tree for this game, starting with a move of nature, which chooses a card for each player. (Hint: nature has 4 possible moves) (b) What is the Bayesian Nash equilibrium of the game? 15. Also from Roy Gardner: Liar’s Poker. There are two players, 1 and 2, and two cards in the deck, an ace and a king. An ace ranks higher than a king. Player 1 is dealt a card, face down, which he or she can look at but which player 2 never sees. All player 2 knows is that with probability .5 the card is an ace; with probability .5 a king. Now player 1 can make an announcement about the card she holds. If the card is an ace, player 1 can’t lie about having a better card and must say that it is an ace. If the card is a king, player 1 can say “ace”or “king.”Player 2 hears what player 1 says. If player 1 says “ace,”then player 2 can either call or fold. If player 2 calls and player 1’s card is not an ace, then player 1 loses and pays $1 to player 2. If player 2 calls and player 1’s card is an ace, then player 2 loses and pays $1 to player 1. If player 2 folds, the player 1 automatically wins $0.50 from player 2. Finally, if player 1 says “king,”then the game ends and both players break even.

(a) Write down a game tree for this game. (b) What is the Perfect Bayesian equilibrium of the game? 16. An employer considers hiring a worker. The worker’s productivity is given by 2 f h ; l g, with h > l > 0. It is common knowledge that the worker’s productivity is high ( = h ) with probability and low ( = l ) with probability (1 ). The worker knows his productivity, but the employer does not. The worker’s next best alternative to working for the employer is to work for himself, where he earns 34 ; this is the worker’s opportunity cost of working for the employer. If the employer hires the worker and pays him a wage w, the employer’s payo¤ is w and the worker’s payo¤ is w 34 . Now consider the following signaling game. Nature moves …rst and chooses the worker’s productivity . The worker observes his productivity and then chooses between working for himself and making a wage demand w to the employer. If the worker decides to work for himself he earns 43 and the employer earns 0. If he demands wage w, the employer can hire the worker and pay him w, or not hire the worker. If the employer hires the worker, she earns w and the worker 3 earns w 4 . If the employer does not hire the worker, she earns 0 and the worker earns 34 . (a) Draw the game tree. (b) Does there exist a perfect Bayesian separating equilibrium (in pure strategies)? If yes, state one and show which additional assumptions you have to make. If not, explain why not. (c) Does this game have a perfect Bayesian pooling equilibrium (in pure strategies)? If yes, state one and show which additional assumptions you have to make. If not, explain why not. 17. Consider a market characterized by the inverse demand function p = ai Q, where Q is industry output and ai > 22. This market is served by an incumbent …rm and a potential entrant, with the incumbent possessing private information about parameter ai . The game between the two players can is as follows: At the beginning of the game nature chooses the value of ai . Demand is high (ai = aH = 22) or low (ai = aL = 10), each with probability 1=2. The incumbent who observes the true value of ai then chooses her output. The entrant observes the incumbent’s output choice but not the value of ai and must then choose whether or not to enter the market. If the entrant does not enter, the incumbent remains a monopolist and earns the pro…t associated with her output choice, while the entrant receives a payo¤ of zero. If the entrant enters, he immediately learns the true demand function and the two …rms simultaneously choose output as in Cournot competition (that is, the incumbent may revise her initial output choice). The cost functions of the two …rms in this case are identical and given by C(q) = 4q, where q is the output

of an individual …rm. In addition, the entrant pays a …xed cost of entry equal to 5. (a) If there were complete information about ai , what would the incumbent’s monopoly output and pro…t be? (Hint: pro…t is simply output squared in this case.) (b) If there were complete information about ai and the entrant did enter, what would the Cournot outputs and pro…ts of the two …rms be? (Hint: pro…t is simply output squared in this case.) Does the entrant make positive pro…ts under both realizations of ai given his entry cost? (c) Does there exist a separating perfect Bayesian equilibrium in which an incumbent of type H initially chooses output qH = 9, an incumbent of type L initially chooses output ql = 3? Explain your answer. (d) Does there exist a pooling perfect Bayesian equilibrium in which both types of incumbent choose qi = 3, i = H; L? Explain your answer. (Hint: assume that at information sets that are not reached in equilibrium the entrant believes that ai = aH with probability 1). 18. Firm A is a monopolist in the local computer market. The market is a natural monopoly, so only one …rm can survive. It is common knowledge that A’s operating cost is 20 with probability 0.75 and 30 with probability 0.25. Firm A knows its cost realization, but a potential entrant, …rm B, does not. A can price Low, losing 40 in pro…ts, or High, losing nothing. Firm B observes A’s choice of price before deciding whether to Enter or N ot to enter. Firm B can enter at a cost of 100, and its operating cost of 25 is common knowledge. The …rm with the highest operating cost immediately drops out if it has a competitor, and the survivor earns the monopoly revenue of 150. (a) There are no separating equilibria in this game. Show why this is true. (b) There are two pooling sequential equilibria. State the two equilibria (strategies and beliefs) and show that they are indeed equilibria. 19. Consider yourself a hard-working student looking for a job. There is a large number of competitive employers who might hire you. But the employers do not know whether you are truly hard-working or lazy. Suppose there are only two levels of laziness given by 2 f h ; l g, with h > l > 0. The proportion of the lazy type ( = h ) in the student population is given by . Fortunately for you, there is a way of signaling your type to potential employers, namely by taking e hours of economics. Economics does not make you a better employee. The point simply is that economics is tough and especially so for lazy students. The cost for a student of type of taking e hours of economics is given by C(e; ) = 2 e2 : The utility of a student who chooses e hours of economics and receives wage w is equal to the wage minus the cost of studying economics:

u(w; ej ) = w C(e; ). The pro…t of an employer who hires a student of type is equal to w, where parameter > h . Now consider the following signalling game. Nature moves …rst and chooses the student’s type . The student observes his type and chooses a combination of wage demand and economics education level (w; e). The employer observes (w; e) but not and then chooses whether to hire the student at the proposed wage or not. (a) Suppose …rst that there is symmetric information. Argue that in the competitive equilibrium no one studies economics and the wage for each type is such that employers earn zero pro…ts. Draw a diagram with e on the horizontal axis and w on the vertical axis. In this diagram, add the employer’s zero-pro…t lines for type h and type l (label these lines), and indicate the competitive equilibrium. Label the competitive wages as whc and wlc . (b) What is the slope of type ’s indi¤erence curves? Show that the singlecrossing property is satis…ed. (c) Now draw a new digram with e on the horizontal axis and w on the vertical axis. Suppose that type h chooses a wage/education level h (wh ; eh ). Mark this point in the diagram and draw an indi¤erence curve for each type through this point. Indicate the set of possible wage/education levels for type l, l , such that type h prefers h to l and type l prefers l to h . Label this region S. (d) Draw another diagram with e on the horizontal axis and w on the vertical axis. In this diagram, add the employer’s zero-pro…t lines for type h and type l . Argue that there are perfect Bayesian separating equilibria in which type h proposes the competitive solution ch (whc ; 0) and type l proposes l (wl ; el ) with el > 0, such that l is in the set S indicated in your diagram. What are the beliefs that support these separating equilibria. (e) Suppose that both types choose the same wage/education level 0 (w0 ; e0 ). Characterize su¢ cient conditions for the existence of a perfect Bayesian pooling equilibrium in which the two types choose 0 . Take care to specify beliefs for the employer even o¤ the equilibrium path. (f) Draw a new digram with e on the horizontal axis and w on the vertical axis. In the diagram indicate the set of possible wage/education levels that can be sustained in a pooling equilibrium. Label this region P . (g) Which of these pooling equilibria, if any, satisfy the intuitive criterion?