# Universidad Carlos III de Madrid Game Theory Problem set on Repeated Games and Bayesian Games

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1 Session 1: 1, 2, 3, 4 Session 2: 5, 6, 8, 9 Universidad Carlos III de Madrid Game Theory Problem set on Repeated Games and Bayesian Games 1. Consider the following game in the normal form: Player 1 Player 2 C N P C 6, 6 0, 7 0, 0 N 7, 0 3, 3 0, 0 P 0, 0 0, 0 1, 1 a) Find all the pure strategy Nash equilibria. Suppose now that the game from part a is played twice. Before playing the second time, the players observe what happened in the first time the game was played, so they can make their strategies for the second round contingent to what occurred in the first round. The final payoffs are the sum of the payoffs from each round. b) Find a subgame perfect equilibrium where the players play (C,C) in the first round. 2. In the telecommunications market of a country there are two Örms that face a market demand of P(q a + q b ) =160 q a q b The two firms have the same costs of production C(q) = 40q. a) Find the Nash equilibria in this game when the two firms simultaneously select their quantities of production just one time. b) Find a discount rate and a subgame perfect nash equilirium such that the firms colude (that they obtain the maximum possible joint benefit) if the game is repeated infinitely. 3. In the following game, the government has to decide between high taxes T or low taxes t while the householders choose between keeping high savings S or low savings s. The payoff matrix is given by: S s T 6, 6 3, 3 t 8, 3 4, 4 a) If the taxes policy and the saving decisions are taken once a year, and the government term is four years, what is the subgame perfect Nash equilibrium of this game?

2 b) If the government and the householders live forever, what is the minimum discount factor such that the payo s (6,6) may result from a subgame perfect Nash equilibrium? Give a strategy profile that supports this equilibrium. 4. There are two firms who are repeating infinitely the game of Bertrand duopoly. In each period the firms set their prices simultaneously; the demand for the product of Firm i is a p! if p! < p!, is 0 fi p! > p! and is (a p! )/2 if p! = p! ; their marginal costs are c < a. Show that if the dicount rate is δ 1 2 then there is a subgame perfect equilibrium in trigger strategies that allows the monopoly price to be maintained. 5. Calculate all of the pure strategy Bayesian Nash equilibria of the following static bayesian game. It is determined randomly whether the payoffs of the players, A and B, are those of game 1 or of game 2, with the probability of each being equal. Player A is informed which of the games, 1 or 2, was chosen, but Player B does not know which of the games is being played. Player A chooses x or y; simultaneously, Player B chooses m or n. Juego 1 Juego 2 m n m n x 1, 1 0, 0 x 0, 0 0, 0 y 0, 0 0, 0 y 0, 0 2, 2 6. There are two suspects who face the problem of the prisoners dilemma, with the added complication that neither suspect knows if the other is a man of honour. Say it is known with certainty that Suspect 1 is not a man of honour, but it is not clear whether or not Suspect 2 is also. If Suspect 2 is not a man of honour, the payoffs have their usual form in this game: Suspect 2 Confess Not to Confess Suspect 1 Confess 1, 1 15, 0 Not to Confess 0, 15 10, 10 On the other hand if Suspect 2 is a man on honour, then he prefers to spend years in jail before he would rat on his colleage. More over, even Suspect 1 would feel bad betraying someone so honourable. For these reasons, if Suspect 2 is a man of honour the payoffs are: Suspect 2 Confess Not to Confess Suspect 1 Confess 1, 1 5, 20 Not to Confess 0, 15 10, 30 Denote the probability that Suspect 2 is a man of honour by p. a) Identify the strategies that are strictly dominant for Suspect 2 in this game of imperfect information. b) Identify the Nash equilibria in this game for each p.

3 7. A firm (Player 1) is already established in a market and must wheter or not to construct a new factory. The potential benefits of this action depend on whether another firm (Player 2) enters or does not enter the market. Player 2 is uncertain of the cost faced by Player 1 of constructing the factory, which Player 2 believes may be high or low with probabilities 1/3 and 2/3 respectively. Player 1 knows the construction costs. The decisions to construct (C) or not to construct (NC) and to enter (E) or not to enter (NE) are taken simultaneously. The payoffs are given by: Find the Bayesian equilibria of this game. High costs Low costs E NE E NE C 0, -1 15, -1 C 1,5, -1 3,5, 0 NC 2, 1 3, 0 NC 2, 1 3, 0 8. Eva Bernard are the only participants in a simultaneous auction (with sealed bid) for an object. The auction is conducted according to the following rules: Each player can only bid either 100 or 200 euros. Once they have made bids the object goes to the the player that made the highest bid, and in case of a draw they toss a coin to see who gets it. The winner (the player who gets the object) pays what he or she bid. Both players know that Eva values the object at 300 euros. But Bernard s type is determined at random, and so only Bernardo knows his own valuation, which could be 150 euros or 250 euros. The probability of each type is 1/2. Calculate the Bayesian Nash equilibria of this game. 9. Consider a Cournot duopoly in a market with inverse demand function given by p q = 10 q, where q = q! + q! is the market total. Firm 1 s costs are c! q! = 2q! with probability 1/3, and c! q! = 0 with probability 2/3. Firm 2 s costs are c! q! = (q! )!. Firm 1 knows its costs, but not Firm 2, who knows Firm 1 s posible types and probabilities. All this is common knowledge. a) Represent the situation as a Bayesian game. I.e., show the set of players, the set of types, beliefs, and utilities. b) Calculate the Bayesian Nash equilibrium of the game. Calculate also the equilibrium profits. c) Consider Firm 1 s low cost type (c! q! = 0), does it gain if it can inform to Firm 2 of its type in a credible way? What about the high cost type? How does this possibility affect the equilibrium? Note: When Firm 1 inform about its type, it provides proof. d) Assume that Firm 1 only has its word (no proof) to convince Firm 2 of its type. Has Firm 1 an incentive to lie about its type? 10. Consider a in a market with inverse demand function given by p q = a q, where q = q! + q! is total market quantity. Both firms have cero costs. Demand is uncertain: it can be high (a = 12) or low (a = 6) with equal probabilities. Information is asymmetric: Firm 1 knows whether the demand is high or low, but Firm 2 does not. All this is common knowledge. Firms choose quantity simultaneously.

4 a) Represent this situation as a Bayesian game. b) Compute the Bayesian equilibrium and the equilibrium profits. Other Exercises 9. Consider the following game in the normal form: A B C A 3, 3 x, 0-1, 0 B 0, x 4, 4-1, 0 C 0, 0 0, 0 1, 1 a) Find the Nash equilibria in pure strategies for the di erent values of x. Suppose now that the game is played twice. After playing the first time, the players observe what happened before playing the second time. The final payoffs are the sum of the payoffs from each round. b) For x = 5 is (B, B) a Nash equilibrium for the game when it is played just once? Is there a subgame perfect Nash equilibrium where (B, B) is played in the first round? If yes, write the strategies for both players for such equilibrium. If no, explain why not, using the definition of subgame perfect Nash equilibrium. c) For x = 7 is there a subgame perfect Nash equilibrium where (B, B) is played in the first round? If yes, write the strategies for both players for such equilibrium. If no, explain why no. 10. Given the following Normal form game: L R A 1, 1 8, 0 B 0, 5 3, 3 Find the smallest discount factor neccessary to get average payoffs equal to (3,3) in a SPNE if the game is repeated an infinite number of times. Describe the strategies that allows us to sustain such an equilibrium 11. Carlos decided to have lunch at the restaurant Casa Pepe. The waiter of the restaurant has to decide whether to attend Carlos with a good service or with a bad service. Carlos, after observing the service received from the waiter, decides if he tips the waiter or not. The waiter likes to receive a tip, but he has a cost to provide a good service. Carlos, on the other hand, likes to receive a good service, although he does not like to tip waiters. Each one of them wants to maximize his own expected value. Suppose that the only possible tips are 2 or 0 euros. For Carlos, a good service is worth 6, while a bad service is worth nothing. For the waiter, a good service costs 1, while a bad service has no cost. a) Draw the extensive form of this game. b) Which are the pure strategies for Carlos?

5 c) Is there a Nash equilibrium where Carlos pays the tip only if he receives a good service, and the waiter gives a good service? Explain. d) Represent this game in the normal form. e) Which of the following phrases is correct?: i) For the waiter, to give a good service is a dominant strategy. ii) Never give a tip, no matter the quality of the service, is a dominant strategy for Carlos. iii) For Carlos, it is better to give a tip if the service is good, and do not if it is bad. iv) None of the above is correct. f) Find the pure strategy Nash equilibrium. g) Does this equilibrium (from part f) result in a Pareto effcient allocation? If not, indicate an strategy profile that results in a Pareto superior allocation. Suppose now that Carlos goes to this restaurant every week, and he is always served by the same waiter. Each one maximizes expected value, and no one discounts the future (i.e., one euro today is worth exactly the same as one euro in the future). In that case, they could reach the following verbal agreement: the waiter starts providing a good service, and will keep doing so in the future if he always receives a tip. If in some week Carlos does not give a tip, the waiter will never give a good service again. And Carlos will always give tips, as long as he receives a good service. If the waiter fails once, and give him a bad service, Carlos will stop tipping the waiter forever. If everybody knows that the waiter is leaving the restaurant the first day of next month, will them follow the agreement? Find the subgame perfect Nash equilibrium. 12. Calculate all of the Bayesian Nash equilibria in pure of the following static bayesian game: Player 1 can choose between two actions A and B. Player 2 can choose between two actions, I and D. The payoffs depend on the players types. Player 1 has just one type and this is known by Player 2. Player 2 can be of either type x or y. Player 2 knows her own type but Player 1 does not know for certain the type of Player 2. Player 1 thinks that Player 2 is Type x with probability 2/3, and Type y with probability 1/3. The payoffs are those that are given in the game that is randomly determined. Type x Type y I D I D A 4, 1 3, 3 A 3, 6 1, 3 B 3, 6 2, 3 B 1, 1 5, The game of chicken will be familiar to those who have seen West Side Story or Rebel without a cause. One simplified version of this game is as follows. Two players drive there cars at each other at full speed. They can take one of two possible actions, swerve aside or continue straight on. The payo s are those following. James Dean Continue Swerve

6 Bad guy Continue -3, -3 2, 0 Swerve 0, 2 1, 1 a) Find all the Nash equilibria in pure strategies and in mixed strategies of this game. Now suppose that everyone knows that the preferences of Dean are those that are given above, but there exist serious doubts about the sanity of the Bad guy, to the point that some people believe that he will never swerve, not even if it might cost him his life. Put more formally, the payoffs are the same as above with probablility p and are as follows with probability 1 p. Bad guy James Dean Continue Swerve Continue 2, -3 2, 0 Swerve 0, 2 1, 1 b) Find the Bayesian equilibrium of this games. 14. Two bidders participate in an auction to buy a painting. Their values for the painting are either 20 or 100, both being equally likely. Each individual knows her valuation but does not the valuation of the other player. If the acceptable bids are {10, 30, 50} and bidder cannot employ weakly dominated strategies. a) Find the Bayes equilibria of the first-price auction. b) Find the Bayes equilibria of the second-price auction. c) Compare the auctioneerís revenue in each auction. 17. Consider the following Bertrand duopoly with differentiated products and asymmetric information. The demand for Firm i is q! p!, p! = 10 p! b! p!. Both firms have zero costs. The effect of the price of the good produced by Firm 2 in the demand for Firm 1 may be high or low. More specifically, b! takes 0 and 1 with probabilities 2/3 and 1/3, respectively. Further, b! = 0. Eacn firm knows its own parameter b!, Firm 1 knows, in addition, b!, but Firm 2 does not know b!. All this is common knowledge. a) Which are the strategies for each firm? b) Compute the Bayesian equilibrium.

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