Player 2 L R M H a,a 7,1 5,0 T 0,5 5,3 6,6

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1 Question 1 : Backward Induction L R M H a,a 7,1 5,0 T 0,5 5,3 6,6 a R a) Give a definition of the notion of a Nash-Equilibrium! Give all Nash-Equilibria of the game (as a function of a)! (6 points) b) Assume that a = 1 and that the game is played sequentially. moves first and player 1 can observe his move. Give the extensive form of the sequential game! Derive the subgame perfect Nash-Equilibrium of the game! (7 points) c) Give the normal form of the game in b). Derive all Nash-Equilibria. Are there non subgame perfect (SP) Nash-Equilibria? Explain why they are not SP! (7 points) 1

2 Question 2 : General Equilibrium Theory a) Define and interpret the notion of a competitive equilibrium used as a solution concept in general equilibrium theory! (6 points) b) Does anyone - consumer or producer - have an incentive to unilaterally deviate from his/her equilibrium plan? Discuss why or why not in detail! (6 points) c) Why does your answer in b) not qualify a competitive equilibrium as a Nash equilibrium? What are the wider implications of this fact for general equilibrium theory as a predictive tool? (8 points) 2

3 Question 3 : Repeated Games D C D a, a -1,5 C 5,-1 0,0 a R a) Give all Nash-Equilibria of the game above (as a function of a)! (5 points) b) For what values of a do we have a prisoners dilemma? (1 point) c) Suppose that a = 1 and that the game is repeated more than once. (2 points) What is the subgame perfect Nash-Equilibrium if there are a finite number of repetitions? Why? d) Suppose that there are an infinite number of repetitions and that the grim (trigger) strategy is employed. For what values of the discount factor δ does a deviation in period T (T 1) not pay? (8 points) e) δ = 0.9. A player dies with probability p > 0 in each round (after the game was played). For what values of p is (D) in all periods still the best choice? Give also an intuition of your answer! (4 points) 3

4 Question 4 : Auction Rhistian C. likes collecting coins very much. A few days ago he learned about ebay. And now he is trying to win an auction for a Thaler struck in Stolberg in Stolberg is about 30 km away from the city where Rhistian was born. Accordingly he has an extra incentive to win the auction. Including this extra incentive, his valuation for the coin is 200 Euro. He assumes that there are n 1 other bidders and according to the literature the possibly highest valuation for the coin is 300 Euro. The valuations for the coin are i.i.d on the interval [0,300]. Assume that ebay uses a second price auction and that the payoff is 0 if the auction is lost. a) Give a precise definition of players, strategies and payoffs. Is there a dominant strategy? If yes, what is the dominant strategy? Explain your answer! (10 points) b) What is the probability that Rhistian wins the auction? What is his expected payoff? (3 points) He lost the auction in the last second. Therefore he decides to go to the Pub for a beer. There he met another collector. This collector has another copy of the coin and invites Rhistian to make an offer. But he can only make one offer. If the offer exceeds the valuation v P of the collector he gets the coin and has to pay v p. If the offer is lower he does not get the coin and has a payoff of 0. v P is drawn independently and at random from the interval 0 and 300. Rhistian does not know v P. c) What is the probability that his valuation is higher than v P? What is Rhistian going to offer? What is the probability that his offer is greater than v P? What is his expected payoff? Explain your answers! (7 points) 4

5 Question 5 : Evolutionary Game Theory Given is the Hawk-Dove game specified by: Hawk Dove Hawk -1,-1 2,0 Dove 0,2 1,1 a) Derive all Nash-Equilibria of this game (pure and mixed)! Define the notion of evolutionary stable strategies (ESS) and the relatedness to Nash-Equilibria! (6 points) b) Is there an evolutionary stable strategy? Why? Give an intuition what that means for a given population! Hint: Argue that no pure strategy equilibrium can be an ESS! (6 points) Assume that a mutation took place. The so called bourgeois strategy appeared: If a bird appears first, he behaves like a hawk but if he is second he acts like a Dove. c) Is this strategy an ESS (against Hawk and Dove)? Why? (5 points) d) Give a graphical sketch of the Evolutionary Dynamics with and without bourgeois strategy! (3 points) 5