Remarks: The problem set contains 6 exercises. Always motivate your solutions! It is often helpful to look up the way we solved similar problems in our lecture notes. The maximum number of points you can achieve in this problem set is 120 points. To get the full score of 30% for your final course grade, however, you need only 100 points. All points above the total of 200 points needed from both problem sets that you achieve will count already for your final exam (if you need some or all of these extra points, because you may get full points in the final anyway). For example, if you score 115 points in the first problem set and 105 points in the second problem set, thus 220 points in total, you have already achieved 20 out of 100 points of the final exam. 1) A governor needs to hire a bureaucrat to evaluate an issue. With probability p the bureaucrat has the same ideology as the governor and, if hired, prefers making an effort to evaluate the issue over not making an effort. And, with probability 1-p both players have different ideologies and, if hired, the bureaucrat prefers making no effort to shed light on the issue. The governor's preferences are known to both players. a) What are the pure Nash equilibria and the subgame-perfect equilibria of the two extensive games if the governor knows with certainty that the bureaucrat has the same ideology as herself (i.e., p = 1; the extensive game on the left hand side) and if the governor knows with certainty that the bureaucrat has a different ideology (i.e., p = 0; the extensive game on the right hand side)? [15 points] Governor and bureaucrat have the same ideology. Governor and bureaucrat have different ideologies. Governor Governor Hire Not hire Hire Not hire Bureaucrat Bureaucrat 0,0 0,0 Effort No effort Effort No effort 4,4 -2,-2 4,2 -2,4 b) Draw an extensive game that includes "nature" as the first player, where nature determines with probability p that the bureaucrat has the same ideology as the governor and with probability 1-p that the bureaucrat has a different ideology than the governor. Whereas the bureaucrat knows her own ideology, the governor does not know the ideology of the bureaucrat. Find all sequential equilibria (i.e., beliefs and optimal strategies) of this extensive game! [15 points] 1 2) Consider an ambitious party member who decides on whether or not to challenge the party leader's predictions about an important issue with uncertain outcomes ("nature" randomly determines whether the leader's prediction is correct, LC, or incorrect, LI, where each outcome occurs with probability 1/2). If the challenger's prediction turns out to be true, her reputation within the party will increase by 1. But if her prediction turns out to be wrong, her reputation will decrease by 1. Finally, if she does not challenge the leader her reputation remains unchanged. "Nature" p(LC)= = 1/2 p(LI)=1/2 Challenge -1 1 Party member Don't challenge a) What are the expected payoffs of the ambitious party member for each of her actions "challenge' and don't challenge"? Will she challenge the leader? [5 points] b) Assume that the ambitious party member has the opportunity to search for more information about the issue's outcome. The search costs are equal to 0.1. Specifically, if she searches she will either receive a signal that "the leader's prediction is correct" or that "the leader's prediction is incorrect". Her signal is true with probability .75 and it is not true with probability .25. Given the search costs, will she search for more information? [15 points] 3) Consider the following 3-player step-level public goods game. Each player i can make up to two units of costly effort, or ei € {0,1,2}. If at least k = 2 units of effort are made, irrespective of who of the players makes these efforts, the public good is provided and each player gets a revenue of 3. If less than k = 2 units of effort are made the public good is not provided and each player gets a revenue of 0. The payoff function for each i is given by 3 if > k = 2 IT; = + 0 if > j=1,2,3 ej ej <k =="" 2.<br="">j=1,2,3 a) Describe all pure strategy Nash equilibria of the game! [10 points] b) What are the individual payoffs of each player in each of the pure strategy Nash equilibria? And what are the respective aggregate (or group) payoffs? [10 points] c) What is the socially optimal outcome (i.e., the highest possible aggregate payoff) when only pure strategies are allowed? Is the social optimum achieved by any of the pure strategy Nash equilibria? Is the social optimum 'fair' in the sense that all players have the same individual payoff? [5 points] 2 </k>5) In the extensive game with imperfect information of the Prisoner's Dilemma (PD), "nature" first chooses a "pro-social' type of player 1 with probability p and a "selfish' type of player 1 with probability 1- p. Player 1 can observe her own type and knows that player 2 is 'pro-social'. On the other hand, player 2 knows her own type 'pro- social' but is not certain about player l's type. Find all sequential equilibria (consisting of beliefs and actions) of this game! [15 points] Nature p 1-p Player 1: Player 1: pro-social type selfish type C D C D Pl. 2 Pl. 2 C D C D C D C D 3,3 0,1 1,0 2,2 2,3 0,1 3,0 1,2 6) Firm A wants to acquire Firm T. Firm A is not informed about Firm T's exact value, X. However, Firm A knows that X is uniformly distributed on the real line [0,100], that is, the minimum value is 0, the maximum value is 100, and all values from 0 to 100 are equally likely to occur. Under the management of Firm A the value of Firm T is Mx, where M 1. Firm A makes a bid, b E [0,100], which Firm T either accepts or rejects. If the bid is accepted, Firm T earns b and Firm A earns Mx - b. If the bid is rejected, Firm T earns X and Firm A earns 0. Determine the threshold M such that Firm A's expected payoff for any b > 0 is strictly positive for M > M and strictly negative for M < M. [20 points]