HW6

(1) Please show ALL of your work!!!!

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1) Stackelberg games: For the following 2-player utility matrix (see below)

(a) Identify all pure-strategy Nash equilibria and mixed-strategy Nash equilibria viewing this as a normal-form game. (10 points)

(b) Identify an optimal pure strategy to commit to (Stackelberg pure equilibrium) assuming player 1 is the leader. (10 points)

(c) Identify an optimal mixed strategy to commit to (Stackelberg mixed equilibrium) assuming player 1 is the leader. You only need to show the LPs. (10 points)

2) Chicken Revisited: Consider the game of chicken in Section 12.2.1 with the parameters R = 8, H = 16, and L = 0 as described there. A preacher, who knows some game theory, decides to use this model to claim that moving to a society in which all parents are lenient will have detrimental effects on the behavior of teenagers. Does equilibrium analysis support this claim? (10 points)

What if R=8, H =0,and L=16? (10 points)

3) Armed Conflict: Consider the following strategic situation: Two rival armies plan to seize a disputed territory. Each army’s general can choose either to attack (A) or to not attack (N). In addition, each army is either strong (S) or weak (W) with equal probability, and the realizations for each army are independent. Furthermore, the type of each army is known only to that army’s general. An army can capture the territory if either (i) it attacks and its rival does not or (ii) it and its rival attack, but it is strong and the rival is weak. If both attack and are of equal strength then neither captures the territory. As for payoffs, the territory is worth m if captured and each army has a cost of fighting equal to s if it is strong and w if it is weak, where s < w. If an army attacks but its rival does not, no costs are borne by either side. Identify all the pure-strategy Bayesian Nash equilibria of this game for the following two cases, and briefly describe the intuition for your results:

a. m=3, w=2, s=1  (10 points)

b. m=3, w=4, s=2 (10 points)

4) Complete Information: Consider a set of N players participating in a sealed-bid second-price auction, but assume that there is complete information so that each player knows the valuation of every other player. 

a. Model the complete information sealed-bid second-price auction as a game. (10 points)

b. Provide a 3-player game of the complete information sealed-bid second-price auction (including setting their actual values). (10 points)

c. Show that each player bidding his valuation is a weakly dominant strategy (using the arguments presented in class). (10 points)