HW4

Submit a single .pdf (or handcopy) for the written problems to me via email (or in person) by the date of the deadline (11:59 pm)

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

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1) 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)

2) 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)

3) Second-Price Auctions: Show that in a second-price sealed-bid auction bidding your valuation weakly dominates bidding above your valuation (following similar arguments of the case 1-3 outlined in the book or the presentation). (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) Formally define the auction. (5 points)

(b) Is it still true that each player bidding his valuation is a weakly dominant strategy? Explain why in detail. (5 points)

5) Define a complete information first-price sealed-bid auction. Provide a concrete example (with actual values and a fixed number of bidders) to show that bidding your valuation is a dominated strategy in a complete information first-price sealed-bid auction. (10 points)