14.123 Microeconomic Theory III Problem Set 3 The due date for this assignment is Th ursday Marc h 11 1. Lecture Notes; Chap ter 6.4, Exer cise 8. 2. Alice and Bob seek eac h other. Simultaneou sly, Alice puts effort sA and Bob puts effort sB to search. The probabilit y of meeting is sAsB; the value of the meeting is vA ≥ 0 for Alice and vB ≥ 0 for Bob, and the searc h costs s3 to Alice and s3 to Bob. A B (a) Compute the set of all rationalizable strategies. (b) How do the rationaliza ble search efforts change with vA and vB? 3. Consider a game w ith a finite set N = {1,...,n} of play ers and a finite set S = S1×· · ·× Sn of strategy profiles. A general inform ation structure is a list (Ω,I1,...,In,p1,...,pn) where Ii is the information partition of i and pi ∈ ∆ (Ω) is the prior belief of i for eve ry i ∈ N . For ev ery rationaliza ble strategy si ∗ ∈ Si∞ of every play er i, show that there exist a general information structure (Ω,I1,...,In,p1,...,pn) and an adapted strategy profile (s1,...,sn) such that • si (ω∗)= s∗ for som e ω∗ ∈ Ω and i • sj (ω) ∈ arg maxsj ∈Sj Epj [uj (sj, s−j) |Ij (ω)] for every ω ∈ Ω and j ∈ N. [Hint: For every i ∈ N and every si ∈ Si∞, si is best reply a belief μisi on S−∞i.Take Ω = S∞.] 4. Characteruze the set of all correlated equilibrium distributions for the following game: L R U 3, 1 0, 0 D 0, 0 1, 3 1MIT OpenCourseWarehttp://ocw.mit.edu 14.123 Microeconomic Theory III Spring 2010 For information about citing these materials or our Terms of Use, visit:
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