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# week_7 - 122 Game Theory Professor Giacomo Bonanno PRACTICE...

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Problems for Week 7 122 - Game Theory - Professor Giacomo Bonanno PRACTICE PROBLEMS for WEEK 7 Topic: information partitions, knowledge and common knowledge VERY IMPORTANT : do not look at the answers until you have made a VERY serious effort to solve the problem. If you turn to the answers to get clues or help, you are wasting a chance to test how well you are prepared for the exams. I will not give you more practice problems later on. 1. Let the set of states be W = {a, b, c, d, e, f, g, h, k, m, n} and the information partition of an individual be: {{a,b,c}, {d}, {e, f, g, h}, {k,m}, {n}} . Consider the following event: E = {a,b,d,k,n}} . (1) Does the individual know E at state a ? (2) Does the individual know E at state c ? (3) Does the individual know E at state d ? (4) Does the individual know E at state h ? (5) Does the individual know E at state k ? (6) Does the individual know E at state n ? (7) Let KE denote the event that the individual knows E (that is, the set of states where the individual knows E ). What is KE ? (8) Given an event, say F, denote by ¬ F be the complement of F, that is, the set of states that are not in F. What is the event ¬ KE , that is the event that the individual does not know E ? What is the event K KE , that is, the event that the individual knows that she does not know E ? 2. Let the set of states be W = {a, b, c, d, e, f, g, h}. There are three individuals with the following information partitions: Ann Bob Carol a b c d e f g h a b c d e f g h a b c d e f g h (1) Consider the event E = {a,b,c,f,g} . Find the following events: K Ann E (that is, the event that Ann knows E ), K Bob E, K Carol E, K Carol K Ann E (that is, the event that Carol knows that Ann knows E ), K Bob K Carol K Ann E, K Ann ¬ K Bob K Carol E (that is, the event that Ann knows that Bob does not know that Carol knows E ). (2) What is the common knowledge partition? is common knowledge.

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week_7 - 122 Game Theory Professor Giacomo Bonanno PRACTICE...

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