please help me to solve six questions on the PDF, I need answers of the questions

Revision Exercises 1. Reconsider Theorem 1' in Kreps' article. Assume in addition that % satisfies Set-Betweenness as defined in Gul and Pesendorfer. never exhibits a strict preference for flexibility. a) Show that % has a representation following Kreps' preference for flexb) Show that % ibility approach with a singleton state space (so the state space = {!} contains a single state !). c) How does your answer to the preceding two questions change if you drop assumption (1.5.)? 2. Say an agents' preference over subsets of {A, B, C, D, E} is represented by the utility function: 1 U (S) = max(u(X) + v1 (X)) 2 X2S . 1 max v1 (X) + max(u(X) + v2 (X)) X2S 2 X2S max v2 (X) X2S a) How can this representation be interpreted as a random temptation model? b) Let u(A) = 10, u(B) = 8, u(C) = 6, u(D) = 4, u(E) = 2, v1 (E) = v2 (D) = 5, and vi (X) = 0 for all values not yet specified. Show that the preferences represented by U violate Set-Betweenness. c) Change v2 such that keeping all other values specified in part b) constant so that U does satisfy Set-Betweenness. d) Does U (as defined in b) satisfy Kreps' 1.3 and 1.5? satisfies Set-Betweenness. Show that for each S 3. a) Assume that % there exist two objects X, Y 2 S such that S {X, Y }. b) Assume that a preference has a representation following Kreps' Theorem 1. Assume furthermore that for each S there exist two objects X, Y 2 S such that S {X, Y }. Can you always find preference for flexibility-type representation following Kreps with only two states? 1 4. Clarice can't resist chocolate. At the same time Clarice believes that chocolate is not good for her. There are two coee shops on campus. The Jane is close by, the Joe is further away. The Jane has a wider selection. It in particular has excellent chocolate. The Joe has a smaller selection. It does not carry any good chocolate. Clarice usually gets her coee at the Joe (even though the Jane, which is closer oers the exact same coee). a) model Clarice's behavior using the model. b) model Clarice's behavior following Gul and Pesendorfer's model of temptation. c) which of Kreps' axioms does Clarice's behavior violate? 5. Consider a preference % on timed consumption bundles (x, t) where x 2 X Rn for some fixed n and t 2 {1, 2, 3, 4, 5}. Say % is represented by U (x, t) = u(x) t with u : X ! [0, 1] a function with u(x) = 1 for some x 2 X and u(x0 ) = 0 for a dierent x0 2 X. a) Is time costly in the sense that (x, t) (x, t0 ) holds for all x and t