4. Suppose a subjective learning DM has prior belief that assigns equal weight, 1/3, to the three states, wi, w2, and w3, of the world (i.e., good, bad, and ugly weather). Suppose the DM expects to receive information about the true state. In particular, let be the DM's subjective information structure which puts equal weight on each posterior belief, p = (2/3, 1/6, 1/6) and q = (1/6,2/3, 1/6) and r = (1/6, 1/6,2/3), but zero weight on any other posterior belief. Let f = (12,6,0) and g = (0,12,6) and h = (6,0,12) be three acts available to the DM and suppose the DM receives payoff from each outcome a by the function u(a) = a. Answer the following questions. (a) Show that the DM strictly prefers (f, g} over 1/2{g} + 1/2{h} (b) Show that the DM strictly prefers (f, g} over {f, 1/2g + 1/2h} (c) Show that the DM strictly prefers {f, g, h} over {f, 1/2g+ 1/2h, h}
4. Suppose a subjective learning DM has prior belief that assigns equal weight, 1/3, to the three states, wi, w2, and w3, of the world (i.e., good, bad, and ugly weather). Suppose the DM expects to receive information about the true state. In particular, let be the DM's subjective information structure which puts equal weight on each posterior belief, p = (2/3, 1/6, 1/6) and q = (1/6,2/3, 1/6) and r = (1/6, 1/6,2/3), but zero weight on any other posterior belief. Let f = (12,6,0) and g = (0,12,6) and h = (6,0,12) be three acts available to the DM and suppose the DM receives payoff from each outcome a by the function u(a) = a. Answer the following questions. (a) Show that the DM strictly prefers (f, g} over 1/2{g} + 1/2{h} (b) Show that the DM strictly prefers (f, g} over {f, 1/2g + 1/2h} (c) Show that the DM strictly prefers {f, g, h} over {f, 1/2g+ 1/2h, h}
Chapter7: Uncertainty
Section: Chapter Questions
Problem 7.11P
Question
![4. Suppose a subjective learning DM has prior belief that assigns equal weight, 1/3, to the
three states, wi, w2, and w3, of the world (i.e., good, bad, and ugly weather). Suppose
the DM expects to receive information about the true state. In particular, let be the
DM's subjective information structure which puts equal weight on each posterior belief,
p = (2/3, 1/6, 1/6) and q = (1/6,2/3, 1/6) and r = (1/6, 1/6,2/3), but zero weight on
any other posterior belief. Let f = (12,6,0) and g = (0,12,6) and h = (6,0,12) be
three acts available to the DM and suppose the DM receives payoff from each outcome
a by the function u(a) = a. Answer the following questions.
(a) Show that the DM strictly prefers (f, g} over 1/2{g} + 1/2{h}
(b) Show that the DM strictly prefers (f, g} over {f, 1/2g + 1/2h}
(c) Show that the DM strictly prefers {f, g, h} over {f, 1/2g+ 1/2h, h}](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F59419ab1-a6d3-4478-b94c-6db0f2625ecb%2F28999b9c-ba61-4b53-adc7-e84a676fd882%2Fmkmr00v_processed.jpeg&w=3840&q=75)
Transcribed Image Text:4. Suppose a subjective learning DM has prior belief that assigns equal weight, 1/3, to the
three states, wi, w2, and w3, of the world (i.e., good, bad, and ugly weather). Suppose
the DM expects to receive information about the true state. In particular, let be the
DM's subjective information structure which puts equal weight on each posterior belief,
p = (2/3, 1/6, 1/6) and q = (1/6,2/3, 1/6) and r = (1/6, 1/6,2/3), but zero weight on
any other posterior belief. Let f = (12,6,0) and g = (0,12,6) and h = (6,0,12) be
three acts available to the DM and suppose the DM receives payoff from each outcome
a by the function u(a) = a. Answer the following questions.
(a) Show that the DM strictly prefers (f, g} over 1/2{g} + 1/2{h}
(b) Show that the DM strictly prefers (f, g} over {f, 1/2g + 1/2h}
(c) Show that the DM strictly prefers {f, g, h} over {f, 1/2g+ 1/2h, h}
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