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Giskard is right lexicographic preferences would make the job being dicsontinuous, but the problem is to find relations with closed contour sets. Maybe an idea would be to start from constructing a preference relation for which the only possible upper and lower contour sets are $X$ and $\emptyset$. In that case, you take the indiscrete topology, and all ...

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The signal structure must specify what the DM learns in all possible states of the world, so I think your question should read: "Suppose that the DM discovers whether the state of the world $v\in V$ is in $[a,b]\subset V$ or not. Can this be written as an information structure?" or should read: "Suppose that the DM discovers the state of the world \$v\in ...

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For decision theory: Gilboa, I. (2009). Theory of decision under uncertainty. Cambridge University Press. Parmigiani, G., & Inoue, L. (2009). Decision theory: Principles and approaches. John Wiley & Sons. Peterson, M. (2009). An introduction to decision theory. Cambridge University Press. For social choice theory, see my answer to this question : ...

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