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Two remarks on Blackwell’s theorem. (English) Zbl 1141.62004

Summary: In a decision problem with uncertainty a decision maker receives partial information about the actual state via an information structure. After receiving a signal, he is allowed to withdraw and gets zero profit. We say that one structure is better than another when a withdrawal option exists if it may never happen that one structure guarantees a positive profit while the other structure guarantees only zero profit. This order between information structures is characterized in terms that are different from those used by Blackwell’s comparison of experiments. We also treat the case of a malevolent nature that chooses a state in an adverse manner. It turns out that Blackwell’s classical characterization also holds in this case.

MSC:

62C10 Bayesian problems; characterization of Bayes procedures
62C99 Statistical decision theory
62B15 Theory of statistical experiments
62C20 Minimax procedures in statistical decision theory
91B06 Decision theory
91B08 Individual preferences
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References:

[1] Blackwell, D. (1953). Equivalent comparisons of experiments. Ann. Math. Statist. 24 , 265–272. · Zbl 0050.36004
[2] Le Cam, L. (1996). Comparison of experiments—a short review. In Statistics, Probability and Game Theory (IMS Lecture Notes Monogr. Ser. 30 ), eds T. Ferguson and L. Shapley, Institute of Mathematical Statistics, Hayward, CA, pp. 127–138.
[3] Torgersen, E. N. (1991). Comparison of Statistical Experiments. Cambridge University Press. · Zbl 0732.62009
[4] Tversky A. and Kahneman, D. (1981). The framing of decisions and the psychology of choice. Science 211 , 453–458. · Zbl 1225.91017
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