Suppose two people are facing the same situation. However, depending on each person's level of information each one's return might be different. By how much?
In the present paper, the concept of "situation" translates into one of a "stochastic environment", and three levels of information are systematically studied (minimum, sequential, maximum). Traditionally, the difference between a statistician (possessing sequential information) and a prophet (possessing maximum information) has been investigated a lot, resulting in so-called prophet regions, characterizing the information gulf between these persons.
The aim of the present contribution is threefold. First, the difference between minimum and maximum information is studied, leading to two new theorems, characterizing the information gap between a person with only minimal information and a prophet. Second, prophet regions are systematically extended to so-called "information sets" that lead to new concepts such as the "overall information difference" and "typical differences and ratios". Leaving the standard viewpoint, the paper also considers "inverse problems" in general, and manages to invert the function x - x ln(x), in particular. Third, the "comprehensive analysis" just described is applied to the information sets derived in the first part of this contribution.
All in all, the paper proposes a new viewpoint, based on the concept of information. Embedding prophet-type results into this larger perspective underlines their importance and brings them closer towards applications.
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Reference: Saint-Mont U et al (2017). Comparing different information levels, The Open Statistics & Probability Journal, DOI: 10.2174/1876527001708010007