Comparing Different Information Levels

Uwe Saint-Mont*
Nordhausen University of Applied Sciences, Nordhausen, Germany

© 2017 Saint-Mont

open-access license: This is an open access article distributed under the terms of the Creative Commons Attribution 4.0 International Public License (CC-BY 4.0), a copy of which is available at: This license permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

* Address correspondence to this author at the Nordhausen University of Applied Sciences, Weinberghof 4, D- 99734 Nordhausen, Germany; Tel: 0049-3631-420-512; Fax: 0049-3631-420-817; E-mail:



Given a sequence of random variables X = X1, X2, . . .suppose the aim is to maximize one’s return by picking a ‘favorable’ Xi. Obviously, the expected payoff crucially depends on the information at hand. An optimally informed person knows all the values Xi = xi and thus receives E(sup Xi).


We will compare this return to the expected payoffs of a number of gamblers having less information, in particular supi(EXi), the value of the sequence to a person who only knows the random variables’ expected values.

In general, there is a stochastic environment, (F.E. a class of random variables C), and several levels of information. Given some XϵC, an observer possessing information j obtains rj(X). We are going to study ‘information sets’ of the form.

characterizing the advantage of k relative to j. Since such a set measures the additional payoff by virtue of increased information, its analysis yields a number of interesting results, in particular ‘prophet-type’ inequalities.

Keywords: Stochastic comparisons, Information levels, Prophet regions, Inverse functions.