Abstract
Objective: 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). Method: 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.
Highlights
That there is no sequential unfolding of information, this partially informed gambler may use the values known to him to update his knowledge on the variables not observed, i.e. he may refer to conditional expectations
Since y may assume any value in the interval [0, 1/n) we have shown that hn(x) = nx is the upper boundary function if x < 1/n
We focus on the prophet, other comparisons, in particular involving the statistician, would be interesting too
Summary
Given a sequence of random variables X = X1, X2, . Method: 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. There is a stochastic environment, (F.E. a class of random variables C), and several levels of information. RCj,k = {(x, y)|x = rj(X), y = rk(X), X ∈ C}, 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
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