Moshe Shaked and J. George Shanthukumar, Stochastic orders and their applications, Academic Press, San Diego, CA, 1994.

In many real-life problems, e.g., in economics, reliability theory, medicine, etc., we must choose between two alternatives whose consequences are not completely known. The books considers the case when we know the {\it probabilities} of different results; hence, each alternatives is represented by a {\it probability distribution} on the set of possible results, i.e., as a {\it random variable} How can we compare two random variables $X$ and $Y$? One possibility (called {\it stochastic order}) is as follows:

In probability theory, a random variable $X$ is usually defined as a probability measure on the set of all real numbers (that describes the probability of different values of this variable). The common-sense understanding of a random variable is better described by an alternative (equivalent) definition: a random variable is a mapping from a set $\Omega$ with a probability measure $\mu$ on it to the set of real numbers (for which $x(\omega)$ has the desired probabilities).

We say that a random variable $X$ is {\it smaller} than a random variable $Y$ in the sense of stochastic ordering (and denote it by $X\le_stY$) if there exists a set $(\Omega,\mu)$ and two mappings $x,y:\Omega\to R$ that represent, correspondingly, variables $X$ and $Y$, and for which $x(\omega)\le y(\omega)$ for all $\omega$. The main result of stochastic ordering theory is a condition necessary and sufficient for $X\le_st Y$: this condition is the inequality between distribution functions: $P\{X\le u\}\ge P\{Y\le u\}$ for all real numbers $u$.

There also exist more complicated modifications of this definition.

The results presented in this book are based on the assumption that we know the probabilities; in many real-life situations, we do not know them. Many methods and ideas presented in the book can be naturally extended to this more general type of uncertainty.

For example, a similar choice problem occurs when we only know intervals $X=[x^-,x^+]$ and $Y=[y^-,y^+]$ of possible values of $x$ and $y$ that correspond to two alternatives. In this case, we can use the above-defined idea: Namely:

Each interval $X$ can be represented as a mapping $x:\Omega\to R$ from some set $\Omega$ to the set of real numbers, for which the set of possible values of $x(\omega)$ is exactly this interval $X$. \item We can say that $X$ is smaller than $Y$ (and denote it by $X\le_ st Y$) iff there exist two mappings $x,y:\Omega\to R$ for which $x$ represents $\bf X$, $y$ represents $Y$, and $x(\omega)\le y(\omega)$ for all $\omega\in \Omega$.

A direct analogue of the main theorem mentioned above can be easily proven for intervals:

PROPOSITION. $[x^-,x^+]\le_ st [y^-,y^+]$ iff $x^-\le y^-$ and $x^+\le y^+$.

{\it Proof.} If $x^-\le y^-$ and $x^+\le y^+$, then we can take $\Omega=[0,1]$, $x(\omega)=\omega\cdot x^+ +(1-\omega)\cdot x^-$, and $y(\omega)=\omega\cdot y^+ +(1-\omega)\cdot y^-$. Vice versa, if $[x^-,x^+]\le_ st[y^-,y^+]$, i.e., if there exists a joint representation $x,y:\Omega\to R$, then $x^+=x(\omega)$ for some $\omega\in \Omega$. For this $\omega$, we have $x^+=x(\omega)\le y(\omega)$; but $y(\omega)\le y^+$; hence, $x^+\le y^+$. Similarly, there exists an $\omega$ for which $y^-=y(\omega)$. For this $\omega$, $y^-=y(\omega)\ge x(\omega)\ge x^-$. Q.E.D.

The book's theory is thus extendible to the case when we do not know probabilities at all. It is desirable to extend the book's results to the intermediate cases when we know some but not all probabilities, e.g., to the case when we know the intervals of possible values of probabilities.

In the majority of applications presented in the book, we do not really know all the probabilities, so, it looks like this generalization will be not technically difficult and very practically useful.

Hung T. Nguyen and V. Kreinovich