### Discussion Paper

## Abstract

Almost stochastic dominance is a relaxation of stochastic dominance, which allows small violations of stochastic dominance rules to avoid situations where most decision makers prefer one alternative to another but stochastic dominance cannot rank them. The authors first discuss the relations between almost first-degree stochastic dominance (AFSD) and the second-degree stochastic dominance (SSD), and demonstrate that the AFSD criterion is helpful to narrow down the SSD efficient set. Since the existing AFSD criterion is not convenient to rank transformations of random variables due to its relying heavily on cumulative distribution functions, the authors propose the AFSD criterion for transformations of random variables by means of transformation functions and the probability function of the original random variable. Moreover, they employ this method to analyze the transformations resulting from insurance and option strategy.

## Comments and Questions

The authors really offer novel and original research in their work, especially with regards to:

1. They focus on transformations of random variables, which have received little attention in the existing literature.

2. They base their measures on the properties of the functions that define the transformations of the random ...[more]

... variables, and not on their probabilistic features.

3. They propose the new almost first-degree stochastic dominance criteria, which is very different with Leshno and Levy’s definition and all other works related to almost stochastic dominance.

The theoretical result is correct. The results obtained in this paper are important for the ranking transformations on a random variable. In my opinion, the paper deserves to be published on Economics.

The paper addresses an interesting topic. There are many situations where one has to compare two different payoff structures which all depend on the outcome of the same random variable. As an example, in finance applications one could have different structured products written on the same underlying and the question ...[more]

... is whether one of those products is dominated by another.

The authors discuss the ranking transformations on a random variable. Some theoretical results are given in the paper and s simple numerical example is given. The mathematical results are well proved and illustrated. We find no flaw on this.

To the best of our knowledge, the authors' contribution is original, meaningful and will become useful. We thus believe that it deserves to be published.

The importance of ordering probability distributions according to first-order stochastic dominance (FSD) lies in the fact that any decision maker with increasing preferences over outcomes must prefer the FSD dominant distribution. Unfortunately, this order is highly incomplete and other criteria such as risk preferences have to be considered in order ...[more]

... to obtain rankings of probability distributions which are not related by FSD. Hence, economists consider higher orders of stochastic dominance. In particular, second order stochastic dominance (SSD) which can be associated with risk aversion has become an important criterion for ranking probability distributions.

Since Leshno and Levy (2002) introduce the concept of almost stochastic dominance, many papers have been devoted to the theory and application research of ASD, and the study of ASD is becoming the hot research point. Unlike all the existing literature, the atuthors generalize Leshno and Levy’s definition of AFSD from a new perspective. They propose the AFSD criterion for transformations of random variables by means of transformation functions and the probability function of the original random variable. In fact, they create a new AFSD rule, which is quite distinct from Leshno and Levy’s definition.

Moreover, they discuss the relations between almost first-degree stochastic dominance (AFSD) and SSD, and they shows that the AFSD criterion may lead to better efficient set than the SSD efficient set in some sense. We agree with this opinion, and this fact makes up for the shortfalls in the study of higher orders stochastic dominance to some extent. We also approve that the second-degree and higher-degree ASD criterion for transformations should be discussed in another paper.

In my opinion the paper is interesting and it should be published in Economics.

Thank you for approving my job. Leshno and Levy (2002) propose the definition of almost stochastic dominance, and many papers have been devoted to the theory and application research of this issue. However, we notice that the ASD criterion in the existing literature is not convenient to deal transformations of ...[more]

... random variables. Just as Levy (1992) says, the SD research of transformations of random variables is an important branch is the SD research. So, we want to find the ASD criteria for transformations of random variables. When we manage to find the AFSD criterion, we find that the second-degree and higher-order ASD rules are extremely difficult to be expressed by transformation functions and the probability function of the original random variable. Note that there are still enormous controversy about the definition of second-degree ASD, and the second-degree and higher-degree ASD criteria are much complicated than the AFSD criterion, and we have to create an entirely new technique. Based on the above considerations, we’d better investigate them in another paper.

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