A New Approach of Stochastic Dominance for Ranking Transformations on the Discrete Random Variable

This paper presents some new stochastic dominance (SD) criteria for ranking transformations on a random variable, which is the first time that this is done for transformations under the discrete framework. By using the expected utility theory, the authors first propose a sufficient condition for general transformations by first degree SD (FSD), and further develop it into the necessary and sufficient condition for monotonic transformations. For the second degree SD (SSD) case, they divide the monotonic transformations into increasing and decreasing categories, and further derive their necessary and sufficient conditions, respectively. For two different discrete random variables with the same possible states, the authors obtain the sufficient and necessary conditions for FSD and SSD, respectively. The new SD criteria have the following features: each FSD condition is represented by the transformation functions and each SSD condition is characterized by the transformation functions and the probability distributions of the random variable. This is different from the classical SD approach where FSD and SSD conditions are described by cumulative distribution functions. Finally, a numerical example is provided to show the effectiveness of the new SD criteria. JEL C51 D81


Introduction
In real world, many human activities in insurance and financial fields induce risk transformations. For example, we assume that an investor owns a house where X denotes the value of the house (a random variable). The investor can insure the house with various levels of deductions. By choosing two different deduction policies, the investor creates different transformations () mX and () nX . Then an interesting question occurs: which deduction policy (transformation) dominates the other? In other words, how to find an effective approach for ranking these transformations so as to choose the beneficial one?
Stochastic dominance (SD) is the most famous approach to compare pairs of prospects. Presented in the context of expected utility theory, SD approach has the advantage that it requires no restrictions on probability distributions. Well-known specifications of SD are first degree SD (FSD) and second degree SD (SSD), which attract by far the most attention in SD research. Due to the advantage mentioned above, SD approach has been proved to be a powerful tool for ranking random variables and employed in various areas of finance, decision analysis, economics and statistics (See e.g., Meyer, 1989;Levy, 1992Levy, , 2006Chiu 2005;Li 2009;Blavaskyy, 2010Blavaskyy, , 2011Tzeng et al.,2013;Loomes et al., 2014;Tsetlin et al., 2015). Unfortunately, SD approach for ranking random variables is inefficient to rank transformations on random variables because it relies on the cumulative distribution functions (CDFs) of random variables, which are hard to calculate. In other words, SD approach cannot be used directly to rank transformations.
To the best of the authors' knowledge, there are only three papers studying SD rules for transformations on the continuous random variables under the conditions of increasing, continuous, and piecewise differentiable transformations (see, Meyer,1989;Brooks and Levy, 1989;Levy,1992).
However, there is little research which focuses on ranking transformations on the discrete framework.
It should be pointed out that the outcomes of transformations for continuous random variables cannot be extended directly to the discrete system. In real life, we notice that the discrete random variables are ubiquitous and even the continuous random variables should be discretely handled in many cases, so it is significant to find new SD criteria for ranking transformations on the discrete random variable.
The paper aims to develop some new SD rules for ranking transformations on the discrete random variable, which is the first time to investigate the ranking approach for the discrete system. In order to construct such theoretical paradigm, we start from the FSD rule by applying the expected utility theory, and derive a sufficient condition (See Theorem 1). We further extend the sufficient condition into the sufficient and necessary condition by introducing the monotonicity of transformations, this FSD relation is determined only by the difference between the compared transformation functions (See Theorem 2). For the case of SSD, we first divide the monotonic transformation functions into increasing case and decreasing situation, respectively. For the increasing transformation functions, we respectively develop a sufficient condition and a sufficient and necessary condition for SSD by means of the transformation functions and the probability distributions of the random variable (See Theorem 3 and Theorem 4). For the decreasing transformation functions, we also respectively obtain a sufficient condition and a sufficient and necessary condition for SSD, while these dominant conditions are different from the increasing situations (See Theorem 5 and Theorem 6). In addition, for two different discrete random variables with the same possible states, we provide the sufficient and necessary conditions for FSD and SSD, respectively (See Theorem 7).
Compared with the existing SD rules, the advantages of the new SD rules we derived are as follows: (1) the new SD rules can rank transformations on a discrete random variable, while the existing SD rules do not work; (2) the new SD rules make us avoid the tedious computation of CDFs, whereas this can not be done in the existing SD rules. In this sense, the new theoretical paradigm we derived can be regarded as a useful complement to the existing SD theory. Finally, a numerical example is provided to show the effectiveness of the new SD rules.
The rest of this paper is organized as follows. Section 2 reviews the existing SD rules. Section 3 and Section 4 present the SD rules of transformations by FSD and SSD, respectively. Section 5 makes a comparison between the new SD rules and the existing SD rules. Section 6 gives a numerical example to show the efficiency of the new SD method and Section 7 draws the conclusions.

Preliminaries
This section introduces the definition of stochastic dominance, and the SD rules for transformation on the continuous random variable.
Let X and Y be two random variables with support in the finite interval [ , ] ab , and their CDFs will be denoted by () Fx and () Gx , respectively. Define ( ) , and define () () n Gx similarly. Moreover, we denote n U as the class containing all the functions u with Definition 1. (Levy, 1992) for any real number x ; for any real number x ; The SD rules and the relevant class of preferences k U are related in the following way: Integral conditions (1) and (2) mean that SD approach relies on CDFs of the random variables, and it is inefficient to rank transformations on the random variable. To overcome this shortcoming, Meyer (1989) proposes the following results.
Lemma 1. (Meyer, 1989) Given a continuous random variable X with the density () fxand support in the interval [ , ] ab . If () mx and () nx are non-decreasing, continuous and piecewise differential functions, then (ii) the transformed random variable () Lemma 1 provides the FSD and SSD rules, which are only valid for non-decreasing, continuous and piecewise differential functions, and these SD rules cannot be directly applied to ranking transformations on the discrete random variable. However, in the real world, the discrete random variables are ubiquitous and even the continuous random variables should be discretely handled in many cases, so it is significant to find SD criteria for ranking transformations on the discrete random variable. Considering that FSD and SSD have more practical implication than higher degree SD rules, this paper will focus on FSD and SSD rules in the remaining part of the paper.

Dominance Conditions for FSD
Let X be a discrete random variable whose prospects are characterized by 11 In order to better understand the meaning of Theorem 1, Fig. 1 shows its graphical illustration.
Different from the existing SD rules, the areas 1 , S 2 , S , n S in Fig. 1    Theorem 1 presents a sufficient condition for determining FSD relations which only involves the transformation function. Apparently, it is much easier to compare the transformation functions than to compare the CDFs of transformed random variables.
However, we see that condition (5) is only a sufficient condition for () mX dominating () nX by FSD, rather than the necessary and sufficient condition. A natural question is whether condition (5) Table 1 shows that 11 1 0 ( ) x n x     , which means that condition (5) is not necessary when the dominated transformation () nx is nonmonotonic.
(c) We notice that from Table 1  x    , which indicates that condition (5) is still not necessary when one transformation is increasing and the other is decreasing.
Based on the above analysis, we conclude that condition (5) is not necessary if () mxand () nx are not comonotonic. The following theorem shows that condition (5) will be sufficient and necessary when the transformation functions are comonotonic. Proof. The sufficiency is obvious from Theorem 1. We then only need to prove the necessity, and this process is divided into the following three steps.
Step 1. Suppose that () mxand () nx are both increasing, we will use the reduction to absurdity to prove this conclusion.
If there exists a number j such that1 jn  and jj mn  , then we can define the utility function with the following form: From the function (6), it is easy to see that Therefore, from the analyses of (a) and (b), we can conclude that ( ) ( ) ii u m u n  for all1 in  and ij  . Combining it with the assumption jj mn  , we have that which is a contradiction with the assumption of () mX dominating () nX by FSD.
Step 2. We will prove that for any two random variables X and Y with the corresponding CDFs Step 3. When the transformations () mxand () nx are both decreasing, it is obvious that () mx  and () nx  are both increasing. Then from Step 1 and Step 2, we derive that Fig.1 can also illustrate the graphical presentation of Theorem 2, that is, () mX dominating () nX by FSD is equivalent to the situation that the n areas 12 , , , n S S S are non-negative.
Remark 1. Theorem 2 provides a sufficient and necessary condition by introducing the monotonic condition of transformations, which only depends on the transformation functions, and in this way it presents a simple way for determining FSD relations between two comonotonic transformations. This situation is different from the case of the existing SD rules for FSD in which we need to take a tedious calculation to get the CDFs of the transformed random variables. Therefore, Theorem 2 plays an active part in dealing with realistic problems via the new SD rules we derived. The assumption of monotonous condition is appropriate because it seems to be a common feature of transformations in the fields of insurance and decision analysis (see Meyer 1989).

Dominance Conditions for SSD
In this section, we try to find some dominant rules for SSD. It is well known that SSD condition in the existing SD approach is more complicated than the case of FSD. In order to find SSD rules for ranking transformed discrete random variables, we will divide the monotonic transformation functions into increasing and decreasing ones, respectively.
Theorem 3. If () mxis increasing and Similarly, from Substituting Eqs. (9) and (10) into (11), we obtain that 12 2 2 2 1 11 11 11 For any utility function 2 () u x U  , by using the differential mean value theorem, we have that   Similar to Theorem 1, the condition (8) in Theorem 3 only indicates the sufficient condition rather than the necessary and sufficient condition. What we focus on is the sufficient and necessary condition, then, is the condition (8) necessary? The following example will answer this question.
Example 2. We assume that a random variable X yields the outcomes 1,  By analyzing the data in Table 2, we can make the following statements.
(a) We find that X dominates () nX by SSD via the hierarchical property of the SD rules. However, x n x p     , which implies that condition (8) According to the definition of () ux and the monotonicity of () mx , () nx and () ux , we conclude that because () ir u n n  and r n is the maximum of () ux . So, which is a contradiction with () mX dominating () nX by SSD. Theorem 4 can also be illustrated by Fig.2. Recall that in Fig. 2  Compared with Theorem 2, Theorem 4 reduces the requirement of the n rectangular areas. That is, any of the n rectangular areas except for the first one may take negative value, but the cumulative sum of the first ( 1, 2, , ) k k n  rectangular areas (from left to right) must be non-negative. It also means that if () mX dominates () nX by FSD, then () mX dominates () nX by SSD, which is in accordance with the hierarchical property of SD rules. Recall that when () mxand () nx are comonotonic, there exists a unified necessary and sufficient condition for FSD case (see Theorem 2). However, this statement is not valid for SSD. That is, we need to seek a necessary and sufficient condition for SSD if () mx and () nx are both decreasing. We first provide a sufficient condition as follows.
which is a contradiction with the assumption that () mX dominates () nX by SSD.
The graphical illustration about Theorem 6 can also be expressed by Fig.3. We investigate these n rectangular areas in Fig. 3 from right to left. If the cumulative sum of the first ( 1, 2, , ) k k n  rectangular areas is non-negative (resp. non-positive), then () mX dominates () nX (resp. () nX dominates () mX ) by SSD. Otherwise, there is no SSD relation between () mX and () nX . In addition, from Fig.1, Fig.2 and Fig.3, we find that Theorem 6 and Theorem 4 both reduce the condition requirement of Theorem 2 in which all the n rectangular areas are non-negative, that is, Theorem 6 and Theorem 4 argue that these n rectangular areas can take negative values, but the precondition is that the cumulative sum of the first ( 1, 2, , ) k k n  rectangular areas should be non-negative. The difference between Theorem 6 and Theorem 4 is that Theorem 6 sums up the rectangular areas from right to left, while Theorem 4 does it from left to right. Undoubtedly, Theorem 6 and Theorem 4 are homogeneous in essence except that the former deals with increasing transformations while the latter copes with decreasing ones.

Comparison of the new SD Rules and the existing SD Rules
Notice that the existing SD rules only rank transformations of the continuously distributed random variables with the piecewise differentiable transformations, while the new SD rules we developed can rank transformations of the discrete random variable. Therefore, the new SD rules we developed can be regarded as an extension to the existing SD rules. In particular, the new SD rules are used to rank the transformed random variables, while the existing SD rules are applied to compare the risk of any two random variables, and the existing SD rules is ineffective when dealing with transformations on the same random variable. In this sense, the new SD rules developed in Section 3 and Section 4 remedy the weakness of the existing SD approach.
With regard to the expression form, the major difference is that the existing SD approach is presented in the framework of CDFs while the new SD rules are expressed by transformation functions and the probability function of the original random variable, so it avoids the tedious computation of CDFs and their integral.
To better understand these two types SD rules, Table 3 shows their main differences from three aspects: application scope, expression approach and the specific criteria. Remark 5. It should be pointed out that as a partial order relation, the existing SD can not rank all the random variables. However, as a screening device, the existing SD rules can divide the whole decision making set into an efficient set and an inefficient one, and then the decision maker can make decision under the efficient set (See, e.g., Li 2009;Blavaskyy, 2010Blavaskyy, , 2011Tzeng et al.,2013;Loomes et al., 2014;Tsetlin et al., 2015). Such statements are also suitable to our new SD rules.
We further study the intrinsic links between the two types SD rules in the following. Note that the two transformations () mX and () nX on a random variable can also be regarded as two special random variables, then there should exist the corresponding random variables Theorem 7 shows that the new SD rules can rank not only the transformations on a discrete random variable, but also different random variables with the same possible future states.

Numerical Example
This section provides a numerical example to illustrate how to determine the SD relations for transformations by using the new SD rules and further determine the efficient set of decision making set.
With the accelerating trend of population aging, the pension fund gap of China is becoming increasingly wide. For example, the World Bank stated that the size of China's 2001 to 2075 pension fund gap is 9.15 trillion Yuan (Wang et al. 2014). To effectively control the pension fund gap, one of the important approaches is to increase the investment return of the pension fund. However, in real world, the government pension sector of China can not directly invest the pension fund, but authorize several institutional investors to invest. Then, there is a principal-agent relationship between the government pension sector of China and each institutional investor. For a given return rate on investment, different institutional investors may provide different revenue-sharing proposals except for the common commission for the agency. Notice that as a screening device, the new SD rules can divide the whole decision set into an efficient and inefficient sets. Then, how to determine the efficient set of these revenue-sharing proposals is an interesting question.
Owing to the uncertainty of the stock market, there is an enormous risk for investing money in the stock market. Let X denote the rate of return on investing money in stock market and its probability distribution is shown as in Table 4. Assume that there are four institutional investors who can provide revenue-sharing proposals denoted by X , () mX , () nX and () rX , respectively. Table 5 shows the probability distributions of the four different revenue-sharing proposals. We assume that revenue-sharing proposals () mX , () nX and () rX are as follows: Here, () mX denotes that the institutional investor A will allocate four fifths of investment return rate to the government pension sector of China if the return rate is positive, and he will allocate two fifths of the the return rate to the government pension sector if the return rate is negative.
While () nX represents the institutional investor B will distribute the total return rate to the government pension sector if the return rate is positive (meaning he only takes the commission in this situation), otherwise, he will distribute four fifths to the government pension sector if the return rate is negative. The meaning of () rX is that institutional investor C will allocate three fifths of the return rate plus five percent (regarded as risk-free interest rate) to the government pension sector. In addition, the implication of the revenue-sharing proposal X is that the institutional investor D will assign the total return rate to the government pension sector and he only takes the commission. proposals in this step will be deleted from the efficient set.
(a3) There is no FSD relation between () nX and () rX because 11 nr  but 55 nr  . Therefore, in this step, there will be no revenue-sharing proposals removed from the efficient set.
(b1) In this step, we will determine the SSD relations between () mX and () nX .
Since 1  Therefore, according to Theorem 4, we can conclude that () mX dominates () nX by SSD. That is, () nX should be ruled out from the efficient set, and only () mX and () rX are kept in the efficient set. (b2) In this step, we will determine the SSD relations between () mX and () rX .
On the one hand, from rm  implies that () rX does not dominate () mX by SSD.
Therefore, according to Theorem 2 and Theorem 4, we can determine the FSD relations and SSD relations among the four revenue-sharing proposals. The above computational procedure shows that () nX dominates X by FSD, and () mX dominates () nX by SSD, so X and () nX are ruled out from the efficient set. That is, () mX and () rX are kept in the efficient set, and X and () nX are included in the inefficient set.

Conclusion
Very often insurance activities induce transformations of an initial risk, which results in a new problem of how to rank transformations on the same random variable. This paper developed the new FSD and SSD rules for ranking transformations on a discrete random variable, which is the first time to consider the ranking approach for transformations on the discrete system. We start from the FSD rule by applying the expected utility theory, and derive the sufficient condition by FSD, and further extend the sufficient condition into the sufficient and necessary condition by introducing the monotonicity of transformations. For the case of SSD, we first divide the transformations into increasing and decreasing ones, and then respectively derive the necessary and sufficient conditions for the increasing and decreasing situations. For two different discrete random variables with the same possible states, we present the sufficient and necessary conditions for FSD and SSD, respectively.
The feature of our new SD rules lies in that each FSD condition is represented by the transformation functions and each SSD condition is characterized by the transformation functions and the probability distributions of the original random variable. This feature is different from the existing SD rules where FSD and SSD are described by CDFs. We notice that the SD rules proposed by Meyer (1989) are only suitable for the continuous random variables under the conditions with the piecewise differentiable transformations. Therefore, the existing SD rules can not be applied directly to the discrete system, in addition, it is very difficult for the existing SD rules to get the CDFs of the transformed random variables. However, the new SD rules we derived just overcame the above limitations. In this sense, the new theoretical paradigm we derived can be regarded as a useful complement for the existing SD approach.
The new SD rules we derived only study transformations on the same random variable. In the future study, it is significant to extend the ranking transformations on different random variables, and it would be interesting to consider higher-degree SD rules for transformations.