The non-integer higher-order Stochastic dominance
Introduction
The study on higher-order risk attitudes and their economic implications attracts a lot of attention [5]. An important class of risk attitudes are “mixed risk aversion” [1]. Specifically, we call decision makers having mixed risk aversion up to degree if their utility function satisfies (), where represents the th-order derivative of . Fox example, when , decision makers are insatiable, risk-averse, prudent and temperate. With respect to the decision implications of mixed risk aversion, it is well known that lottery dominates in the sense of th-degree stochastic dominance (NSD) () if and only if all decision makers with mixed risk aversion up to degree prefer to [6]. Recently, the paper [2] considers risk lovers and defines that decision makers are mixed risk loving up to degree if their utility function satisfies (), corresponding to th-degree risk seeking stochastic dominance (NRSD). When , the decision makers with utility function are insatiable, risk loving, prudent and intemperate.
One important concern for NSD (and/or NRSD) is that there is a substantial jump from lower-order stochastic dominance (SD) to higher-order SD. For example, first-order SD (FSD) requires increasing utility while second-order SD (SSD) requires that utility be increasing and concave, and risk seeking second-order SD (RSSD) requires that utility be increasing and convex. Admitting convex segments in otherwise concave utility functions, Muller et al. [10] develop SD of order for , a continuum between FSD and SSD. Parallelly, by incorporating concave segments in otherwise convex functions, they propose risk seeking SD of order for , a continuum between FSD and RSSD.
There is experimental evidence that both risk averse and risk loving decision makers exist. The paper [11] finds that the majority of individuals’ decisions are consistent with risk aversion while there is still a significant proportion of risk lovers (about 15 percent). In the experiments, [11] also finds that some individuals exhibit imprudent and/or intemperate behavior although most individuals are prudent and temperate. The paper [3] finds that the majority of individuals are even intemperate. More importantly, [11] shows that risk aversion, prudence, and temperance are positively correlated but not completely correlated, inconsistent with the assumption of mixed risk aversion. They also find that most risk-seeking individuals are also imprudent and intemperate, contradicting the assumption of mixed risk lovingness. Overall, these observations challenge the thoughts that individuals are mixed risk averse or mixed risk loving and call for the reconsideration for the higher-order SD.
Motivated by these concerns, we establish a continuum of higher-order SD rules for risk averters and risk lovers, respectively. One is (risk averse) SD of order and the other is risk seeking SD of order for , a continuum from order to order where . These rules extend the results of [10] to the higher-order cases.
Our paper is organized as follows. In Section 2, we develop SD of order for both risk averse and risk loving decision makers, and illustrate these SD rules with some examples. Next we establish SD of order for in Section 3. In Section 4, we show an application of SD of order .
Section snippets
Stochastic dominance of order
In this section we will establish two types of -SD, where . One is a continuum between SSD and the third-order SD (TSD), termed as -risk averse SD since this type of SD rules requires decision makers being risk averse. The other is a continuum between RSSD and the risk seeking third-order SD (RTSD), termed as -risk seeking SD because this type of SD rules requires decision makers being risk loving.
We first introduce some notation. The finite random variables and with
Stochastic dominance of order
This section generalizes the previous results and establishes a continuum of higher-order SD rules. For , define where is the th derivative of utility function and .
Definition 3.1 For ,
-stochastically dominates , , if for all utility functions .
Let be the class of utility functions such that () and
Application: expectation dependence
Consider an investor with initial wealth and utility function . He allocates in a riskless asset and a single risky asset in one period. The riskless return is and random return of the risky asset is . Let denote the amount invested in the risky asset. Then the value at the end of the period is . Let , which represents the investment risk, and . Let denote an additive background risk. The investor’s decision problem is to find the optimal
Acknowledgments
The authors thank the editor, associate editor, and anonymous reviewer. H. Bi is supported by National Natural Science Foundation of China (No. 11801075). W. Zhu is supported by the Humanity and Social Science Foundation of the Ministry of Education in China (No. 16YJA790072), Beijing Municipal Social Science Foundation (No. 17YJC039), the Fundamental Research Funds for the Central Universities in UIBE (CXTD8-03) and the Program for Young Excellent Talents, UIBE (17YQ13).
References (12)
- et al.
Mixed risk aversion
J. Econom. Theory
(1996) - et al.
Increasing risk decreasing absolute risk aversion and diversification
J. Math. Econom.
(1995) The demand for a risky asset in the presence of a background risk
J. Econom. Theory
(2011)- et al.
Even (mixed) risk lovers are prudent
Amer. Econ. Rev.
(2013) - et al.
Exploring higher order risk effects
Rev. Econom. Stud.
(2010) - et al.
Smooth generators of integral stochastic orders
Ann. Appl. Probab.
(2002)
Cited by (4)
Nonmonotonic risk preferences over lottery comparison
2022, European Journal of Operational ResearchFractional-Degree Expectation Dependence
2023, Communications in Mathematics and StatisticsFraction-Degree Reference Dependent Stochastic Dominance
2022, Methodology and Computing in Applied Probability