The non-integer higher-order Stochastic dominance

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Abstract

We establish a continuum of higher-order stochastic dominance rules. Depending on whether decision makers are risk averse or risk loving, two types of continuum higher-order stochastic dominance rules are established and their properties are illustrated with some examples.

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 n if their utility function u satisfies (1)k1u(k)0 (k=1,,n), where u(k) represents the kth-order derivative of u. Fox example, when n=4, 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 F dominates G in the sense of nth-degree stochastic dominance (NSD) (n=1,2,) if and only if all decision makers with mixed risk aversion up to degree n prefer F to G [6]. Recently, the paper [2] considers risk lovers and defines that decision makers are mixed risk loving up to degree n if their utility function v satisfies v(k)0 (k=1,,n), corresponding to nth-degree risk seeking stochastic dominance (NRSD). When n=4, the decision makers with utility function v 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 1+γ for 0γ1, a continuum between FSD and SSD. Parallelly, by incorporating concave segments in otherwise convex functions, they propose risk seeking SD of order 1+γ for 0γ1, 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 n+γ and the other is risk seeking SD of order n+γ for 0γ1, a continuum from order n to order n+1 where n2. 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 2+γ for both risk averse and risk loving decision makers, and illustrate these SD rules with some examples. Next we establish SD of order n+γ for n3 in Section 3. In Section 4, we show an application of SD of order n+γ.

Section snippets

Stochastic dominance of order 2+γ

In this section we will establish two types of (2+γ)-SD, where 0γ1. One is a continuum between SSD and the third-order SD (TSD), termed as (2+γ)-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 (2+γ)-risk seeking SD because this type of SD rules requires decision makers being risk loving.

We first introduce some notation. The finite random variables X and Y with

Stochastic dominance of order n+γ

This section generalizes the previous results and establishes a continuum of higher-order SD rules. For 0γ1, define Un,γ={u|(1)k+1u(k)0fork=1,,n,γ(1)n+1u(n)(y)(1)n+1u(n)(x)for anyxy}, where u(k) is the kth derivative of utility function u and n3.

Definition 3.1

For 0γ1, Y (n+γ)-stochastically dominates X, X(n+γ)SDY, if E[u(X)]E[u(Y)] for all utility functions uUn,γ.

Let Un,γ be the class of utility functions u such that (1)k+1u(k)0 (k=1,,n) and γ(1)n1u(n1)(x4)u(n1)(x3)x4x3(1)n1u(n1

Application: expectation dependence

Consider an investor with initial wealth w and utility function u(x). He allocates w in a riskless asset and a single risky asset in one period. The riskless return is rf and random return of the risky asset is r̃. Let α denote the amount invested in the risky asset. Then the value at the end of the period is w(1+rf)+α(r̃rf). Let X=r̃rf, which represents the investment risk, and w0=w(1+rf). Let Y 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).

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