#  Multi-fractional Stochastic Dominance: Mathematical Foundations

Ehsan Azmoodeh and Ozan Hür University of Liverpool, Department of
Mathematical Sciences, L69 7ZL Liverpool, United Kingdom E-mail:
<EMAIL_ADDRESS>of Liverpool, Department of
Mathematical Sciences, L69 7ZL Liverpool, United Kingdom E-mail:
<EMAIL_ADDRESS>
###### Abstract

In the landmark article [MSTW17], Müller et. al. introduced the notion of
fractional stochastic dominance (SD) to interpolate between first and second
SD relations. In this article, we introduce a novel family of multi-fractional
stochastic orders that generalizes fractional SD in a natural manner. The
family of multi-fractional SD is parametrized by an arbitrary non-decreasing
function $\boldsymbol{\upgamma}$ ranging between $0$ and $1$ which provides
the feature of local interpolation rather than a global one. We show that the
multi-fractional $(1+\boldsymbol{\upgamma})$-SD is generated by a class of
increasing utility functions allowing local non-concavity where the steepness
of the non-concavity depends on its location and it is controlled by function
$\boldsymbol{\upgamma}$. We also introduced the notion of local greediness
that allows us, among other things, to systematically study multi-fractional
utility class. The multi-fractional utility class is well-suited for
representing a decision maker’s preferences in terms of risk aversion and
greediness at a local level. Several basic properties as well as illustrating
examples are presented.

Keywords: (Fractional) Stochastic dominance; Utility function; Local
greediness
MSC2020 subject classifications: Primary 60E15; Secondary 90B50; 91B06; 91B16

###### Contents

  1. 1 Introduction
    1. 1.1 Overview and motivation
    2. 1.2 Plan
  2. 2 Multi-fractional Stochastic Dominance (MFSD)
    1. 2.1 Definition and basic examples
    2. 2.2 Further illustrative examples
    3. 2.3 The universe of multi-fractional stochastic dominance
  3. 3 MFSD: Utility Theory
    1. 3.1 Towards a utility class
    2. 3.2 Utility space $\mathscr{U}^{\textsc{mf}}_{\upgamma}$ and the main result
    3. 3.3 Examples
  4. 4 MFSD: Economical Aspects
    1. 4.1 Utility space $\mathscr{U}^{\textsc{cv}}_{\upgamma}$
    2. 4.2 Local Greediness
    3. 4.3 An illustrating (counter) example
    4. 4.4 $\mathscr{U}^{\textsc{mf}}_{\upgamma}$ and the local greediness
    5. 4.5 Average risk aversion
  5. 5 MFSD: Further Mathematical Aspects
  6. 6 Applications
    1. 6.1 Local risk aversion curve
    2. 6.2 Maximal feature of local risk aversion
    3. 6.3 Location-scale families
    4. 6.4 Sure gain and the Omega ratio
  7. 7 Appendix

##  1 Introduction

###  1.1 Overview and motivation

Stochastic dominance (SD) is a popular tools for decision making under
uncertainty. It allows us to rank different prospects (random variables) by
examining their cumulative distribution functions (CDF) through specific (most
often) integral conditions. The equivalence of ranking with the expected
utility theory can be established through the use of dual utility conditions.
This involves comparing the expected values of a class of suitable (utility)
functions which reflect the preferences of decision-makers who share common
attitudes towards risk. We refer the reader to the excellent textbooks [MS02,
SS07, Lev16] for a comprehensive treatment of the general theory as well as
applications.

Undebatable, first and second stochastic dominance relations serve as the two
most well-known stochastic dominance rules. For every two prospects $X\sim F$
and $Y\sim G$ supported on the real line, we say that $G$ dominates $F$ in the
first stochastic order (FSD) denoted by $F\preceq_{\textsc{FSD}}G$ provided
that $F(x)\geq G(x)$ for all $x\in\mathbb{R}$. In the language of the utility
theory, such a preference can be expressed through the utility condition:

$\mathbb{E}[u(X)]\leq\mathbb{E}[u(Y)],\quad\forall\,u\in\mathscr{U}_{0}$

where the utility class $\mathscr{U}_{0}$ consists of increasing continuous
functions, hence reflecting the preferences of non-satiable individuals i.e.,
who prefer "more" to "less". Meanwhile, second order stochastic dominance
(SSD) pertains to decision makers who are also risk averse and represents
their preferences by the utility condition on the set of increasing and
concave utility functions denoted by $\mathscr{U}_{1}$ namely that we say $G$
dominates $F$ in the SSD sense denoted by $F\preceq_{\textsc{SSD}}G$ iff
$\mathbb{E}[u(X)]\leq\mathbb{E}[u(Y)]$ for all $u\in\mathscr{U}_{1}$. It is
well known that SSD can be equivalently stated through the integral condition
111The differentiability of utility functions is not crucial for developing
the theory of SD. We refer to [HL69] for the equivalence between utility and
integral conditions of FSD and SSD.:

$\int_{-\infty}^{t}\left(F(x)-G(x)\right)_{-}dx\leq\int_{-\infty}^{t}\left(F(x)-G(x)\right)_{+}dx,\quad\forall
t\in\mathbb{R}.$

The jump between FSD and SSD is substantial. The strict assumption of global
risk aversion in SSD can exclude decision makers with mixed risk attitudes
i.e. who are mostly risk averse but also exhibit some degree of risk seeking
behavior. The preferences of such decision makers can be captured by utility
functions that are increasing functions with both concave and non-concave
segments, such as those in [FS48, Mar52]. On the other hand, the lack of
assumptions on decision makers’ risk attitudes in FSD results in a strong
integral condition that has low discriminatory power. As a result, even minor
violations in the integral condition can prevent given cumulative distribution
functions (CDF) from being ordered in FSD sense. To overcome these
limitations, numerous stochastic order relations have been proposed in the
literature that fall between FSD and SSD. In [Fis76, Fis80], Fishburn defined
a class of SD relations using fractional integration while [LL02] introduced
almost stochastic dominance, which enables the ordering of distribution
functions with a slight deviation from FSD. In the seminal work [MSTW17],
Müller et. al. introduced the concept of fractional $(1+\gamma)$-SD which
employ a fixed parameter $\gamma\in[0,1]$ to interpolate between FSD and SSD.
The key advantage of the fractional SD rule is that the corresponding utility
class (generator) is known and it provides attractive characteristics from a
decision theoretic perspective. Hence, it is gathered a considerable attention
in recent literature [HTZ20, MW22, LB20].

In this study, we introduce a new family of multi-fractional stochastic
dominances that extends the notion of fractional stochastic dominance. Several
well known stochastic orders in the literature are defined through the use of
utility (generator) counterpart. We refer the reader to [Mül97] for a unified
approach towards integral stochastic orders. Despite this, due to the nature
of our generalization, we formulate multi-fractional stochastic dominance by
its integral condition rather than its generator. Hence, for the sake of
consistency, we first recall the definition of fractional SD expressed in
terms of integral condition.

######  Definition 1.1 (Fractional $(1+\gamma)$-SD).

[MSTW17] For fixed $\gamma\in[0,1]$, we say $G$ dominates $F$ in the sense of
fractional $(1+\gamma)$-SD, denoted by $F\preceq_{(1+\gamma)\textsc{-SD}}G$,
if

$\int_{-\infty}^{t}\left(F(x)-G(x)\right)_{-}dx\leq\gamma\int_{-\infty}^{t}\left(F(x)-G(x)\right)_{+}dx,\quad\forall
t\in\mathbb{R}.$ (1.1)

It is clear that for $\gamma=0$, fractional $(1+\gamma)$-SD coincides with FSD
whereas for $\gamma=1$ it boils down to SSD. Therefore, fractional SD
establishes an interpolation between integer degree SD rules. This
interpolation is global in a sense that, for a fixed $\gamma\in[0,1]$, given
distribution functions are ordered by $(1+\gamma)$-SD over their entire
supports. Our motivation towards generalization of fractional SD is mainly
inspired by local interpolation idea where given distribution functions are
simultaneously can be ordered from first to second SDs on different portions
of their supports. In order to achieve such phenomenon, we relax the condition
of parameter $\gamma$ being constant by replacing it with an arbitrary non-
decreasing function $\boldsymbol{\upgamma}:\mathbb{R}\to[0,1]$ in the integral
condition (1.1). As we discuss in the sequel, this enables the corresponding
integral condition of multi-fractional SD to make use of additional
information about the local relationship (captured by $\boldsymbol{\upgamma}$)
between given distribution functions, allowing for a more accurate and further
interpolation between FSD and SSD. The fractional $(1+\gamma)$-SD can be
generated by the utility class $\mathscr{U}_{\gamma}$ consisting of smooth
utility functions $u$ so that $0\leq\gamma u^{\prime}(y)\leq u^{\prime}(x)$
for every $x\leq y$. From the mathematical point of view, parameter $\gamma$
bounds how much marginal utility can decrease as $x$ decreases. Setting
$\gamma$ to 0 results in $\mathscr{U}_{0}$ and setting $\gamma=1$ leads to
$\mathscr{U}_{1}$ given in above. From a decision theoretic perspective,
parameter $\gamma$ globally controls the degree of risk loving behaviour,
where the higher values of $\gamma$ corresponds to lower degrees of risk
lovingness. When $\gamma$ is between 0 and 1, the utility class
$\mathscr{U}_{\gamma}$ contains individuals who are generally risk averse but
are willing to accept some risk depending on the value of $\gamma$. Unlike
fractional $(1+\gamma)$-SD, function $\boldsymbol{\upgamma}$ plays a crucial
role in controlling the local risk loving behaviour of decision makers in
terms of average risk aversion and greediness at a local level. The non-
decreasing property of $\boldsymbol{\upgamma}$ allows that multi-fractional SD
represent the preferences of decision makers who are mostly risk averse but
exhibit a higher degree of risk lovingness when they possess lower levels of
wealth.

###  1.2 Plan

The paper is organised as follows. Section 2 contains definition of multi-
fractional stochastic dominance and several illustrating examples towards
understanding the multi-fractional universe. Section 3 focusses on utility
theory of multi-fractional stochastic dominance, whereas Section 4 deals with
a few economical aspects of the order. Section 5 gathers basic properties of
multi-fractional stochastic order. Finally, Section 6 contains a few
applications.

##  2 Multi-fractional Stochastic Dominance (MFSD)

In this section, we introduce the natural generalization of fractional SD as
given in Definition 1.1 to its multi-fractional form. Following the definition
of MFSD, we present a series of examples to motivate our generalization.

###  2.1 Definition and basic examples

######  Definition 2.1 (Multi-fractional $(1+\boldsymbol{\upgamma})$-SD).

Let $\boldsymbol{\upgamma}:\mathbb{R}\to[0,1]$ be an arbitrary non-decreasing
function. Let $F$ and $G$ be two arbitrary distribution functions. We say that
$G$ dominates $F$ in the sense of the multi-fractional
$(1+\boldsymbol{\upgamma})$-SD, denote by
$F\preceq^{\textsc{mf}}_{(1+\boldsymbol{\upgamma})\textsc{-SD}}G$ if

$\int_{-\infty}^{t}\left(F(x)-G(x)\right)_{-}dx\leq\boldsymbol{\upgamma}(t)\int_{-\infty}^{t}\left(F(x)-G(x)\right)_{+}dx,\quad\forall
t\in\mathbb{R}.$ (2.1)

######  Remark 2.1.

  * (i)

Apart from fractional SD, another source of inspiration behind generalization
(2.1) is based on the notion of multi-fractional Brownian motion allowing the
Hurst parameter $H\in(0,1)$ varies with time unlike the classical fractional
Brownian motion, see [PV95, ST06] for details.

  * (ii)

Clearly, the multi-fractional $(1+\boldsymbol{\upgamma})$-SD coincides with
the fractional $(1+\gamma)$-SD as soon as
$\boldsymbol{\upgamma}(x)=\gamma\in[0,1]$ for every $x$. Furthermore, when for
some $\gamma\in(0,1)$, function $\boldsymbol{\upgamma}(t)\leq\gamma$ for every
$t\in\mathbb{R}$, then the multi-fractional $(1+\boldsymbol{\upgamma})$-SD
implies the fractional $(1+\gamma)$-SD and, similarly, if function
$\boldsymbol{\upgamma}(t)\geq\gamma$ for every $t\in\mathbb{R}$, then the
fractional $(1+\gamma)$-SD implies the multi-fractional
$(1+\boldsymbol{\upgamma})$-SD.

  * (iii)

One of our primary purposes is to ponder on the remarkable subject of
interpolating between the first and second order stochastic dominances. In
this regard, our extension has a significant different feature compare to the
major works [MSTW17, HTZ20]. Recall that, we require function
$\boldsymbol{\upgamma}$ being non-decreasing. In fact, the notion of the
multi-fractional $(1+\boldsymbol{\upgamma})$-SD provides us with a local
interpolation phenomenon rather than global. This is well explained via
Example 2.2 below.

  * (iv)

Let $F$ and $G$ be two probability distribution functions. Then, it is known
that the assumption of equal means ($\mu_{F}=\mu_{G}$) leads to the given
distribution functions being indistinguishable in the sense of the fractional
$(1+\gamma)$-SD for any constant $\gamma\in[0,1)$. Nonetheless, as
demonstrated in Example 2.3, the multi-fractional
$(1+\boldsymbol{\upgamma})$-SD can effectively order distributions with equal
means, provided that an appropriate function i.e. non-constant and satisfying
$\boldsymbol{\upgamma}(t)=1$ for some $t\in\mathbb{R}$, is chosen. In
practical scenarios, such as estimating distribution parameters using the
method of moments, the need to compare distributions with identical means may
arise [DDGK05]. In these cases, the multi-fractional SD can prove valuable.

  * (v)

By our assumptions, function $\boldsymbol{\upgamma}$ is non-decreasing and
bounded, hence it has at most countably many discontinuity of the first kind
and clearly, $\boldsymbol{\upgamma}(t)\leq\boldsymbol{\upgamma}(t^{+})$ for
each $t$. Therefore, due to continuity of the integral operator, without of
loss of generality, one can always assume that function
$\boldsymbol{\upgamma}$ is always right continuous. This desirable feature
allows us to add one extra layer of randomness to multi-fractional stochastic
dominance framework by considering $\boldsymbol{\upgamma}$ as a distribution
function (see Example 3.3). We expect that such a desirable viewpoint provides
us extra strengths that can be useful in certain applications.

  * (vi)

For the given non-decreasing function $\boldsymbol{\upgamma}$, let
$t_{0}:=\sup\\{t\,:\,\boldsymbol{\upgamma}(t)=0\\}$ stands for the last exist
time of the set $\\{0\\}$ and $\boldsymbol{\upgamma}(t_{0})>0$. This
immediately implies that on the segment $(-\infty,t_{0})$, the multi-
fractional $(1+\boldsymbol{\upgamma})$-SD reduces to that of the FSD and
therefore, the value of $\boldsymbol{\upgamma}$ becomes irrelevant in the
sense that if one glues any non-decreasing function $\boldsymbol{\upbeta}$ so
that $\boldsymbol{\upbeta}>0$ on the segment $(-\infty,t_{0})$ to
$\boldsymbol{\upgamma}$ resulting in the new non-decreasing function
$\widetilde{\boldsymbol{\upgamma}}$ then,
$F\preceq_{(1+\boldsymbol{\upgamma})\textsc{{-SD}}}G$ implies the validity of
$F\preceq_{(1+\widetilde{\boldsymbol{\upgamma}})\textsc{{-SD}}}$.

  * (vii)

In a similar way, an extension can be accomplished for the combined order
concave-convex stochastic dominance, see [MSTW17, Section 4] for detail.
However, for the ease of development, we defer it to a separate study.

######  Example 2.1 (The role of non-decreasing assumption).

The assumption that the function $\boldsymbol{\upgamma}:\mathbb{R}\to[0,1]$ is
non-decreasing is crucial for Definition 2.1 to be meaningful. In fact, let
$\boldsymbol{\upgamma}$ be a non-constant, non-increasing function. Then, any
value other than the minimum value of $\boldsymbol{\upgamma}$ will not impact
the ordering rule of the given distribution functions. This is because smaller
values of $\boldsymbol{\upgamma}$ lead to stronger dominance rules, and
therefore, resulting in, the smallest value of function
$\boldsymbol{\upgamma}$ determines the ordering relation throughout the entire
support. Let us to demonstrate this with a simple example. Let
$t_{0}\in\mathbb{R}$. Consider the (non-constant, non-increasing) function
$\boldsymbol{\upgamma}:\mathbb{R}\to[0,1]$,

$\boldsymbol{\upgamma}(t)=\begin{cases}1&\text{ if }t<t_{0}\\\ 0&\text{ if
}t_{0}\leq t.\end{cases}$

Now, assume that
$F\preceq^{\textsc{mf}}_{(1+\boldsymbol{\upgamma})\textsc{{-SD}}}G$. Then, by
Definition 2.1, for all $t\in(-\infty,t_{0})$ (note that
$\boldsymbol{\upgamma}(t)=1$ over the interval), we have

$\int_{-\infty}^{t}\left(F(x)-G(x)\right)_{-}dx\leq\int_{-\infty}^{t}\left(F(x)-G(x)\right)_{+}dx,$
(2.2)

and on the other hand, for all $t\geq t_{0}$, inequality (2.1) implies that
$G(t)\leq F(t)$ for all $t\in\mathbb{R}$. Clearly, the latter implies that
$(F(t)-G(t))_{-}=0$ for all $t\in\mathbb{R}$, thus eliminating the impact of
value of $\boldsymbol{\upgamma}(t)=1$ on the integral condition (2.2) and
hence on the ordering of $F$ and $G$ as well. The importance of non-decreasing
assumption can be observed in more general situations in a similar way.

######  Example 2.2 (Local interpolation).

The purpose of this example is to illustrate one of the main features of MFSD,
namely that, the integral condition 2.1 equips us with local interpolation
between the first and second order stochastic dominance rules, rather than the
global. Let $\gamma\in(0,1)$ be a constant. Consider the following non-
decreasing function

$\boldsymbol{\upgamma}(t)=\begin{cases}0&\text{ if }t\leq t_{1}\\\
\gamma&\text{ if }t_{1}<t\leq t_{2}\\\ 1&\text{ if }t>t_{2}\end{cases}$

where $t_{1}<t_{2}<t_{3}$ are real numbers. Assume that
$F\preceq^{\textsc{mf}}_{(1+\boldsymbol{\upgamma})\textsc{{-SD}}}G$. Then,
$\boldsymbol{\upgamma}\left(t_{1}\right)=0$ entails that $G(x)\leq F(x)$ on
the portion $\left(-\infty,t_{1}\right)$ of the full supports. In other words,
we have $F\preceq_{\textsc{FSD}}G$ on the interval
$\left(-\infty,t_{1}\right)$. Similarly, the condition
$\boldsymbol{\upgamma}\left(t_{2}\right)=\gamma$ yields that

$\int_{-\infty}^{t}\left(F(x)-G(x)\right)_{-}dx\leq\gamma\int_{-\infty}^{t}\left(F(x)-G(x)\right)_{+}dx,\quad\forall
t\leq t_{2},$

i.e., $F\preceq_{(1+\gamma)\textsc{{-SD}}}G$ on the interval
$\left(-\infty,t_{2}\right]$ (this directly follows from the non-decreasing
property of function $\boldsymbol{\upgamma}$). Lastly,
$\boldsymbol{\upgamma}\left(t\right)=1$ for all $t\in(t_{2},\infty)$ implies
that

$\int_{-\infty}^{t}\left(F(x)-G(x)\right)_{-}dx\leq\int_{-\infty}^{t}\left(F(x)-G(x)\right)_{+}dx,\quad\forall
t\in\mathbb{R},$

i.e., $G$ dominates $F$ in second order stochastic dominance sense,
$F\preceq_{\textsc{SSD}}G$. Hence, with appropriately chosen non-decreasing
function $\boldsymbol{\upgamma}$, one can simultaneously order given $F$ and
$G$ from first (strongest) to second (weakest) order on different portions
(capture with function $\boldsymbol{\upgamma}$) of the supports. Later on, we
discuss the reflections and impacts of the local interpolation phenomenon in
terms of utility classes (see Remarks 3.5) and 4.6).

######  Example 2.3 (The equal-mean scenario).

The purpose of this example is to construct two distribution functions $F$ and
$G$ with the identical means $(\mu_{G}=\mu_{F})$ in which it is impossible to
distinguish them according to the fractional $(1+\gamma)$-SD for any given
constant $\gamma\in[0,1)$, however distinguishable based on a (more
informative) function $\boldsymbol{\upgamma}$. We exclude the limiting case
$\gamma=1$ due to the fact that it corresponds to (less interesting) SSD. We
introduce

$\displaystyle\boldsymbol{\upgamma}(t)=\begin{cases}0&\mbox{ if }t<0,\\\
t&\mbox{ if }0\leq t\leq 1,\\\ 1&\mbox{ if }t>1.\end{cases}$ (2.3)

We also consider the following distribution functions with identical means,
$\mu_{F}=\mu_{G}=23/32$:

$F(x)=\begin{cases}0&\mbox{ if }x<0,\\\ x&\mbox{ if }0\leq x<1/4\\\ 1/4&\mbox{
if }1/4\leq x<3/4\\\ 1/2&\mbox{ if }3/4\leq x<1\\\ 1&\mbox{ if }x\geq
1\end{cases}\quad\text{ and }\quad G(x)=\begin{cases}0&\mbox{ if }x<1/4\\\
x-1/4&\mbox{ if }1/4\leq x<1\\\ 1&\mbox{ if }x\geq 1.\end{cases}$

$0$$0.2$$0.4$$0.6$$0.8$$1$$0$$0.2$$0.4$$0.6$$0.8$$1$ Figure 1: The plots of
$F$ (blue) and $G$ (red)

The straightforward calculation shows that the integral condition (2.1) holds,
namely that

$\int_{0}^{t}\left(F(x)-G(x)\right)_{-}dx\leq\boldsymbol{\upgamma}(t)\int_{0}^{t}\left(F(x)-G(x)\right)_{+}dx,\quad\forall\,t\in\mathbb{R}.$

Hence, $F\preceq^{\textsc{mf}}_{(1+\upgamma)\textsc{{-SD}}}G$ . However,
$F\npreceq_{(1+\gamma)\textsc{-SD}}G$ for any constant $\gamma\in[0,1)$. The
reason for this is that the inequality (1.1) fails at $t=1$, since

$\displaystyle 0=\mu_{G}-\mu_{F}$
$\displaystyle=\int_{0}^{1}\left(F(x)-G(x)\right)dx$
$\displaystyle=\int_{0}^{1}\left(F(x)-G(x)\right)_{+}dx-\int_{0}^{1}\left(F(x)-G(x)\right)_{-}dx$

however observe that $\boldsymbol{\upgamma}(1)=1$. Lastly, to complete the
example we require to demonstrate that with function $\boldsymbol{\upgamma}$
given in 4.2, the multi-fractional $(1+\boldsymbol{\upgamma})$-SD does not
coincide to SSD. To accomplish that, we introduce the following distribution
functions:

$F(x)=\begin{cases}0&\mbox{ if }x<0\\\ 1/4&\mbox{ if }0\leq x<1/2\\\ 1&\mbox{
if }x\geq 1/2\end{cases}\quad\text{ and }\quad G(x)=\begin{cases}0&\mbox{ if
}x<1/4\\\ 1/2&\mbox{ if }1/4\leq x<1/2\\\ 1&\mbox{ if }x\geq 1/2.\end{cases}$

It is clear that $F\preceq_{\textsc{SSD}}G$, however, note that (at $t=1/2$
with $\boldsymbol{\upgamma}(1/2)=1/2$):

$\displaystyle\int_{0}^{1/2}\left(F(x)-G(x)\right)_{-}dx\nleq
1/2\,\int_{0}^{1/2}\left(F(x)-G(x)\right)_{+}dx.$

Hence, integral condition (2.1) fails and therefore,
$F\npreceq^{\textsc{mf}}_{(1+\boldsymbol{\upgamma})\textsc{{-SD}}}G$ although
$F\preceq_{\textsc{SSD}}G$. In general, for any given non-decreasing
$\boldsymbol{\upgamma}\not\equiv 1$, one can always construct two distribution
functions $F$ and $G$ such that $F\preceq_{\textsc{SSD}}G$ but
$F\not\preceq^{\textsc{mf}}_{(1+\boldsymbol{\upgamma})\textsc{{-SD}}}G$.

###  2.2 Further illustrative examples

In this section, with the help of a couple of illustrating examples, we aim to
justify not only the usefulness of multi-fractional stochastic dominance rule
given in Definition 2.1 but also we discuss the non-trivial relationship
between two notions of fractional and multi-fractional stochastic dominance
rules.

######  Example 2.4.

This example generalizes Example 2.3 beyond the identical mean scenario. More
precisely, we show the following phenomenon that for every arbitrary given
constant $\gamma\in(0,1)$ there exist two distribution functions $F$ and $G$
with the non-identical means $(\mu_{G}>\mu_{F})$ 222Note that without loss of
generality, one can also assume $(\mu_{G}\geq\mu_{F})$. Our motivation for
excluding the case $\mu_{G}=\mu_{F}$ is to demonstrate that the assumption of
identical means is sufficient, but not necessary, for given distributions
$F\preceq^{\textsc{mf}}_{(1+\boldsymbol{\upgamma})\textsc{{-SD}}}G$ but
$F\not\preceq_{(1+\gamma)\textsc{{-SD}}}G$. such that we cannot distinguish
them based on the fractional $(1+\gamma)$-SD however, there is (at least) one
suitable function $\boldsymbol{\upgamma}:\mathbb{R}\rightarrow[0,1]$ which
makes $F$ and $G$ distinguishable based on the multi-fractional
$(1+\boldsymbol{\upgamma})$-SD. Let $\gamma\in(0,1)$ be fixed. Choose
$N\in\mathbb{N}$ so that $2\gamma/N<1$. We introduce the following
distribution functions:

$F(x)=\begin{cases}0&\mbox{ if }x<0\\\ \gamma/N&\mbox{ if }0\leq
x<(\gamma+\gamma^{3/2})/N\\\ 1&\mbox{ if
}x\geq(\gamma+\gamma^{3/2})/N\end{cases}\quad\mbox{ and }\quad
G(x)=\begin{cases}0&\mbox{ if }x<\gamma/N\\\ 2\gamma/N&\mbox{ if }\gamma/N\leq
x<(\gamma+\gamma^{3/2})/N\\\ 1&\mbox{ if
}x\geq(\gamma+\gamma^{3/2})/N.\end{cases}$

First, note that we have
$\gamma/N+(1-2\gamma/N)(\gamma^{3/2}/N)=\mu_{G}>\mu_{F}=(1-\gamma/N)((\gamma+\gamma^{3/2})/N)$.
In addition, at $x=(\gamma+\gamma^{3/2})/N$, we obtain

$\displaystyle\gamma^{5/2}/N^{2}=\int_{-\infty}^{(\gamma+\gamma^{3/2})/N}\left(F(x)-G(x)\right)_{-}dx\leq\int_{-\infty}^{(\gamma+\gamma^{3/2})/N}\left(F(x)-G(x)\right)_{+}dx=\gamma^{2}/N^{2}$

Hence, the integral requirement (1.1) fails at point
$x=(\gamma+\gamma^{3/2})/N$ because $\gamma<\sqrt{\gamma}<1$ for all
$\gamma\in(0,1)$. Hence, $F\not\preceq_{(1+\gamma)\textsc{{-SD}}}G$. One has
to note that $F\preceq_{\textsc{SSD}}G$ since
$\gamma^{5/2}/N^{2}<\gamma^{2}/N^{2}$. Next, consider the following function:

$\displaystyle\boldsymbol{\upgamma}(t)=\begin{cases}\boldsymbol{\upgamma}_{1}(t)&\mbox{
if }t<\gamma/N\\\ \boldsymbol{\upbeta}(t)&\mbox{ if }\gamma/N\leq
t\leq(\gamma+\gamma^{3/2})/N\\\ \boldsymbol{\upgamma}_{2}(t)&\mbox{ if
}t>(\gamma+\gamma^{3/2})/N\end{cases}$

where $\boldsymbol{\upgamma}_{1}$ and $\boldsymbol{\upgamma}_{2}$ can be any
non-decreasing functions that do not violate the non-decreasing assumption of
$\boldsymbol{\upgamma}$ and $\boldsymbol{\upbeta}(t)\in(0,1)$ can be taken any
non-decreasing function satisfying:

$\boldsymbol{\upbeta}(t)\geq\frac{\int_{-\infty}^{t}\left(F(x)-G(x)\right)_{-}dx}{\gamma^{2}/N^{2}}=\frac{\gamma/N(t-\gamma/N)}{\gamma^{2}/N^{2}},\quad\forall
t\in[\gamma/N,(\gamma+\gamma^{3/2})/N].$

Clearly, one obvious choice of such a $\boldsymbol{\upgamma}$ function is:

$\displaystyle\boldsymbol{\upgamma}(t)=\begin{cases}0&\mbox{ if }t<\gamma/N\\\
\frac{\gamma/N(t-\gamma/N)}{\gamma^{2}/N^{2}}&\mbox{ if }\gamma/N\leq
t\leq(\gamma+\gamma^{3/2})/N\\\ 1&\mbox{ if
}t>(\gamma+\gamma^{3/2})/N.\end{cases}$

Then, it is straightforward to check that
$F\preceq^{\textsc{mf}}_{(1+\boldsymbol{\upgamma})\textsc{{-SD}}}G$ holds.

######  Example 2.5.

This example is rather technical. The purpose of the example is to show that
for any given constant $\gamma\in(0,1)$ and every (non-decreasing) function
$\boldsymbol{\upgamma}$ satisfying
$\boldsymbol{\upgamma}(t^{0})<\gamma<\boldsymbol{\upgamma}(t_{0})$ for some
$t^{0}<t_{0}$ (see Remark 2.1, part (ii)), there exists distribution functions
$F$ and $G$ so that either one the followings holds:

  * (i)

$F\preceq^{\textsc{mf}}_{(1+\boldsymbol{\upgamma})\textsc{{-SD}}}G$ but
$F\npreceq_{(1+\gamma)\textsc{{-SD}}}G$

  * (ii)

$F\npreceq^{\textsc{mf}}_{(1+\boldsymbol{\upgamma})\textsc{{-SD}}}G$ but
$F\preceq_{(1+\gamma)\textsc{{-SD}}}G$.

First, $F\preceq^{\textsc{mf}}_{(1+\boldsymbol{\upgamma})\textsc{{-SD}}}G$ but
$F\npreceq_{(1+\gamma)\textsc{{-SD}}}G$. We discuss separately the following
two possible cases:

Case 1: There is a left-side neighbourhood
$N_{\delta}(t_{0})=(t_{0}-\delta,t_{0})$, $\delta>0$ such that function
$\boldsymbol{\upgamma}(t)=\boldsymbol{\upgamma}(t_{0})$ for all $t\in
N_{\delta}(t_{0})$. Choose $\ell<\delta$ small enough (in the sense that
$\boldsymbol{\upgamma}(t_{0})-\ell>\gamma$). Now, consider the square with the
length $\ell$ and with the right-up corner at
$(t_{0},\boldsymbol{\upgamma}(t_{0}))$. Let’s denote the square with
$S(\ell)$. Next, there are two natural numbers $M,N\in\mathbb{N}$ so that
$\gamma<M/N<\boldsymbol{\upgamma}(t_{0})$ (rational numbers are dense in
$\mathbb{R}$). Without loosing of generality, we can assume that
$(M+N)/MN<\ell^{-1}$ (otherwise we work with $KM$, and $KN$ with large $K$
integer. This keeps the ratio unchanged however note that
$KN+KM/(KM)(KN)=(1/K)\times(M+N)/(MN)\to 0$ as $K\to\infty$). Now, we equally
divide $S(\ell)$ to $(MN)^{2}$ little squares each one has the length
$\ell/(MN)$. Let’s denote the $t$-coordinate of the left-bottom corner of
$S(\ell)$ by $t^{*}$ such that $t^{*}+2\ell/MN\in N_{\delta}(t_{0})$ and
define

$\displaystyle F(t)=$ $\displaystyle\begin{cases}0&\quad\quad\quad\mbox{ if
}t<t^{*}\\\ N(\ell/MN)&\quad\quad\quad\mbox{ if }t^{*}\leq t<t^{*}+2\ell/MN\\\
1&\quad\quad\quad\mbox{ if }t\geq t^{*}+2\ell/MN\end{cases}$ $\displaystyle
G(t)=$ $\displaystyle\begin{cases}0&\mbox{ if }t<t^{*}+\ell/MN\\\
(M+N)\ell/(MN)&\mbox{ if }t^{*}+\ell/MN\leq t<t^{*}+2\ell/MN\\\ 1&\mbox{ if
}t\geq t^{*}+2\ell/MN.\end{cases}$

Then, we have

$\displaystyle
A_{+}:=\int_{t^{*}}^{t^{*}+\ell/MN}\left(F(x)-G(x)\right)_{+}dx$
$\displaystyle=N(\ell/MN)(\ell/MN)$ $\displaystyle
A_{-}:=\int_{t^{*}+\ell/MN}^{t^{*}+2\ell/MN}\left(F(x)-G(x)\right)_{-}dx$
$\displaystyle=M(\ell/MN)(\ell/MN)$

Therefore, we have
$F\preceq^{\textsc{mf}}_{(1+\boldsymbol{\upgamma})\textsc{-SD}}G$ but
$F\not\preceq_{(1+\gamma)\textsc{{-SD}}}G$ since

$\gamma<\frac{A_{-}}{A_{+}}=\frac{M}{N}<\boldsymbol{\upgamma}(t^{*}+\ell/MN)=\boldsymbol{\upgamma}(t^{*}+2\ell/MN)=\boldsymbol{\upgamma}(t_{0}).$

$\gamma$$t_{0}$$\boldsymbol{\upgamma}(t_{0})$$N_{\delta}(t_{0})=(t_{0}-\delta,t_{0})$$\ell$$\ell$$S(\ell)$$S(\ell)$$(t_{0},\boldsymbol{\upgamma}(t_{0}))$$t^{*}$$N\frac{\ell}{MN}$$\frac{\ell}{MN}$$\frac{\ell}{MN}$$(M+N)\frac{\ell}{MN}$$A_{+}$$A_{-}$$\boldsymbol{\upgamma}(t)$$t$$G(t)$$F(t)$$t^{*}$

Case 2: Function $\boldsymbol{\upgamma}$ is not flat at any left-neighbourhood
of $t_{0}$. If $\boldsymbol{\upgamma}$ is left-continuous at $t_{0}$, then the
construction boils down to that of the previous case with two natural numbers
$M,N\in\mathbb{N}$ where $\gamma<M/N<\inf_{t\in
N_{\delta}(t_{0})}\boldsymbol{\upgamma}(t)$ so that we have

$\gamma<\frac{A_{-}}{A_{+}}=\frac{M}{N}<\inf_{t\in
N_{\delta}(t_{0})}\boldsymbol{\upgamma}(t)\leq\boldsymbol{\upgamma}(t^{*}+\ell/MN)\leq\boldsymbol{\upgamma}(t^{*}+2\ell/MN)\leq\boldsymbol{\upgamma}(t_{0}).$

$\gamma$$t_{0}$$\boldsymbol{\upgamma}(t_{0})$$N_{\delta}(t_{0})=(t_{0}-\delta,t_{0})$$\ell$$\ell$$S(\ell)$$\inf_{\\{t\in
N_{\delta}(t_{0})\\}}\boldsymbol{\upgamma}(t)$$\gamma$$\boldsymbol{\upgamma}(t_{0})$$\ell$$\ell$$S(\ell)$$t_{0}$$N_{\delta}(t_{0})=(t_{0},t_{0}+\delta)$$\boldsymbol{\upgamma}(t)$$t$$\boldsymbol{\upgamma}(t)$$t$$t^{*}$

Otherwise, namely that $\boldsymbol{\upgamma}(t)<\gamma$ on a left
neighbourhood $N_{\delta}(t_{0})$. In this scenario, since function
$\boldsymbol{\upgamma}$ is non-decreasing, we can assume that on a tiny right
neighbourhood of $t_{0}$ function $\boldsymbol{\upgamma}$ is flat. Now, we can
similarly construct two distribution functions $F$ and $G$ in the very tiny
square $S(\ell)$ on the right hand side of $t_{0}$ (the left-bottom corner is
$t_{0}$ instead of $t^{*}$). Note that this time constructed $F$ and $G$ take
$0$ on $(-\infty,t_{0})$ thus, the non-pleasant values of
$\boldsymbol{\upgamma}$ i.e. $\boldsymbol{\upgamma}(t)<\gamma$ for $t<t_{0}$
become irrelevant.

Second, $F\preceq_{(1+\gamma)\textsc{-SD}}G$ but
$F\npreceq_{(1+\boldsymbol{\upgamma})\textsc{-SD}}G$. Recall that there exists
(at least one) $t^{0}$ such that $\boldsymbol{\upgamma}(t^{0})<\gamma$. Again
the construction is the same as above, meaning that after $t^{0}$ we define
$F$ and $G$ the same and discuss in a left neighbourhood of $t^{0}$ whether
$\boldsymbol{\upgamma}$ is flat or not and continue as before.

The conclusion is that in both non-trivial scenarios, one can always construct
two distribution functions $F$ and $G$ so that there are not distinguishable
by the fractional $(1+\gamma)$-SD but possible to order them with the multi-
fractional $(1+\boldsymbol{\upgamma})$-SD and vice versa.

###  2.3 The universe of multi-fractional stochastic dominance

In this section, we formalize all prior observations within a structured
framework. Let $\mathscrbf{D}(\mathbb{R})$ denote the set of all distribution
functions (of the real-valued random variables). For every constant
$\gamma\in[0,1]$ and, non-decreasing function
$\boldsymbol{\upgamma}:\mathbb{R}\to[0,1]$, let introduce

$\displaystyle\mathscrbf{E}_{\textsc{f}}(\gamma)$
$\displaystyle:=\Big{\\{}\left(F,G\right)\in\mathscrbf{D}(\mathbb{R})\times\mathscrbf{D}(\mathbb{R})\,:\,F\preceq_{(1+\upgamma)\textsc{{-SD}}}G\Big{\\}},$
$\displaystyle\mathscrbf{E}_{\textsc{mf}}(\boldsymbol{\upgamma})$
$\displaystyle:=\Big{\\{}\left(F,G\right)\in\mathscrbf{D}(\mathbb{R})\times\mathscrbf{D}(\mathbb{R})\,:\,F\preceq^{\textsc{mf}}_{(1+\boldsymbol{\upgamma})\textsc{{-SD}}}G\Big{\\}}.$

Also, we denote $\mathscrbf{E}_{\textsc{f}}(1)=\mathscrbf{E}_{\textsc{ssd}}$.
Note that, clearly we have the set identity
$\mathscrbf{E}_{\textsc{mf}}(\boldsymbol{\upgamma})=\mathscrbf{E}_{\textsc{f}}(\gamma)$
as soon as $\boldsymbol{\upgamma}\equiv\gamma$ is the constant function, and
furthermore, when $\boldsymbol{\upgamma}(t)\geq\gamma$
$(\boldsymbol{\upgamma}(t)\leq\gamma)$ for every $t\in\mathbb{R}$, the set
inclusion
$\mathscrbf{E}_{\textsc{f}}(\gamma)\,\subsetneqq\,\mathscrbf{E}_{\textsc{mf}}(\boldsymbol{\upgamma})$
$(\mathscrbf{E}_{\textsc{mf}}(\boldsymbol{\upgamma})\,\subsetneqq\,\mathscrbf{E}_{\textsc{f}}(\gamma))$
holds. Lastly, Example 2.4 shows that for every constant $\gamma\in(0,1)$ and
every (non-decreasing) function $\boldsymbol{\upgamma}$ satisfying
$\boldsymbol{\upgamma}(t^{0})<\gamma<\boldsymbol{\upgamma}(t_{0})$ for some
$t^{0}<t_{0}$, we have

$\mathscrbf{E}_{\textsc{mf}}(\boldsymbol{\upgamma})\setminus\mathscrbf{E}_{\textsc{f}}(\gamma)\neq\emptyset\quad\text{
and
}\quad\mathscrbf{E}_{\textsc{f}}(\gamma)\setminus\mathscrbf{E}_{\textsc{mf}}(\boldsymbol{\upgamma})\neq\emptyset.$

The next result aims to summarise the non-trivial relations between the
corresponding universes.

######  Theorem 2.1.

Let $\boldsymbol{\upgamma}:\mathbb{R}\rightarrow[0,1]$ be an arbitrary non-
decreasing function and, define
$\accentset{\rule{4.20004pt}{1.5pt}}{\boldsymbol{\upgamma}}:=\lim_{t\to+\infty}\boldsymbol{\upgamma}(t)=\sup_{t\in\mathbb{R}}\boldsymbol{\upgamma}(t)$
and
$\stackunder[1pt]{$\boldsymbol{\upgamma}$}{\rule{6.00006pt}{1.5pt}}:=\lim_{t\to-\infty}\boldsymbol{\upgamma}(t)=\inf_{t\in\mathbb{R}}\boldsymbol{\upgamma}(t)$.
Then, the following statements hold:

  * (a)

The inclusions
$\mathscrbf{E}_{\textsc{f}}(\stackunder[1pt]{$\boldsymbol{\upgamma}$}{\rule{6.00006pt}{1.5pt}})\subseteq\mathscrbf{E}_{\textsc{mf}}(\boldsymbol{\upgamma})\subseteq\mathscrbf{E}_{\textsc{f}}(\accentset{\rule{4.20004pt}{1.5pt}}{\boldsymbol{\upgamma}})$
holds and are strict as soon as $\boldsymbol{\upgamma}$ is a non-constant
function.

  * (b)

The corresponding universes associated to fractional and multi-fractional
stochastic dominances coincide with $\mathscrbf{E}_{\textsc{ssd}}$, namely
that

$\displaystyle\bigcup_{\gamma\in[0,1]}\mathscrbf{E}_{\textsc{f}}(\gamma)\quad=\quad\bigcup_{\mathclap{\begin{subarray}{c}\boldsymbol{\upgamma}:\mathbb{R}\to[0,1]\\\
\text{non-
decreasing}\end{subarray}}}\,\mathscrbf{E}_{\textsc{mf}}(\boldsymbol{\upgamma})\quad=\quad\mathscrbf{E}_{\textsc{ssd}}.$
(2.4)

In addition, if the non-interesting case i.e.
$\boldsymbol{\upgamma}\equiv\gamma=1$ is excluded from both sides, then the
relation (2.4) reduces to

$\bigcup_{\gamma\in[0,1)}\mathscrbf{E}_{\textsc{f}}(\gamma)\quad\mathrel{\vbox{\hbox{\scalebox{1.2}{$\subsetneqq$}}}}\quad\bigcup_{\mathclap{\begin{subarray}{c}\boldsymbol{\upgamma}:\mathbb{R}\to[0,1]\\\
\text{non-decreasing}\\\ \boldsymbol{\upgamma}\not\equiv
1\end{subarray}}}\,\mathscrbf{E}_{\textsc{mf}}(\boldsymbol{\upgamma})\quad=\quad\mathscrbf{E}_{\textsc{ssd}}.$
(2.5)

######  Remark 2.2.

Part (b) of Theorem 2.1 demonstrates a significant feature of multi-fractional
stochastic dominance in the sense that by removing the constant function
$\boldsymbol{\upgamma}\equiv\gamma=1$ (corresponding to SSD case), still the
universe of multi-fractional stochastic dominance remains as large as that of
SSD whilst this is not the case for fractional stochastic dominance.

###### Proof.

(a) The inclusions are immediate consequence of the monotonicity of integral
condition (2.1)(see Remark 2.1 (ii)). We only prove the strict inclusion case.
Suppose that $\boldsymbol{\upgamma}$ is not a constant function and so that
there exists at least one $t\in\mathbb{R}$ with
$\boldsymbol{\upgamma}(t)>\stackunder[1pt]{$\boldsymbol{\upgamma}$}{\rule{6.00006pt}{1.5pt}}$.
Then, we can always find a constant $c\in(0,1)$ satisfying
$\sqrt{c}+\sqrt{c}\,\boldsymbol{\upgamma}(t)<1$. Fix such $c$ and construct
the following pair of distribution functions $(F^{*},G^{*})$:

$F^{*}(x)=\begin{cases}0&\mbox{ if }x<t-\sqrt{c}\\\ \sqrt{c}&\mbox{ if
}t-\sqrt{c}\leq x<t+\sqrt{c}\\\ 1&\mbox{ if }t+\sqrt{c}\leq
x\end{cases}\quad\mbox{and}\quad G^{*}(x)=\begin{cases}0&\mbox{ if }x<t\\\
\sqrt{c}+\sqrt{c}\,\boldsymbol{\upgamma}(t)&\mbox{ if }t\leq x<t+\sqrt{c}\\\
1&\mbox{ if }t+\sqrt{c}\leq x\end{cases}.$

Then, $F^{*}$ and $G^{*}$ have a single crossing at $t$ and

$\displaystyle A_{+}:=\int_{-\infty}^{t}(F^{*}(x)-G^{*}(x))_{+}dx=c$
$\displaystyle
A_{-}:=\int_{t}^{t+\sqrt{c}}(F^{*}(x)-G^{*}(x))_{-}dx=\boldsymbol{\upgamma}(t)c.$

Note that
$\stackunder[1pt]{$\boldsymbol{\upgamma}$}{\rule{6.00006pt}{1.5pt}}<\frac{A_{-}}{A_{+}}=\boldsymbol{\upgamma}(t)\leq\boldsymbol{\upgamma}(x),\,\forall
x\geq t$. Thus,
$(F^{*},G^{*})\in\mathscrbf{E}_{\textsc{mf}}(\boldsymbol{\upgamma})$ but
$(F^{*},G^{*})\notin\mathscrbf{E}_{\textsc{f}}(\stackunder[1pt]{$\boldsymbol{\upgamma}$}{\rule{6.00006pt}{1.5pt}})$.
This shows that the first inclusion is strict. Next, we show that the second
inclusion
$\mathscrbf{E}_{\textsc{mf}}(\boldsymbol{\upgamma})\subsetneqq\mathscrbf{E}_{\textsc{f}}(\accentset{\rule{4.20004pt}{1.5pt}}{\boldsymbol{\upgamma}})$
is strict as well. Similarly, for a given non-constant
$\boldsymbol{\upgamma}$, first we fix $t\in\mathbb{R}$ such that
$\accentset{\rule{4.20004pt}{1.5pt}}{\boldsymbol{\upgamma}}>\boldsymbol{\upgamma}(t)$
holds. Then, we can always find a constant $c\in(0,1)$ satisfying
$\sqrt{c}+\sqrt{c}\,\accentset{\rule{4.20004pt}{1.5pt}}{\boldsymbol{\upgamma}}<1$.
Fix such $c$ and construct the following pair of distribution functions
$(F^{*},G^{*})$:

$F^{*}(x)=\begin{cases}0&\mbox{ if }x<t-2\sqrt{c}\\\ \sqrt{c}&\mbox{ if
}t-2\sqrt{c}\leq x<t\\\ 1&\mbox{ if }t\leq x\end{cases}\quad\mbox{and}\quad
G^{*}(x)=\begin{cases}0&\mbox{ if }x<t-\sqrt{c}\\\
\sqrt{c}+\sqrt{c}\,\accentset{\rule{4.20004pt}{1.5pt}}{\boldsymbol{\upgamma}}&\mbox{
if }t-\sqrt{c}\leq x<t\\\ 1&\mbox{ if }t\leq x\end{cases}.$

Again, $F^{*}$ and $G^{*}$ have a single crossing at $t-\sqrt{c}$ and
moreover,

$\displaystyle A_{+}=:\int_{-\infty}^{t-\sqrt{c}}(F^{*}(x)-G^{*}(x))_{+}dx=c$
$\displaystyle
A_{-}=:\int_{t-\sqrt{c}}^{t}(F^{*}(x)-G^{*}(x))_{-}dx=\accentset{\rule{4.20004pt}{1.5pt}}{\boldsymbol{\upgamma}}c.$

This time, observe that
$(F^{*},G^{*})\in\mathscrbf{E}_{\textsc{f}}(\accentset{\rule{4.20004pt}{1.5pt}}{\boldsymbol{\upgamma}})$
but $(F^{*},G^{*})\notin\mathscrbf{E}_{\textsc{mf}}(\boldsymbol{\upgamma})$
since
$\boldsymbol{\upgamma}(t)<\frac{A_{-}}{A_{+}}=\accentset{\rule{4.20004pt}{1.5pt}}{\boldsymbol{\upgamma}}$
meaning that the integral condition (2.1) fails at $t$.

(b) First, the set equalities in (2.4) are obvious. Next, note that the first
strict inclusion in relation (2.5) is a direct application of Example 2.3.
Finally, we show that the second set equality in relation (2.5). First, we
have

$\bigcup_{\mathclap{\begin{subarray}{c}\boldsymbol{\upgamma}:\mathbb{R}\to[0,1]\\\
\text{non-decreasing}\\\ \boldsymbol{\upgamma}\not\equiv
1\end{subarray}}}\,\mathscrbf{E}_{\textsc{mf}}(\boldsymbol{\upgamma})\quad\subseteq\quad\mathscrbf{E}_{\textsc{ssd}}.$
(2.6)

In order to demonstrate the converse, let $(F,G)$ be an arbitrary element of
$\mathscrbf{E}_{\textsc{ssd}}$. Then, w.l.o.g, there exists the first crossing
point $x^{*}\in\mathbb{R}$ (otherwise, we have $F\preceq_{\textsc{FSD}}G$ and
the result becomes trivial) for the pair $(F,G)$ meaning that
$\int_{-\infty}^{x^{*}}\left(F(x)-G(x)\right)_{+}dx>0$,
$\int_{-\infty}^{x^{*}}\left(F(x)-G(x)\right)_{-}dx=0$ and
$F(x^{*})=G(x^{*})$. Now, we define the following function:

$\boldsymbol{\upgamma}(F,G)(t):=\begin{cases}0&\mbox{ if }t\leq x^{*}\\\
\sup_{x^{*}\leq s\leq
t}\frac{\int_{x^{*}}^{s}\left(F(x)-G(x)\right)_{-}dx}{\int_{-\infty}^{s}\left(F(x)-G(x)\right)_{+}dx}&\mbox{
if }x^{*}\leq t.\end{cases}$ (2.7)

Clearly, $\boldsymbol{\upgamma}(F,G)$ is a non-negative, non-decreasing, non-
constant, continuous function and it is bounded by one. Therefore,

$\boldsymbol{\upgamma}(F,G)\in\\{\boldsymbol{\upgamma}:\boldsymbol{\upgamma}:\mathbb{R}\to[0,1],\mbox{
non-decreasing and }\boldsymbol{\upgamma}\not\equiv 1\\},$ (2.8)

and moreover, we have
$F\preceq^{\textsc{mf}}_{(1+\boldsymbol{\upgamma}(F,G))\textsc{-SD}}G$. This
completes the proof. ∎

##  3 MFSD: Utility Theory

In this section, first we introduce the set of utility functions that
generates the multi-fractional $(1+\boldsymbol{\upgamma})$-SD. Then, we prove
the if-and-only-if theorem between the introduced utility class and the
integral condition (2.1).

###  3.1 Towards a utility class

We start with the following simple lemma motivated by the excellent work
[MSTW17, page 2943]. In fact, the lemma below somehow is encoded the structure
of a suitable utility class for multi-fractional stochastic dominance.

######  Lemma 3.1.

Let $\boldsymbol{\upgamma}:\mathbb{R}\rightarrow[0,1]$ be an arbitrary non-
decreasing function ($\boldsymbol{\upgamma}(x)\leq\boldsymbol{\upgamma}(y)$
for all $x\leq y$). For each $t\in\mathbb{R}$, we consider the following
functions space:

$\displaystyle\mathscr{U}^{t}_{\boldsymbol{\upgamma}(t)}:$
$\displaystyle=\Bigg{\\{}u:\mathbb{R}\to\mathbb{R}\,\,:\,u(x)=u(t),\,\forall\,x\geq
t\text{ and, }$ (3.1) $\displaystyle\hskip 56.9055pt\forall\,x_{1}<x_{2}\leq
x_{3}<x_{4},\,0\leq\boldsymbol{\upgamma}(t)\frac{u(x_{4})-u(x_{3})}{x_{4}-x_{3}}\leq\frac{u(x_{2})-u(x_{1})}{x_{2}-x_{1}}\Bigg{\\}}.$

Let $F$ and $G$ be two arbitrary distribution functions. For every
$t\in\mathbb{R}$, define the (utility) function $u_{t}$ via its right
derivative as

$u_{t}^{\prime}(x)=\begin{cases}\boldsymbol{\upgamma}(t)&\text{ if }G(x)\leq
F(x)\text{ and }x\leq t\\\ 1&\text{ if }F(x)<G(x)\text{ and }x\leq t\\\
0&\text{ if }t<x.\end{cases}$ (3.2)

Then, $u_{t}\in\mathscr{U}^{t}_{\boldsymbol{\upgamma}(t)}$ for every
$t\in\mathbb{R}$.

###### Proof.

First, we avoid of the trivial case $\boldsymbol{\upgamma}(t)=0$ (since by
very definition, function $u$ is monotone). On the other hand, since $F$ and
$G$ are right continuous and have countably many discontinuity points (of the
first kind since there are monotone) hence there is a countable set of points
$(x_{i})$ so that $x_{i-1}<x_{i}$, and $u^{\prime}_{t}$ is a constant function
on each subinterval $(x_{i-1},x_{i})$. Therefore, by the mean value theorem,
for every $x$ and $y$ on each subinterval, we have

$\frac{u_{t}(y)-u_{t}(x)}{y-x}=1\quad\text{ or
}\quad\frac{u_{t}(y)-u_{t}(x)}{y-x}=\boldsymbol{\upgamma}(t).$

This, in particular implies that for all $x,y\in\mathbb{R}$,

$\boldsymbol{\upgamma}(t)\leq\frac{u_{t}(y)-u_{t}(x)}{y-x}\leq 1.$ (3.3)

Thus, for all $x_{1}<x_{2}\leq x_{3}<x_{4}$, the relation

$\boldsymbol{\upgamma}(t)\frac{u_{t}(x_{4})-u_{t}(x_{3})}{x_{4}-x_{3}}\leq\frac{u_{t}(x_{2})-u_{t}(x_{1})}{x_{2}-x_{1}}$

holds true. In addition, $u_{t}$ is a constant function on region $(t,\infty)$
and hence, by continuity we have $u(x)=u(t)$ for every $x\geq t$. Therefore,
$u_{t}\in\mathscr{U}^{t}_{\boldsymbol{\upgamma}(t)}$ for every
$t\in\mathbb{R}$.

∎

######  Proposition 3.1.

Let $X\sim F$ and $Y\sim G$ be two arbitrary probability distributions having
finite first moments and, $\boldsymbol{\upgamma}:\mathbb{R}\rightarrow[0,1]$
be a non-decreasing function. Then, relation

$\mathbb{E}\left[u_{t}(X)\right]\leq\mathbb{E}\left[u_{t}(Y)\right],\quad\forall\,t\in\mathbb{R}$
(3.4)

implies that, for every $t\in\mathbb{R}$:

$\int_{-\infty}^{t}\left(F(x)-G(x)\right)_{-}dx\leq\boldsymbol{\upgamma}(t)\int_{-\infty}^{t}\left(F(x)-G(x)\right)_{+}dx.$

###### Proof.

Let $t\in\mathbb{R}$ and $u_{t}$ be the function defined as (3.2). Then, the
assumption of having finite means and the form of $u_{t}$ function imply that
$\mathbb{E}[u_{t}(X)],\mathbb{E}[u_{t}(Y)]>-\infty$. In addition, one has to
note that $u_{t}(x)\leq ax+b$, $a>0,b\in\mathbb{R}$ for each $x\in\mathbb{R}$.
Hence, the assumption of finite means, once again, imply that
$\mathbb{E}[u_{t}(X)],\mathbb{E}[u_{t}(Y)]<\infty$ and therefore all the
quantities below are well-defined. Hence, by integration by parts formula (see
[Tes76, Lemma 1*]), we obtain

$\displaystyle
0\leq\mathbb{E}\left[u_{t}(Y)\right]-\mathbb{E}\left[u_{t}(X)\right]$
$\displaystyle=\int_{\mathbb{R}}u_{t}(x)d(F-G)(x)=\int_{\mathbb{R}}\left(F(x)-G(x)\right)du_{t}(x)$
$\displaystyle=\int_{\mathbb{R}}u^{\prime}_{t}(x)\left(F(x)-G(x)\right)dx$
$\displaystyle=\int_{-\infty}^{t}\boldsymbol{\upgamma}(t)\
\left(F(x)-G(x)\right)_{+}dx-\int_{-\infty}^{t}\left(F(x)-G(x)\right)_{-}dx$
$\displaystyle=\boldsymbol{\upgamma}(t)\int_{-\infty}^{t}\left(F(x)-G(x)\right)_{+}dx-\int_{-\infty}^{t}\left(F(x)-G(x)\right)_{-}dx.$

It is worth to mention that the very last integral in the right hand side of
the first line above can be understood as the Lebesgue-Stieltjes and Riemann-
Stieltjes integral. In addition, note that $u_{t}$ is right continuous and
$u^{\prime}_{t}$ has only countably many discontinuities and therefore, the
second line follows from the fact that $u_{t}$ is defined as the Lebesgue
integral of $u^{\prime}_{t}$ (fundamental theorem of calculus).
∎

Next, we seek for an appropriate utility class for the multi-fractional
$(1+\boldsymbol{\upgamma})$-SD. We take a naive approach. Inspired by Lemma
3.1, a desirable utility class should contain those "basis" functions
$(u_{t}:t\in\mathbb{R})$ that are defined in (3.2). Observe that these
functions enjoy two important features:

  * (i)

they are constant over the region $(t,+\infty)$ and hence their derivatives
vanish (this in particular implies that no contribution in the expected
utility through the integration by parts formula),

  * (ii)

in addition, on the complementary range $(-\infty,t)$ they should manifest the
fractional $(1+\gamma)$-SD utilities with $\boldsymbol{\upgamma}(t)$ instead
of a fixed parameter $\gamma$.

Motivated by these observations, we introduce the following function spaces:
for real $s,t\in\mathbb{R}$, define

$\displaystyle\mathscr{U}^{s}_{\boldsymbol{\upgamma}(t)}:$
$\displaystyle=\Bigg{\\{}u:\mathbb{R}\to\mathbb{R}\,\,:\,u(x)=u(s),\,\forall\,x\geq
s\text{ and, }$ (3.5) $\displaystyle\hskip 56.9055pt\forall\,x_{1}<x_{2}\leq
x_{3}<x_{4},\,0\leq\boldsymbol{\upgamma}(t)\frac{u(x_{4})-u(x_{3})}{x_{4}-x_{3}}\leq\frac{u(x_{2})-u(x_{1})}{x_{2}-x_{1}}\Bigg{\\}}$

$\displaystyle\mathscr{U}^{s}_{\boldsymbol{\upgamma}(t)}(C^{1}(\mathbb{R})):$
$\displaystyle=\Bigg{\\{}u:\mathbb{R}\to\mathbb{R}\in
C^{1}(\mathbb{R}):\,u^{\prime}(x)=0,\forall x>s\,\text{ and
}0\leq\boldsymbol{\upgamma}(t)u^{\prime}(y)\leq u^{\prime}(x),\,\forall x\leq
y\Bigg{\\}}.$ (3.6)

Let us recall the following function spaces introduced in [MSTW17]: for each
real number $\gamma\in[0,1]$,

$\displaystyle\mathscr{U}^{*}_{\gamma}:$
$\displaystyle=\Bigg{\\{}u:\mathbb{R}\to\mathbb{R}\,\,:\,0\leq\gamma\frac{u(x_{4})-u(x_{3})}{x_{4}-x_{3}}\leq\frac{u(x_{2})-u(x_{1})}{x_{2}-x_{1}},\forall\,x_{1}<x_{2}\leq
x_{3}<x_{4}\Bigg{\\}}$ (3.7) $\displaystyle\mathscr{U}_{\gamma}:$
$\displaystyle=\Bigg{\\{}u:\mathbb{R}\to\mathbb{R}\in
C^{1}(\mathbb{R}):\,0\leq\gamma u^{\prime}(y)\leq u^{\prime}(x),\,\forall
x\leq y\Bigg{\\}}.$ (3.8)

Characteristic (i) allows the utility class (3.5) to incorporate additional
information about decision makers’ preferences that $\mathscr{U}^{*}_{\gamma}$
does not capture. This results in more informative utility classes, meaning
that
$\mathscr{U}^{s}_{\boldsymbol{\upgamma}(t)}\subsetneqq\mathscr{U}^{*}_{\boldsymbol{\upgamma}(t)}$
for all $s\leq t$. However, it is important to note that this information does
not necessarily indicate a meaningful behavior of decision makers in terms of
utility theory. In fact, the utility functions in (3.5) are considered to be
pathological and contain extreme preferences that do not accurately represent
realistic decision maker behavior, as discussed in [LL02]. Therefore, while we
use this set as the foundation for constructing the utility class of MFSD, our
main focus is on the mathematical aspects of the functions in (3.5).
Understanding the mathematical properties of these functions will be essential
in proving several key results later in this paper.

Functions spaces $\mathscr{U}^{t}_{\boldsymbol{\upgamma}(t)}$ and
$\mathscr{U}^{t}_{\boldsymbol{\upgamma}(t)}(C^{1}(\mathbb{R}))$ (namely that
when $s=t$) play important role in our development. The functions space
$\mathscr{U}^{t}_{\boldsymbol{\upgamma}(t)}$ is particularly important for us
as it contains utility functions with jump discontinuities in their first
derivative (kink). Note that, the relation between two spaces
$\mathscr{U}^{s}_{\boldsymbol{\upgamma}(s)}$ and
$\mathscr{U}^{t}_{\boldsymbol{\upgamma}(t)}$ for $s<t$ is not straightforward.
The reason is that the set $\mathscr{U}^{t}_{\boldsymbol{\upgamma}(t)}$ is
equipped with two conditions which give rise to two opposite inclusion
relations. To be more precise, for $s\leq t$ (and hence
$\boldsymbol{\upgamma}(s)\leq\boldsymbol{\upgamma}(t)$), we have

$\displaystyle u^{\prime}(x)=0,\mbox{ }\forall x>s$
$\displaystyle\quad\text{implies that }\quad u^{\prime}(x)=0,\mbox{ }\forall
x>t,$ $\displaystyle 0\leq\boldsymbol{\upgamma}(t)u^{\prime}(y)\leq
u^{\prime}(x),\,\forall x\leq y$ $\displaystyle\quad\text{implies that }\quad
0\leq\boldsymbol{\upgamma}(s)u^{\prime}(y)\leq u^{\prime}(x),\,\forall x\leq
y.$

######  Lemma 3.2.

Let $s\leq t$. Then, the following set-inclusions are in place:

  * (a)

$\mathscr{U}^{t}_{\boldsymbol{\upgamma}(t)}(C^{1}(\mathbb{R}))\mathrel{\vbox{\hbox{\scalebox{1.4}{$\subseteq$}}}}\mathscr{U}^{t}_{\boldsymbol{\upgamma}(t)}$.

  * (b)

$\mathscr{U}^{s}_{\boldsymbol{\upgamma}(t)}\mathrel{\vbox{\hbox{\scalebox{1.4}{$\subseteq$}}}}\mathscr{U}^{t}_{\boldsymbol{\upgamma}(t)}$,
and
$\mathscr{U}^{s}_{\boldsymbol{\upgamma}(t)}\mathrel{\vbox{\hbox{\scalebox{1.4}{$\subseteq$}}}}\mathscr{U}^{s}_{\boldsymbol{\upgamma}(s)}$.

  * (c)

$\mathscr{U}^{s}_{\boldsymbol{\upgamma}(t)}(C^{1}(\mathbb{R}))\mathrel{\vbox{\hbox{\scalebox{1.4}{$\subseteq$}}}}\mathscr{U}^{t}_{\boldsymbol{\upgamma}(t)}(C^{1}(\mathbb{R}))$,
and
$\mathscr{U}^{s}_{\boldsymbol{\upgamma}(t)}(C^{1}(\mathbb{R}))\mathrel{\vbox{\hbox{\scalebox{1.4}{$\subseteq$}}}}\mathscr{U}^{s}_{\boldsymbol{\upgamma}(s)}(C^{1}(\mathbb{R}))$.

The following proposition summarizes important mathematical properties of the
functions space $\mathscr{U}^{t}_{\boldsymbol{\upgamma}(t)}$.

######  Proposition 3.2.

Let $\boldsymbol{\upgamma}:\mathbb{R}\rightarrow[0,1]$ be an arbitrary non-
decreasing function and $t\in\mathbb{R}$.

  * (a)

Assume that $\boldsymbol{\upgamma}(t)\neq 0$. Then, every
$u\in\mathscr{U}^{t}_{\boldsymbol{\upgamma}(t)}$ is non-decreasing function
that is right and left differentiable at every point $x\in\mathbb{R}$. When
$\boldsymbol{\upgamma}(t)=0$, then every
$u\in\mathscr{U}^{t}_{\boldsymbol{\upgamma}(t)}$ is an increasing function,
bounded from above (in fact $u(x)=u(t)$ for all $x\geq t$), and almost
everywhere differentiable.

  * (b)

The class $\mathscr{U}_{\boldsymbol{\upgamma}(t)}^{t}$ is a convex cone, i.e.,
it is closed under multiplication with positive scalars and being a convex
set.

  * (c)

Limit
$\displaystyle{\lim_{t\rightarrow-\infty}\mathscr{U}^{t}_{\boldsymbol{\upgamma}(t)}}$
as $t\to-\infty$ exists and, moreover,

$\displaystyle{\lim_{t\rightarrow-\infty}\mathscr{U}^{t}_{\boldsymbol{\upgamma}(t)}}=:\mathscr{U}^{-\infty}_{\stackunder[1pt]{\boldsymbol{\upgamma}}{\rule{3.5pt}{1.1pt}}}=\\{\text{
constant functions }\\}.$ (3.9)

  * (d)

Limit
$\displaystyle{\lim_{t\rightarrow\infty}\mathscr{U}^{t}_{\boldsymbol{\upgamma}(t)}}$
as $t\to+\infty$ exists and, moreover,

$\displaystyle{\lim_{t\rightarrow\infty}\mathscr{U}^{t}_{\boldsymbol{\upgamma}(t)}}=\mathscr{U}^{+\infty}_{\accentset{\rule{3.5pt}{1.2pt}}{\boldsymbol{\upgamma}}}$
(3.10)

where
$\mathscr{U}^{+\infty}_{\accentset{\rule{3.5pt}{1.2pt}}{\boldsymbol{\upgamma}}}:=\\{u\,:\,0\leq\accentset{\rule{3.5pt}{1.2pt}}{\boldsymbol{\upgamma}}\frac{u(x_{4})-u(x_{3})}{x_{4}-x_{3}}\leq\frac{u(x_{2})-u(x_{1})}{x_{2}-x_{1}},\,\forall\,x_{1}<x_{2}\leq
x_{3}<x_{4}\\}.$

###### Proof.

(a) When $\boldsymbol{\upgamma}(t)\neq 0$, it is elementary to show that every
functions $u\in\mathscr{U}^{t}_{\boldsymbol{\upgamma}(t)}$ is right and left
differentiable at every point whereas, the case $\boldsymbol{\upgamma}(t)=0$
follows from Lebesgue Theorem [RF18, page 112]. (b) Obvious. (d) Let $(t_{n})$
be a sequence of real numbers diverging to infinity. First note that by
inclusion relations ((b)) and ((b)) we have

$\bigcap_{k=n}^{\infty}\mathscr{U}^{t_{k}}_{\boldsymbol{\upgamma}(t_{k})}=\mathscr{U}^{t_{n}}_{\accentset{\rule{3.5pt}{1.2pt}}{\boldsymbol{\upgamma}}}.$

Thus,
$\displaystyle{\liminf_{n\rightarrow\infty}\mathscr{U}^{t_{n}}_{\boldsymbol{\upgamma}(t_{n})}}=\displaystyle{\bigcup_{n\geq
1}}\mathscr{U}^{t_{n}}_{\accentset{\rule{3.5pt}{1.2pt}}{\boldsymbol{\upgamma}}}=\mathscr{U}^{+\infty}_{\accentset{\rule{3.5pt}{1.2pt}}{\boldsymbol{\upgamma}}}$.
Similarly, we can show that
$\displaystyle{\limsup_{n\rightarrow\infty}\mathscr{U}^{t_{n}}_{\boldsymbol{\upgamma}(t_{n})}}=\bigcap_{n\geq
1}\bigcup_{k\geq
n}^{\infty}\mathscr{U}^{t_{k}}_{\boldsymbol{\upgamma}(t_{k})}=\mathscr{U}^{+\infty}_{\accentset{\rule{3.5pt}{1.2pt}}{\boldsymbol{\upgamma}}}$.
Therefore,
$\displaystyle{\limsup_{t_{n}\rightarrow\infty}\mathscr{U}^{t_{n}}_{\boldsymbol{\upgamma}(t_{n})}}=\displaystyle{\liminf_{t_{n}\rightarrow\infty}\mathscr{U}^{t_{n}}_{\boldsymbol{\upgamma}(t_{n})}}=\mathscr{U}^{+\infty}_{\accentset{\rule{3.5pt}{1.2pt}}{\boldsymbol{\upgamma}}}$.
This completes the proof. (c) Similar to part (d). ∎

######  Remark 3.1.

  * (i)

Similar statements as in Proposition 3.2 hold true for smooth functions spaces
$\mathscr{U}^{t}_{\boldsymbol{\upgamma}(t)}(C^{1}(\mathbb{R}))$.

  * (ii)

The limiting sets
$\mathscr{U}^{+\infty}_{\accentset{\rule{3.5pt}{1.2pt}}{\boldsymbol{\upgamma}}}$
and
$\mathscr{U}^{+\infty}_{\accentset{\rule{3.5pt}{1.2pt}}{\boldsymbol{\upgamma}}}(C^{1}(\mathbb{R}))$
coincide with the utility class
$\mathscr{U}^{*}_{\accentset{\rule{3.5pt}{1.2pt}}{\boldsymbol{\upgamma}}}$ and
$\mathscr{U}_{\accentset{\rule{3.5pt}{1.2pt}}{\boldsymbol{\upgamma}}}$
corresponding to the fractional
$(1+\accentset{\rule{3.5pt}{1.2pt}}{\boldsymbol{\upgamma}})$-SD introduced in
[MSTW17]. In particular, when
$\accentset{\rule{3.5pt}{1.2pt}}{\boldsymbol{\upgamma}}=1$ it consists of non-
decreasing concave functions associated to the second order stochastic
dominance.

  * (iii)

One has to note that
$\mathscr{U}^{-\infty}_{\stackunder[1pt]{$\boldsymbol{\upgamma}$}{\rule{3.5pt}{1.1pt}}}\neq\mathscr{U}^{*}_{\stackunder[1pt]{$\boldsymbol{\upgamma}$}{\rule{3.5pt}{1.1pt}}}$
(in fact, we have the crystal strict inclusion
$\mathscr{U}^{-\infty}_{\stackunder[1pt]{$\boldsymbol{\upgamma}$}{\rule{3.5pt}{1.1pt}}}\subset\mathscr{U}^{*}_{\stackunder[1pt]{$\boldsymbol{\upgamma}$}{\rule{3.5pt}{1.1pt}}}$
) unlike
$\mathscr{U}^{+\infty}_{\accentset{\rule{3.5pt}{1.2pt}}{\boldsymbol{\upgamma}}}=\mathscr{U}^{*}_{\accentset{\rule{3.5pt}{1.2pt}}{\boldsymbol{\upgamma}}}$.

###  3.2 Utility space $\mathscr{U}^{\textsc{mf}}_{\upgamma}$ and the main
result

For each $t\in\mathbb{R}$, the class
$\mathscr{U}^{t}_{\boldsymbol{\upgamma}(t)}$ contains (utility) functions with
limited variety of characteristics. Therefore, by gathering spaces
$\mathscr{U}^{t}_{\boldsymbol{\upgamma}(t)}$ for all $t\in[-\infty,\infty]$
one can create a larger class which contains a broader range of utility
functions with different attributes. Define

$\mathscr{U}^{\text{Union}}_{\boldsymbol{\upgamma}}:=\bigcup_{t\in[-\infty,\infty]}\mathscr{U}^{t}_{\boldsymbol{\upgamma}(t)},\quad\text{and}\quad\mathscr{U}^{\text{Union}}_{\boldsymbol{\upgamma}}(C^{1}(\mathbb{R})):=\bigcup_{t\in[-\infty,\infty]}\mathscr{U}^{t}_{\boldsymbol{\upgamma}(t)}(C^{1}(\mathbb{R})).$
(3.11)

In above, for the technical reasons (see Remark 3.2), when
$\boldsymbol{\upgamma}(t)=0$, we restrict the utility space
$\mathscr{U}^{t}_{\boldsymbol{\upgamma}(t)}$ to consist of non-decreasing
continuous functions. As it follows from Proposition 3.2 (b), although for
each $t\in[-\infty,\infty]$, the class
$\mathscr{U}^{t}_{\boldsymbol{\upgamma}(t)}$ is a convex cone, however,
$\mathscr{U}^{\text{Union}}_{\boldsymbol{\upgamma}}$ is not necessarily closed
under (arbitrary) addition. Since the sum of utility functions would naturally
result in a new utility function, and expected utilities are linear, hence,
the class $\mathscr{U}^{\text{Union}}_{\boldsymbol{\upgamma}}$ alone cannot
serve as a candidate for utility class of multi-fractional stochastic order.
However, it is simple to overcome the latter issue by considering all the
possible positive (finite) linear combinations. Hence, we finally introduce
the following (utility) class.

######  Definition 3.1 (Utility class of MFSD).

Let $\boldsymbol{\upgamma}:\mathbb{R}\rightarrow[0,1]$ be an arbitrary non-
decreasing function. A (utility) class for the multi-fractional
$(1+\boldsymbol{\upgamma})$-SD is defined as

$\mathscr{U}^{\textsc{mf}}_{\boldsymbol{\upgamma}}:=\operatorname{span}_{\mathbb{R}_{+}}\Big{\\{}u\,:\,u\in\mathscr{U}^{\text{Union}}_{\boldsymbol{\upgamma}}\Big{\\}}=\Big{\\{}u=\sum_{\text{finite}}\lambda_{k}u_{k},\,\lambda_{k}\geq
0,\,u_{k}\in\mathscr{U}^{\text{Union}}_{\boldsymbol{\upgamma}}\Big{\\}}.$
(3.12)

Finally, we are ready to state the main theorem of this section. First, we
state the following simple lemma that we will make use of it in the proof of
the main theorem.

######  Lemma 3.3.

Let $\boldsymbol{\upgamma}:\mathbb{R}\rightarrow[0,1]$ be an arbitrary non-
decreasing function. Let
$F\preceq^{\textsc{mf}}_{(1+\boldsymbol{\upgamma})\textsc{{-SD}}}G$. Fix
$t\in\mathbb{R}$. Then, we have

$\int_{-\infty}^{s}\left(F(x)-G(x)\right)_{-}dx\leq\boldsymbol{\upgamma}(t)\int_{-\infty}^{s}\left(F(x)-G(x)\right)_{+}dx,\quad\forall
s\leq t.$ (3.13)

###### Proof.

This is a direct consequence of the fact that $\boldsymbol{\upgamma}$ is a
non-decreasing functions. ∎

######  Theorem 3.1.

Let $\boldsymbol{\upgamma}:\mathbb{R}\rightarrow[0,1]$ be an arbitrary non-
decreasing function. For every two distribution functions $X\sim F$ and $Y\sim
G$, the following statements are equivalent:

  * (a)

$F\preceq^{\textsc{mf}}_{(1+\boldsymbol{\upgamma})\textsc{{-SD}}}G$.

  * (b)

$\mathbb{E}[u(X)]\leq\mathbb{E}[u(Y)]$ for every
$u\in\mathscr{U}^{\textsc{mf}}_{\boldsymbol{\upgamma}}$.

######  Remark 3.2.

Some parts of proof of Theorem 3.1 involve rather mathematical technicalities
that are discussed in a great care in [Tes76]. In turns out that assumptions
of finite means together with the well-definiteness of the quantity
$\mathbb{E}[u(X)]-\mathbb{E}[u(Y)]$ provide us with the most transparent
scenario. Hence, in the proof below we assume that those hypotheses are
available without mentioning in each required instance.

###### Proof.

The implication (b) to (a) is just Proposition 3.1. Hence, we prove (a)
implies (b). Due to the linearity of the mathematical expectation and the
structure of the elements in the class
$\mathscr{U}^{\textsc{mf}}_{\boldsymbol{\upgamma}}$ it is enough to show that

$\mathbb{E}[u(X)]\leq\mathbb{E}[u(Y)]\quad\text{ for every }\quad
u\in\mathscr{U}^{t}_{\boldsymbol{\upgamma}(t)}\quad\text{ where }\quad
t\in(-\infty,\infty].$ (3.14)

Let assume that $t=\infty$. Pick
$u\in\mathscr{U}^{+\infty}_{\accentset{\rule{3.5pt}{1.2pt}}{\boldsymbol{\upgamma}}}=\mathscr{U}^{*}_{\accentset{\rule{3.5pt}{1.2pt}}{\boldsymbol{\upgamma}}}$.
Now, observe that the integral condition (2.1) clearly yields that the
integral condition

$\int_{-\infty}^{t}(F(x)-G(x))_{-}dx\leq\accentset{\rule{3.5pt}{1.2pt}}{\boldsymbol{\upgamma}}\int_{-\infty}^{t}(F(x)-G(x))_{+}dx,\quad\forall
t\in\mathbb{R},$

that is
$F\preceq_{(1+\accentset{\rule{3.5pt}{1.2pt}}{\boldsymbol{\upgamma}})\textsc{-SD}}G$
and therefore, the result directly follows from [MSTW17]. Next, let
$t\in\mathbb{R}$ and pick an arbitrary
$u\in\mathscr{U}^{t}_{\boldsymbol{\upgamma}(t)}$. Since $u$ is constant after
$t$, hence, we need to show that

$\displaystyle\mathbb{E}\left[u(Y)\right]-\mathbb{E}\left[u(X)\right]=\int_{-\infty}^{\infty}u(x)dG(x)-\int_{-\infty}^{\infty}u(x)dF(x)=\int_{-\infty}^{t}\left(F(x)-G(x)\right)du(x)\geq
0.$

First, we discuss the case $\boldsymbol{\upgamma}(t)\neq 0$. Hereafter, and in
the lights of Lemma 3.3, we follow the arguments provided in the proof of
Theorem 2 in [HL69] (see also proof of Theorem 2* in [Tes76] for more detailed
discussion). Now, define transformation $T$ as

$Ts:=\inf\left\\{s^{\prime}:\boldsymbol{\upgamma}(t)\int_{-\infty}^{s^{\prime}}\left(F(x)-G(x)\right)_{+}dx=\int_{-\infty}^{s}\left(F(x)-G(x)\right)_{-}dx\right\\}$
(3.15)

such that

$\boldsymbol{\upgamma}(t)\int_{-\infty}^{Ts}\left(F(x)-G(x)\right)_{+}dx=\int_{-\infty}^{s}\left(F(x)-G(x)\right)_{-}dx,\quad\forall
s\leq t.$ (3.16)

Then, the following holds:

  * (i)

Transformation $T:s\leq t\mapsto T(s)$ as in (3.15) exists since the mapping
$s\mapsto\boldsymbol{\upgamma}(t)\int_{-\infty}^{s}\left(F(x)-G(x)\right)_{+}dx$
is continuous and the integrand $\left(F(x)-G(x)\right)_{+}\geq 0$. However,
$T$ may take value $-\infty$. In fact, let $x^{*}$ be the first crossing point
of $F$ and $G$ (w.l.o.g we can assume exists, otherwise, it means that we are
in the first order stochastic dominance setup). Then, clearly, $T(s)=-\infty$
on the interval $(-\infty,x^{*})$.

  * (ii)

Given the integral condition in Lemma 3.3, it is clear that in order for the
equality in (3.16) to hold, $Ts\leq s$ for all $s\leq t$.

  * (iii)

$T$ is a non-decreasing transformation of $s$ since
$\int_{-\infty}^{s}\left(F(x)-G(x)\right)_{-}dx$ is a non-decreasing function
of $s$. Hence, it is differentiable almost everywhere over the set $\\{s\leq
t:Ts>-\infty\\}$.

Differentiating equation (3.16), one obtains:

$\boldsymbol{\upgamma}(t)\left(F(Ts)-G(Ts)\right)_{+}\widetilde{T}(s)=\left(F(s)-G(s)\right)_{-}\text{
for almost all }s\leq t$ (3.17)

where

$\displaystyle\widetilde{T}(s):=\begin{cases}T^{\prime}(s)&\mbox{ if
}s\in\\{s\leq t:Ts>-\infty\\}\\\ 1&\mbox{ if }s\in\\{s\leq
t:Ts=-\infty\\}.\end{cases}$

The equality (3.17) can be verified as follows: if $s\in\\{s\leq
t:Ts>-\infty\\}$ then it holds immediately since $T^{\prime}(s)$ exist for
almost any $s$, if $s\in\\{s\leq t:Ts=-\infty\\}$ then the right hand side
$\left(F(s)-G(s)\right)_{-}=0$ is equal to the left hand side since by
definition

$\left(F(Ts)-G(Ts)\right)_{+}=\left(F(-\infty)-G(-\infty)\right)_{+}=0.$

Moreover, $T$ is strictly increasing over $\\{s:F(s)<G(s)\mbox{ and
}F(Ts)\not=G(Ts)\\}$ since

$T^{\prime}(s)=\frac{\left(F(s)-G(s)\right)_{-}}{\boldsymbol{\upgamma}(t)\left(F(Ts)-G(Ts)\right)_{+}}>0,\quad\text{a.e}.$

Also, it can be verified from (3.15) and (3.16), $T$ is constant
($T^{\prime}(s)=0$) over $\\{s:F(s)\geq G(s)\\}\cap\\{s\leq t:Ts>-\infty\\}$.
Now, we turn back to our main statement. In fact, we shall prove stronger
positivity requirement $\int_{-\infty}^{s}\left(F(x)-G(x)\right)du(x)\geq 0$
for all $s\leq t$. Note that,

$\int_{-\infty}^{s}\left(F(x)-G(x)\right)du(x)=\int_{-\infty}^{s}\left(F(x)-G(x)\right)_{+}du(x)-\int_{-\infty}^{s}\left(F(x)-G(x)\right)_{-}du(x).$
(3.18)

Substituting (3.17) in the integrand of the second integral on the right hand
side (3.18), we obtain

$\int_{-\infty}^{s}\left(F(x)-G(x)\right)du(x)=\int_{-\infty}^{s}\left(F(x)-G(x)\right)_{+}du(x)-\boldsymbol{\upgamma}(t)\int_{-\infty}^{s}\left(F(Tx)-G(Tx)\right)_{+}\widetilde{T}(x)du(x).$
(3.19)

Recall that $T$ is a non-decreasing function, and hence, the set
$\mathcal{X}_{1}=\\{x:T^{\prime}(x)\text{ does not exits and }Tx>-\infty\\}$
have Lebesgue measure zero. Let $\mathcal{X}_{2}=\\{x:Tx=-\infty\text{ or
}T^{\prime}(x)=0\\}$. Then,

$\boldsymbol{\upgamma}(t)\left(F(Tx)-G(Tx)\right)_{+}\widetilde{T}(x)=0,\quad\forall\,x\in\mathcal{X}_{2}.$

Now, for every $s\leq t$, let $S=\\{x:x\leq
s\\}\cap(\mathcal{X}_{1}\cup\mathcal{X}_{2})^{c}$. Intuitively, set $S$
removes the constant parts and the jump discontinues of $T$, hence, $T$
becomes invertible over $S$. Moreover, the following holds:

$\boldsymbol{\upgamma}(t)\int_{-\infty}^{s}\left(F(Tx)-G(Tx)\right)_{+}\widetilde{T}(x)du(x)=\boldsymbol{\upgamma}(t)\int_{S}\left(F(Tx)-G(Tx)\right)_{+}T^{\prime}(x)du(x).$

Consider the term on the right side. Since the integrand is non-negative,
$Tx\leq x$ and, $u\in\mathscr{U}^{t}_{\boldsymbol{\upgamma}(t)}$, we obtain

$\boldsymbol{\upgamma}(t)\int_{S}\left(F(Tx)-G(Tx)\right)_{+}T^{\prime}(x)du(x)\leq\int_{S}\left(F(Tx)-G(Tx)\right)_{+}du(Tx).$
(3.20)

Recall that $T$ is invertible over $S$. The change of variable $z=Tx$ on the
right hand side of (3.20) gives:

$\int_{T(S)}\left(F(z)-G(z)\right)_{+}du(z)\leq\int_{-\infty}^{Ts}\left(F(z)-G(z)\right)_{+}du(z).$

Note that $S$ excludes the points before the first crossing, the points where
$T$ is constant and the points $T$ has upwards jumps, and thus,
$T(S)\subset(-\infty,Ts)$. Next, we are collecting the observation above back
in (3.18) to write

$\displaystyle\int_{-\infty}^{s}\left(F(x)-G(x)\right)du(x)\geq$
$\displaystyle\int_{-\infty}^{s}\left(F(x)-G(x)\right)_{+}du(x)-\int_{-\infty}^{Ts}\left(F(x)-G(x)\right)_{+}du(x)$
$\displaystyle=$
$\displaystyle\int_{Ts}^{s}\left(F(x)-G(x)\right)_{+}du(x)\geq 0.$

When $\boldsymbol{\upgamma}(t)=0$, this implies that
$u\in\mathscr{U}^{t}_{\boldsymbol{\upgamma}(t)}$ is a non-decreasing function
over the interval $(-\infty,t)$ and becomes constant on the interval
$(t,\infty)$. However, this scenario reduces to the first order stochastic
dominance on the interval $(-\infty,t)$, for details see [HL69, Theorem 1] . ∎

######  Remark 3.3.

  * (i)

Clearly, if the non-decreasing function
$\boldsymbol{\upgamma}\equiv\gamma\in[0,1]$ be constant, then
$\mathscr{U}^{\textsc{mf}}_{\boldsymbol{\upgamma}}=\mathscr{U}^{*}_{\gamma}$
given by relation (3.7). In addition, for a non-decreasing function
$\boldsymbol{\upgamma}$, the inclusions

$\mathscr{U}^{*}_{\accentset{\rule{3.5pt}{1.2pt}}{\boldsymbol{\upgamma}}}\subseteq\,\mathscr{U}^{\textsc{mf}}_{\boldsymbol{\upgamma}}\,\subseteq\mathscr{U}^{*}_{\stackunder[1pt]{\boldsymbol{\upgamma}}{\rule{3.5pt}{1.1pt}}}$

hold and, both inclusions being strict as soon as $\boldsymbol{\upgamma}$ is a
non-constant (see Theorem 2.1, part (a)). From the utility theory perspective,
the second set inclusion means that the utility class
$\mathscr{U}^{\textsc{mf}}_{\boldsymbol{\upgamma}}$ provides us a more
informative class compare to utility class
$\mathscr{U}^{*}_{\stackunder[1pt]{$\boldsymbol{\upgamma}$}{\rule{3.5pt}{1.1pt}}}$.
However, the first inclusion implies that the utility class
$\mathscr{U}^{\textsc{mf}}_{\boldsymbol{\upgamma}}$ represents the preferences
of a broader class of decision makers.

  * (ii)

Due to the linear structure of
$\mathscr{U}^{\textsc{mf}}_{\boldsymbol{\upgamma}}$ and, for every $t$,
observing that
$\mathscr{U}^{t}_{\boldsymbol{\upgamma}(t)}\subsetneqq\mathscr{U}^{*}_{\boldsymbol{\upgamma}(t)}$,
while multi-fractional $(1+\boldsymbol{\upgamma})$-SD establishes stochastic
dominance over $\mathscr{U}^{\textsc{mf}}_{\boldsymbol{\upgamma}}$ that might
possibly there is no dominance (in the sense of fractional dominance) in any
$\mathscr{U}^{*}_{\gamma}$ for all
$\gamma<\accentset{\rule{4.20004pt}{1.5pt}}{\boldsymbol{\upgamma}}$ (also see
Example 2.3).

  * (iii)

Therefore, for a given parameter $\gamma$, one can choose a function
$\boldsymbol{\upgamma}$ that satisfies $\boldsymbol{\upgamma}(t)\leq\gamma$
for all $t\in\mathbb{R}$ to further expand the group of investors from
$\mathscr{U}^{*}_{\gamma}$ to
$\mathscr{U}^{\textsc{mf}}_{\boldsymbol{\upgamma}}$. This will ensure that
individuals in $\mathscr{U}^{*}_{\gamma}$ also agree on the ordering of random
variables under $(1+\boldsymbol{\upgamma})$-SD, since
$\mathscr{U}^{*}_{\gamma}$ is a subset of
$\mathscr{U}^{\textsc{mf}}_{\boldsymbol{\upgamma}}$. Similarly, choosing a
function $\boldsymbol{\upgamma}$ with $\boldsymbol{\upgamma}(t)\geq\gamma$ for
all $t\in\mathbb{R}$ will result in a smaller set of decision makers,
$\mathscr{U}^{\textsc{mf}}_{\boldsymbol{\upgamma}}\subset\mathscr{U}^{*}_{\gamma}$,
who unanimously agree on the ordering of random variables in the
$(1+\gamma)$-SD sense. Hence, MFSD can be utilised to further interpolate
between FSD and $(1+\gamma)$-SD by choosing a function $\boldsymbol{\upgamma}$
that satisfies $\boldsymbol{\upgamma}(t)\leq\gamma$ for all $t\in\mathbb{R}$,
or between $(1+\gamma)$-SD and SSD by choosing a function
$\boldsymbol{\upgamma}$ that satisfies $\boldsymbol{\upgamma}(t)\geq\gamma$
for all $t\in\mathbb{R}$.

######  Remark 3.4 (Maximal/Small generators).

In this remark we discuss possible utility classes that might generate multi-
fractional stochastic dominance. In order to do this, first, we adjust ourself
within the unified approach introduced in [Mül97]. Since, Proposition 3.1
requires existence of the finite first moments, hence, a natural function
$b:\mathbb{R}\to[1,\infty)$ to consider is given by $b(s)=|s|+1$. This means
that the set $\mathcal{B}_{b}$ consists of measurable functions with linear
growth and $\mathbb{P}_{b}$ is the set of all the probability measures on the
real line with finite first moment. Let
$\boldsymbol{\upgamma}:\mathbb{R}\to[0,1]$ be a non-decreasing function and
consider the set
$\mathfrak{F}=\mathscr{U}^{\textsc{mf}}_{\boldsymbol{\upgamma}}\cap\mathcal{B}_{b}$
(for every given arbitrary function $\boldsymbol{\upgamma}$, it is always
possible to construct a function
$u\in\mathscr{U}^{\textsc{mf}}_{\boldsymbol{\upgamma}}$ which is not of linear
growth). For every two $F,G\in\mathbb{P}_{b}$ define

$F\preceq_{\mathfrak{F}}G\quad\text{ if }\quad\int udF\leq\int udG,\quad\text{
for all }\quad u\in\mathfrak{F}.$

Clearly, $\mathfrak{F}$ is a convex cone containing the constant functions and
being closed under the topology of pointwise convergence, hence, the maximal
generator $\mathfrak{R}_{\mathfrak{F}}$ coincides with that of the generator
$\mathfrak{F}$. Typically, larger generators (i.e. maximal generator) are
favoured for applications. However, when it comes to studying the mathematical
properties of stochastic dominance rules, a smaller set that produces the same
ordering as the original generator is more desirable. There are two common
approaches to construct smaller generators (a) smooth generator and (b) base
generators. It is well-known that (see [DM02, Theorem 2.1]) being closed under
convolution is a sufficient condition for a stochastic integral dominance to
admit a smooth generator. However, later on, we will show that the utility
class $\mathscr{U}^{\textsc{mf}}_{\boldsymbol{\upgamma}}$ is not closed under
translations. Furthermore, according to [MS02, Theorem 2.5.1], each base (for
definition, see Section 2.5, page 75, [MS02]) of $\mathfrak{F}$ also generates
stochastic dominance $\preceq_{\mathfrak{F}}$. However, at this moment, it is
not clear to us how to find such a base for given $\mathfrak{F}$ as above. It
is worth to mention that the sets
$\\{\boldsymbol{1}_{[t,\infty)}:t\in\mathbb{R}\\}$ and
$\\{(x-t)_{-}:t\in\mathbb{R}\\}$ constitute a basis for the first and second
stochastic dominances respectfully. This can be demonstrated through their
expectations, which can be expressed for each given $t\in\mathbb{R}$ as
follows:

$\mathbb{E}_{G}[\boldsymbol{1}_{[t,\infty)}]-\mathbb{E}_{F}[\boldsymbol{1}_{[t,\infty)}]=F(t)-G(t)\geq
0,\quad\forall t\in\mathbb{R}$

and

$\mathbb{E}_{G}[(x-t)_{-}]-\mathbb{E}_{F}[(x-t)_{-}]=\int_{-\infty}^{t}[F(x)-G(x)]dx\geq
0,\quad\forall t\in\mathbb{R}.$

###  3.3 Examples

Next, we continue with a few illustrating examples to gain better
understanding of the utility class.

######  Example 3.1.

(a) Assume that $\boldsymbol{\upgamma}(t)=\gamma$ is a constant function.
Then, we have,

$\displaystyle\mathscr{U}^{\text{Union}}_{\boldsymbol{\upgamma}}$
$\displaystyle=\mathscr{U}^{*}_{\gamma}=\left\\{u:\mathbb{R}\to\mathbb{R}\,:\,0\leq\gamma\frac{u(x_{4})-u(x_{3})}{x_{4}-x_{3}}\leq\frac{u(x_{2})-u(x_{1})}{x_{2}-x_{1}},\,\forall\,x_{1}<x_{2}\leq
x_{3}<x_{4}\right\\}.$

coincides with the utility space $\mathscr{U}^{*}_{\gamma}$ introduced in
[MSTW17]. This is a direct consequence of inclusion
$\mathscr{U}^{t}_{\boldsymbol{\upgamma}(t)}\subseteq\mathscr{U}^{*}_{\gamma}$
for every $t$.

(b) Let $\gamma_{1}<\gamma_{2}$ be two real numbers belong to the interval
$(0,1)$. Consider the step function

$\boldsymbol{\upgamma}(t)=\begin{cases}\gamma_{1}&\mbox{ if }t\leq t_{1}\\\
\gamma_{2}&\mbox{ if }t>t_{1}\end{cases}$

We have,
$\mathscr{U}^{\text{Union}}_{\boldsymbol{\upgamma}}=\mathscr{U}^{*}_{\gamma_{2}}\cup\mathscr{U}^{t_{1}}_{\gamma_{1}}$.

(c) This time, let $\gamma_{1}<\gamma_{2}<\gamma_{3}$ be three real numbers
belong to the interval $(0,1)$. Consider the step function

$\boldsymbol{\upgamma}(t)=\begin{cases}\gamma_{1}&\mbox{ if }t\leq t_{1}\\\
\gamma_{2}&\mbox{ if }t_{1}<t\leq t_{2}\\\ \gamma_{3}&\mbox{ if
}t>t_{2}.\end{cases}$ (3.21)

Then, we have
$\mathscr{U}^{\text{Union}}_{\boldsymbol{\upgamma}}=\mathscr{U}^{*}_{\gamma_{3}}\cup\mathscr{U}^{t_{2}}_{\gamma_{2}}\cup\mathscr{U}^{t_{1}}_{\gamma_{1}}$.

$\mathscr{U}^{*}_{\gamma_{1}}$$\mathscr{U}{{}_{\gamma_{2}}^{t_{2}}}$$\mathscr{U}{{}_{\gamma_{1}}^{t_{1}}}$$\mathscr{U}^{*}_{\gamma_{3}}$$\mathscr{U}^{*}_{\gamma_{2}}$
Figure 2: Illustration of
$\mathscr{U}^{\text{Union}}_{\boldsymbol{\upgamma}}$.

######  Remark 3.5.

This example highlights the local interpolation phenomenon that we previously
discussed in Example 2.2, particularly within the structure of the
corresponding utility class
$\mathscr{U}^{\textsc{mf}}_{\boldsymbol{\upgamma}}$. Specifically, for the
function $\boldsymbol{\upgamma}$ given as in (3.21), the utility class
$\mathscr{U}^{t_{1}}_{\gamma_{1}}$ contains the most non-concave functions in
$\mathscr{U}^{\text{Union}}_{\boldsymbol{\upgamma}}$ that are captured by the
fractional $(1+\gamma_{1})$-SD over the interval $(-\infty,t_{1}]$. Similarly,
utility functions that belong to $\mathscr{U}^{t_{2}}_{\gamma_{2}}$ display a
higher degree of non-concavity than those in $\mathscr{U}^{*}_{\gamma_{3}}$.
Consequently, while $(1+\gamma_{2})$-SD over $(-\infty,t_{2}]$ captures
$u\in\mathscr{U}^{t_{2}}_{\gamma_{2}}$, those
$u\in\mathscr{U}^{*}_{\gamma_{3}}$ corresponds to the somewhat weaker order
$(1+\gamma_{3})$-SD over the whole real line $(-\infty,\infty)$.

In general, we have the following:

######  Example 3.2.

Let $\gamma_{1}<\gamma_{2}<\cdots<\gamma_{n}$ be $n$ real numbers belonging to
the interval $[0,1]$. Consider function

$\boldsymbol{\upgamma}(t)=\begin{cases}\gamma_{1}&\mbox{ if }t\leq t_{1}\\\
\gamma_{2}&\mbox{ if }t_{1}<t\leq t_{2}\\\ \vdots\\\ \gamma_{n-1}&\mbox{ if
}t_{n-2}<t\leq t_{n-1}\\\ \gamma_{n}&\mbox{ if }t>t_{n-1}.\\\ \end{cases}$
(3.22)

Then, we have

$\displaystyle\mathscr{U}^{\text{Union}}_{\boldsymbol{\upgamma}}=\bigg{(}\bigcup_{k=1}^{n-1}\mathscr{U}^{t_{k}}_{\boldsymbol{\upgamma}(t_{k})}\bigg{)}\bigcup\mathscr{U}^{+\infty}_{\accentset{\rule{3.5pt}{1.2pt}}{\boldsymbol{\upgamma}}}=\bigg{(}\bigcup_{k=1}^{n-1}\mathscr{U}^{t_{k}}_{\gamma_{k}}\bigg{)}\bigcup\mathscr{U}^{*}_{\gamma_{n}}.$

######  Example 3.3.

The purpose of this example is to provide yet another capability of multi-
fractional stochastic dominance through its utility space. In certain
occasions, one may require a given family of utility functions sharing
particular features belongs to a single utility space corresponding to a given
stochastic dominance rule. Next, we construct a parametrized family
$\mathscr{U}_{\Theta}$ of utility functions such that the only single utility
space $\mathscr{U}^{*}_{\gamma}$, containing $\mathscr{U}_{\Theta}$,
associated to the fractional $(1+\gamma)$ stochastic dominance corresponds to
first order stochastic dominance, namely that $\gamma=0$ whereas one can
capture $\mathscr{U}_{\Theta}$ with a multi-fractional utility space
$\mathscr{U}^{\textsc{mf}}_{\boldsymbol{\upgamma}}$ that is a strict subset of
the utility space $\mathscr{U}_{\textsc{FSD}}$ and hence resulting in a
stochastic dominance rule in which a larger set of distribution functions can
be ordered. Let $\theta^{*}>0$ be fixed. Let $\Theta=(\theta^{*},\infty)$ and
consider a function $\widetilde{\boldsymbol{\upgamma}}:\Theta\to(0,1]$ so that
(i) it is non-decreasing, (ii)
$\widetilde{\boldsymbol{\upgamma}}(\theta)<\theta$ for every
$\theta\in\Theta$, and (iii)
$\lim_{\theta\to\theta^{*}}\widetilde{\boldsymbol{\upgamma}}(\theta)=0$. For
every $\theta\in\Theta$, we define a utility function $u_{\theta}$ via its
right derivative as

$u^{\prime}_{\theta}(x)=\begin{cases}2\widetilde{\boldsymbol{\upgamma}}(\theta)&\mbox{
if }x\leq\widetilde{\boldsymbol{\upgamma}}(\theta),\\\
1+\widetilde{\boldsymbol{\upgamma}}(\theta)&\mbox{ if
}\widetilde{\boldsymbol{\upgamma}}(\theta)<x\leq\theta,\\\
\widetilde{\boldsymbol{\upgamma}}(\theta)&\mbox{ if }x>\theta.\end{cases}$

Then, it is simple to observe that
$u_{\theta}\in\mathscr{U}^{*}_{\widetilde{\boldsymbol{\upgamma}}(\theta)}$
(see (3.7) for definition) for each $\theta\in\Theta$. Let
$\mathscr{U}_{\Theta}:=\left\\{u_{\theta}\,:\,\theta\in\Theta\right\\}$. Now,
for a moment assume that for a given constant $\gamma\in[0,1]$ the set
inclusion $\mathscr{U}_{\Theta}\subseteq\mathscr{U}^{*}_{\gamma}$ holds. Then,
property (iii) above dictates that $\gamma=0$ must be. In other words, the
only possibility to capture all the utility functions $u_{\theta}$ in a single
utility space $\mathscr{U}^{*}_{\gamma}$ associated to the fractional
$(1+\gamma)$-SD corresponds to the first stochastic dominance. On the other
hand, define the non-decreasing function

$\boldsymbol{\upgamma}(\theta)=\begin{cases}0&\mbox{ if
}\theta\leq\theta^{*},\\\ \widetilde{\boldsymbol{\upgamma}}(\theta)&\mbox{ if
}\theta>\theta^{*}.\end{cases}$

Now, we claim that
$\mathscr{U}_{\Theta}\subseteq\mathscr{U}^{\textsc{mf}}_{\boldsymbol{\upgamma}}\subset\mathscr{U}_{\textsc{FSD}}$
where clearly the second inclusion is strict. For each $\theta\in\Theta$,
define utility functions $v_{\theta}$ and $w_{\theta}$ via their right
derivatives as
$v^{\prime}_{\theta}(x)=\widetilde{\boldsymbol{\upgamma}}(\theta)$, and

$w^{\prime}_{\theta}(x)=\begin{cases}\widetilde{\boldsymbol{\upgamma}}(\theta)&\mbox{
if }x\leq\widetilde{\boldsymbol{\upgamma}}(\theta),\\\ 1&\mbox{ if
}\widetilde{\boldsymbol{\upgamma}}(\theta)<x\leq\theta,\\\ 0&\mbox{ if
}x>\theta.\end{cases}$

Then, clearly,
$v_{\theta},w_{\theta}\in\mathscr{U}^{\textsc{mf}}_{\boldsymbol{\upgamma}}$,
and therefore,
$u_{\theta}=v_{\theta}+w_{\theta}\in\mathscr{U}^{\textsc{mf}}_{\boldsymbol{\upgamma}}$
for each $\theta$ in which proves the validity of the first inclusion. To
finalize this example through visualization, we shall specify
$\widetilde{\boldsymbol{\upgamma}}(\theta)$. We set $\theta^{*}=1$ and choose
$\widetilde{\boldsymbol{\upgamma}}(\theta)=1-e^{(1-\theta)}$. It is apparent
that it is a non-decreasing function, where $1-e^{(1-\theta)}<\theta$ for
every $\theta\in\Theta$, $\lim_{\theta\to\theta^{*}}1-e^{(1-\theta)}=0$, and
$\lim_{\theta\to 1}1-e^{(1-\theta)}=1$. Now, consider exponential random
variable $E\sim\text{Exp}(1)$ and let $X=E+1$ with distribution function given
as

$F_{X}(\theta)=\boldsymbol{\upgamma}(\theta)=\begin{cases}0&\mbox{ if
}\theta<1,\\\ 1-e^{(1-\theta)}&\mbox{ if }\theta\geq 1.\end{cases}$

and

$u^{\prime}_{\theta}(x)=\begin{cases}2(1-e^{(1-\theta)})&\mbox{ if }x\leq
1-e^{(1-\theta)},\\\ 2-e^{(1-\theta)}&\mbox{ if
}1-e^{(1-\theta)}<x\leq\theta,\\\ 1-e^{(1-\theta)}&\mbox{ if
}x>\theta.\end{cases}$

$-1$$-3$$-5$$\theta^{*}$$3$$5$$7$$0$$0.2$$0.4$$0.6$$0.8$$\theta$$\boldsymbol{\upgamma}(\theta)$$-4$$-2$$0$$2$$4$$6$$8$$0$$0.5$$1$$1.5$$2$$x$$u^{\prime}_{1.01}(x)$$-4$$-2$$0$$2$$4$$6$$8$$0$$0.5$$1$$1.5$$2$$x$$u^{\prime}_{1.1}(x)$$-4$$-2$$0$$2$$4$$6$$8$$0$$0.5$$1$$1.5$$2$$x$$u^{\prime}_{3}(x)$
Figure 3: Graphs of $u^{\prime}_{\theta}$ for values $\theta=1.01$,
$\theta=1.1$ and $\theta=3$ (blue) and $\boldsymbol{\upgamma}$ (red)

##  4 MFSD: Economical Aspects

Definition 3.1 provides us with a rather abstract form for the utility class,
making it rather challenging to determine which type of utility functions it
encompasses and moreover how it should be understood within the context of
utility theory. Therefore, in the current section, we introduce a larger
(utility) space containing multi-fractional utility class and is particularly
useful in the various decision making applications.

###  4.1 Utility space $\mathscr{U}^{\textsc{cv}}_{\upgamma}$

First, we introduce a derivative-relation property that it’s root goes to the
basis utility functions $u_{t}$ introduced in Lemma 3.1.

######  Definition 4.1 ($\mathscrbf{D}(\boldsymbol{\upgamma})$-property).

Let $\boldsymbol{\upgamma}:\mathbb{R}\to[0,1]$ be a non-decreasing function.
We say that a continuous function $u:\mathbb{R}\to\mathbb{R}$ satisfies in
$\mathscrbf{D}(\boldsymbol{\upgamma})$ property if either left or right one-
sided derivatives exist at any point and, moreover

$0\leq\boldsymbol{\upgamma}(y)u^{\prime}_{\pm}(y)\leq
u^{\prime}_{\pm}(x),\,\forall\,x\leq y.$ (4.1)

(Clarify that relation (4.1) is a compact writing form of relations $\quad
0\leq\boldsymbol{\upgamma}(y)u^{\prime}_{-}(y)\leq u^{\prime}_{-}(x)$, or
$0\leq\boldsymbol{\upgamma}(y)u^{\prime}_{+}(y)\leq u^{\prime}_{+}(x)$ for all
$x\leq y$ depending on whether left or right derivatives are under
consideration.) Also, we set

$\mathscr{U}^{\textsc{cv}}_{\boldsymbol{\upgamma}}:=\Big{\\{}u:\mathbb{R}\to\mathbb{R}\,:\,\mathscrbf{D}(\boldsymbol{\upgamma})\text{
property holds}\Big{\\}}.$ (4.2)

######  Convention 4.1.

Before we continue, let us emphasize the following technical point. Given an
arbitrary non-decreasing function $\boldsymbol{\upgamma}$, according to
Proposition 3.2 item (a) every function
$u\in\mathscr{U}^{t}_{\boldsymbol{\upgamma}(t)}$ has left and right
derivatives at every point (and hence continuous) as soon as
$\boldsymbol{\upgamma}(t)\neq 0$. However, this is not true for all functions
in $\mathscr{U}^{\textsc{mf}}_{\boldsymbol{\upgamma}}$, and in fact, for the
utility functions in
$\bigcup_{t\in(-\infty,t_{0})}\mathscr{U}^{t}_{\boldsymbol{\upgamma}(t)}$
where $t_{0}:=\sup\\{t\,:\,\boldsymbol{\upgamma}(t)=0\\}$, the existence of
left and right derivatives at each point is not necessary guaranteed.
Therefore, for the practical purposes, throughout this section, we make the
convention that each function $u\in\mathscr{U}^{t}_{\boldsymbol{\upgamma}(t)}$
with $\boldsymbol{\upgamma}(t)=0$ is either left or right differentiable at
every point in $\mathbb{R}$ (although, based on Proposition 3.2, we know that
those utility functions are almost everywhere differentiable). While this
assumption may not always hold in general, from a practical perspective, it
does not affect the validity or usefulness of the forthcoming discussion.

######  Proposition 4.1.

$\mathscr{U}^{\textsc{mf}}_{\boldsymbol{\upgamma}}\subseteq\mathscr{U}^{\textsc{cv}}_{\boldsymbol{\upgamma}}$.

###### Proof.

Due to the linear structure of the elements in
$\mathscr{U}^{\textsc{mf}}_{\boldsymbol{\upgamma}}$, it is enough to show that
$\mathscr{U}^{t}_{\boldsymbol{\upgamma}(t)},\mathscr{U}^{\infty}_{\accentset{\rule{3.5pt}{1.2pt}}{\boldsymbol{\upgamma}}}\subseteq\mathscr{U}^{\textsc{cv}}_{\boldsymbol{\upgamma}}$
for every real $t$. Let $u\in\mathscr{U}^{t}_{\boldsymbol{\upgamma}(t)}$.
There are two possibilities: (i) $\boldsymbol{\upgamma}(t)\neq 0$. In this
case, $u$ admits one-sided derivatives at every point, and moreover, by very
definition, we have $0\leq\boldsymbol{\upgamma}(t)u^{\prime}_{+}(y)\leq
u^{\prime}_{+}(x)$ for all $x\leq y$. Since $\boldsymbol{\upgamma}$ is non-
decreasing, and $u^{\prime}_{+}(x)=0$ for each $x\geq t$, we have
$0\leq\boldsymbol{\upgamma}(y)u^{\prime}_{+}(y)\leq u^{\prime}_{+}(x)$ for all
$x\leq y$. (ii) the argument for the case $\boldsymbol{\upgamma}(t)=0$ is the
same as in (i) under Convention 4.1.The case
$u\in\mathscr{U}^{\infty}_{\accentset{\rule{3.5pt}{1.2pt}}{\boldsymbol{\upgamma}}}$
is similar. ∎

######  Remark 4.1.

  * (i)

Although, the space $\mathscr{U}^{\textsc{cv}}_{\boldsymbol{\upgamma}}$ is the
natural choice towards extension of the utility space
$\mathscr{U}^{*}_{\gamma}$ corresponding to the fractional $(1+\gamma)$-SD to
the context of the multi-fractional $(1+\boldsymbol{\upgamma})$-SD, however,
one has to note that, in general,
$\mathscr{U}^{\textsc{cv}}_{\boldsymbol{\upgamma}}\nsubseteq\mathscr{U}^{\textsc{mf}}_{\boldsymbol{\upgamma}}$.
In fact, we constructed an insightful example (see Section 4.3) of a function
$u\in\mathscr{U}^{\textsc{cv}}_{\boldsymbol{\upgamma}}$ so that
$u\notin\mathscr{U}^{\textsc{mf}}_{\boldsymbol{\upgamma}}$.

  * (ii)

Always $\mathscr{U}^{\textsc{cv}}_{\boldsymbol{\upgamma}}\neq\emptyset$ is a
non-empty set and in fact it includes every increasing globally concave
function. Moreover, $\mathscr{U}^{\textsc{cv}}_{\boldsymbol{\upgamma}}$ is a
positive cone but it is not a ring under the composition operator (consider as
multiplication).

  * (iii)

One advantage of Proposition 4.1 is that it provides us with simple tools to
check whether a given function $u$ is not a utility function of multi-
fractional stochastic dominance, although the question of finding easy-to-
check tests to verify a given function is a utility function in the context of
multi-fractional stochastic dominance remains valid. Later, in Section 4.4 we
will discuss and provide economical criteria (through the notion of the local
greediness) to verify a given function $u$ belongs to utility space
$\mathscr{U}^{\textsc{mf}}_{\boldsymbol{\upgamma}}$.

  * (iv)

In fact, a careful investigation of the basis utility functions $u_{t}$ and
the utility space $\mathscr{U}^{t}_{\boldsymbol{\upgamma}(t)}$ reveal that the
desirable utility space consists of functions satisfying in a more restrictive
$\mathscr{D}(\boldsymbol{\upgamma})$ property, namely that

$\displaystyle 0\leq\boldsymbol{\upgamma}(y)u^{\prime}_{+}(y)\leq
u^{\prime}_{+}(x),$ $\displaystyle\quad
0\leq\boldsymbol{\upgamma}(y)u^{\prime}_{-}(y)\leq u^{\prime}_{-}(x),$ (4.3)
$\displaystyle 0\leq\boldsymbol{\upgamma}(y)u^{\prime}_{-}(y)\leq
u^{\prime}_{+}(x),$ $\displaystyle\quad
0\leq\boldsymbol{\upgamma}(y)u^{\prime}_{+}(y)\leq
u^{\prime}_{-}(x),\quad\forall\,x<y$ $\displaystyle
0\leq\boldsymbol{\upgamma}(y)u^{\prime}_{+}(x)\leq u^{\prime}_{-}(x),$
$\displaystyle\quad\forall\,x.$

However, the utility space consisting of those functions satisfying in
relations (4.3) is rather restrictive in order to pay the way to build a
general theory. Hence, we will work within the extended framework of the
utility space $\mathscr{U}^{\textsc{cv}}_{\boldsymbol{\upgamma}}$.

######  Definition 4.2 (Multi-fractional concave stochastic dominance
(MFSD(cv))).

Let $\boldsymbol{\upgamma}:\mathbb{R}\to[0,1]$ be a non-decreasing function.
Assume that $X\sim F$ and $Y\sim G$. We say that $G$ dominates $F$ in the
sense of concave multi-fractional $(1+\boldsymbol{\upgamma})$-SD, denote by
$F\preceq^{\textsc{mf-cv}}_{(1+\boldsymbol{\upgamma})\textsc{{-SD}}}G$ if

$\mathbb{E}\left[u(X)\right]\leq\mathbb{E}\left[u(Y)\right],\quad\forall\,u\in\mathscr{U}^{\textsc{cv}}_{\boldsymbol{\upgamma}}.$

######  Remark 4.2.

(i) It is clear that
$\mathscr{U}^{*}_{\accentset{\rule{3.5pt}{1.2pt}}{\boldsymbol{\upgamma}}}\subseteq\mathscr{U}^{\textsc{mf}}_{\boldsymbol{\upgamma}}\subseteq\mathscr{U}^{\textsc{cv}}_{\boldsymbol{\upgamma}}\subseteq\mathscr{U}^{*}_{\stackunder[1pt]{$\boldsymbol{\upgamma}$}{\rule{3.5pt}{1.1pt}}}$.
Hence, in general, multi-fractional concave SD is stronger than multi-
fractional SD. In the terminologies of Section 2.3, we have the set inclusion
$\mathscrbf{E}_{\textsc{mf-
cv}}(\boldsymbol{\upgamma})\subseteq\mathscrbf{E}_{\textsc{mf}}(\boldsymbol{\upgamma})$.

(ii) Up to now, whether multi-fractional concave stochastic dominance admits
an integral form remains as an open problem, and it is under further
investigation.

######  Example 4.1.

The purpose of this example is to provide a parametrized family
$\mathscr{U}_{\Theta}$ so that the only possibility that
$\mathscr{U}_{\Theta}\subseteq\mathscr{U}^{*}_{\gamma}$ for some constant
$\gamma\in[0,1]$ is to have $\gamma=0$ (hence, corresponding to the utility
class of the FSD). However, one can find a suitable non-decreasing function
$\boldsymbol{\upgamma}$ so that

$\mathscr{U}_{\Theta}\subseteq\mathscr{U}^{\textsc{cv}}_{\boldsymbol{\upgamma}}\neq\mathscr{U}_{\textsc{FSD}}\quad\text{
and more importantly
}\quad\mathscr{U}_{\Theta}\not\subseteq\mathscr{U}^{\textsc{mf}}_{\boldsymbol{\upgamma}}.$

Let $\Theta=(0,1)$ and consider a function
$\widetilde{\boldsymbol{\upgamma}}:\Theta\to(0,1)$ with $\lim_{\theta\to
0}\widetilde{\boldsymbol{\upgamma}}(\theta)=0$. For every $\theta\in\Theta$,
we define a utility function $u_{\theta}$ via its right derivative as

$u^{\prime}_{\theta}(x)=\begin{cases}u_{\theta,1}^{\prime}(x)&\mbox{ if
}-\infty<x<\theta,\\\ x&\mbox{ if }\theta\leq x\leq\sqrt{\theta},\\\
u_{\theta,2}^{\prime}(x)&\mbox{ if }\sqrt{\theta}<x<\infty,\\\ \end{cases}$

where $u_{\theta,1}^{\prime}$ and $u_{\theta,2}^{\prime}$ are arbitrary non-
negative and non-increasing functions, provided they meet the requirements of
$u_{\theta}$ being continuous. Additionally,
$\inf_{x\in\mathbb{R}}u_{\theta,1}^{\prime}(x)=\sqrt{\theta}$ while
$\sup_{x\in\mathbb{R}}u_{\theta,2}^{\prime}(x)=\theta$. Thus, for a given
$\theta$, it is straightforward to see that
$u_{\theta}\in\mathscr{U}^{*}_{\gamma}$ for some constant $\gamma$ (i.e, we
have $0\leq\gamma u^{\prime}_{\theta}(y)\leq u^{\prime}_{\theta}(x),\,\forall
x\leq y$) if and only if the condition

$\sqrt{\theta}\geq\gamma$ (4.4)

holds true. Let
$\mathscr{U}_{\Theta}:=\left\\{u_{\theta}\,:\,\theta\in\Theta\right\\}$. Now,
observe that, for some constant $\gamma$, the validity of the set inclusion
$\mathscr{U}_{\Theta}\subseteq\mathscr{U}^{*}_{\gamma}$ dictates that
$\gamma=0$ must be. This conclusion directly follows from (4.4) when
$\theta\to 0$. Hence, the only scenario where $u_{\theta}$ for every
$\theta\in\Theta$ included within a single utility space associated with the
fractional $(1+\gamma)$-SD corresponds to the utility class
$\mathscr{U}_{\textsc{FSD}}=\mathscr{U}^{*}_{\gamma=0}$. Next, consider the
continuos non-decreasing function

$\displaystyle\boldsymbol{\upgamma}(x)=\begin{cases}0&\mbox{ if }x<0,\\\
x&\mbox{ if }0\leq x\leq 1,\\\ 1&\mbox{ if }x>1.\end{cases}$

We first show that
$\mathscr{U}_{\Theta}\subseteq\mathscr{U}^{\textsc{cv}}_{\boldsymbol{\upgamma}}$.
In order to do this, note that, for each $\theta\in\Theta$, the condition

$\theta\geq u^{\prime}_{\theta}(y)\boldsymbol{\upgamma}(y)=y\times
y,\quad\forall y\in[\theta,\sqrt{\theta}]$ (4.5)

is both necessary and sufficient for the validity of
$u_{\theta}\in\mathscr{U}^{\textsc{cv}}_{\boldsymbol{\upgamma}}$ in which
obviously holds true. Lastly, we show that
$\mathscr{U}_{\Theta}\not\subseteq\mathscr{U}^{\textsc{mf}}_{\boldsymbol{\upgamma}}$.
Fix $\theta\in\Theta$. First, when $y=\sqrt{\theta}$ and $x=\theta<y$, it
follows from (4.5) that

$\frac{1}{\boldsymbol{\upgamma}(y)}=\frac{1}{\sqrt{\theta}}=\frac{u_{\theta}^{\prime}(y)}{u_{\theta}^{\prime}(x)}.$
(4.6)

Next, we discuss that
$u_{\theta}\notin\mathscr{U}^{\text{Union}}_{\boldsymbol{\upgamma}}$. Note
that $u_{\theta}$ is not globally concave function and hence
$u_{\theta}\notin\mathscr{U}^{+\infty}_{\accentset{\rule{3.5pt}{1.2pt}}{\boldsymbol{\upgamma}}=1}$.
Now, by contradiction, assume that there exists $t\in\mathbb{R}$ such that
$u_{\theta}\in\mathscr{U}^{t}_{\boldsymbol{\upgamma}(t)}$. Therefore, based on
the equality in (4.6), this is only possible when $t=\sqrt{\theta}$. However,
this leads to a contradiction, as the derivative of the subsequent concave
segment $u^{\prime}_{2,\theta}$ of $u_{\theta}$ is not necessarily equal to 0
after $\sqrt{\theta}$. Hence, $u_{\theta}$ can only belong to
$\mathscr{U}^{\textsc{mf}}_{\boldsymbol{\upgamma}}$ if it can be represented
as a linear combination of functions that belong to
$\mathscr{U}^{\text{Union}}_{\boldsymbol{\upgamma}}$. For a moment, assume
that $u_{\theta}=\lambda_{1}v_{1}+\lambda_{2}v_{2}$ where
$v_{1}\in\mathscr{U}^{t=\sqrt{\theta}}_{\boldsymbol{\upgamma}(t)=\sqrt{\theta}}$
and $v_{2}\in\mathscr{U}^{*}_{1}$. Then, from the mediant inequity (4.14) we
obtain

$\frac{u_{\theta}^{\prime}(\sqrt{\theta})}{u_{\theta}^{\prime}(\theta)}=\frac{\lambda_{1}v_{1}(\sqrt{\theta})+\lambda_{2}v_{2}(\sqrt{\theta})}{\lambda_{1}v_{1}(\theta)+\lambda_{2}v_{2}(\theta)}=\frac{\lambda_{1}\sqrt{\theta}+\lambda_{2}v_{2}(\sqrt{\theta})}{\lambda_{1}\theta+\lambda_{2}v_{2}(\theta)}<\frac{1}{\sqrt{\theta}}$

since $\frac{v_{2}(\sqrt{\theta})}{v_{2}(\theta)}\leq 1$ for any
$v_{2}\in\mathscr{U}^{*}_{1}$ by definition in which violates (4.6). The
general case can be discussed in a similar way relying on mediant inequality
and the facts that any linear combination should contain at least one piece
from the utility space $\mathscr{U}^{\sqrt{\theta}}_{\sqrt{\theta}}$ and
another piece of $\mathscr{U}^{*}_{1}$.

$\theta$$\theta$$\sqrt{\theta}$$\sqrt{\theta}$$\inf_{x\in\mathbb{R}}u^{\prime}_{1,\theta}$$u_{\theta}^{\prime}(x)$$x$$\sup{}_{x\in\mathbb{R}}u^{\prime}_{2,\theta}$

###  4.2 Local Greediness

In this section, we introduce the novel notion of local greediness of a
utility function at an arbitrary given point $x\in\mathbb{R}$ that provides a
meaning to the function $\boldsymbol{\upgamma}$ from a decision-making
perspective in the multi-fractional stochastic dominance framework. The notion
of local greediness (local non-concavity) at the location $x\in\mathbb{R}$
characterizes the greediness behaviour of decision-makers over half-open
intervals $(x,\infty)$ rather than the entire real line as in [CCM05]. They
define the index of global greediness (non-concavity) for a strictly
increasing function $u:\mathbb{R}\to\mathbb{R}$ as follows:

$G_{u}:=\sup_{x_{1}<x_{2}\leq
x_{3}<x_{4}}\left(\frac{u\left(x_{4}\right)-u\left(x_{3}\right)}{x_{4}-x_{3}}/\frac{u\left(x_{2}\right)-u\left(x_{1}\right)}{x_{2}-x_{1}}\right).$
(4.7)

It is known that always $G_{u}\geq 1$ and $G_{u}=1$ if and only if $u$ is a
concave function. Moreover, when $u$ is differentiable, it is true that
$G_{u}=\sup_{x\leq y}u^{\prime}(y)/u^{\prime}(x)$ (see [CCM05]). In this
regard, the index $G_{u}$ globally determines the maximal magnitude of the
increase in the slope of the utility function (marginal utility) over its
entire domain. Let $\gamma\in[0,1]$ be a constant. Consider the utility class
$\mathscr{U}^{*}_{\gamma}$ corresponding to fractional $(1+\gamma)$-SD. It can
be easily seen that the parameter $\gamma$ bounds the level of greediness
(non-concavity) $G_{u}$ for all $u\in\mathscr{U}^{*}_{\gamma}$ through the
inequality $G_{u}\leq 1/\gamma$. Inspired from [CCM05], we define a local
version of the greediness index as the following.

######  Definition 4.3 (Discrete local greediness).

Let $u$ be a strictly increasing function. The discrete local greediness at
location $x\in\mathbb{R}$ is defined as:

$G^{d}_{\text{loc}}(u;x):=\sup_{x<x_{1}<x_{2}\leq
x_{3}<x_{4}}\left(\frac{u(x_{4})-u(x_{3})}{x_{4}-x_{3}}/\frac{u(x_{2})-u(x_{1})}{x_{2}-x_{1}}\right).$
(4.8)

######  Remark 4.3.

  * (i)

In the context of decision theory, $G_{u}$ serves as a global measure,
capturing the maximal proportional increase in a decision maker’s valuation of
an additional cent as wealth transitions from lower to higher values. Hence,
$G_{u}$ takes into account all possible values of wealth. This makes it a
natural notion for measuring the global greediness of a decision maker over
$\mathbb{R}$. Building on the concept of $G_{u}$, for a fixed wealth level
$x\in\mathbb{R}$, the measure $G^{d}_{\text{loc}}(u;x)$ quantifies the maximum
amplification of a decision maker’s valuation of an extra cent within a
specific region — specifically when their wealth exceeds $x$, i.e., for
$(x,\infty)\subset\mathbb{R}$. We direct the reader to [MY19] for related
concepts of local greediness within the framework of cumulative prospect
theory.

  * (ii)

The discrete local greediness $G^{d}_{\text{loc}}$ can be well-defined for
functions not necessary being strictly increasing. For example, it fully makes
sense to talk about $G^{d}_{\text{loc}}(u;\cdot)$ for functions
$u\in\mathscr{U}^{\textsc{cv}}_{\boldsymbol{\upgamma}}$ which contains
functions which allows for non strictly increasing segments. We will discuss
it in a great details later in Remark 4.5.

######  Proposition 4.2.

  * (a)

Clearly, $G^{d}_{\text{loc}}(u;\cdot)\leq G_{u}$ where the global index of the
greediness $G_{u}$ is given by (4.7).

  * (b)

Function $x\mapsto G^{d}_{\text{loc}}(u;x)$ is decreasing, i.e.,
$G^{d}_{\text{loc}}(u;x_{2})\leq G^{d}_{\text{loc}}(u;x_{1})$ for $x_{1}\leq
x_{2}$. Moreover, $G^{d}_{\text{loc}}(u;x)\to G_{u}$ as $x\to-\infty$.

  * (c)

For every $x$, $G^{d}_{\text{loc}}(u;x)\geq 1$ and,
$G^{d}_{\text{loc}}(u;x)=1$ if and only if $u$ is concave on $(x,\infty)$. In
addition, $u$ is concave if and only if $G^{d}_{\text{loc}}(u;x)=1$ for every
$x$.

  * (d)

Function $x\mapsto G^{d}_{\text{loc}}(u;x)$ is right continuous.

###### Proof.

Items (a) and (b) are obvious. Next, we prove (c). Since $u$ is strictly
increasing, hence for every given $x$, there is a point $y=y_{x}\in(x,\infty)$
that $u$ is differentiable at $y$. Hence, by letting $x_{1}\to y^{-}$ and
$x_{4}\to y^{+}$ we arrive to the claim. The validity of the second statement
is just rely on the basic definition of concave function in terms monotonicity
of quotients. Lastly, we show part (d). Fix $x_{0}\in\mathbb{R}$. Let
$x_{n}\to x^{+}_{0}$. Then, based on parts (b), (c) and moving to a
subsequence, we have $\lim_{n\to\infty}G^{d}_{\text{loc}}(u;x_{n})\leq
G^{d}_{\text{loc}}(u;x_{0})$. Now, let $\varepsilon>0$ be arbitrary. Then,
there exist four points $x_{0}<x_{1}<x_{2}\leq x_{3}<x_{4}$ such that

$G^{d}_{\text{loc}}(u;x_{0})-\varepsilon<\frac{u(x_{4})-u(x_{3})}{x_{4}-x_{3}}/\frac{u(x_{2})-u(x_{1})}{x_{2}-x_{1}}.$

On the other hand side, there exists $n_{0}\in\mathbb{N}$ such that
$x_{n}<x_{1}$ for every $n\geq n_{0}$. Therefore,

$\sup_{n\geq
n_{0}}G^{d}_{\text{loc}}(u;x_{n})\geq\frac{u(x_{4})-u(x_{3})}{x_{4}-x_{3}}/\frac{u(x_{2})-u(x_{1})}{x_{2}-x_{1}}.$

Hence, by part (b), we obtain
$\lim_{n\to\infty}G^{d}_{\text{loc}}(u;x_{n})>G^{d}_{\text{loc}}(u;x_{0})-\varepsilon$.
Letting $\varepsilon\to$ we achieve the claim. ∎

Although the discrete local greediness $G^{d}_{\text{loc}}$ provides us with a
measure of greediness attached to the location however it is not well suited
to work with functions of our interest in the space
$\mathscr{U}^{\textsc{cv}}_{\boldsymbol{\upgamma}}$. This is due to the fact
that functions in the latter space are defined via properties of the one-sided
derivatives. Hence, we take a different route to define a novel notion of the
local nature which is more compatible with the
$\mathscr{U}^{\textsc{cv}}_{\boldsymbol{\upgamma}}$ structure. First, we need
the following simple result taken from [Min07b] that opens the door whenever
continuity assumption is available.

######  Proposition 4.3.

Let $u$ is a continuous function with one-sided left derivatives
$u^{\prime}_{-}$ exist at each point $x\in\mathbb{R}$. Then, $u$ is concave if
and only if

$u_{-}^{\prime}(x)\geq u_{-}^{\prime}(y),\text{ for all }x\leq y.$

The similar conclusion holds true with the right derivative $u^{\prime}_{+}$.

Proposition 4.3 propels us to define the following local measure of greediness
that provides us with a continuous version of that given in Definition 4.3.

######  Definition 4.4 (Continuous local greediness).

Let $u$ be a strictly increasing function with either left or right
derivatives exist at each point $x\in\mathbb{R}$. We define the continuous
local greediness at location $x$ as

$G^{c}_{\text{loc}}(u;x):=\sup_{x<y<z}\frac{u^{\prime}_{\pm}(z)}{u^{\prime}_{\pm}(y)}.$
(4.9)

######  Remark 4.4.

  * (i)

There is no ambiguity in Definition 4.4 in the sense that when function $u$
admits both left and right derivatives at every point, then one can
straightforwardly show that the definition of continuous local greediness
$G^{c}_{\text{loc}}(u;\cdot)$ is independent of the choice of one-sided
derivatives.

  * (ii)

One has to note that the continuous local greediness is invariant under any
linear transformation.

Now, we are ready to state that two notions of discrete and continuous local
greediness coincide in our framework.

######  Proposition 4.4.

Let $u$ be a continuous, strictly increasing and either left or right
derivatives exist at each point. Then

$G^{d}_{loc}(u,x)=G^{c}_{loc}(u,x),\quad\text{ for all }x\in\mathbb{R}.$

###### Proof.

It is clear that $G^{c}_{\text{loc}}(u;x)\leq G^{d}_{\text{loc}}(u;x)$. Thus,
to complete the proof, it remains to show $G^{c}_{loc}(u;x)\geq
G^{d}_{loc}(u;x)$. We assume that $u$ is right differentiable at any points.
The case of left differentiable is the same. Let $\varepsilon>0$. Then, by the
definition of $G^{d}_{loc}(u,x)$, there exist four points, $x\leq
x_{1}<x_{2}\leq x_{3}<x_{4}$ such that

$G:=\left(\frac{u(x_{4})-u(x_{3})}{x_{4}-x_{3}}/\frac{u(x_{2})-u(x_{1})}{x_{2}-x_{1}}\right)>G^{d}_{loc}(u;x)-\varepsilon.$

On the other hand, by Theorem 7.1, there exist $c_{1},c_{2}\in(x_{1},x_{2})$
and $c_{3},c_{4}\in(x_{3},x_{4})$ such that

$u^{\prime}_{+}(c_{1})\leq\frac{u(x_{2})-u(x_{1})}{x_{2}-x_{1}}\leq
u^{\prime}_{+}(c_{2})\quad\text{ and }\quad
u^{\prime}_{+}(c_{3})\leq\frac{u(x_{4})-u(x_{3})}{x_{4}-x_{3}}\leq
u^{\prime}_{+}(c_{4})$

Hence,

$G^{c}_{\text{loc}}(u;x)\geq\sup_{x<y<z}\frac{u^{\prime}_{+}(z)}{u^{\prime}_{+}(y)}\geq\frac{u^{\prime}_{+}(c_{4})}{u^{\prime}_{+}(c_{1})}\geq
G>G^{d}_{loc}(u,x)-\varepsilon.$

By letting $\varepsilon\to 0$, we complete the proof. ∎

Hereafter, in our framework, Proposition 4.4 allows us to use notation
$G_{\text{loc}}(u;\cdot)$ interchangeably without any ambiguity.

######  Proposition 4.5.

Let $u_{1},\dots,u_{n}$ be $n$ strictly increasing continuous functions with
either left or right derivatives exist at each point $x\in\mathbb{R}$. Then,
for non-negative constants $\lambda_{1},\dots,\lambda_{n}$, the following
statements hold:

  * (a)

For every x, we have
$G_{\text{loc}}(\lambda_{1}u_{1}+\cdots+\lambda_{n}u_{n};x)\leq\max_{1\leq
k\leq n}G_{\text{loc}}(u_{k};x)$.

  * (b)

For every $x$, $G_{\text{loc}}(\cdot;x)$ is a sub-linear functional in the
sense that

$G_{\text{loc}}(\lambda_{1}u_{1}+\cdots+\lambda_{n}u_{n};x)\leq\sum_{k=1}^{n}G_{\text{loc}}(u_{k};x).$

###### Proof.

(a) It is a direct application of the generalized mediant inequality, see
Lemma 7.1. (b) It is also a direct application of part (a) and the fact that
$\max\\{a,b\\}\leq a+b$ for every two non-negative real numbers. ∎

######  Remark 4.5.

It is worth to highlight that one can talk about the both notions of local
greediness in the context of the (utility) space
$\mathscr{U}^{\textsc{cv}}_{\boldsymbol{\upgamma}}$ where in general the
assumption of strict increasing may lost. For example, note that from the set
condition

$0\leq\boldsymbol{\upgamma}(y)u_{\pm}^{\prime}(y)\leq
u_{\pm}^{\prime}(x),\,\quad\forall\,x\leq y$ (4.10)

one can easily see that if $u_{\pm}^{\prime}(x)=0$ then
$u_{\pm}^{\prime}(y)=0$ for all $y>x$. Therefore, when considering the ratio
$u_{\pm}^{\prime}(y)/u_{\pm}^{\prime}(x)$, $x\leq y$, the only problematic
scenario we may encounter is when we have a $0/0$ case. Hence, the convention
$0/0=1$ removes all the ambiguities. However, this convention does not affect
the measurement of non-concavity, as the lower bound, $1$ is reached if (only
if) the function is concave and constant functions are concave. In addition,
due to convention above, Proposition 4.5 remains valid for the functions
chosen from the utility space
$\mathscr{U}^{\textsc{cv}}_{\boldsymbol{\upgamma}}$. We will make use of this
several instance later on in Section 4.

######  Corollary 4.1.

Let $u\in\mathscr{U}^{\textsc{cv}}_{\boldsymbol{\upgamma}}$. Then, for every
$x$, $G^{d}_{loc}(u;x)=G^{c}_{loc}(u;x)=:G_{\text{loc}}(u;x)$ and moreover,

$G_{\text{loc}}(u;x)\leq\frac{1}{\boldsymbol{\upgamma}(x)}.$ (4.11)

###### Proof.

The bound in (4.11) follows directly from the set condition (4.1) and the fact
that $\boldsymbol{\upgamma}$ is a non-decreasing function. ∎

######  Remark 4.6.

  * (i)

In contrast to fractional $(1+\gamma)$-SD, the multi-fractional
$(1+\boldsymbol{\upgamma})$-SD provides local upper bounds to the steepness of
the convex segments depending on the location, let’s $x$. Under the assumption
that $\boldsymbol{\upgamma}$ is a non-decreasing, the utility function
$u\in\mathscr{U}^{\textsc{mf}}_{\boldsymbol{\upgamma}}$ (and hence
$u\in\mathscr{U}^{\textsc{cv}}_{\boldsymbol{\upgamma}}$) is allowed to have
less steeper convex segments as one moves to the right hand side of the real
axis. This is perfectly in line with the local interpolation property of MFSD
(see Example 2.2). Essentially, the non-decreasing $\boldsymbol{\upgamma}$
function enforces a stronger dominance (closer to FSD) on the lower values of
the real line, while the dominance progressively weakens (closer to SSD) as we
move towards the higher values on the real line.

  * (ii)

Analogous to the fact that fractional $(1+\upgamma)$-SD controls the global
index of greediness (see (4.7)), in our context, the multi-fractional
$(1+\boldsymbol{\upgamma})$-SD controls from above the local greediness
according to Corollary 4.1.

  * (iii)

The multi-fractional $(1+\boldsymbol{\upgamma})$-SD more informatively
(locally) bounds the size of upward jumps in the derivative $u^{\prime}$ (of
the utility function $u\in\mathscr{U}^{\textsc{mf}}_{\boldsymbol{\upgamma}}$)
at kink points based on the position of the kink. For example, let
$x^{*}_{1}<x^{*}_{2}$ be two points where $u$ has kinks, then

$\frac{u_{+}^{\prime}(x^{*}_{1})}{u_{-}^{\prime}(x^{*}_{1})}\leq\frac{1}{\boldsymbol{\upgamma}(x^{*}_{1})}\quad\text{
and
}\quad\frac{u_{+}^{\prime}(x^{*}_{2})}{u_{-}^{\prime}(x^{*}_{2})}\leq\frac{1}{\boldsymbol{\upgamma}(x^{*}_{2})},$

where we have
$\boldsymbol{\upgamma}(x^{*}_{2})\geq\boldsymbol{\upgamma}(x^{*}_{1})$. Such a
feature is not available in the context of fractional $(1+\gamma)$-SD and the
ration $1/\gamma$ provides an universal bound independent of the location of
kinks.

Therefore, in summary, we can conclude that the MFSD approach makes use of the
additional information about the location of the non-concave segments in the
utility function $u$ to carry out a more precise interpolation between first
and second-order stochastic dominance rules. This, in turn, results in a more
informative decision making rule.

We end this section with the introducing the greediness metric on the utility
space $\mathscr{U}^{\textsc{cv}}_{\boldsymbol{\upgamma}}$. We expect that such
metric provides us with new machinery to study multi-fractional utility
functions. This subject is under further investigation.

######  Definition 4.5.

Let $u,v\in\mathscr{U}^{\textsc{cv}}_{\boldsymbol{\upgamma}}$. We define the
greediness distance $d_{\textbf{G}}$ as follows:

$d_{\text{G}}(u,v):=\sup_{x\in\mathbb{R}}\Big{|}G_{\text{loc}}(u;x)-G_{\text{loc}}(v;x)\Big{|}.$
(4.12)

######  Proposition 4.6.

$(\mathscr{U}^{\textsc{cv}}_{\boldsymbol{\upgamma}},d_{\text{G}})$ is a
pseudometric space.

###### Proof.

Clearly, $d_{\text{G}}(u,v)\geq 0$, $d_{\text{G}}(u,u)=0$ and
$d_{\text{G}}(u,v)=d_{\text{G}}(v,u)$. Also, the triangular inequality
$d_{\text{G}}(u,v)\leq d_{\text{G}}(u,w)+d_{\text{G}}(w,v)$ is a direct
consequence of the triangular inequality on the real line. Lastly, one has to
clearly note that $d_{\text{G}}(u,v)=0$ does not necessary imply that $u=v$ (a
simple scenario is to take $v(x)=u(L(x))$ where $L$ is a any linear
transformation). ∎

###  4.3 An illustrating (counter) example

In this section, we construct a non-decreasing function
$\boldsymbol{\upgamma}$ and a function
$u\in\mathscr{U}^{\textsc{cv}}_{\boldsymbol{\upgamma}}$ so that
$u\notin\mathscr{U}^{\textsc{mf}}_{\boldsymbol{\upgamma}}$. In addition, we
build two distribution functions $F$ and $G$ so that for the constructed
function $u$ the non-negativity requirement
$\mathbb{E}_{G}[u]-\mathbb{E}_{F}[u]\geq 0$ fails. This observation has a
direct consequence that the class
$\mathscr{U}^{\textsc{cv}}_{\boldsymbol{\upgamma}}$ cannot serve (although as
one may naturally expect) as a utility class for the multi-fractional
$(1+\boldsymbol{\upgamma})$-SD.

######  Example 4.2.

Consider the following non-decreasing step function

$\displaystyle\boldsymbol{\upgamma}(x)=\begin{cases}0&\mbox{ if }x<0,\\\
1/4&\mbox{ if }0\leq x\leq 2/4,\\\ 2/3&\mbox{ if }2/4<x\leq 1,\\\ 1&\mbox{ if
}1<x,\\\ \end{cases}$

and the distribution functions:

$\displaystyle F(x)=\begin{cases}0&\mbox{ if }x<0,\\\ 4/10&\mbox{ if }0\leq
x<2/4,\\\ 7/10&\mbox{ if }2/4\leq x<1,\\\ 1&\mbox{ if }1\leq x,\\\
\end{cases}\quad\text{ and }\quad G(x)=\begin{cases}0&\mbox{ if }x<1/4,\\\
5/10&\mbox{ if }1/4\leq x<3/4,\\\ 1&\mbox{ if }3/4\leq x,\\\ \end{cases}$

The graphical representations of $F$, $G$, and $\boldsymbol{\upgamma}$ are
given below:

$1/4$$2/4$$3/4$$1$$c$$4c$$3/2\
c$$u^{\prime}(x)$$1/4$$2/4$$3/4$$1$$\boldsymbol{\upgamma}(x)$$1/4$$2/3$$1/4$$2/4$$3/4$$1$$1$$7/10$$4/10$$5/10$$+$$+$$-$$-$$F(x)$$G(x)$

It is straightforward to check that
$F\preceq^{\textsc{mf}}_{(1+\boldsymbol{\upgamma})\textsc{{-SD}}}G$, i.e., the
integral condition (2.1) holds. In fact, $F$ and $G$ have crossing points at
$1/4$, $2/4$ and $3/4$ and the following integral condition is in place

$\displaystyle\int_{1/4}^{2/4}\left(F(x)-G(x)\right)_{-}dx=\int_{1/4}^{2/4}(1/10)dx\leq\underbrace{1/4}_{\boldsymbol{\upgamma}(2/4)}\int_{0}^{1/4}4/10dx=1/4\int_{0}^{1/4}\left(F(x)-G(x)\right)_{+}dx$

$\displaystyle\int_{0}^{1}\left(F(x)-G(x)\right)_{-}dx$
$\displaystyle=\int_{1/4}^{2/4}1/10dx+\int_{3/4}^{1}3/10dx$
$\displaystyle\leq\boldsymbol{\upgamma}(1)\int_{0}^{1}\left(F(x)-G(x)\right)_{+}dx=2/3\Big{\\{}\int_{0}^{1/4}4/10dx+\int_{2/4}^{3/4}2/10dx\Big{\\}}.$

Next consider the following continuous function defined through its right
derivative:

$u^{\prime}(x)=\begin{cases}c&\mbox{ if }0\leq x<1/4,\\\ 4c&\mbox{ if }1/4\leq
x<2/4,\\\ c&\mbox{ if }2/4\leq x<3/4,\\\ 3c/2&\mbox{ if }3/4\leq x\leq 1\\\
\end{cases}$ (4.13)

where $c$ can be taken any positive real number (its value is immaterial).
Note that $u$ satisfies the condition
$\boldsymbol{\upgamma}(y)u^{\prime}(y)\leq u^{\prime}(x)$, for every $x\leq
y$, where hereafter all the derivatives stand for the right derivative. To see
this, observe that,

  * (i)

If $0\leq x<1/4$, then we have

$\frac{u^{\prime}(x)}{u^{\prime}(y)}=\begin{cases}1&\mbox{ if }0\leq y<1/4,\\\
1/4&\mbox{ if }1/4\leq y<2/4,\\\ 1&\mbox{ if }2/4\leq y<3/4,\\\ 2/3&\mbox{ if
}3/4\leq y\leq 1\\\ \end{cases}$

therefore, $\frac{u^{\prime}(x)}{u^{\prime}(y)}\geq 1/4$ on $[0,2/4)$ and,
$\frac{u^{\prime}(x)}{u^{\prime}(y)}\geq 2/3$ on $[2/4,1].$

  * (ii)

If $1/4\leq x<2/4$, then it is straightforward to see that the condition holds
since $4c>c$ and $4c>3c/2$.

  * (iii)

If $2/4\leq x<3/4$, then

$\frac{u^{\prime}(x)}{u^{\prime}(y)}=\begin{cases}1&\mbox{ if }2/4\leq
y<3/4,\\\ 2/3&\mbox{ if }3/4\leq y\leq 1\\\ \end{cases}$

thus, $\frac{u^{\prime}(x)}{u^{\prime}(y)}\geq 2/3$ on $[2/4,1]$.

However, we have

$\displaystyle\mathbb{E}_{G}[u]-\mathbb{E}_{F}[u]=$ $\displaystyle
c\left[\int_{0}^{1/4}4/10dx+\int_{2/4}^{3/4}2/10dx-4\int_{1/4}^{2/4}1/10dx-3/2\int_{3/4}^{1}3/10dx\right]=-c/16<0.$

The reason is that although function $u$ is continuous and it is
differentiable everywhere except countably many points, it cannot be written
as a non-negative linear combination of the elements in
$\mathscr{U}^{\text{Union}}_{\boldsymbol{\upgamma}}$. To see this first note
that

$\mathscr{U}^{\text{Union}}_{\boldsymbol{\upgamma}}=\mathscr{U}^{t=\frac{2}{4}}_{\boldsymbol{\upgamma}(2/4)=\frac{1}{4}}\cup\mathscr{U}^{+\infty}_{\accentset{\rule{3.5pt}{1.2pt}}{\boldsymbol{\upgamma}}}=\mathscr{U}^{2/4}_{1/4}\cup\mathscr{U}_{2/3}.$

Then, clearly $u\not\in\mathscr{U}^{\frac{2}{4}}_{\frac{1}{4}}$ since
$u^{\prime}(x)\not=0$ for $x>2/4$. Also, $u\not\in\mathscr{U}_{\frac{2}{3}}$,
since $\frac{u^{\prime}(x)}{u^{\prime}(y)}=1/4\not\geq 2/3$ on $[0,2/4)$.
Thus, $u\not\in\mathscr{U}^{\text{Union}}_{\boldsymbol{\upgamma}}.$ Next, we
show that $u\notin\mathscr{U}^{\textsc{mf}}_{\boldsymbol{\upgamma}}$. For now,
let’s assume that $u=\lambda_{1}u_{1}+\lambda_{2}u_{2}$ where (without loss of
generality) $u_{1}\in\mathscr{U}^{\frac{2}{4}}_{\frac{1}{4}}$ and
$u_{2}\in\mathscr{U}_{\frac{2}{3}}$ and $\lambda_{1},\lambda_{2}>0$. Then by
the linearity of differentiation we have the following system of equations:

$\displaystyle\frac{u^{\prime}(x)}{u^{\prime}(y)}$
$\displaystyle=\frac{1}{4}=\frac{\lambda_{1}u_{1}^{\prime}(x)+\lambda_{2}u_{2}^{\prime}(x)}{\lambda_{1}u_{1}^{\prime}(y)+\lambda_{2}u_{2}^{\prime}(y)},\quad
x<1/4\leq y<2/4$ $\displaystyle\frac{u^{\prime}(x)}{u^{\prime}(y)}$
$\displaystyle=4=\frac{\lambda_{1}u_{1}^{\prime}(x)+\lambda_{2}u_{2}^{\prime}(x)}{\lambda_{2}u_{2}^{\prime}(y)},\quad
1/4\leq x<2/4\leq y<3/4$ $\displaystyle\frac{u^{\prime}(x)}{u^{\prime}(y)}$
$\displaystyle=\frac{2}{3}=\frac{u_{2}^{\prime}(x)}{u_{2}^{\prime}(y)},\quad
2/4\leq x<3/4\leq y.$

Let $x\leq y<2/4$. If
$\frac{u_{1}^{\prime}(x)}{u_{1}^{\prime}(y)}\neq\frac{u_{2}^{\prime}(x)}{u_{2}^{\prime}(y)}$,
then by the mediant inequality (see Lemma 7.1), we have

$\frac{\lambda_{1}u_{1}^{\prime}(x)+\lambda_{2}u_{2}^{\prime}(x)}{\lambda_{1}u_{1}^{\prime}(y)+\lambda_{2}u_{2}^{\prime}(y)}>\min\left\\{\frac{u_{1}^{\prime}(x)}{u_{1}^{\prime}(y)},\frac{u_{2}^{\prime}(x)}{u_{2}^{\prime}(y)}\right\\}\geq\frac{1}{4},$
(4.14)

and otherwise, if
$\frac{u_{1}^{\prime}(x)}{u_{1}^{\prime}(y)}=\frac{u_{2}^{\prime}(x)}{u_{2}^{\prime}(y)}$,
then the mediant inequality yields that

$\frac{u_{1}^{\prime}(x)}{u_{1}^{\prime}(y)}=\frac{u_{2}^{\prime}(x)}{u_{2}^{\prime}(y)}=\frac{\lambda_{1}u_{1}^{\prime}(x)+\lambda_{2}u_{2}^{\prime}(x)}{\lambda_{1}u_{1}^{\prime}(y)+\lambda_{2}u_{2}^{\prime}(y)}=1/4$

which is a contradiction with the fact that
$u_{2}\in\mathscr{U}_{\frac{2}{3}}$. Later on in Section 4.4, Theorem 4.1, we
provide an elegant device that allows one to check whether a given function
$u\in\mathscr{U}^{\textsc{mf}}_{\boldsymbol{\upgamma}}$.

######  Remark 4.7.

(Would the continuity assumption on $\boldsymbol{\upgamma}$ help?) It is
natural that one think of that the continuity assumption on the function
$\boldsymbol{\upgamma}$ would eliminate the pathological members of
$\mathscr{U}^{\textsc{cv}}_{\boldsymbol{\upgamma}}$ such as $u$ given in
(4.13) that results to negative expected utilities to achieve the set equality
$\mathscr{U}^{\textsc{cv}}_{\boldsymbol{\upgamma}}=\mathscr{U}^{\textsc{mf}}_{\boldsymbol{\upgamma}}$.
However, the short answer is no. In fact, the properties of
$\boldsymbol{\upgamma}$ only matters on the negative intervals i.e.,
$\\{x:G(x)\geq F(x)\\}$. Hence, as long as the $\boldsymbol{\upgamma}$ is kept
same on $[1/4,2/4]$ and $[3/4,1]$ replacing $\boldsymbol{\upgamma}$ with any
arbitrary function such as the following,

$1/4$$2/4$$3/4$$1$$\boldsymbol{\upgamma}(x)$$1/4$$2/3$

would not be enough to remove (4.13) from the class
$\mathscr{U}^{\textsc{cv}}_{\boldsymbol{\upgamma}}$. Thus, one can conclude
that the space $\mathscr{U}^{\textsc{cv}}_{\boldsymbol{\upgamma}}$ even cannot
serve as a utility class for the multi-fractional
$(1+\boldsymbol{\upgamma})$-SD under the extra continuity assumptions on
$\boldsymbol{\upgamma}$.

######  Remark 4.8.

In this remark, we discuss a fruitful observation that possibly every function
$u\in\mathscr{U}^{\textsc{cv}}_{\boldsymbol{\upgamma}}$ can be brought as
closely as possible to a function in the utility space
$\mathscr{U}^{\textsc{mf}}_{\boldsymbol{\upgamma}}$ in terms of the local
greediness metric. More precisely, for every given $\epsilon>0$, we construct
a function $w_{\epsilon}\in\mathscr{U}^{\textsc{mf}}_{\boldsymbol{\upgamma}}$
so that for the function
$u\in\mathscr{U}^{\textsc{cv}}_{\boldsymbol{\upgamma}}\setminus\mathscr{U}^{\textsc{mf}}_{\boldsymbol{\upgamma}}$
given as in (4.13) we have

$d_{G}(u,w_{\epsilon})<\epsilon.$

To this end, first we compute $G_{\text{loc}}(u;x)$ from (4.13) as the
following,

$G_{\text{loc}}(u;x)=\begin{cases}4&\mbox{ if }0\leq x<1/4,\\\ 3/2&\mbox{ if
}1/4\leq x<2/4,\\\ 3/2&\mbox{ if }2/4\leq x<3/4,\\\ 1&\mbox{ if }3/4\leq x\leq
1.\\\ \end{cases}$

As a building blocks of such
$w_{\epsilon}\in\mathscr{U}^{\textsc{mf}}_{\boldsymbol{\upgamma}}$ we
introduce the following functions:

$u_{1}^{\prime}(x)=\begin{cases}c&\mbox{ if }0\leq x<1/4,\\\ 4c&\mbox{ if
}1/4\leq x<2/4,\\\ 0&\mbox{ if }2/4\leq x<3/4,\\\ 0&\mbox{ if }3/4\leq x\leq
1\\\ \end{cases}\quad\text{ and }\quad u_{2}^{\prime}(x)=\begin{cases}c&\mbox{
if }0\leq x<1/4,\\\ 3c/2&\mbox{ if }1/4\leq x<2/4,\\\ c&\mbox{ if }2/4\leq
x<3/4,\\\ 3c/2&\mbox{ if }3/4\leq x\leq 1.\\\ \end{cases}$

Clearly, $u_{1}\in\mathscr{U}^{\frac{2}{4}}_{\frac{1}{4}}$ with
$\mathbb{E}_{G}[u_{1}]-\mathbb{E}_{F}[u_{1}]=0$ and
$u_{2}\in\mathscr{U}_{\frac{2}{3}}$ with
$\mathbb{E}_{G}[u_{2}]-\mathbb{E}_{F}[u_{2}]=0$. Let

$w(x)=\lambda_{1}u_{1}(x)+\lambda_{2}u_{2}(x)\in\mathscr{U}^{\textsc{mf}}_{\boldsymbol{\upgamma}},$

thus by the linearity of differentiation we have

$w^{\prime}(x)=\begin{cases}\lambda_{1}c+\lambda_{2}c&\mbox{ if }0\leq
x<1/4,\\\ \lambda_{1}4c+\lambda_{2}3c/2&\mbox{ if }1/4\leq x<2/4,\\\ c&\mbox{
if }2/4\leq x<3/4,\\\ 3c/2&\mbox{ if }3/4\leq x\leq 1\\\ \end{cases}$ (4.15)

and similarly by the linearity of expectation, we have

$\mathbb{E}_{G}[w]-\mathbb{E}_{F}[w]=\mathbb{E}_{G}[\lambda_{1}u_{1}+\lambda_{2}u_{2}]-\mathbb{E}_{F}[\lambda_{1}u_{1}+\lambda_{2}u_{2}]=0\quad\forall\lambda_{1},\lambda_{2}\geq
0$

as expected. To see the differences between $w$ and $u$, we examine the
following values of $G_{\text{loc}}(w;x)$ : $(\lambda_{2}\neq 0)$

$G_{\text{loc}}(w;x)=\begin{cases}\frac{\lambda_{1}4c+\lambda_{2}3c/2}{\lambda_{1}c+\lambda_{2}c}&\mbox{
if }0\leq x<1/4,\\\ 3/2&\mbox{ if }1/4\leq x<2/4,\\\ 3/2&\mbox{ if }2/4\leq
x<3/4,\\\ 1&\mbox{ if }3/4\leq x\leq 1.\\\ \end{cases}$

The computation above can be easily seen from below. Recalling the values of
$w^{\prime}(x)$ from (4.15) we have

$\displaystyle
G_{\text{loc}}(w;x)=\begin{cases}\max\left[1,\frac{\lambda_{1}4c+\lambda_{2}3c/2}{\lambda_{1}c+\lambda_{2}c},\frac{c}{\lambda_{1}c+\lambda_{2}c},\frac{3c/2}{\lambda_{1}c+\lambda_{2}c}\right]=\frac{\lambda_{1}4c+\lambda_{2}3c/2}{\lambda_{1}c+\lambda_{2}c}&\mbox{
if }0\leq x<1/4,\\\
\max\left[3/2,\frac{c}{\lambda_{1}4c+\lambda_{2}3c/2},\frac{3c/2}{\lambda_{1}4c+\lambda_{2}3c/2}\right]=3/2&\mbox{
if }1/4\leq x<2/4.\end{cases}$

Now, note that, by simple computation (or by the mediant inequality (4.14)),
for $\lambda_{1},\lambda_{2}>0$ we have

$G_{\text{loc}}(w;x)=\frac{\lambda_{1}4+\lambda_{2}3/2}{\lambda_{1}+\lambda_{2}}<4=G_{\text{loc}}(u;x),\quad
x\in[0,1/4)$ (4.16)

and $G_{\text{loc}}(w;x)=G_{\text{loc}}(u;x)$ for all $x\in[1/4,1]$. Since,
$G_{\text{loc}}(w;x)=\frac{4\lambda_{1}+3/2\lambda_{2}}{\lambda_{1}+\lambda_{2}}$
is independent of $x\in[0,1/4)$ and it converges to $4$ as
$\lambda_{1}\to+\infty$ ($\lambda_{2}>0$ being fixed), hence the claim
follows.

###  4.4 $\mathscr{U}^{\textsc{mf}}_{\upgamma}$ and the local greediness

In this section, we provide several tools in terms of the local greediness
$G_{\text{loc}}(u;\cdot)$ to verify a given function
$u\in\mathscr{U}^{\textsc{cv}}_{\boldsymbol{\upgamma}}$ belongs to the multi-
fractional utility space $\mathscr{U}^{\textsc{mf}}_{\boldsymbol{\upgamma}}$.
The main inspiration source is the example given in Section 4.3.

######  Lemma 4.1.

Let $\boldsymbol{\upgamma}:\mathbb{R}\to[0,1]$ be an arbitrary non-decreasing
function. Let
$u=\sum_{k=1}^{n}\lambda_{k}u_{k}\in\mathscr{U}^{\textsc{mf}}_{\boldsymbol{\upgamma}}$,
$n\in\mathbb{N}$, $\lambda_{k}\geq 0$ and
$u_{k}\in\mathscr{U}^{\text{Union}}_{\boldsymbol{\upgamma}}$, $k=1,...,n$.
Assume that there exists $x_{0}\in\mathbb{R}$ such that
$\boldsymbol{\upgamma}(x_{0})\neq 0$ and moreover

$G_{\text{loc}}(u;x_{0})=\frac{1}{\boldsymbol{\upgamma}(x_{0})}.$ (4.17)

Then, there exists an index $1\leq j\leq n$ so that

$G_{\text{loc}}(u;x_{0})=G_{\text{loc}}(u_{j};x_{0})=\max_{1\leq k\leq
n}G_{\text{loc}}(u_{k};x_{0})=\frac{1}{\boldsymbol{\upgamma}(x_{0})}.$

###### Proof.

Recall that by Proposition 4.5 and Corollary 4.1 we have
$G_{\text{loc}}(u;x_{0})\leq\max_{1\leq k\leq n}G_{\text{loc}}(u_{k};x_{0})$
and $G_{\text{loc}}(u_{k};x_{0})\leq\frac{1}{\boldsymbol{\upgamma}(x_{0})}$
for each $k=1,...,n$. Now, relation (4.17) yields that

$\frac{1}{\boldsymbol{\upgamma}(x_{0})}=G_{\text{loc}}(u;x_{0})\leq\max_{1\leq
k\leq
n}G_{\text{loc}}(u_{k},x_{0})\leq\frac{1}{\boldsymbol{\upgamma}(x_{0})}.$

Hence, the result follows. ∎

######  Proposition 4.7.

Let $\boldsymbol{\upgamma}:\mathbb{R}\to[0,1]$ be an arbitrary non-decreasing
function. Assume that $u\in\mathscr{U}^{\textsc{mf}}_{\boldsymbol{\upgamma}}$
and there exists $x_{0}\in\mathbb{R}$ such that
$\boldsymbol{\upgamma}(x_{0})\neq 0$ and moreover

$G_{\text{loc}}(u;x_{0})=\frac{1}{\boldsymbol{\upgamma}(x_{0})}.$ (4.18)

Then, there exists an open interval $I=I_{x_{0}}=(x_{0},x_{0}+\delta)$ for
some $\delta>0$ such that $\boldsymbol{\upgamma}$ is a constant function on
$I$.

###### Proof.

If $\boldsymbol{\upgamma}(x_{0})=1$, then it is obvious that
$\boldsymbol{\upgamma}$ is a constant function over the interval
$(x_{0},\infty)$. Hence, hereafter, we assume that
$\boldsymbol{\upgamma}(x_{0})<1$. Next, let $u$ takes the form
$u=\sum_{k=1}^{n}\lambda_{k}u_{k}\in\mathscr{U}^{\textsc{mf}}_{\boldsymbol{\upgamma}}$,
$n\in\mathbb{N}$, $\lambda_{k}\geq 0$ and
$u_{k}\in\mathscr{U}^{\text{Union}}_{\boldsymbol{\upgamma}}$, $k=1,...,n$.
Now, by Lemma 4.1, there exists an index $1\leq j\leq n$ so that

$\frac{1}{\boldsymbol{\upgamma}(x_{0})}=G_{\text{loc}}(u;x_{0})=G_{\text{loc}}(u_{j},x_{0}).$

Recall that $u_{j}\in\mathscr{U}^{\text{Union}}_{\boldsymbol{\upgamma}}$, so
there exists $t\in[-\infty,\infty]$ with
$u_{j}\in\mathscr{U}^{t}_{\boldsymbol{\upgamma}}$. The case $t=-\infty$
obviously not possible. If $t=\infty$, then clearly $\boldsymbol{\upgamma}$ is
a constant function over the interval $(x_{0},\infty)$. Now, let
$t\in\mathbb{R}$. If $t\leq x_{0}$, then we must have
$G_{\text{loc}}(u_{j};x_{0})=1$ that contradicts our assumption. Therefore,
$t>x_{0}$ must be. By definition, we have

$\frac{1}{\boldsymbol{\upgamma}(x_{0})}=G_{\text{loc}}(u_{j};x_{0})\leq\frac{1}{\boldsymbol{\upgamma}(t)}\leq\frac{1}{\boldsymbol{\upgamma}(x_{0})}.$

Since, $\boldsymbol{\upgamma}$ is non-decreasing, therefore must be a constant
function over the interval $(x_{0},t)$, hence the claims follows. ∎

######  Corollary 4.2.

Let $\boldsymbol{\upgamma}:\mathbb{R}\rightarrow[0,1]$ be an arbitrary non-
decreasing function such that there exists an open subset
$U\subseteq\mathbb{R}$ where $\boldsymbol{\upgamma}$ is strictly increasing on
$U$. Assume that $u\in\mathscr{U}^{\textsc{cv}}_{\boldsymbol{\upgamma}}$ and
there exists $x_{0}\in U$ such that

$G_{\text{loc}}\left(u;x_{0}\right)=\frac{1}{\boldsymbol{\upgamma}(x_{0})}.$
(4.19)

Then, $u\notin\mathscr{U}^{\textsc{mf}}_{\boldsymbol{\upgamma}}$.

######  Theorem 4.1.

Let $\boldsymbol{\upgamma}:\mathbb{R}\to[0,1]$ be an arbitrary non-decreasing
function. Assume that $u\in\mathscr{U}^{\textsc{cv}}_{\boldsymbol{\upgamma}}$
and there exist $x_{0},x_{1}\in\mathbb{R}$ such that
$0<\boldsymbol{\upgamma}(x_{0})\neq\boldsymbol{\upgamma}(x_{1})<1$. Let

$G_{\text{loc}}(u;x_{k})=\frac{1}{\boldsymbol{\upgamma}(x_{k})},\quad k=0,1.$
(4.20)

Then, $u\notin\mathscr{U}^{\textsc{mf}}_{\boldsymbol{\upgamma}}$.

###### Proof.

We proceed by contradiction. Assume that
$u\in\mathscr{U}^{\textsc{mf}}_{\boldsymbol{\upgamma}}$ takes the form,

$u=\sum_{k=1}^{n}\lambda_{k}u_{k},\quad\lambda_{k}\geq
0,\,u_{k}\in\mathscr{U}_{\boldsymbol{\upgamma}}^{\text{Union
}},n\in\mathbb{N}.$

Now, Lemma 4.1 implies that there at least exist two indexes $1\leq
j_{0},j_{1}\leq n$, $j_{0}\neq j_{1}$ such that

$G_{\text{loc}}(u_{j_{k}},x_{k})=\frac{1}{\boldsymbol{\upgamma}(x_{k})},\quad
k=0,1.$

Let’s assume that $x_{0}<x_{1}$ (hence, by our assumption
$\boldsymbol{\upgamma}(x_{0})<\boldsymbol{\upgamma}(x_{1})$). Let
$A_{0}=\\{1\leq k\leq
n\,:\,G_{\text{loc}}(u_{k};x_{0})=\frac{1}{\boldsymbol{\upgamma}(x_{0})}\\}$.
Note that $j_{0}\in A_{0}$ and let

$\mathcal{E}_{0}=\\{\varepsilon>0\,:\,G_{\text{loc}}(u_{j_{0}},x_{0})=\frac{1}{\boldsymbol{\upgamma}(x_{0})}>\max_{k\in
A^{c}_{0}}G_{\text{loc}}(u_{k},x_{0})+\varepsilon\\}.$

Next, for $\varepsilon\in\mathcal{E}_{0}$, by definition, there exist four
points with $x_{0}<x^{\varepsilon}_{1}<x^{\varepsilon}_{2}\leq
x^{\varepsilon}_{3}<x^{\varepsilon}_{4}$ such that

$\frac{u(x^{\varepsilon}_{4})-u(x^{\varepsilon}_{3})}{x^{\varepsilon}_{4}-x^{\varepsilon}_{3}}/\frac{u(x^{\varepsilon}_{2})-u(x^{\varepsilon}_{1})}{x^{\varepsilon}_{2}-x^{\varepsilon}_{1}}=:\frac{u[x^{\varepsilon}_{4},x^{\varepsilon}_{3}]}{u[x^{\varepsilon}_{2},x^{\varepsilon}_{1}]}>\underbrace{G_{\text{loc}}(u,x_{0})}_{=1/\boldsymbol{\upgamma}(x_{0})\text{(By
assumption)}}-\quad\varepsilon\,>\,\max_{k\in
A^{c}_{0}}G_{\text{loc}}(u_{k},x_{0})\geq\frac{u_{k}[x^{\varepsilon}_{4},x^{\varepsilon}_{3}]}{u_{k}[x^{\varepsilon}_{2},x^{\varepsilon}_{1}]}$

for every $k\in A^{c}_{0}$. This means that (with
$\bar{x}^{\varepsilon}=(x^{\varepsilon}_{1},x^{\varepsilon}_{2},x^{\varepsilon}_{3},x^{\varepsilon}_{4})$),

$u[\bar{x}^{\varepsilon}]:=\frac{u[x^{\varepsilon}_{4},x^{\varepsilon}_{3}]}{u[x^{\varepsilon}_{2},x^{\varepsilon}_{1}]}=\frac{\sum_{k=1}^{n}\lambda_{k}u_{k}[x^{\varepsilon}_{4},x^{\varepsilon}_{3}]}{\sum_{k=1}^{n}\lambda_{k}u_{k}[x^{\varepsilon}_{2},x^{\varepsilon}_{1}]}>u_{k}[\bar{x}^{\varepsilon}],\quad\forall
k\in A^{c}_{0}.$

On the other hand, by the mediant inequality (Lemma 7.1) we have
$u_{k}[\bar{x}^{\varepsilon}]>u[\bar{x}^{\varepsilon}]$, $k\in A_{0}$. This
leads to the following final inequality that for every index $k\in A_{0}$,

$\frac{1}{\boldsymbol{\upgamma}(x_{0})}\geq
u_{k}[\bar{x}^{\varepsilon}]>u[\bar{x}^{\varepsilon}]>G_{\text{loc}}(u;x_{0})-\varepsilon,$

Let $\varepsilon\to 0$ to obtain that

$\lim_{\varepsilon\to 0}u_{k}[\bar{x}^{\varepsilon}]=\lim_{\varepsilon\to
0}u[\bar{x}^{\varepsilon}]=\frac{1}{\boldsymbol{\upgamma}(x_{0})},\quad k\in
A_{0}.$

Let $u_{A_{0}}=\sum_{k\in A_{0}}\lambda_{k}u_{k}$, and similarly
$u_{A^{c}_{0}}=\sum_{k\in A^{c}_{0}}\lambda_{k}u_{k}$. First, by the mediant
inequality $\lim_{\varepsilon\to
0}u_{A_{0}}[\bar{x}^{\varepsilon}]=\frac{1}{\boldsymbol{\upgamma}(x_{0})}$.
This, implies that,

$\displaystyle\frac{1}{\boldsymbol{\upgamma}(x_{0})}=\lim_{\varepsilon\to
0}u_{A_{0}}[\bar{x}^{\varepsilon}]$ $\displaystyle=\lim_{\varepsilon\to
0}u[\bar{x}^{\varepsilon}]=\lim_{\varepsilon\to
0}\frac{u_{A_{0}}[x^{\varepsilon}_{4},x^{\varepsilon}_{3}]+u_{A^{c}_{0}}[x^{\varepsilon}_{4},x^{\varepsilon}_{3}]}{u_{A_{0}}[x^{\varepsilon}_{2},x^{\varepsilon}_{1}]+u_{A^{c}_{0}}[x^{\varepsilon}_{2},x^{\varepsilon}_{1}]}$
$\displaystyle=\lim_{\varepsilon\to