\section{Introduction}

The expected utility theory (\cite{vNM47}), the dual utility theory (\cite{Y87}),  and the rank-dependent utility theory (\cite{Q82})  are among the most popular probabilistic preference models, and they are closely related to several large classes of  law-based risk measures (\cite{MFE15} and \cite{FS16}).

These decision models and risk measures can be equivalently formulated on either a set of distributions   or a set of random variables.
The most popular operation on a set of distributions is a probabilistic mixture; for instance, the independence axiom of \cite{vNM47} is formulated using probabilistic mixtures.  Quasi-convexity in probabilistic mixtures is a  useful property in decision models and risk measures, and it means that
if the first distribution is preferred over the second one, then the first distribution is also preferred to a mixture of them. The property of having both quasi-convexity and quasi-concavity   is called betweenness (\cite{D86}), which was introduced to weaken the independence axiom; see \cite{W94} for the importance of these properties in decision theory.

It is well known that the expected utility model is linear with respect to mixtures, thus both convex and concave (we omit ``in mixtures" below unless it is contrasted to another sense of convexity).
Whereas convexity is well understood for dual utilities and rank-dependent utility models (e.g., \cite{W94}), quasi-convexity is not completely characterized for these models. Although being weaker, quasi-convexity is similar to convexity, and \cite{W94} showed that
they are equivalent  for dual utilities under a strict monotonicity condition.
  Nevertheless, if we remove strict monotonicity, there are commonly used functionals in decision theory, statistics and risk management, such as left and right quantiles  which are quasi-convex   but not convex.

The main aim of this paper is to understand  quasi-convexity for a large class of mappings, called the \emph{generalized rank-dependent functions}, which include dual utilities and rank-dependent utilities as special cases.
Our main result is a full characterization of quasi-convexity for this class,
and this characterization is built on a corresponding result on \emph{signed Choquet integrals}, a class of non-monotone mappings studied recently by \cite{WWW20a,WWW20}.

 Although most preference models in decision theory are monotone (either with respect to some notions of stochastic dominance or other orders),
there are three main advantages of working with non-monotone mappings, justifying the relevance of the study in this paper.
First, with monotonicity relaxed, results on convexity and concavity, or those on maxima and minima, are symmetric; we only need to analyze one of them, and the other is clear automatically. This is particularly helpful when we switch between the world of risk measures (a smaller value is preferred) and that of utilities (a larger value is preferred).
Second, signed Choquet integrals include many popular non-monotone objects in risk management, such as the mean-median deviation, the Gini deviation,   the inter-quantile range, and the inter-Expected Shortfall range; see the examples in \cite{WWW20}. Note that variability measures in the sense of \cite{FWZ17} are never monotone with respect to first-order stochastic dominance.
Third, removing monotonicity from the analysis allows us to have a deeper understanding of the essence of   important properties, such as quasi-convexity, by disentangling monotonicity from them.
That being said, it is not our intention to argue against   monotonicity  in decision making; openning up the discussions on non-monotone mappings indeed helps to better understand monotone ones.  All   results in this paper are new with or without the additional assumption of monotonicity.

  To fully characterize quasi-convexity for generalized rank-dependent functions,
we begin by collecting  definitions and some preliminaries in Section \ref{sec:2}, where  a unifying equivalence result on signed Choquet integrals (Theorem \ref{th-grand}) is   presented (without a proof): If a distortion function is continuous, then quasi-convexity  is equivalent to six other equivalent conditions in Theorem 3 of \cite{WWW20}.  This equivalence result relies on our main result (Theorem \ref{th-main})  in Section \ref{sec:mqc} on
the characterization all quasi-convexity signed Choquet integrals. The   class  turns out to contain slightly more than the quantiles and those with convex  distortion functions: A signed Choquet integral is quasi-convex  in mixtures if and only if it is either convex in mixtures or it is a \emph{scaled quantile-spread mixture}.
The result relies on delicate technical analysis, and uses some results in \cite{DK82}, \cite{W94}, \cite{WWW20} and \cite{WW20}.  The technical challenges may explain why the result was not available before, given the prominence of both concepts of quasi-convexity and rank-dependent utilities.
Some   implications of Theorem \ref{th-main} for quasi-linearity and dual utilities  are reported in Section \ref{sec:special}.
The full characterization for generalized rank-dependent functions (Theorem \ref{coro:rdu}), including rank-dependent utilities, is presented in Section \ref{sec:RDU}, extending the results of \cite{W94}.    In Section \ref{app:A}, we present
a conflict
between convexity in mixtures and convexity in risk pooling among the class of constant-additive mappings.
Section \ref{sec:7} concludes the paper.

\section{Preliminaries}
\label{sec:2}

\subsection{Convexity and  quasi-convexity}

Let
 $\mathcal M$ be a set of distributions on $\mathbb{R}$.
In this paper, distributions are always represented by their cumulative distribution functions.
A mapping $\rho:\mathcal{M}\to \mathbb{R}$ is \emph{m-convex}  if
$$
\rho(\lambda F+(1-\lambda)G) \le \lambda \rho(F) +(1-\lambda) \rho(G) \mbox{~~~for all $F,  G\in \mathcal{M}$ and $\lambda \in [0,1]$},
$$
and it is \emph{m-quasi-convex} if
$$
\rho(\lambda F+(1-\lambda)  G) \le \max\{\rho(F), \rho(G) \} \mbox{~~~for all $F,G\in \mathcal{M}$ and $\lambda \in [0,1]$.}
$$
As usual, m-concavity and m-quasi-concavity  are defined by flipping the inequalities above, respectively.
The reason that we emphasize ``m"  (which stands for ``mixture") for these properties will be explained soon, as another form of   convexity and concavity will   appear and be contrasted.

Quasi-convexity is an ordinal property, whereas convexity is not. Indeed,
for a partial order $\preceq$ on $\mathcal{M}$ numerically represented by $\rho$, i.e., $F\preceq G \Longleftrightarrow \rho(F)\le \rho(G)$,
quasi-convexity of $\rho$ corresponds to the following  property, sometimes called concavity of $\preceq$ (see e.g., \cite{WY19, WY21}),
$$
\Big (F  \succeq  G  ~\Longrightarrow~  F \succeq  \lambda F +(1-\lambda)G \Big)\mbox{~~~for all $F,G\in \mathcal{M}$ and $\lambda \in [0,1]$.}
$$
For the above reason,
convexity or concavity of a mapping $\rho$  is commonly used in risk management, where the value of $\rho$, typically representing a monetary value, is the primitive; see e.g., \cite{FS16}.   Quasi-convexity or quasi-concavity of $\rho$ is commonly used in decision theory, where the preference relation $\preceq$ is the primitive; see e.g., \cite{W10} and \cite{CMMM11}.
To unify both literatures, we will formulate all properties on the numerical representation $\rho$.

Let  $\mathcal{X}$ be a set of   random variables in a fixed probability space $(\Omega,\mathcal F,\mathbb{P})$ having the distributions in $\mathcal{M}$.  Let $\mathcal{M}_c$ be the set of compactly supported distributions on $\mathbb{R}$.
A mapping $\rho$ from $\mathcal{M}$ to $\mathbb{R}$ can be equivalently formulated as a mapping $\rho$ from $\mathcal{X}$ to $\mathbb{R}$ via $\rho(X):=\rho(F)$ where $F$ is the distribution of $X$; here we slightly abuse the notation to use   $\rho$ to represent both, and this should   be clear from the context. Such a mapping  $\rho$ on $\mathcal{X}$ is \emph{law-based}, that is, $X\buildrel \mathrm{d} \over = Y$ $\Longrightarrow$ $\rho(X)=\rho(Y)$, where $\buildrel \mathrm{d} \over =$ represents equality in distribution.\footnote{In decision theory, this property is called probabilistic sophistication by \cite{MS92}.}
We need both versions of the same mapping to make some interesting contrasts.
A mapping $\rho$ is \emph{o-convex} (where ``o" stands for ``outcome") if it is convex on $\mathcal{X}$,
and it is \emph{o-quasi-convex} if it is quasi-convex on $\mathcal{X}$. O-concavity and o-quasi-concavity are defined similarly.

For the same law-based mapping $\rho$,  o-convexity
and m-convexity have different interpretations, and sometimes they conflict each other.
For instance, the variance is \emph{o-convex and  m-concave}.
This conflict is indeed intuitive, because a mixture of  two random losses, representing diversification,  reduces variability, whereas a mixture of two  distributions, representing   throwing a die to determine between two models, increases variability.\footnote{In the risk measure literature, o-quasi-convexity on $\mathcal{X}$ is  argued by \cite{CMMM11} to better represent the consideration of diversification.}
A mapping may be both o-convex and m-convex, and an example is the expected utility, $F\mapsto  \int u \,\mathrm{d} F$ for a convex function $u$; this mapping   is indeed o-convex and m-linear (i.e., both m-convex and m-concave). Nevertheless, Proposition \ref{pr:conflict} in Section \ref{app:A} shows a conflict between o-convexity and m-convexity; that is, among constant-additive mappings, only a scaled expected value satisfies both properties.

\subsection{Generalized rank-dependent functions}
\label{sec:22}

 We first introduce Signed Choquet integrals and dual utilities, and then use them to define generalized rank-dependent functions and rank-dependent utilities.
Signed Choquet integrals are law-based mappings which are additive for comonotonic random variables (Theorem 1  of \cite{WWW20}, based on a result of \cite{S86}), but not necessarily monotone.
Denote  the set of all \emph{distortion functions} by $ \mathcal{H} ^{\rm BV}$,  $$\mathcal{H}^{\rm BV}=\{h:  [0,1]\to\mathbb{R} \mid \mbox{$h$ is of bounded variation and }h(0)=0 \},$$
 and  its subset of increasing normalized distortion functions by $ \mathcal{H}^{\rm DT}$,\footnote{DT stands for the dual theory of choice under risk of \cite{Y87}.}
 $$\mathcal{H}^{\rm DT}=\{h:  [0,1]\to\mathbb{R}  \mid \mbox{$h$ is increasing, $h(0)=0$ and $h(1)=1$} \}.$$
 Throughout, terms like ``increasing" and ``decreasing" are in the non-strict sense.
			A \emph{signed Choquet integral} $I_h: \mathcal{M}_c \rightarrow \mathbb{R}$ is defined as
\begin{align}
I_h(F)=\int_0^\infty h(1-F(x))\,\mathrm{d} x + \int_{-\infty}^{0} (h(1-F(x))-h(1) )\,\mathrm{d} x,
\label{eq:1}
\end{align}
			  where $h\in \mathcal{H}^{\rm BV}$ is its distortion function.  If $h\in \mathcal{H}^{\rm DT}$, then $I_h$ is called a \emph{dual utility} of \cite{Y87}.
			  As explained earlier, $I_h$ is also formulated on $\mathcal{X}_c$, the set of bounded random variables,   via
\begin{align}
I_h(X)=\int_0^\infty h(\mathbb{P}(X>x))\,\mathrm{d} x + \int_{-\infty}^{0} (h(\mathbb{P}(X>x))-h(1) )\,\mathrm{d} x.
\label{eq:1-prime}\end{align}
The mapping $I_h$ can be defined on domains other than $\mathcal{M}_c$ and $\mathcal{X}_c$ via  \eqref{eq:1} and \eqref{eq:1-prime} whenever the integrals are finite.  In some results, we treat $I_h$ as a mapping from $\mathcal{M}$ to $\mathbb{R}$ for a general $\mathcal{M}$.

As a main subject of the paper,
a \emph{generalized rank-dependent function}
is a mapping   $R_{h,v}:\mathcal{M}_c\to \mathbb{R}$ defined by, for  $h\in \mathcal H^{\rm BV}$ and $v:\mathbb{R}\to \mathbb{R}$,
$$
R_{h,v}(F)=  I_{h}(F\circ v^{-1}),$$
where
   $v^{-1}$ is the set-valued inverse of $v$, or equivalently formulated on $\mathcal{X}_c$ via $ R_{h,v}(X) = I_{h}(v(X))$.
If $h\in \mathcal H^{\rm DT}$ and $v$ is increasing, then $R_{h,v}$ corresponds to a \emph{rank-dependent utility} of \cite{Q93}.

In what follows, we will take the convexity of $h$ as the primary property to consider,
as the preference represented by $I_h$ is strongly risk averse if $h$ is increasing and convex (Theorem 2 of \cite{Y87}). Changing convexity to concavity makes no real mathematical difference since all results can be written for concavity via a sign change; recall that this is an advantage of working with non-monotone mappings such as generalized rank-dependent functions.

\subsection{A new equivalence result}

Before discussing the   characterization of m-quasi-convexity, we first present a unifying equivalence result on signed Choquet integrals  to illustrate the power of our characterization.
In what follows, $I_h$ is \emph{o-superadditive} if $I_h(X+Y)\ge I_h(X)+I_h(Y)$ for all $X,Y\in \mathcal{X}_c$, and $I_h$ is \emph{concave-order monotone}
if   $I_h(F)\le I_h(G)$ for all $F,G\in \mathcal{M}_c$ such that $F\le_{\rm cv} G$ where $\le_{\rm cv}$ represents concave order between distributions, i.e., $F\le_{\rm cv}G$ means that $ \int f \,\mathrm{d} F \le \int f\,\mathrm{d} G$ for all concave $f:\mathbb{R}\to \mathbb{R}$.

\begin{theorem}\label{th-grand} If $h\in\mathcal{H}^{\rm BV}$
	is continuous or strictly monotone, then
	the following are equivalent:
	(i) $h$ is convex;
	(ii) $I_h$ is concave-order monotone;
	(iii) $I_h$ is o-superadditive;
	(iv) $I_h$ is o-concave;
	(v) $I_h$ is o-quasi-concave;
	(vi) $I_h$ is m-convex;
	(vii)  $I_h$ is m-quasi-convex.
\end{theorem}

Theorem \ref{th-grand} can be broken down to a few pieces. The first six properties,  (i)-(vi) in Theorem \ref{th-grand}, are shown to be equivalent by \citet[Theorem 3]{WWW20} without any assumption on $h\in\mathcal{H}^{\rm BV}$.\footnote{\cite{WWW20} used the   narrative of risk measures, where results are stated for concave $h$; we seamlessly translate these results by a sign change.} We summarize this fact in the following lemma, which will be used repeatedly in the paper.

\begin{lemma}[\cite{WWW20}]\label{th:www20} For $h\in\mathcal{H}^{\rm BV}$,  (i)-(vi) in Theorem \ref{th-grand} are equivalent.
\end{lemma}
Lemma \ref{th:www20}  does not include m-quasi-convexity.
Indeed, m-quasi-convexity is not always equivalent to the above six conditions.
In particular, any left or right quantile
is both m-quasi-convex and m-quasi-concave, but it is not o-concave or o-convex.\footnote{A left or right quantile has \emph{convex level sets}, i.e., $\rho(F)=\rho(G)$ implies $\rho(\lambda F+(1-\lambda)G)=\rho(F)$ for all $\lambda \in [0,1]$ and $F,G\in \mathcal{M}_c$. This property and monotonicity together imply   m-quasi-convexity and m-quasi-concavity.  All signed Choquet integrals with convex level sets are characterized by \cite{WW20}, which are slightly more than linear transformations of the quantiles and the mean.}
Nevertheless, m-quasi-convexity and m-convexity are not too far away from each other.
\citet[Theorem 24]{W94} showed that (i) and (vii) in Theorem \ref{th-grand} are equivalent if $h\in\mathcal H^{\rm DT}$ is strictly increasing.\footnote{
Under the same assumption,  \citet[Corollary 8]{WY21}
showed the equivalence of (v) quasi-concavity  and (vii) m-quasi-convexity, which can also be seen from combining Lemmas \ref{th:www20}  and \ref{th:w94}.}

\begin{lemma}[\cite{W94}]\label{th:w94} For strictly increasing $h\in\mathcal{H}^{\rm DT}$,
(i) and
(vii) in Theorem \ref{th-grand} are equivalent.
\end{lemma}
It is obvious that the normalization $h(1)=1$ does not matter, and thus $\mathcal{H}^{\rm DT}$ can be safely replaced by $\mathcal{H}^{\rm BV}$ in Lemma \ref{th:w94}. Note that $I_h=-I_{-h}$ for all $h\in\mathcal H^{\rm BV}$. Hence, for any strictly decreasing $h\in \mathcal{H}^{\rm BV}$, (i) and (vii) are also equivalent. Combining these arguments and Lemma \ref{th:www20}, we obtain the equivalence in Theorem \ref{th-grand} under the assumption that $h$ is strictly monotone.

The remaining part of Theorem \ref{th-grand}, where $h$ is continuous, relies on the main result in this paper, Theorem \ref{th-main} in Section \ref{sec:mqc}, which characterizes all m-quasi-convex signed Choquet integrals. \cite{Y87} originally imposed continuity of $h$ instead of strict monotonicity. From a decision theoretical standpoint, a continuous distortion function $h$ is empirically plausible as in, e.g., the cumulative prospect theory of \cite{TK92}.
From a technical standpoint,  strict monotonicity rules out any non-trivial considerations for signed Choquet integrals over dual utilities.

We collect  some notation used later in the paper. Denote by $\delta_x$ the point-mass at $x\in\mathbb{R}$.
For a real-valued function $f$, denote by $f(x-)$ and $f(x+)$ the left and right limits of $f$ at $x\in\mathbb{R}$, that is $f(x-)=\lim_{y\uparrow x}f(y)$ and $f(x-)=\lim_{y\downarrow x}f(y)$. The function $\mathds{1}_{A}$ is the indicator function of an event $A$.
The mapping $I_h$ will be restricted to a domain $\mathcal{M}$ that is possibly smaller than $\mathcal{M}_c$, and all properties of $I_h$ are confined to the domain.
The following assumption on the set $\mathcal{M}$ will be useful for our characterization results.

 \renewcommand\theassumption{3P}

\begin{assumption}\label{assm:1}
The set $\mathcal{M} $ contains all three-point distributions on three specific points; that is, there exist $x>y>z$ such that
\begin{align}\label{eq-setassumption}
\left\{p\delta_x+(q-p)\delta_y+(1-q)\delta_z: 0\le p\le q\le1\right\}\subseteq \mathcal{M} \subseteq\mathcal M_c.
\end{align}
\end{assumption}
Assumption \ref{assm:1} is very weak and harmless for any practical purpose.
For the characterization of signed Choquet integrals in Theorem \ref{th-main}, one may safely take $\mathcal{M}=\mathcal{M}_c$. Nevertheless, it is important to state  Theorem \ref{th-main}  under Assumption \ref{assm:1}, because this assumption allows
us to analyze a general non-monotone function $v:\mathbb{R}\to \mathbb{R}$ that may take only a few values when we extend Theorem \ref{th-main}
to  generalized rank-dependent functions in Theorem \ref{coro:rdu}.

\section{Signed Choquet integrals with m-quasi-convexity}
\label{sec:mqc}

\subsection{Three examples of distortion functions}

This section is the main part of the paper where we investigate which functions $h\in\mathcal H^{\rm BV}$ yield m-quasi-convexity of $I_h$.
We assume $\mathcal{M}\subseteq\mathcal{M}_c$ in all results, and we will mention Assumption \ref{assm:1} if it is needed in a result.

We first define a few important special cases of signed Choquet integrals.
All mappings below are defined on both $\mathcal{M}_c$ and $\mathcal{X}_c$, and we will use whichever is more convenient in different places.
The \emph{left-} and \emph{right-quantile} at level $\alpha\in(0,1)$ are respectively defined by
\begin{align*}
Q^{-}_{\alpha}(F)=\inf\{x: F(x)\ge \alpha\}~~{\rm and}~~Q^{+}_{\alpha}(F)=\inf\{x: F(x)> \alpha\}.
\end{align*}
For the level $\alpha=0$ and $1$, left- and right-quantiles   coincide, and they are defined by
\begin{align*}
Q_0^+(F)=Q_0^-(F)=Q_0(F)=\inf\{x: F(x)> 0\}
\end{align*}
and
\begin{align*}
Q_1^+(F)=Q_1^-(F)=Q_{1}(F)=\inf\{x: F(x)\ge 1\}.
\end{align*}
For some $c\in[0,1]$ and $\alpha\in[0,1]$, the \emph{mixed quantile} is defined by $Q_\alpha^{c}=cQ_\alpha^{+}+(1-c)Q_\alpha^{-}$.
An \emph{asymmetric (negative) spread} is the mapping
$$
S_{a,b}= a Q_0-b Q_1
$$
where $a,b \ge 0$. Note that  $-S_{1,1}=Q_1-Q_0$ is the usual spread of the support of the distribution. We omit ``negative" below for simplicity and   call $S_{a,b}$ an asymmetric spread.
Finally, we consider the \emph{scaled quantile-spread mixture}, given by
\begin{align}
S_{a,b}+kQ_\alpha^
,~~~~ \mbox{for some~} a,b\ge0, ~\alpha,c\in [0,1],~k\in \mathbb{R}. \label{eq-H2}
\end{align}
A scaled quantile-spread mixture is
 the sum of an asymmetric spread  $S_{a,b}$ and a scaled mixed quantile $kQ_\alpha^c$, and it includes mixed quantiles and asymmetric spreads as special cases by setting $(a,b)=(0,0)$ or $k=0$.
Note that we allow $\alpha=0$ or $\alpha=1$ in \eqref{eq-H2}, which leads to
$
aQ_0- bQ_1$ where $a\ge 0$ or $b\ge 0$, but it does not include the case  $a,b<0$.
 Figure \ref{fig:distortion} reports an example of the distortion functions of an asymmetric spread, a scaled mixed quantile and a scaled quantile-spread mixture.

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	\caption{Distortion functions of an asymmetric spread (left), a scaled mixed quantile (middle) and scaled quantile-spread mixture (right) with $\alpha ,c\in (0,1)$ and $k>0$.}\label{fig:distortion}
\end{figure}

We denote by  $\mathcal H^{\rm CX}$ the set of all convex distortion functions $h\in \mathcal H^{\rm BV}$ and  by $\mathcal H^{\rm QSM}$ the set of distortion functions of scaled quantile-spread mixtures. The distortion functions in $ \mathcal H^{\rm QSM}$ are given by
\begin{align}\label{eq:hQSM} h(p)=
-b\mathds{1}_{\{0<p<\alpha\}}+(-b+kc)\mathds{1}_{\{p=\alpha\}}
+(-b+k)\mathds{1}_{\{\alpha<p<1\}}+(a-b+k)\mathds{1}_{\{p=1\}} ,  ~p\in[0,1],
\end{align}
for some parameters in \eqref{eq-H2}.

We have seen in Lemma \ref{th:w94}  that for strictly increasing distortion functions $h$, only the convex ones are possible for $I_h$ to be m-quasi-convex.
 Without monotonicity, the class of signed Choquet integrals is much larger than the class of strictly increasing ones,
 and this includes the scaled quantile-spread mixtures.
 In the following, we first propose a lemma which
 says that the asymmetric spread $S_{a,b}$ can be safely added to any $I_h$ without changing its m-quasi-convexity, thus highlighting the special role of $S_{a,b}$.

 \begin{lemma}\label{lm-dis01}
 	Let $h,\widetilde{h}\in\mathcal H^{\rm BV}$ satisfying $I_h=S_{a,b}+I_{\widetilde{h}}$ for some $a,b\ge 0$. If $I_{\widetilde{h}}$ is m-quasi-convex, then $I_h$ is also m-quasi-convex.
 \end{lemma}
 \begin{proof}

 	Suppose that $I_{\widetilde{h}}$ is m-quasi-convex.

 	For $F,G\in\mathcal M_c$ and $\lambda\in(0,1)$, we have $Q_1(\lambda F+(1-\lambda) G))=\max\{Q_1(F), Q_1(G)\}$ and $ Q_0(\lambda F+(1-\lambda) G))=\min\{ Q_0(F), Q_0(G)\}$. Hence, by the m-quasi-convexity of $I_{\widetilde{h}}$, we have
 	\begin{align*}
 		I_h(\lambda F+(1-\lambda) G))&=a Q_0(\lambda F+(1-\lambda) G))-bQ_1(\lambda F+(1-\lambda) G))+I_{\widetilde{h}}(\lambda F+(1-\lambda) G))\\
 		&\le \min\{a Q_0(F), a Q_0(G)\}-\max\{b Q_1(F), bQ_1(G)\}+\max\{I_{\widetilde{h}}(F),I_{\widetilde{h}}(G)\}\\
 		&\le \max\left\{aQ_0(F)-bQ_1(F)+I_{\widetilde{h}}(F),aQ_0(G)-bQ_1(G)
 		+I_{\widetilde{h}}(G)\right\}\\
 		&=\max\{ I_{{h}}(F),I_{{h}}(G)\},
 	\end{align*}
 	which implies the m-quasi-convexity of $I_h$.
 \end{proof}

Using Lemma \ref{lm-dis01}, we can
verify that scaled quantile-spread mixtures are m-quasi-convex. Hence, we have that both   convex distortion functions and distortion functions in \eqref{eq:hQSM} leads to m-quasi-convexity of $I_h$. Later we will show that they are the only possibilities.

\begin{proposition}\label{prop:trivial}
If $h\in \mathcal{H}^{\rm CX}\cup H^{\rm QSM}$, then $I_h$ is m-quasi-convex.
\end{proposition}
\begin{proof}
The case that $h$ is convex is obvious by applying the definition of $I_h$ in \eqref{eq:1}.
For $h\in\mathcal H^{\rm QSM}$, we can let $I_{\widetilde{h}}=k Q_{1-\alpha}^{c}$ for some $\alpha\in[0,1]$, $k\ge 0$ and $c\in[0,1]$.   Since $I_{\widetilde{h}}$ is   monotone and has convex level sets (e.g., \cite{WW20}), we know that $I_{\widetilde{h}}$ is both  m-quasi-convex and m-quasi-concave, which in turn implies that $I_h$ is m-quasi-convex by applying Lemma \ref{lm-dis01}.
\end{proof}

 \subsection{The main result and two corollaries}
The following theorem, which is the main result in the paper, establishes that the only possible signed Choquet integrals with m-quasi-convexity  are the ones in Proposition \ref{prop:trivial}, i.e., those with a convex distortion function and the scaled quantile-spread mixtures.
\begin{theorem}\label{th-main}
Under Assumption \ref{assm:1}, for $h\in\mathcal H^{\rm BV}$, $I_h$ is m-quasi-convex on $\mathcal M$ if and only if $h\in \mathcal H^{\rm CX}\cup\mathcal H^{\rm QSM}$.
\end{theorem}

Because of Theorem \ref{th-main}, we  will denote by $\mathcal H^{\rm QCX}=\mathcal H^{\rm CX}\cup\mathcal H^{\rm QSM}$,  which is the set of all $h\in \mathcal H^{\rm BV}$ for $I_h$ to be m-quasi-convex.
Besides the class of convex distortion functions, the only other choices are  $h\in\mathcal H^{\rm QSM}$.
One immediately observes that all distortion functions in $\mathcal H^{\rm QSM}$ are neither continuous nor strictly monotone.
Therefore, we conclude that for a continuous or strictly increasing $h$, $I_h$ is m-quasi-convex if and only if $h$ is convex.  This yields to the equivalence  statement  for (vii) in  Theorem \ref{th-grand} in addition to the six conditions in Lemma \ref{th:www20}.

The following proposition illustrates that
if the domain $\mathcal{M}$ of $I_h$ is chosen to be the set of
two-point distributions on two specific points, then the m-quasi-convexity of $I_h$ is equivalent to the quasi-convex of $h$ (hence, there are more m-quasi-convexity $I_h$ than Theorem \ref{th-main}). Hence, Assumption \ref{assm:1} on $\mathcal M$ is needed for the result in Theorem \ref{th-main} to hold.  \

\begin{proposition}
	Fix $x>y$ and let $\mathcal M=\{p\delta_x+(1-p)\delta_y: p\in[0,1]\}$. For $h\in\mathcal H^{\rm BV}$, $I_h$ is  m-quasi-convex on $\mathcal M$ if and only if $h$ is quasi-convex.
\end{proposition}
\begin{proof}
The	m-quasi-convexity of $I_h$ on $\mathcal M$ is equivalent to
$$
I_h((\lambda p+(1-\lambda) q)\delta_x+(\lambda(1-p)+(1-\lambda)(1-q))\delta_y)\le \max \{I_h(p\delta_x+(1-p)\delta_y),I_h(q\delta_x+(1-q)\delta_y)\}
$$
for any $p,q,\lambda\in[0,1]$. This is equivalent to
$$
h(\lambda p+(1-\lambda)q)\le \max\{h(p),h(q)\}
$$
for any $p,q,\lambda\in[0,1]$.
Hence, we complete the proof.
\end{proof}

Below we present a corresponding result to Theorem \ref{th-main} on a general space that contains unbounded random variables.
Denote by $\mathcal L$ a set of random variables which containing all three-point random variables on three specific points, such that there exists $X\in\mathcal L$ such that $X$ is both unbounded both from below and from above. We use $\mathcal M_{\mathcal L}$ to represent the set of distributions of all random variables in $\mathcal L$, and we assume that $\mathcal M_{\mathcal L}$ is convex. Denote by
\begin{align*}
\mathcal H^{\mathcal L}=\{h\in\mathcal H^{\rm BV}: |I_h(X)|<\infty~{\rm for~all}~X\in\mathcal L\},
\end{align*}
which is the set of distortion functions suitable for $I_h$ to be real-valued on $\mathcal L$.

\begin{corollary}
Let $h\in\mathcal H^{\mathcal L}$. The mapping  $I_h: \mathcal{M}_{\mathcal L}\to \mathbb{R}$ is m-quasi-convex if and only if $h$ is convex and continuous or $I_h=kQ_{1-\alpha}^c$ for some $\alpha\in(0,1)$, $c\in[0,1]$ and $k\in\mathbb{R}$. \label{coro:gen}
\end{corollary}

 Corollary \ref{coro:gen} formalizes the observation that, since asymmetric  spreads are not finite-valued on a general space $\mathcal L$ with unbounded random variables, we are left with only the class of convex distortion functions and those that correspond  to scaled mixed quantiles.
As a consequence of Corollary \ref{coro:gen}, on $\mathcal L$,
except for the case of a scaled mixed quantile,
  (i)-(vii) in  Section \ref{sec:2} are all equivalent.

  A special case of $\mathcal L=L^q$, $q\in [1,\infty)$, i.e., the normed space of random variables with finite $q$-th moment, is summarized in the corollary below.

\begin{corollary}
Let $h\in\mathcal H^{  L^q}$, $q\in [1,\infty)$.
For a norm-continuous mapping $I_h:\mathcal{M}_{L^q}\to \mathbb{R}$, (i)-(vii) in  Theorem \ref{th-grand} are equivalent.
\label{coro:gen2}
\end{corollary}
Corollary \ref{coro:gen2} follows  from Corollary \ref{coro:gen} and the fact that mixed quantiles are not
norm-continuous.
Different from  the other results, Corollary \ref{coro:gen2} imposes a continuity condition on $I_h$, instead of $h$,  to guarantee the equivalence of (i)-(vii).

 \subsection{Proof of Theorem \ref{th-main} and its corollaries}

This section is dedicated to a proof of Theorem \ref{th-main} which will further lead to the results in Corollaries  \ref{coro:gen} and \ref{coro:gen2}.
Several technical lemmas are needed, as the proof is much more involved than the case of strictly increasing ones.
Recall the set $\mathcal M$ satisfies Assumption \ref{assm:1}. We define a bivariate function as follows
\begin{align}\label{eq-biquasifunction}
	\pi(p,q)= (x-y)h(p)+(y-z)h(q), ~~(p,q)\in T_2:=\{(a,b)\in [0,1]:a\le b\}
\end{align}
The first lemma shows that the m-quasi-convexity of $I_h$ on $\mathcal M$ implies that $\pi(p,q)$ is quasi-convex on $T_2$.

\begin{lemma}\label{prop:quasi-pi}
Suppose that Assumption \ref{assm:1} holds, $h\in\mathcal H^{\rm BV}$ and $I_h$ is m-quasi-convex on $\mathcal M$.
	Then the function $\pi$ defined in \eqref{eq-biquasifunction} is quasi-convex on $T_2$.  In particular, $h$ is  quasi-convex on $[0,1]$.
\end{lemma}

\begin{proof}
	For any $0\le p\le q\le 1$, we have

	\begin{align*}
		I_h(p\delta_x+(q-p)\delta_y+(1-q)\delta_z)&=xh(p)+y(h(q)-h(p))+z(h(1)-h(q))\\
		&=zh(1)+(x-y)h(p)+(y-z)h(q)\\
		&=zh(1)+\pi(p,q).
	\end{align*}

	One can verify that the m-quasi-convexity of $I_h$ implies
	\begin{align}\label{eq-quasieq1}
		\pi(\lambda p_1+(1-\lambda)p_2,\lambda q_1+(1-\lambda)q_2)\le \max\{\pi(p_1,q_1), \pi(p_2,q_2)\}
	\end{align}
	for any $\lambda,p_1,p_2,q_1,q_2\in[0,1]$ with $p_1\le q_1$ and $p_2\le q_2$.

	This is equivalent to the quasi-convexity of $\pi$ on $T_2$. Let $p=0$ in \eqref{eq-biquasifunction},   the quasi-convexity of $\pi$ implies the quasi-convexity of $h$ on $[0,1]$. This completes the proof.
\end{proof}

The second lemma is a generalization of Lemma 26 of \cite{W94} to distortion functions in $\mathcal H^{\rm BV}$ and signed Choquet integral on $\mathcal M$. For $h\in\mathcal H^{\rm BV}$ and $0\le p<q\le 1$ such that $h(p)\neq h(q)$,
we define $\lambda_h(p,q)$ as
\begin{align}\label{eq-lambda_h}
	\lambda_h(p,q)=\frac{h(p)/2+h(q)/2-h(p/2+q/2)}{|h(q)-h(p)|}.
\end{align}
The value of this function can be interpreted as a measure of local convexity of $h$.

\begin{lemma}\label{lm-generalmain2}
Suppose that Assumption \ref{assm:1} holds, $h\in\mathcal H^{\rm BV}$ and $I_h$ is m-quasi-convex on $\mathcal M$, and let $0\le p<q\le s<t\le1$. If $h(p)\neq h(q)$, $h(s)\neq h(t)$ and $|h(q)-h(p)|(x-y)=|h(t)-h(s)|(y-z)$, then we have $\lambda_h(p,q)+\lambda_h(s,t)\ge 0$.
\end{lemma}

\begin{proof}

	Let $0\le p<q\le s<t\le1$ and $h(p)\neq h(q)$ and $h(s)\neq h(t)$. By Lemma \ref{prop:quasi-pi}, it holds that $h$ is quasi-convex on $[0,1]$. Thus,
	one can check that only one of the following three cases will happen. \underline{Case 1}: $h(p)<h(q)\le h(s)<h(t)$; \underline{Case 2}: $h(p)>h(q)\ge h(s)>h(t)$; \underline{Case 3}: $h(p)>h(q)$ and $h(s)<h(t)$.

	Let us first verify Cases 1 and 2. Suppose that $0\le p<q\le s<t\le 1$,
	and $h(p)<h(q)\le h(s)<h(t)$ or $h(p)>h(q)\ge h(s)>h(t)$. All distributions in this proof are of the form $p\delta_{x_1}+\frac{q-p}{2}\delta_{x_2}+\frac{q-p}{2}\delta_{x_3}+
	(s-q)\delta_{x_4}+\frac{t-s}{2}\delta_{x_5}+\frac{t-s}{2}\delta_{x_6}
	+(1-t)\delta_{x_7}$ with fixed probabilities and $x_1,\dots,x_7\in\{x,y,z\}$ with $x_1\ge\dots \ge x_7$.   Define two distributions $F,G$  such that the outcomes $(x_1,\dots,x_7)$ of them equal to $(x,y,y,y,y,y,z)$ and $(x,x,x,y,z,z,z)$, respectively. One can verify that
	$$
	I_h(F)-I_h(G)=(y-z)(h(t)-h(s))-(x-y)(h(q)-h(p)).
	$$
	Since $h(q)-h(p)$ and $h(t)-h(s)$ have the same sign, and $x>y>z$, the condition $|h(q)-h(p)|(x-y)=|h(t)-h(s)|(y-z)$ in the lemma implies $I_h(F)=I_h(G)$.
	By the m-quasi-convexity of $I_h$, we have $I_h((F+G)/2)\le(I_h(F)+I_h(G))/2$ where the outcome of
	$(F+G)/2$ is $(x,x,y,y,y,z,z)$. This leads to
	$$
	\left(\frac{h(p)+h(q)}{2}-h\left(\frac{p+q}{2}\right)\right)(x-y)
	+\left(\frac{h(s)+h(t)}{2}-h\left(\frac{s+t}{2}\right)\right)(y-z)\ge 0.
	$$
	Dividing by the positive factors in  $|h(q)-h(p)|(x-y)=|h(t)-h(s)|(y-z)$, we have
	$$
	\frac{1}{|h(q)-h(p)|}	\left(\frac{h(p)+h(q)}{2}-h\left(\frac{p+q}{2}\right)\right)
	+\frac{1}{|h(t)-h(s)|}\left(\frac{h(s)+h(t)}{2}-h\left(\frac{s+t}{2}\right)\right)\ge 0.
	$$
	This gives $\lambda_h(p,q)+\lambda_h(s,t)\ge 0$. To see Case 3,
	we only need to replace the outcomes of constructed distributions $F,G$ by $(x,y,y,y,z,z,z)$ and $(x,x,x,y,y,y,z)$, respectively, and then, it follows from a similar proof of Cases 1 and 2 in the previous arguments.
\end{proof}

The third Lemma is divided into three cases to consider nonconvex distortion functions. We will apply Lemmas \ref{prop:quasi-pi}, \ref{lm-generalmain2} and some results in \cite{DK82} to prove this lemma.

\begin{lemma}\label{lm-generalupdown}
Suppose that Assumption \ref{assm:1} holds, $h\in\mathcal H^{\rm BV}$, and $I_h$ is m-quasi-convex on $\mathcal M$. The following three statements hold.
	\begin{itemize}
		\item[(i)]	If $h$ is nonconvex on $(0,1)$, then there is a point $\alpha\in(0,1)$ such that	$h$ is
		a constant on both intervals $(0,\alpha)$ and $(\alpha,1)$.

		\item[(ii)] If $h$ is nonconvex on $[0,1)$  and convex on $(0,1)$, then $h(0+)>h(0)=0$, $h(p)=h(0+)$ for all $p\in(0,1)$ and $h(1-)\le h(1)$.

		\item[(iii)] If $h$ is nonconvex on $(0,1]$ and convex on $(0,1)$, then $h(1-)>h(1)$, $h(p)=h(1-)$ for all $p\in(0,1)$ and $h(0+)\le h(0)=0$.
	\end{itemize}
\end{lemma}

\begin{proof}

	By Lemma \ref{prop:quasi-pi}, the m-quasi-convexity of $I_h$ implies the quasi-convexity of $\pi$ defined in \eqref{eq-biquasifunction}. Hence, it is sufficient to prove this lemma based on the quasi-convexity of $\pi$.

\underline{(i)} We prove this statement by two steps (see Figure \ref{fig-mainlm1} for an illustration). First, we will show that there is at most one nonconvexity kink of $h$ on $(0,1)$. Second, we will show that $h$ should be a constant both on the left and the right of the nonconvexity kink.
\begin{figure}
{\centering
		\begin{tikzpicture}
			\draw[-] (0,5) -- (0,0) -- (8,0);
			\draw [black] (0,0) .. controls (1,0.05) and (3,0.4).. (4,2.5);
			\draw [black] (4,2.5) .. controls (5,2.6) and (6,3).. (7,4.5);
			\draw  [dash pattern={on 1.5pt off 2pt}] (3.5,1.7) -- (4.5,2.58);
\draw  [dash pattern={on 0.84pt off 2.51pt}] (3.45,1.2) -- (4,2.5)--(4.55,3.8)
node[above] {\footnotesize Slope is $K_L$};
\draw  [dash pattern={on 0.84pt off 2.51pt}] (2,2.2)--(4,2.5)--(6,2.8)
node[right] {\footnotesize Slope is $K_R$};

\node at (4,2.5) { $\bigcdot$};
\node at (3.3,2.85) {\footnotesize $(\alpha,h(\alpha))$};

\node at (2.5,0.75) { $\bullet$};
\node at (1.85,1.1) {\footnotesize $(t_0,h(t_0))$};

\node at (1.8,0.4) {$\bullet$};
\node at (1,0.65) {\footnotesize $(s_n,h(s_n))$};

\node at (3.5,1.7) {$\bullet$};
\node at (2.6,1.9) {\footnotesize $(p_n,h(p_n))$};

\node at (4.5,2.58) {$\bullet$};
\node at (5.5,2.25) {\footnotesize $(q_n,h(q_n))$};

\node at (4,2.12) {$\bullet$};
\draw[very thick, dashed, <-](4.05,2.05)--(4.5,1.5)
node[right] {\footnotesize $\left(\alpha,\frac{h(p_n)+h(q_n)}{2}\right)$};
\end{tikzpicture}
~\\[5pt]
}
\begin{center}
\footnotesize{
Step 1. There exists only one nonconvexity kink $\alpha$; $h$ is convex on $[0,\alpha)$ and on $(\alpha,1]$.
\\ Step 2. Contradiction from being nonconstant on $(0,\alpha)$. Here, $p_n=\alpha-1/n$,  $q_n=\alpha+1/n$,
\\
$|h(t_0)-h(s_n)|(x-y)=|h(p_n)-h(q_n)|(y-z)$,  $K_{L}=h'_-(\alpha)$ and $K_{R}=h'_+(\alpha)$.
}
\end{center}
\caption{Graphic illustration of the proof of Lemma \ref{lm-generalupdown} (i).}\label{fig-mainlm1}
\end{figure}

Step 1: Suppose that $h$ is nonconvex on $(a,b)\subseteq(0,1)$,
	and we will show that there is at most one nonconvexity kink on $(a,b)$. To see this,

	note that the function $\pi$ defined in \eqref{eq-biquasifunction} is quasi-convex both on $(0,a)\times(a,b)$ and  $(a,b)\times(b,1)$. Since $h$ is nonconvex on $(a,b)$, it follows from \citet[Theorem 2]{DK82}
	that $h$ is convex both on $(0,a)$ and $(b,1)$. Applying this result to smaller and smaller  subintervals of $(a,b)$, there is one point $\alpha\in(a,b)$ (the nonconvexity kink) such that $h$ is convex both on $(0,\alpha)$ and $(\alpha,1)$.

Step 2: We aim to verify that $h$ is

	a constant on both intervals $(0,\alpha)$ and $(\alpha,1)$, and this will complete the proof of (i).

	We only show that $h$ is nonconstant on $(0,\alpha)$ as the proof of the other case is similar. Assume now by contradiction that $h$ is nonconstant on $(c,d)\subseteq (0,\alpha)$. Let us consider the bivariate function $\pi$ (see \eqref{eq-biquasifunction}) on $(c,d)\times(d,1)$. Note that $h$ is nonconstant both on $(c,d)$ and $(d,1)$, and $\pi$ is quasi-convex. It follows from
	\citet[Theorem 1]{DK82} that $h$ is continuous on $(d,1)$. Since $\alpha\in(d,1)$ and $h$ is convex on both $(0,\alpha)$ and $(\alpha,1)$, it holds that $h$ is continuous on $(0,1)$.
	Now, recall $\lambda_h(p,q)$ defined in \eqref{eq-lambda_h}, and we
	rewrite   it as
	\begin{align}\label{eq-relambda_h}
		\lambda_h(p,q)&=\frac{h(p)/2+h(q)/2-h(p/2+q/2)}{|h(q)-h(p)|}\notag\\
		&=\frac{1}{4\left|{(h(q)-h(p))}/{(q-p)}\right|}\left(\frac{h(q)-h(p/2+q/2)}{(q-p)/2}
		-\frac{h(p/2+q/2)-h(p)}{(q-p)/2}\right).
	\end{align}
	By the convexity of $h$ on $(c,d)$ and also noting that $h$ is nonconstant on $(c,d)$, there exists $t_0$ in $(c,d)$ such that $h$ has nonzero left derivative at $t_0$. Therefore, it follows from \eqref{eq-relambda_h} that
	\begin{align}\label{eq-leftdert_0}
		\lambda_h(s,t_0)\to0,~~ {\rm as}~ s\uparrow t_0.
	\end{align}
	Denote  $K_{L}=h'_-(\alpha)$ and $K_{R}=h'_+(\alpha)$ by the left and right derivative of $h$ at $\alpha$. Since   $\alpha$ is a point of    nonconvex kink, we have $K_{L}>K_{R}$. Take $p_n=\alpha-1/n$ and $q_n=\alpha+1/n$ for $n\in\mathbb{N}$. We show that $h(p_n)\neq h(q_n)$ for large enough $n$. To see this, if $K_L\le 0$, then $K_{R}<0$ which means that $h(p_n)> h(q_n)$ for large enough $n$. Suppose now $K_L>0$. By Lemma \ref{prop:quasi-pi}, $h$ is quasi-convex on $[0,1]$, and hence, we have $K_R\ge 0$. This implies $h(p_n)< h(q_n)$ for large enough $n$. Therefore, we conclude that $h(p_n)\neq h(q_n)$ for large enough $n$.
	By \eqref{eq-relambda_h}, we have
	\begin{align*}
		\lambda_h(p_n,q_n)

		&=\frac{1}{4\left|{(h(q_n)-h(p_n))}/{(q_n-p_n)}\right|}\left(\frac{h(q_n)-h(p_n/2+q_n/2)}{(q_n-p_n)/2}
		-\frac{h(p_n/2+q_n/2)-h(p_n)}{(q_n-p_n)/2}\right)\\
		&=\frac{1}{4\left|{(h(q_n)-h(p_n))}/{(q_n-p_n)}\right|}\left[n\left(h\left(\alpha+1/n	\right)-h(\alpha)\right)-n\left(h(\alpha)-h\left(\alpha-1/n\right)\right)\right].
	\end{align*}
	Note that $(h(q_n)-h(p_n))/(q_n-p_n)$ is the slope of the line through $(p_n,h(p_n))$ and $(q_n,h(q_n))$. It holds that $(h(q_n)-h(p_n))/(q_n-p_n)\in[K_R,K_L]$ for large enough $n$ (see Figure \ref{fig-mainlm1}). Hence, when $n$ is large enough, we have
	\begin{align}\label{eq-noncvkink}
		\lambda_h(p_n,q_n)&\le \frac{1}{4\max\{|K_L|,|K_R|\}}\left[n\left(h\left(\alpha+1/n	\right)-h(\alpha)\right)-n\left(h(\alpha)-h\left(\alpha-1/n\right)\right)\right]\notag\\
		&\to \frac{K_R-K_L}{4\max\{|K_L|,|K_R|\}}<0,~~{\rm as}~n\to\infty.
	\end{align}
	Recall the definition of $t_0$ in \eqref{eq-leftdert_0}. Define a sequence $\{s_n\}_{n\in\mathbb{N}}$ such that $s_n< t_0$ for all $n\in\mathbb{N}$, $s_n\to t_0$ and $|h(t_0)-h(s_n)|(x-y)=|h(p_n)-h(q_n)|(y-z)$ for large enough $n$. Because $h$ is continuous on $(0,1)$, such a sequence  exists. Noting that $t_0<\alpha$, we have $s_n<t_0<p_n<q_n$ for large enough $n$.
	By Lemma \ref{lm-generalmain2}, we obtain $\lambda_h(s_n,t_0)+\lambda_h(p_n,q_n)\ge 0$ for large enough $n$.
	However, combining  \eqref{eq-leftdert_0} and \eqref{eq-noncvkink}, we have $\lambda_h(s_n,t_0)+\lambda_h(p_n,q_n)<0$ for large enough $n$, and this yields a contradiction.
	Therefore, we conclude that $h$ should be a constant on both intervals $(0,\alpha)$ and $(\alpha,1)$, and this completes the proof of (i).

	\underline{(ii)}: Suppose that $h$ is nonconvex on $[0,1)$ and convex on $(0,1)$. This implies that $h(0+)>h(0)=0$.  It follows from Lemma \ref{prop:quasi-pi} that $h$ is quasi-convex on $[0,1]$. Combining this   with $h(0+)>h(0)$, it holds that $h$ is increasing on $[0,1]$, which implies $h(1-)\le h(1)$.
	It remains to verify that $h(p)=h(0+)$ for all $p\in(0,1)$.

	Assume now by contradiction that $h$ is nonconstant on $(0,1)$. Since $h$ is convex on $(0,1)$, there exists $\beta\in(0,1)$ such that $h$ is strictly increasing on $[\beta,1)$. Let $\epsilon>0$ and $k=2(x-y)/(y-z)$, and denote by $p_1=0$, $p_2=2\epsilon$, $q_1=\beta+2k\epsilon$ and $q_2=\beta$ such that $p_2\le q_2$ and $q_1<1$ (see the left panel of Figure \ref{fig-mainlm2}).
	 \begin{figure}
		\centering
		\begin{tikzpicture}
			\draw[-] (0,3.5) -- (0,0) -- (6,0);
			\draw [black] (0.02,0.5) .. controls (2,0.4) and (3,1.2).. (5,3.25);

			\node at (0,0) {$\bullet$};
			\node at (0,0.5) {$\circ$};

			\node[below] at (-0.2,0)  {\footnotesize $0$ $(p_1)$};

			\node[below] at (3,0) {\footnotesize $\beta$ $(q_2)$};
			\draw[gray,dotted] (3,0) -- (3,1.35);

			\node[below] at (1,0) {\footnotesize $2\epsilon$ $(p_2)$};
			\draw[gray,dotted] (1,0) -- (1,0.55);

			\node[below] at (4.5,0) {\footnotesize $\beta+2k\epsilon$ $(q_1)$};
			\draw[gray,dotted] (4.5,0) -- (4.5,2.78);

			\node at (0.5,0.46) {$\smallcdot$};
			\draw[very thick, dashed, <-](0.5,0.55)--(0.6,1.25)
			node at (0.75,1.6) {\footnotesize $\frac{p_1+p_2}{2}=\epsilon$};

			\node at (3.75,2) {$\smallcdot$};
			\draw[very thick, dashed, <-](3.72,2.1)--(3.4,2.5)
			node at (2.8,2.82) {\footnotesize $\frac{q_1+q_2}{2}=\beta+k\epsilon$};
		\end{tikzpicture}
		~~~
		\begin{tikzpicture}
			\draw[-] (0,-3.5) -- (0,0) -- (6,0);
			\draw [black] (0,0) .. controls (2,-2.45) and (3,-2.75).. (4.98,-2.65);
			\node at (5,-3.25) {$\bullet$};
			\node at (5,-2.65) {$\circ$};

			\node[above] at (5.4,0) {\footnotesize $1$ $(q_1)$};
			\draw[gray,dotted] (5,0) -- (5,-3.25);

			\node[above] at (4,0) {\footnotesize $1-2\epsilon$ $(q_2)$};
			\draw[gray,dotted] (3.8,0) -- (3.8,-2.63);

			\node[above] at (2.5,0) {\footnotesize $\beta$ $(p_2)$};
			\draw[gray,dotted] (2.5,0) -- (2.5,-2.25);

			\node[above] at (1,0) {\footnotesize $\beta-2k\epsilon$ $(p_1)$};
			\draw[gray,dotted] (1,0) -- (1,-1.1);

			\node at (4.4,-2.7) {$\smallcdot$};
			\draw[very thick, dashed, <-](4.38,-2.73)--(4.2,-3.25)
			node at (4.2,-3.6) {\footnotesize $\frac{q_1+q_2}{2}=1-\epsilon$};

			\node at (1.75,-1.83) {$\smallcdot$};
			\draw[very thick, dashed, <-](1.75,-1.85)--(1.85,-2.4)
			node at (1.85,-2.75) {\footnotesize $\frac{p_1+p_2}{2}=\beta-k\epsilon$};

		\end{tikzpicture}
		\caption{Contradiction from $h$ being nonconstant on $(0,1)$ in the proof of (ii) and (iii) of Lemma \ref{lm-generalupdown}: the Left panel illustrates (ii) and the right panel illustrates (iii).}\label{fig-mainlm2}
	\end{figure}
We calculate the following items:
	$$
	A_1:=\pi(p_1,q_1)=(y-z)h(\beta+2k\epsilon),
	$$
	$$
	A_2:=\pi(p_2,q_2)=(x-y)h(2\epsilon)+(y-z)h(\beta),
	$$
	and
	$$
	A:=\pi\left(\frac{p_1+p_2}{2},\frac{q_1+q_2}{2}\right)=(x-y)h\left(\epsilon\right)+(y-z) h\left(\beta+k\epsilon\right).
	$$
	On the one hand,
	since $h(\epsilon)>h(0+)>0$ for any $\epsilon\ge 0$ and $h$ is continuous on $(0,1)$, we have $A>A_1$ for small enough $\epsilon>0$. On the other hand,
	\begin{align*}
		A_2-A&=(x-y)\left(h(2\epsilon)-h\left(\epsilon\right)\right)
		-(y-z)(h(\beta+k\epsilon)-h(\beta))\\
		&=\epsilon(x-y)\left(\frac{h(2\epsilon)-h(\epsilon)}{\epsilon}-\frac{2(h(\beta+k\epsilon)-h(\beta))}{k\epsilon} \right)
		<0,
	\end{align*}
	where the inequality holds because $h$ is convex on $(0,1)$ and strictly decreasing on $[\beta,1)$. Therefore, we conclude that $A>\max\{A_1,A_2\}$ which contradicts the quasi-convexity of $\pi$.
	This completes the proof of (ii).

	\underline{(iii)}: The proof of this statement is similar to (ii). Suppose that $h$ is nonconvex on $(0,1]$ and convex on $(0,1)$. This implies $h(1-)>h(1)$. By Lemma \ref{prop:quasi-pi}, we know that $h$ is quasi-convex on $[0,1]$. Combining with $h(1-)>h(1)$, we conclude that $h$ is decreasing on $[0,1]$. Hence, we have $h(0+)\le h(0)$. It remains to verify that $h$ is a constant on $(0,1)$.
	Assume now by contradiction that $h$ is nonconstant on $(0,1)$. Then, there exists $\beta\in(0,1)$ such that $h$ is strictly decreasing on $(0,\beta]$. Let $\epsilon>0$ and $k=2(y-z)/(x-y)$, and denote by $p_1=\beta-2k\epsilon$, $p_2=\beta$, $q_1=1$ and $q_2=1-2\epsilon$ such that $p_2\le q_2$ and $p_1>0$ (see the right panel Figure \ref{fig-mainlm2}). We calculate the following items:
	$$
	A_1:=\pi(p_1,q_1)=(x-y)h(\beta-2k\epsilon)+(y-z)h(1),
	$$
	$$
	A_2:=\pi(p_2,q_2)=(x-y)h(\beta)+(y-z)h(1-2\epsilon),
	$$
	and
	$$
	A:=\pi\left(\frac{p_1+p_2}{2},\frac{q_1+q_2}{2}\right)=(x-y)h\left(\beta-k\epsilon\right)+(y-z) h\left(1-\epsilon\right).
	$$
	On the one hand, since $h(1-\epsilon)\ge h(1-)>h(1)$ for any $\epsilon>0$ and $h$ is continuous on $(0,1)$, we have $A>A_1$ for small enough $\epsilon>0$. On the other hand,
	\begin{align*}
		A_2-A&=(x-y)(h(\beta)-h(\beta-k\epsilon))-(y-z)(h(1-\epsilon)-h(1-2\epsilon))\\
		&=\epsilon(y-z)\left(\frac{2(h(\beta)-h(\beta-k\epsilon))}{k\epsilon}-\frac{h(1-\epsilon)-h(1-2\epsilon)}{\epsilon}\right)<0,
	\end{align*}
	where the inequality holds because $h$ is convex on $(0,1)$ and strictly increasing on $(0,\beta]$.
	Therefore, we conclude that $A>\max\{A_1,A_2\}$ which implies $\pi$ is not quasi-convex, a contradiction. Hence, we complete the proof of (iii).

	%

	%

	%

	%

\end{proof}

 \begin{proof}[Proof of Theorem \ref{th-main}]
Sufficiency is proved in Proposition \ref{prop:trivial}.
To see necessity, it is sufficient to verify that for nonconvex $h\in\mathcal H^{\rm BV}$, the  m-quasi-convexity of $I_h$ implies $h\in\mathcal H^{\rm QSM}$. To see this,  we divide $h$ into three cases as shown in Lemma \ref{lm-generalupdown}: Case 1. $h$ is nonconvex on $(0,1)$; Case 2. $h$ is nonconvex on $[0,1)$ and convex on $(0,1)$; Case 3. $h$ is nonconvex on $(0,1]$ and convex on $(0,1)$. By Lemma \ref{lm-generalupdown}, if $h$ is the form of Case 2 or Case 3, then one can check that $h\in\mathcal H^{\rm QSM}$. Thus, it remains to consider Case 1.

Suppose that $h$ is nonconcave on $(0,1)$.
By Lemma \ref{lm-generalupdown} (i), there exists $\alpha\in(0,1)$ such that $h$ is a constant both on $(0,\alpha)$ and $(\alpha,1)$ and $h(\alpha-)\neq h(\alpha+)$. Hence, $h$ can be represented as
$$
h(p)=
h(0+)\mathds{1}_{\{0<p<\alpha\}}+h(\alpha)\mathds{1}_{\{p=\alpha\}}
+h(1-)\mathds{1}_{\{\alpha<p<1\}}+h(1)\mathds{1}_{\{p=1\}},~~p\in[0,1].
$$
In order to show that $h\in\mathcal H^{\rm QSM}$, we need to verify two assertions: (a) $h(0+)\le 0$ and $h(1-)\le h(1)$; (b) $(h(\alpha)-h(0+))(h(1-)-h(\alpha))\ge 0$. Both assertions will be proved by counter evidence. For (a), we assume by contradiction that $h(0+)>0$ or $h(1-)>h(1)$. We only consider the case of $h(0+)>0$ as the case of $h(1-)>h(1)$ is similar.
By Lemma \ref{prop:quasi-pi}, we know that $h$ is quasi-convex, and combining with $h(0+)>0$, it holds that $h$ is increasing on $[0,1]$ which implies $h(0+)=h(\alpha-)<h(\alpha+)=h(1-)$.
Let $p_1=0$, $p_2=\alpha/2$, $q_1=\alpha+2\epsilon$ and $q_2=\alpha-\epsilon$ such that $\epsilon>0$, $p_2\le q_2<q_1<1$ (see Figure \ref{fig-mainth1}).   \begin{figure}
	\centering
	\begin{tikzpicture}
		\draw[-] (0,5) -- (0,0) -- (8,0);
		\draw [black] (0.02,2)--(4.47,2);
		\draw [black] (4.52,4)--(7,4);

\node at (0,0) {$\bullet$};
\node at (0,2) {$\circ$};
\node at (4.5,2) {$\circ$};
\node at (4.5,3) {$\bullet$};
\node at (4.5,4) {$\circ$};

\node at (-0.2,-0.3) {\footnotesize $0$ $(p_1)$};

\node at (2,-0.3) {\footnotesize $\frac{\alpha}{2}$ $(p_2)$};
\draw[gray,dotted] (2,0) -- (2,2);

\node at (3.5,-0.3) {\footnotesize $\alpha-\epsilon$ $(q_2)$};
\draw[gray,dotted] (3.5,0) -- (3.5,2);

\node at (4.5,-0.3) {\footnotesize $\alpha$};
\draw[gray,dotted] (4.5,0) -- (4.5,4);

\node at (6,-0.3) {\footnotesize $\alpha+2\epsilon$ $(q_1)$};
\draw[gray,dotted] (6,0) -- (6,4);

\node at (1,1.96) {$\smallcdot$};
\draw[very thick, dashed, <-](1,2.05)--(1,2.55)
node at (1,2.85) {\footnotesize $\left(\frac{\alpha}{4},h(0+)\right)$};

\node at (4.75,3.96) {$\smallcdot$};
\draw[very thick, dashed, <-](4.75,4.05)--(4.75,4.6)
node at (5,4.9) {\footnotesize $\left(\alpha+\frac{\epsilon}{2},h(1-)\right)$};

	\end{tikzpicture}
	\caption{Contradiction from $h(0+)>0$. }\label{fig-mainth1}
\end{figure}
Recall the function $\pi$ defined in \eqref{eq-biquasifunction}, we have
\begin{align*}
	\pi(p_1,q_1)=(x-y)h(0)+(y-z)h(1-),~~\pi(p_2,q_2)=(x-z)h(0+),
\end{align*}
and
$$
\pi\left(\frac{p_1+p_2}{2},\frac{q_1+q_2}{2}\right)
=(x-y)h(0+)+(y-z)h(1-).
$$
One can check that
$
\pi\left((p_1+p_2)/{2},(q_1+q_2)/{2}\right)>\max\{\pi(p_1,q_1),\pi(p_2,q_2)\}
$
which implies $\pi$ is not quasi-convex, and this contradicts Lemma \ref{prop:quasi-pi}. Hence, we have $h(0+)\ge 0$. For (b), we assume by contradiction that $h(\alpha)<\min{\{h(0+),h(1-)\}}$ or $h(\alpha)>\max{\{h(0+),h(1-)\}}$. The case of $h(\alpha)>\max{\{h(0+),h(1-)\}}$ contradicts Lemma \ref{prop:quasi-pi} as $h$ should be quasi-convex. If $h(\alpha)<\min{\{h(0+),h(1-)\}}$,
let $p_1=\alpha/2$, $p_2=q_1=\alpha$ and $q_2=(1+\alpha)/2$ (see Figure \ref{fig-mainth2}).
 \begin{figure}
	\centering
	\begin{tikzpicture}
		\draw (0,-3) -- (0,2);
		\draw (0,0) -- (8,0);
		\draw [black] (0.02,-1.5)--(3.97,-1.5);
		\draw [black] (4.02,1)--(7,1);

		\node at (0,0) {$\bullet$};
		\node at (0,-1.5) {$\circ$};
		\node at (4,-1.5) {$\circ$};
		\node at (4,-2.5) {$\bullet$};
		\node at (4,1) {$\circ$};

		\node at (-0.2,-0.3) {\footnotesize $0$};

		\node at (2,0.3) {\footnotesize $\frac{\alpha}{2}$ $(p_1)$};
\draw[gray,dotted] (2,0) -- (2,-1.5);

		\node at (4,-0.3) {\footnotesize $\alpha$ $(p_2, q_1)$};
\draw[gray,dotted] (4,-2.5) -- (4,1);

		\node at (5.5,-0.3) {\footnotesize $\frac{1+\alpha}{2}$ $(q_2)$};
\draw[gray,dotted] (5.5,0) -- (5.5,1);

		\node at (3,-1.53) {$\smallcdot$};
		\draw[very thick, dashed, <-](3,-1.55)--(3,-2.1)
		node at (3,-2.4) {\footnotesize $\left(\frac{3\alpha}{4},h(0+)\right)$};

	   \node at (4.75,1) {$\smallcdot$};
		\draw[very thick, dashed, <-](4.75,1.07)--(4.75,1.6)
		node at (4.75,1.9) {\footnotesize $\left(\frac{3\alpha}{4}+\frac{1}{4},h(1-)\right)$};
	\end{tikzpicture}
	\caption{Contradiction from $h(\alpha)<\min\{h(0+),h(1-)\}$. }\label{fig-mainth2}
\end{figure}
 We have
$$
\pi(p_1,q_1)=(x-y)h(0+)+(y-z)h(\alpha),~~\pi(p_2,q_2)=(x-y)h(\alpha)+(y-z)h(1-)
$$
and
$$
\pi\left(\frac{p_1+p_2}{2},\frac{q_1+q_2}{2}\right)=(x-y)h(0+)+(y-z)h(1-).
$$
One can check that
$
\pi\left((p_1+p_2)/{2},(q_1+q_2)/{2}\right)>\max\{\pi(p_1,q_1),\pi(p_2,q_2)\}
$, which yields a contradiction to Lemma \ref{prop:quasi-pi}. Hence, we conclude that if $h$ is the form of Case 1, then $h\in\mathcal H^{\rm QSM}$. This completes the proof.
\end{proof}

\begin{proof}[Proof of Corollary \ref{coro:gen}]
	The ``if" statement is straightforward to verify.
	To show the ``only if" statement, we first note that,
	since $\mathcal L$ contains all three-point random variables on three specific points, it follows from Theorem \ref{th-main} that $h\in\mathcal H^{\rm QCX}$.
	Since $\mathcal L$ contains random variables that are unbounded from below and above, we know that $h\in \mathcal{H}^{\mathcal L}$ must be continuous at $0$ and $1$;  otherwise $Q_1(X)=\infty$ and $Q_0(X)=-\infty$ would lead to $I_h(X) \not \in \mathbb{R}$. This continuity condition for $h\in\mathcal H^{\rm QCX}$ leads to the   cases stated in the corollary.
\end{proof}

\begin{proof}[Proof of Corollary \ref{coro:gen2}]
	Based on Corollary \ref{coro:gen}, it suffices to show that mixed quantiles are not
	norm-continuous.  To verify this fact, take $\alpha \in(0,1)$ and $c\in [0,1]$, and a uniform random variable $U$ on $[0,1]$.
	Let $X_n=a\mathds{1}_{\{U\in [\alpha-\epsilon,\alpha+\epsilon]\}}+ b\mathds{1}_{\{U> \alpha+\epsilon\}}  $ and $X=b\mathds{1}_{\{U> \alpha+\epsilon\}}  $
	for some $0<a<b$ with $a\ne cb$.
	It is clear that $Q_\alpha^c(X ) =cb$, $Q_\alpha^c(X_n)  = a $ for each $n\in \mathbb{N}$, and $X_n\to X  $ in $L^q$. This example justifies the non-continuity of $Q_\alpha^c$.
\end{proof}

\section{Consequences of Theorem \ref{th-main}}
\label{sec:special}
\subsection{M-quasi-concavity and m-quasi-linearity}

A considerable convenience to work with non-monotone $h\in \mathcal{H}^{\rm BV}$ is that we can put a negative sign in front of $h$ without leaving the class, which is  not the case for $h\in \mathcal{H}^{\rm DT}$.
In particular, we have
 $I_{h}(X)=-I_{-h}(X)$ for $h\in\mathcal H^{\rm BV}$. Therefore,   $I_h$ is m-quasi-concave if and only if $I_{-h}$ is m-quasi-convex,
 and all results on m-quasi-convexity immediately translate into results on m-quasi-concavity.

\begin{corollary}\label{co-QCX}
Suppose that Assumption \ref{assm:1} holds. For $h\in\mathcal H^{\rm BV}$, $I_h$ is m-quasi-concave on $\mathcal M$ if and only if either $h $ is concave, or $I_{-h}$ is a scaled quantile-spread mixture.
\end{corollary}

Combining   Theorem \ref{th-main} and Corollary \ref{co-QCX}, a characterization of signed Choquet integrals with m-quasi-linearity is obtained.
\begin{corollary}\label{co-linearity}
Suppose that Assumption \ref{assm:1} holds. For $h\in\mathcal H^{\rm BV}$, $I_h$ is m-quasi-linear on $\mathcal M$ if and only if it is one of the following forms.
\begin{itemize}
\item[(i)] $I_h=k \mathbb{E}$ for some $k\in\mathbb{R}$ where $\mathbb{E}$ represents the expectation.
\item[(ii)] $I_h=k(cQ_{1}+(1-c)Q_{0})$ for some $k\in\mathbb{R}$ and $c\in[0,1]$.
\item[(iii)] $I_h=kQ_{1-\alpha}^{c}$ for some $k\in\mathbb{R}$, $c\in[0,1]$ and $\alpha\in(0,1)$.
\end{itemize}
\end{corollary}
\begin{proof}
By Theorem \ref{th-main} and Corollary \ref{co-QCX}, we know that $I_h$ is m-quasi-linear if and only if $h\in\mathcal H^{\rm QCX}\cap\mathcal (-\mathcal H^{\rm QCX})$.
It is straightforward to check that $I_h$  is one of the three forms in the corollary.
\end{proof}
Although m-quasi-linearity by definition is not related to the monotonicity of the distortion function,
all three forms of the distortion functions in Corollary \ref{co-linearity} are monotone. This is not surprising. Because m-quasi-linearity of $I_h$ implies quasi-linearity of $h$ (see Lemma \ref{prop:quasi-pi} and use a parallel result for the case of m-quasi-concavity), and a quasi-linear univariate function must be monotone.
\begin{remark}
\cite{WW20} showed that for $h\in\mathcal H^{\rm BV}$, only the three forms in Corollary \ref{co-linearity} are possible for $I_h$ to have convex level sets (CxLS). As such, Corollary \ref{co-linearity} is not   new   since m-quasi-linearity is   stronger (although only slightly)  than CxLS.
\end{remark}

\subsection{Dual utilities}
In this section, we establish the related results for dual utility which is a classic decision model. To this end, we introduce some subsets of $\mathcal H^{\rm DT}$ that are important for the property of m-quasi-convexity or m-quasi-concavity.
\begin{itemize}
\item[(i)] $ \mathcal H_1$: the set of all convex $h\in \mathcal{H}^{\rm DT}$; $ \mathcal H_1^*$: the set of all concave $h\in \mathcal{H}^{\rm DT}$.
\item[(ii)]
$ \mathcal{H}_2$: the set of all $h$ satisfying,  for some $\alpha, k\in[0,1]$, $a,b\ge 0$ and $a+b\le 1$, $h(p)=
    kc\mathds{1}_{\{p=\alpha\}}+k\mathds{1}_{\{\alpha<p<1\}}+\mathds{1}_{\{p=1\}}$, $p\in[0,1]$. In this case,
\begin{align}
I_h=kQ_{1-\alpha}^{c}+(1-k)Q_0.
 \end{align}

$ \mathcal{H}_2^*$: the set of all $h$ satisfying, for some $\alpha, k\in[0,1]$, $a,b\ge 0$ and $a+b\le 1$, $h(p)=
(1-k)\mathds{1}_{\{0<p\le \alpha\}}+kc\mathds{1}_{\{p=\alpha\}}+\mathds{1}_{\{p>\alpha\}}$, $p\in[0,1]$. In this case,
\begin{align}
	I_h=(1-k)Q_{1}+kQ_{1-\alpha}^{c}.
\end{align}
\end{itemize}

We denote by $\mathcal H_{\rm DT}^{\rm QCX}=\mathcal H_1\cup\mathcal H_2$ and $\mathcal H_{\rm DT}^{\rm QCV}=\mathcal H_1^*\cup\mathcal H_2^*$.

\begin{corollary}\label{co-DT}
Suppose that Assumption \ref{assm:1} holds and $h\in\mathcal H^{\rm DT}$. The following statements hold for $I_h:\mathcal{M}\to \mathbb{R}$.
\begin{itemize}
\item[(i)] $I_h$ is m-quasi-convex if and only if $h\in\mathcal H_{\rm DT}^{\rm QCX}$.
\item[(ii)] $I_h$ is m-quasi-concave if and only if $h\in\mathcal H_{\rm DT}^{\rm QCV}$.
\item[(iii)] $I_h$ is m-quasi-linear if and only if one of the cases holds:
 $I_h$ is the expectation; $I_h=cQ_1+(1-c)Q_0$ for some $c\in[0,1]$; $I_h=Q^c_{\alpha}$ for some $c\in[0,1]$ and $\alpha\in(0,1)$.
\end{itemize}
\end{corollary}
\begin{proof}
One can check that $\mathcal H^{\rm DT}\cap\mathcal H^{\rm QCX}=\mathcal H_{\rm DT}^{\rm QCX}$ and $\mathcal H^{\rm DT}\cap(-\mathcal H^{\rm QCX})=\mathcal H_{\rm DT}^{\rm QCV}$. Hence, Statement (i) and (ii) follow immediately from  Theorem \ref{th-main} and Corollary \ref{co-QCX}, respectively. Statement (iii) holds because of the fact that $h\in\mathcal H_{\rm DT}^{\rm QCX}\cap\mathcal H_{\rm DT}^{\rm QCV}$ implies $I_h$ must be one of the forms in (iii).
\end{proof}

\begin{remark}
For $h\in\mathcal H^{\rm DT}$, the functional $I_h$ is monotone and  translation invariant.\footnote{A functional $\rho: \mathcal{X} \to\mathbb{R}$ is monotone if $\rho(X)\le \rho(Y)$ for all $X,Y\in \mathcal{X}$ satisfying $X\le Y$ $\mathbb{P}$-a.s., and it is translation invariant if $\rho(X+c)=\rho(X)+c$ for all $X\in \mathcal{X}$ and $c\in\mathbb{R}$. }   By Lemma 2.2 of \cite{BB15}, $I_h$ is m-quasi-linearity if and only if it has CxLS.  Hence, we obtain a different proof for the  characterization of dual utilities with CxLS in  Theorem 2 of \cite{KP16}.
\end{remark}

\section{Generalized rank-dependent functions with m-quasi-convexity} \label{sec:RDU}

In this section, we discuss the effect of risk transforms on the property of m-quasi-convexity (m-quasi-concavity, m-quasi-linearity) of a signed Choquet integral.
This will  generalize all our results to rank-dependent utilities.

 For a comprehensive treatment of risk transforms, see \cite{LSW21}.
 We will focus
on a special class of transforms, the utility transforms. For a measurable function $v$, typically monotone in economic applications (e.g., a utility function), a $v$-transform
maps the distribution of a random variable $X$ to the distribution of $v(X)$.\footnote{As shown by \cite{LSW21}, commutation with $v$-transforms essentially characterizes probability distortions among all distributional transforms.} In other words, the probability
measure $F$ is transformed to $F\circ v^{-1}$, where $v^{-1}$ is the set-valued inverse of $v$. For a functional $\rho$ well
defined on  $\mathcal M^{v} = \{F\circ v^{-1} : F\in\mathcal M\}$, let $\rho_{v}$ be defined as
\begin{align}\label{eq-rho_v}
\rho_v(F)=\rho(F\circ v^{-1}), ~~~F\in\mathcal M.
\end{align}
Similar to Proposition 3.5 of \cite{WW20}, we show an invariance property of m-quasi-convexity (m-quasi-concavity, m-quasi-linearity) under $v$-transforms.

\begin{proposition}\label{prop-transform}
For any measurable function $v$ and a set of distributions $\mathcal M$, $\rho_v$ in \eqref{eq-rho_v} is m-quasi-convex (m-quasi-concave, m-quasi-linear, resp.) on $\mathcal M$ if and only if $\rho$ is m-quasi-convex (m-quasi-concave, m-quasi-linear, resp.) on $\mathcal M^v$.
\end{proposition}
\begin{proof}
We only consider the case of m-quasi-concavity as other two cases are similar.
For any $\lambda\in(0,1)$ and $F,G\in\mathcal M$, denote by $F'=F\circ v^{-1} \in \mathcal{M}^v$
and $G'=G\circ v^{-1}\in \mathcal{M}^v$, and  we have
\begin{align*}
(\lambda F+(1-\lambda)G)\circ v^{-1}=\lambda(F\circ v^{-1})+(1-\lambda)(G\circ v^{-1}) = \lambda F'+(1-\lambda) G',
\end{align*}
which follows directly from the definition of probability measures.
Since
\begin{align*}
\rho_v(\lambda F+(1-\lambda)G) =\rho(\lambda F'+(1-\lambda) G') \mbox{~~~and~~~}
 \min\{\rho_v(F),\rho_v(G)\} =  \min\{\rho(F'),\rho(G')\},
\end{align*}
 we obtain $$\rho_v(\lambda F+(1-\lambda)G)\ge  \min\{\rho_v(F),\rho_v(G)\}\mbox{~~if and only if~~}\rho(\lambda F'+(1-\lambda) G') \ge  \min\{\rho(F'),\rho(G')\}. $$
It follows that
 m-quasi-concavity of $\rho$ on $\mathcal{M}^c$ is equivalent to that of $\rho_v$ on $\mathcal{M}$, noting that
 $F, G$ can be arbitrarily chosen from $\mathcal{M}$ and $F',G'$ can be arbitrarily chosen from $\mathcal{M}^c$.
\end{proof}

In economics and finance, $v$-transforms are widely used. In addition to the expected utility or the rank-dependent utility models,
a pricing functional for options can be seen as the (risk-neutral) expectation of a transformed asset price distribution.

Recall that a generalized rank dependent  function  is a $v$-transform of some signed Choquet integrals $I_h$, denoted by $R_{h,v}$, that is,
$
R_{h,v}(X) = I_{h}(v(X))$. The mapping $R_{h,v}$ is a rank-dependent utility if $h\in \mathcal{H}^{\rm DT}$ and $v$ is increasing, but we do not need these assumptions for our next result.
Proposition \ref{prop-transform} and Theorem \ref{th-main} directly lead  to the following characterization of quasi-convexity (m-quasi-concavity, m-quasi-linearity) in this model.
Recall that $\mathcal H^{\rm QCX} = \mathcal H^{\rm CX}\cup\mathcal H^{\rm QSM}$
which contains  distortion functions that are convex and those of scaled quantile-spread mixtures.
\begin{theorem}\label{coro:rdu}
For $h\in\mathcal H^{\rm BV}$ and a function $v$  on $\mathbb{R}$, suppose that Assumption \ref{assm:1} holds where the values of
$v(x)$, $v(y)$ and $v(z)$ are distinct.
The following statements hold for $R_{h,v}:\mathcal{M}\to \mathbb{R}$.
\begin{itemize}
\item[(i)] $R_{h,v}$ is m-quasi-convex  if and only if $h\in\mathcal H^{\rm QCX}$.
\item[(ii)] $R_{h,v}$ is m-quasi-concave  if and only if $ h\in (-\mathcal H^{\rm QCX})$.
\item[(iii)] $R_{h,v}$ is m-quasi-linear   if and only if $h$ is one of the forms in Corollary \ref{co-linearity}.
\end{itemize}
In particular, if $h$ is continuous or strictly monotone, then
$R_{h,v}$ is m-quasi-convex (m-quasi-concave, resp.) if and only if $h$ is convex (concave, resp.).
\end{theorem}

\begin{proof}
Define
$$
\mathcal M^v=\{F\circ v^{-1}:F\in \mathcal M\}.
$$
Since the values of
$v(x)$, $v(y)$ and $v(z)$ are different, $\mathcal M^v$ contains at least all three-point distributions on three specific points $v(x)$, $v(y)$ and $v(z)$. Using Proposition \ref{prop-transform},
(i), (ii) and (iii) in this theorem can be immediatelly verified by applying Theorem
\ref{th-main}, Corollary \ref{co-QCX} and Corollary \ref{co-linearity}, respectively.
\end{proof}

Theorem \ref{coro:rdu} implies that, for a rank-dependent utility   $R_{h,v}$ with a strictly increasing utility function $v$ and a continuous distortion function $h$, which are the most common assumptions, m-quasi-concavity alone characterizes concavity of $h$.
For the corresponding statement on strictly increasing $h$, see Theorem 7 of \cite{WY21}.

\section{A conflict between o-concavity and m-concavity}
\label{app:A}

In this section, we illustrate a conflict between o-concavity and m-concavity for constant-additive mappings.
Let $\mathcal{X}$ be the set of bounded random variables, and recall that o-convexity of a mapping on $\mathcal{M}$ or $\mathcal{X}$ is defined as convexity on $\mathcal{X}$.
A mapping $\rho:\mathcal{X}\to \mathbb{R}$ is \emph{constant additive} if $\rho(X+c)=\rho(X)+\rho(c)$ for $X\in \mathcal{X}$ and $c\in \mathbb{R}$.
On $\mathcal{X}$, continuity is   with respect to the supremum-norm.
Continuous and constant-additive mappings on $\mathcal{X}$ include, but are not limited to, all signed Choquet integrals and normalized monetary risk measures (\cite{FS16}).

 \begin{proposition}\label{pr:conflict}
For a continuous and constant-additive mapping $\rho:\mathcal{X} \to \mathbb{R}$,
 the following are equivalent:
 \begin{enumerate}[(i)]
 \item $\rho$ is o-convex and m-convex;
 \item $\rho$ is o-concave and m-concave;
 \item there exists $c\in \mathbb{R}$ such that $\rho(X)=c \mathbb{E}[X] $ for all $X\in \mathcal{X}$.
 \end{enumerate}
 \end{proposition}
 \begin{proof}
Note that (i) and (ii) are symmetric, and (iii)$\Rightarrow$(i) is trivial.
It suffices to show the direction (i)$\Rightarrow$(iii).
 Since a m-convex mapping is necessarily law-based, we equivalently formula $\rho$ on $\mathcal{M}_c$.
 Denote by $\delta_x\in \mathcal M$   the point-mass at $x\in \mathbb{R}$.
Denote by $c=\rho(1)$. Note that
  $\rho(x)=cx$ for $x\in \mathbb{R}$ since $\rho$ is constant additive and continuous.
Let $ F=\sum_{i=1}^n p_i \delta_{x_i}$ for some numbers $p_1,\dots,p_n\ge 0$ which add up to $1$ and $\{x_1,\dots,x_n\}\subseteq \mathbb{R}$. Let $X\sim F$.
By m-convexity of $\rho$, we have
$  \rho(F) \le \sum_{i=1}^n p_i   \rho(\delta_{x_i} ) = \sum_{i=1}^n p_i c  x_i   = c \mathbb{E}[X]  $.
 Since $\rho$ is  law-based, continuous and o-convex, it is convex-order monotone by e.g., the representation in \citet[Theorem 2.2]{LCLW20}.
Thus, $\rho(X)\ge \rho (\mathbb{E}[X]) = c\mathbb{E}[X]$ for all $X\in \mathcal{X}$.
Putting the above two inequalities together, we have $\rho(X)=c\mathbb{E}[X] $ for all finitely supported random variables $X\in \mathcal{X}$.

For a general $X\in  \mathcal{X} $,
let $X_n=   \lfloor  n X \rfloor /n$ for $n\in \mathbb{N}$, which is an approximation of $X$. It is clear that $X_n\to  X$ as $n\to \infty$, and $|X_n-X|\le 1/n$.
Using continuity of $\rho $ again,
 we have $\rho(X_n)\to \rho(X)$ as $n\to \infty$.
 Since $\rho(X_n) = c \mathbb{E}[X_n]   \to c \mathbb{E}[X]   $ as $n\to\infty$, we obtain   $\rho(X)=c \mathbb{E}[X]  $.
 \end{proof}

 The same conclusion in Proposition \ref{pr:conflict} holds for  $\rho:L^p\to \mathbb{R}$ for $p\in [1,\infty)$ following the same proof.
 \begin{remark}Constant additivity of $\rho$ is essential for Proposition \ref{pr:conflict}.
A mapping $\rho: \mathcal{X} \to \mathbb{R}$ that is monotone, m-convex and o-convex does not need to be m-linear.
For an example, take
$
\rho_1:X\mapsto \mathbb{E}[f(X)]$ and $\rho_2:X\mapsto \mathbb{E}[g(X)]$ where $f$ and $g$ are two increasing  convex functions.
Clearly, $\rho_1$ and $\rho_2$ are both o-convex and m-linear.
Since convexity is preserved under a maximum operation, the mapping $\rho:=\max \{\rho_1,\rho_2\}$ is o-convex and m-convex,
but it is not m-linear unless $f\ge g$ or $g\ge f$.
The reason that the proof does not work in this case is that, by letting $\ell(x)=\rho(x)$ for $x\in \mathbb{R}$, we can show using the argument above that
$\ell(\mathbb{E}[X]) \le \rho(X)\le  \mathbb{E}[\ell(X)]$,
but this does not pin down $\rho$ unless $\ell$ is linear.
 \end{remark}

\section{Conclusion}\label{sec:7}

In this paper, m-quasi-convexity is characterized  for signed Choquet integrals  (Theorem \ref{th-main}) and generalized rank-dependent functions (Theorem \ref{coro:rdu}).
Based on the  characterization for signed Choquet integrals, we obtain in a unified result  (Theorem \ref{th-grand}) containing  seven equivalent conditions for m-quasi-convexity.
Our results require no monotonicity assumption. Although our results are formulated for signed Choquet integrals and generalized rank-dependent functions, the corresponding results for dual utilities and rank-dependent utilities were not known in the literature, and  our results help to understanding classic decision models by disentangling monotonicity from other important properties.
We have chosen to use m-convexity and o-concavity to present our main results, to be consistent with the literature on decision theory (e.g., \cite{Q93,W10}); by a simple sign change, we obtain corresponding results for m-concavity and o-convexity, a convention that is more common in the literature of risk management (e.g., \cite{MFE15, FS16}).

\subsubsection*{Acknowledgements}
The authors thank  Peter Wakker and Jingni Yang
for helpful comments and discussions on an earlier version of the paper.
Ruodu Wang is supported by the Natural Sciences and Engineering Research Council of Canada (RGPIN-2018-03823, RGPAS-2018-522590).

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