2.1. MFSD: Definition

We begin by defining multi-fractional stochastic dominance.

Definition 2. (Multi-fractional $(1+\unicode{x1D6C4})$ -SD.) Let $\boldsymbol{ \gamma}\colon \mathbb{R} \to [0, 1]$ be an arbitrary non-decreasing bounded function. Let F and G be two arbitrary distribution functions. We say that G dominates F in the sense of the multi-fractional $(1+\boldsymbol{ \gamma})$ -SD, denoted by $F \preceq^{\mathrm{MF}}_{(1+\boldsymbol{ \gamma})\textrm{-SD}} G$ , if

(3) \begin{equation}\int_{-\infty}^{t} ( F(x) - G(x) )_{-}{\mathrm{d}} x \le \boldsymbol{ \gamma}(t) \int_{-\infty}^{t} (F(x) - G(x) )_{+} {\mathrm{d}} x \quad \text{for all $t\in \mathbb{R}$.}\end{equation}

As noted, the function $\unicode{x1D6C4}$ encodes local relationships between the distributions. Accordingly, multi-fractional $(1+\unicode{x1D6C4})$ -SD exhibits a form of local interpolation. We now provide an alternative characterization of multi-fractional $(1+\unicode{x1D6C4})$ -SD that rigorously formalizes this local interpolation property and clarifies its precise interpretation motivated by the classical work [Reference Atkinson2].

Definition 3. (Restricted fractional $(1+\gamma)_{t}^r$ -SD.) For fixed $\gamma\in [0, 1]$ and given $t\in \mathbb{R}$ , we say that G dominates F in the sense of restricted fractional $(1+\gamma)_{t}^r$ -SD, denoted by $F \preceq_{(1+\gamma)_{t}^r\textrm{-SD}} G$ , if

\begin{equation*} \int_{-\infty}^{s} ( F(x) - G(x) )_{-}{\mathrm{d}} x \le \gamma \int_{-\infty}^{s} (F(x) - G(x) )_{+} {\mathrm{d}} x \quad \text{for all $s \le t$.}\end{equation*}

Example 1. (Local interpolation.) Let $\gamma\in(0,1)$ be a constant. Consider the following non-decreasing function:

\begin{equation*}\boldsymbol{ \gamma}(t)=\begin{cases}0 & \text{if $ t \le t_1$,} \\\gamma & \text{if $ t_1 < t \leq t_2$,} \\1 & \text{if $ t>t_2$,} \end{cases}\end{equation*}

where $t_1<t_2$ are real numbers. Assume that $F \preceq^{\mathrm{MF}}_{ (1+ \boldsymbol{ \gamma})\textrm{-SD}} G$ . Then $\boldsymbol{ \gamma}(t_1)=0$ entails that $F \preceq_{(1+\boldsymbol{ \gamma}(t_1))^r_{t_1}\textrm{-SD}} G$ , i.e. $F(x) \ge G(x)$ for all $x \le t_1$ . Similarly, the condition $\boldsymbol{ \gamma}(t_2)=\gamma$ yields that $F \preceq_{(1+\gamma)^r_{t_2}\textrm{-SD}} G$ , that is, $F \preceq_{(1+\gamma)\textrm{-SD}} G$ holds on the interval $(\!-\infty, t_2]$ . Lastly, the fact that $\boldsymbol{ \gamma}(t) =1$ , for all $t \in (t_2,\infty)$ , implies that $F \preceq_{\mathrm{SSD}} G$ . Hence, with appropriately chosen non-decreasing function $\boldsymbol{ \gamma}$ , one can simultaneously order given F and G from first (strongest) to second (weakest) order on different portions (capture with function $\boldsymbol{ \gamma}$ ) of the supports.

Now we discuss yet another feature of the MFSD framework. It is known that the assumption of equal means for the distribution functions ( $\mu_F=\mu_G$ ) renders them indistinguishable in terms of the fractional $(1 + \gamma)$ -SD, for any constant $\gamma \in [0, 1)$ . We exclude the limiting case $\gamma=1$ due to the fact that it corresponds to (less interesting) SSD. Nonetheless, as demonstrated in the following example, the multi-fractional $(1+\unicode{x1D6C4})$ -SD can effectively order distributions with equal means, provided that an appropriate function, i.e. non-constant and satisfying $\boldsymbol{ \gamma}(t)=1$ for some $t\in \mathbb{R}$ , is chosen.

Example 2. Consider $G \sim \delta_{\mu} $ (the Dirac measure at the point $\mu\in \mathbb{R}$ ), and the distribution functions

\begin{equation*}F(x)=\begin{cases}0 & \text{if $ x < \mu-\varepsilon$,} \\1/2 & \text{if $ \mu-\varepsilon \le x < \mu+\varepsilon$,}\\1 & \text{if $ \mu+\varepsilon \le x$,}\end{cases} \quad \text{and} \quad \unicode{x1D6C4}(t)=\begin{cases}0 & \text{if $ \mu-\varepsilon\le t \le \mu$,} \\{{(t-\mu)}/{ \varepsilon}} & \text{if $ \mu< t <\mu+\varepsilon$,}\\1 & \text{if $ \mu+\varepsilon\le t$,}\end{cases}\end{equation*}

where $\mu>\varepsilon>0$ . The straightforward calculation shows that the integral condition (3) holds. Hence $F \preceq^{\mathrm{MF}}_{(1+\unicode{x1D6C4})\textrm{-SD}} G$ . However, $F \npreceq_{ (1+ \gamma)\textrm{-SD}} G$ for any constant $\gamma \in [0,1)$ , since the inequality (1) fails at $t=\mu+\varepsilon$ ,

\begin{align*}0=\mu_G-\mu_F=\int_{\mu-\varepsilon}^{\mu+\varepsilon} ( F(x)-G(x) )_+ {\mathrm{d}} x - \int_{\mu-\varepsilon}^{\mu+\varepsilon} ( F(x)-G(x) )_- {\mathrm{d}} x.\end{align*}

In contrast, we have $\boldsymbol{ \gamma}(\mu+\varepsilon)=1$ .

We can readily construct an additional example that extends Example 2 to a scenario beyond identical means. However, to avoid redundancy, we will not pursue this further. The following example, though rather technical, examines the non-trivial relationship between fractional and multi-fractional SD rules. While we provide the example here for context, the explicit constructions are provided in the supplementary material to avoid disrupting the flow.

The conclusion is that in both non-trivial scenarios, one can always construct two distribution functions F and G that are not distinguishable by the fractional $(1+\gamma)$ -SD but it is possible to order them with the multi-fractional $(1+\unicode{x1D6C4})$ -SD, and vice versa.

Example 4. In this example we visualize the function $t\mapsto \unicode{x1D6C4}(t)$ to show how it changes under perturbations of one distribution within a location–scale family; this provides a visual counterpart to Example 2.11 of [Reference Müller, Scarsini, Tsetlin and Winkler23] in our setting. Let F and G be from the same location–scale family,

\[F(x)=H\biggl(\frac{x-\mu_F}{\sigma_F}\biggr)\quad \text{and}\quad G(x)=H\biggl(\frac{x-\mu_G}{\sigma_G}\biggr),\]

where H denotes the standard normal CDF. Write $\Delta\mu\,:\!=\, \mu_G-\mu_F$ and $\Delta\sigma\,:\!=\, \sigma_F-\sigma_G$ . Under $\mu_F\le \mu_G$ and $\sigma_G<\sigma_F$ , the pair (F,G) is single-crossing with crossing point $t_1=\mu_F+\sigma_F{{\Delta\mu}/{\Delta\sigma}}$ . In this regime, the strongest MFSD ordering is obtained by the smallest admissible degree function, denoted $\unicode{x1D6C4}_{\min}(F,G)$ , which one can derive as

\begin{equation*} \unicode{x1D6C4}_{\min}(F,G)(t)=\begin{cases}0, & t\le t_1,\\\dfrac{\Delta \sigma \displaystyle\int_{\Delta \mu/\Delta \sigma}^{(t-\mu_F)/\sigma_F} ( 1 - H (z) )\,{\mathrm{d}} z- \sigma_G \int_{(t-\mu_F)/\sigma_F}^{(t-\mu_G)/\sigma_G} ( 1 - H (z) )\,{\mathrm{d}} z}{\Delta\sigma \displaystyle\int_{-\infty}^{\Delta \mu/\Delta \sigma} H(z)\,{\mathrm{d}} z},& t>t_1.\end{cases}\end{equation*}

For plotting, fix $F=\mathcal N(0,1)$ (so $\mu_F=0$ , $\sigma_F=1$ ), and $G=\mathcal N(\mu_G,\sigma_G^2)$ with $\Delta\mu=\mu_G$ and $\Delta\sigma=1-\sigma_G$ . We perturb G over a grid of $(\mu_G,\sigma_G)$ summarized by the ratio $\Delta\mu/\Delta\sigma=\mu_G/(1-\sigma_G)$ . The resulting surface $(t,\Delta\mu/\Delta\sigma)\mapsto \unicode{x1D6C4}_{\min}(F,G)(t)$ (and its contours/slices) makes the interpretation transparent, $\unicode{x1D6C4}_{\min}(t)=0$ up to the crossing point $t_1=\mu_G/(1-\sigma_G)$ , and larger perturbations (larger $\Delta\mu/\Delta\sigma$ ) shift the crossing point $t_1$ to the right and keep $\unicode{x1D6C4}_{\min}(t)$ closer to 0 over a wider range of t. Holding t fixed, $\unicode{x1D6C4}_{\min}(F,G)(t)$ falls quickly toward zero as $\Delta\mu/\Delta\sigma$ increases, indicating that stronger perturbations push the maximal admissible MFSD degree toward the FSD limit while weaker perturbations move it towards the SSD limit. Higher values of t shift the curve upward, since $\unicode{x1D6C4}_{\min}$ is non-decreasing in t and therefore attains larger values.

Figure 1.

Normal location–scale perturbations with $F=\mathcal N(0,1)$ and $G=\mathcal N\bigl(\mu_G,\sigma_G^2\bigr)$ . (a) Fixed- $\Delta\mu/\Delta\sigma$ slices. (b) Fixed-t slices. (c) Surface.

Figure 1 Long description

Panel A: This line graph shows the relationship between t and gamma_min(F,G)(t) for different values of delta_mu/delta_sigma. The x-axis represents t, and the y-axis represents gamma_min(F,G)(t). The graph includes multiple lines representing different values of delta_mu/delta_sigma (0, 0.5, 1, 2, 3). The lines show how gamma_min(F,G)(t) changes with t for each value of delta_mu/delta_sigma. Panel B: This line graph shows the relationship between delta_mu/delta_sigma and gamma_min(F,G)(t) for different values of t. The x-axis represents delta_mu/delta_sigma, and the y-axis represents gamma_min(F,G)(t). The graph includes multiple lines representing different values of t (0.5, 1, 2, 4). The lines show how gamma_min(F,G)(t) changes with delta_mu/delta_sigma for each value of t. Panel C: This 3D surface plot shows the relationship between delta_mu/delta_sigma, t, and gamma_min(F,G)(t). The x-axis represents delta_mu/delta_sigma, the y-axis represents t, and the z-axis represents gamma_min(F,G)(t). The surface plot illustrates how gamma_min(F,G)(t) varies with both delta_mu/delta_sigma and t.

2.2. The MFSD universe

In this section we formalize all prior observations within a structured framework. For every constant $\gamma \in [0, 1]$ and non-decreasing function $\boldsymbol{ \gamma }\colon \mathbb{R} \to [0, 1]$ , we introduce

\begin{align*}\unicode{x1D4D4}_{{F}}(\gamma) & \,:\!=\, \{ (F,G) \in \unicode{x1D4D3}\,(\mathbb{R}) \times \unicode{x1D4D3}\,(\mathbb{R}) \colon F \preceq_{(1+\gamma)\textrm{-SD}} G \},\\\unicode{x1D4D4}_{\mathrm{MF}}(\boldsymbol{ \gamma}) & \,:\!=\, \bigr\{ (F,G) \in \unicode{x1D4D3}\,(\mathbb{R}) \times \unicode{x1D4D3}\,(\mathbb{R}) \colon F \preceq^{\mathrm{MF}}_{ (1+ \boldsymbol{ \gamma})\textrm{-SD}} G \bigr\},\end{align*}

where $\unicode{x1D4D3}\,(\mathbb{R})$ is the set of all distribution functions supported on $\mathbb{R}$ . Also, we denote $\unicode{x1D4D4}_{{F}}(1) = \unicode{x1D4D4}_{\mathrm{SSD}}$ . Note that, clearly, we have the set identity $ \unicode{x1D4D4}_{\mathrm{MF}}(\boldsymbol{ \gamma}) = \unicode{x1D4D4}_{{F}}(\gamma)$ as soon as $\unicode{x1D6C4} \equiv \gamma$ is the constant function, and furthermore, when $ \boldsymbol{ \gamma} (t ) \ge \gamma$ $(\boldsymbol{ \gamma} (t ) \le \gamma)$ for every $t \in \mathbb{R}$ , the set inclusion $ \unicode{x1D4D4}_{{F}}(\gamma) \, \subsetneq \, \unicode{x1D4D4}_{\mathrm{MF}}(\boldsymbol{ \gamma})$ $( \unicode{x1D4D4}_{\mathrm{MF}}(\boldsymbol{ \gamma}) \, \subsetneq \, \unicode{x1D4D4}_{{F}}(\gamma))$ holds. Lastly, Example 3 shows that for every constant $\gamma \in (0,1)$ and every (non-decreasing) function $\boldsymbol{ \gamma}$ satisfying $\boldsymbol{ \gamma}(t^0) < \gamma < \boldsymbol{ \gamma}(t_0)$ for some $t^0<t_0$ , we have

\begin{equation*}\unicode{x1D4D4}_{\mathrm{MF}}(\boldsymbol{ \gamma}) \setminus \unicode{x1D4D4}_{{F}}(\gamma) \neq \emptyset \quad \text{and} \quad \unicode{x1D4D4}_{{F}}(\gamma) \setminus \unicode{x1D4D4}_{\mathrm{MF}}(\boldsymbol{ \gamma}) \neq \emptyset .\end{equation*}

Before proceeding further, for notational convenience throughout this paper, we use the following notation for the limits of a non-decreasing function $\unicode{x1D6C4}$ . We define

\[{\stackrel{\rule{.6em}{1.5pt}}{\unicode{x1D6C4}}}\,:\!=\, \lim_{t\to +\infty}\boldsymbol{ \gamma}(t)=\sup\nolimits_{t\in\mathbb{R}}\unicode{x1D6C4}(t) \quad \text{and} \quad \raise-4.8pt\hbox{$\buildrel \displaystyle\unicode{x1D6C4} \over {\rule{.6em}{1.5pt}}$} \,:\!=\, \lim_{t\to -\infty}\boldsymbol{ \gamma}(t)=\inf_{t \in \mathbb{R}} \unicode{x1D6C4}(t).\]

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

2.3. Utility class

In this section we first introduce the set of utility functions that generates the multi-fractional SD. Then we prove the ‘if and only if’ theorem between the introduced utility class and the integral condition (3). As mentioned previously, the utility class corresponding to MFSD is not readily available. Therefore the definition requires preliminary steps to be established. As our starting point, we present the following simple lemma and the subsequent proposition, inspired by the fractional SD utility space $\mathscr{U\,}^*_{\gamma}$ and by [Reference Müller, Scarsini, Tsetlin and Winkler23, Theorem 2.4] (specifically, direction (b)).

Lemma 1. Let $\boldsymbol{ \gamma}\,:\, \mathbb{R} \rightarrow [0, 1]$ be an arbitrary non-decreasing function. For each $t \in \mathbb{R}$ , we consider the following function space:

\begin{align*} \mathscr{U\,}^{t}_{\boldsymbol{ \gamma}(t)}&\,:\!=\, \biggl\{ u\colon \mathbb{R} \to \mathbb{R}\colon u(x)=u(t) \ \ {for\, all}\ \, x \ge t \ \, {and} \notag \\&\qquad {for\, all}\,\,\ x_1 < x_2 \le x_3 < x_4, \, 0\leq \boldsymbol{ \gamma}(t) \dfrac{u(x_4) -u(x_3)}{x_4-x_3} \leq \dfrac{u(x_2)-u(x_1)}{x_2-x_1} \biggr\}.\end{align*}

Let F and G be two arbitrary distribution functions. For every $t \in \mathbb{R}$ , define the continuous function $u_t$ via its right-derivative

(5) \begin{equation}u_t^{\prime}(x)=\begin{cases}\boldsymbol{ \gamma}(t) & {if \, G(x) \le F(x)\, and \, x \leq t,} \\1 & {if \, F(x) < G(x)\, and \, x \leq t\,,} \\0 & {if \, t < x\,.}\end{cases}\end{equation}

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

Proposition 2. Let F and G be two arbitrary probability distributions having finite means, and let $\boldsymbol{ \gamma}\colon \mathbb{R} \rightarrow [0, 1]$ be a non-decreasing function. Then relation

\begin{equation*} \mathbb{E}_F[u_t] \le \mathbb{E}_G [u_t] \quad {for\, all \, t \in \mathbb{R}}\end{equation*}

implies that $F \preceq^{\mathrm{MF}}_{(1+\boldsymbol{ \gamma})\textrm{-SD}} G$ .

The conclusion is that any desirable utility class for MFSD should contain the family of those base-type functions $\{u_t \colon t \in \mathbb{R}\}$ given in (5). Observe that these functions enjoy two important characteristics.

1. (i) They are constant over the region $(t,+\infty)$ and hence their derivatives vanish (this in particular implies that there is no contribution to the expected value via the integration by parts formula).

2. (ii) On the complementary range $(\!-\infty,t)$ they should manifest the fractional $(1+\gamma)$ -SD base-type functions with $\unicode{x1D6C4}(t)$ instead of a fixed parameter $\gamma$ .

Based on the discussion thus far, we take the function spaces $\mathscr{U\,}^{t}_{\boldsymbol{ \gamma}(t)}$ for $t \in \mathbb{R}$ as the building blocks of the MFSD utility class. Before proceeding, let us summarize some of the key mathematical properties that are crucial for the subsequent discussion.

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

\begin{equation*} \mathscr{U\,}^{\mathrm{Union}}_{\unicode{x1D6C4}}\,:\!=\, \bigcup_{t\in [\!-\infty,\infty]}\mathscr{U\,}^t_{\unicode{x1D6C4}(t)}.\end{equation*}

In the above, for technical reasons, when $\unicode{x1D6C4}(t)=0$ , we restrict the utility space $\mathscr{U\,}^{t}_{\unicode{x1D6C4}(t)}$ to consist of non-decreasing continuous functions. As follows from Proposition 3(d), although for each $t \in [\!-\infty,\infty]$ the class $\mathscr{U\,}^t_{\unicode{x1D6C4}(t)}$ is a convex cone, however, $\mathscr{U\,}^{\mathrm{Union}}_{\unicode{x1D6C4}}$ is not necessarily closed under (arbitrary) addition. Given that the sum of utility functions naturally produces a new utility function, and considering the linearity of expected utilities, the class $\mathscr{U\,}^{\mathrm{Union}}_{\unicode{x1D6C4}}$ alone is not a useful candidate for the utility class of the multi-fractional $(1+\unicode{x1D6C4})$ -SD. 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 4. (Utility class of MFSD.) Let $\boldsymbol{ \gamma}\colon \mathbb{R} \rightarrow[0, 1]$ be an arbitrary non-decreasing function. A (utility) class for the multi-fractional $(1+\unicode{x1D6C4})$ -SD is defined as

\begin{equation*} \mathscr{U\,}^{\mathrm{MF}}_{\unicode{x1D6C4}} \,:\!=\, \operatorname{span}_{\mathbb{R}_+} \bigl\{u \colon u\in \mathscr{U\,}^{\mathrm{Union}}_{\unicode{x1D6C4}} \bigr\} = \Biggl\{ u = \sum_{\text{finite}} \lambda_k u_k, \, \lambda_k \ge 0, \, u_k \in \mathscr{U\,}^{\mathrm{Union}}_{\unicode{x1D6C4}} \Biggr\}.\end{equation*}

Now we are ready to state the main theorem of this section.

Following the main theorem, we provide the next remark to emphasize several important points.

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

Example 5. Let $\gamma_1 < \gamma_2 < \cdots < \gamma_n $ be n real numbers belonging to the interval [0, 1]. Consider the function

(6) \begin{equation}\unicode{x1D6C4} (t) =\begin{cases}\gamma_1 & \text{if $ t \le t_1$,}\\\gamma_2 & \text{if $ t_1\! \lt t \le t_2$,}\\\vdots\\\gamma_{n-1} & \text{if $ t_{n-2} < t \le t_{n-1}$,}\\\gamma_n & \text{if $ t > t_{n-1}$.}\end{cases}\end{equation}

Then we have

\begin{align*}\mathscr{U\,}^{\mathrm{Union}}_{\unicode{x1D6C4}} = \Biggl( \bigcup_{k=1}^{n-1} \mathscr{U\,}^{\,t_k}_{\unicode{x1D6C4}(t_{k})} \Biggr) \bigcup \mathscr{U\,}^{+\infty}_{\stackrel{\rule{.6em}{1.5pt}}{\unicode{x1D6C4}}} =\Biggl( \bigcup_{k=1}^{n-1} \mathscr{U\,}^{\,t_k}_{\gamma_k} \Biggr) \bigcup \mathscr{U\,}^{*}_{\gamma_n},\end{align*}

where the function class $\mathscr{U\,}^{*}_{\gamma_n}$ corresponds to the fractional $(1+\gamma_n)$ -SD.

The purpose of the following example is to provide yet another capability of multi-fractional SD through its utility space.

Example 6. On certain occasions we require a given family of utility functions sharing particular features to belong to a single utility space, corresponding to a given SD rule. Next we construct a parametrized family $\mathscr{U\,}_{\Theta}$ of utility functions such that the only fractional utility space $\mathscr{U\,}^{*}_{\gamma}$ (associated with fractional $(1+\gamma)$ -SD) that contains $\mathscr{U\,}_{\Theta}$ is the first-order stochastic dominance (FSD) space, i.e. $\gamma=0$ . By contrast, $\mathscr{U\,}_{\Theta}$ can be embedded in a multi-fractional utility space $\mathscr{U\,}^{\mathrm{MF}}_{\unicode{x1D6C4}}$ that is a strict subset of the FSD utility space $\mathscr{U\,}_{0}$ , thereby yielding a stochastic dominance criterion under which a larger class of distribution functions can be ordered.

Let $\theta^* >0$ be fixed. Let $\Theta = (\theta^*,\infty)$ and consider a function $\widetilde{\unicode{x1D6C4}} \colon \Theta \to (0,1]$ so that (i) it is non-decreasing, (ii) $\widetilde{\unicode{x1D6C4}} (\theta) < \theta $ for every $\theta \in \Theta$ , and (iii) $\lim_{\theta \to \theta^*} \widetilde{\unicode{x1D6C4}} (\theta) =0$ . For every $\theta \in \Theta $ , we define a utility function $u_\theta$ via its right-derivative as

\begin{equation*}u'_\theta (x) =\begin{cases}2 \widetilde{\unicode{x1D6C4}} (\theta)& \text{if $ x \le \widetilde{\unicode{x1D6C4}} (\theta)$,}\\1 + \widetilde{\unicode{x1D6C4}} (\theta)& \text{if $ \widetilde{\unicode{x1D6C4}} (\theta) < x \le \theta$,}\\\widetilde{\unicode{x1D6C4}} (\theta) & \text{if $ x > \theta$.}\end{cases}\end{equation*}

Then it is simple to observe that $u_\theta \in \mathscr{U\,}^*_{\widetilde{\unicode{x1D6C4}}(\theta)}$ (see (2) for the definition) for each $\theta \in \Theta$ . Let $\mathscr{U\,}_{\Theta } \,:\!=\, \{ u_\theta \colon \theta \in \Theta \}$ . Now suppose 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 necessarily $\gamma =0$ . In other words, the only way 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

\begin{equation*}\unicode{x1D6C4} (\theta)=\begin{cases}0 & \text{if $ \theta \le \theta^*$,}\\\widetilde{\unicode{x1D6C4}}(\theta) & \text{if $ \theta > \theta^*$.}\end{cases}\end{equation*}

Now we claim that $\mathscr{U\,}_{\Theta} \subseteq \mathscr{U\,}^{\mathrm{MF}}_{\unicode{x1D6C4}} \subset \mathscr{U\,}_{0}$ , where clearly the second inclusion is strict. For each $\theta \in \Theta$ , define utility functions $v_\theta$ and $w_\theta$ via their right-derivatives $v'_\theta (x) = \widetilde{\unicode{x1D6C4}}(\theta)$ and

\begin{equation*}w'_\theta (x) =\begin{cases}\widetilde{\unicode{x1D6C4}}(\theta) & \text{if $ x \le \widetilde{\unicode{x1D6C4}}(\theta)$,}\\1 & \text{if $ \widetilde{\unicode{x1D6C4}}(\theta) < x \le \theta$,}\\0 & \text{if $ x > \theta$.}\end{cases}\end{equation*}

Then, clearly, $v_\theta, w_\theta \in \mathscr{U\,}^{\mathrm{MF}}_{\boldsymbol{\gamma }}$ , and therefore $u_\theta = v_\theta + w_\theta \in \mathscr{U\,}^{\mathrm{MF}}_{\boldsymbol{\gamma }}$ for each $\theta$ , which proves the validity of the first inclusion. To finalize this example through visualization, we shall specify $\widetilde{\unicode{x1D6C4}}(\theta)$ . We set $\theta^*=1$ and choose $\widetilde{\unicode{x1D6C4}}(\theta)=1-{\mathrm{e}}^{(1-\theta)}$ . It is apparent that it is a non-decreasing function, where $1-{\mathrm{e}}^{(1-\theta)}< \theta$ for every $\theta \in \Theta$ , $\lim_{\theta \to \theta^*} 1-{\mathrm{e}}^{(1-\theta)}=0$ , and $\lim_{\theta \to 1} 1-{\mathrm{e}}^{(1-\theta)}=1$ . Now consider an exponential random variable $E \sim \text{Exp}(1)$ and let $X = E+1$ with the distribution function

\begin{equation*}F_X(\theta) =\unicode{x1D6C4} (\theta)=\begin{cases}0 & \text{if $ \theta < 1$,}\\1-{\mathrm{e}}^{(1-\theta)} & \text{if $ \theta \ge 1$.}\end{cases}\end{equation*}

Then the derivative of the utility function $u'_\theta$ becomes

\begin{equation*}u'_\theta (x) =\begin{cases}2 (1-{\mathrm{e}}^{(1-\theta)})& \text{if $ x \le 1-{\mathrm{e}}^{(1-\theta)}$,}\\2-{\mathrm{e}}^{(1-\theta)}& \text{if $ 1-{\mathrm{e}}^{(1-\theta)} < x \le \theta$,}\\1-{\mathrm{e}}^{(1-\theta)} & \text{if $ x > \theta$.}\end{cases}\end{equation*}

To highlight the comprehensiveness of our proposed utility class in modelling diverse risk preferences, it is worth emphasizing the inclusion of both star-shaped utility functions [Reference Landsberger and Meilijson13, Reference Landsberger and Meilijson14] and fourth-degree polynomial functions [Reference Benishay3] within our utility class $\mathscr{U\,}^{\mathrm{MF}}_{\unicode{x1D6C4}}$ . These specific utility functions are of particular interest due to their ability to capture local convexities. Such convex segments represent intervals where individuals might exhibit risk-seeking behaviour, a phenomenon that traditional concave utility functions fail to capture.

Figure 2.

Graphs of $u'_{\theta}$ for values $\theta=1.01$ , $\theta=1.1$ and $\theta=3$ , and $\unicode{x1D6C4}$ .

2.4. Further mathematical aspects

In this we gather several basic properties of multi-fractional stochastic dominance. First we investigate a few interesting properties of the utility class $u\in \mathscr{U\,}^{\mathrm{MF}}_{\unicode{x1D6C4}}$ .

Remark 4. It is worth pointing out that for any non-decreasing functions $\boldsymbol{ \gamma}_1,\boldsymbol{ \gamma}_2\colon \mathbb{R} \rightarrow[0, 1]$ with $\boldsymbol{ \gamma}_1(t)\le \boldsymbol{ \gamma}_2(t)$ for every $t \in \mathbb{R}$ , we have

\begin{equation*} \mathscr{U\,}^{\mathrm{MF}}_{\boldsymbol{ \gamma}_2} \, \subseteq \mathscr{U\,}^{\mathrm{MF}}_{\boldsymbol{ \gamma}_1} \quad \text{and} \quad \unicode{x1D4D4}_{\mathrm{MF}}(\boldsymbol{ \gamma}_2) \supseteq \unicode{x1D4D4}_{\mathrm{MF}}(\boldsymbol{ \gamma}_1).\end{equation*}

As mentioned earlier in Remark 1(iii), incorporating the function $\boldsymbol{ \gamma}$ as a CDF introduces an additional layer of randomness, enriching our framework. In line with this, we now present a probabilistic characterization of Proposition 5(c).

Inspired by [Reference Müller, Scarsini, Tsetlin and Winkler23, Theorem 4.8], we provide the following result, which extends the invariance property to more general transformations.

2.5. Connection with $\gamma$ -transfers

In this section we briefly outline a connection to the notion of $\gamma$ -transfers introduced in [Reference Müller, Scarsini, Tsetlin and Winkler23]. In fact, the concept of restricted fractional $(1+\gamma)^r_t$ -SD, combined with Proposition 1, offers a natural route for formalizing this relationship. Following [Reference Müller, Scarsini, Tsetlin and Winkler23], for two discrete distributions F and G with probability mass functions f and g respectively, we say that:

1. (a) G is obtained from F via an increasing transfer if there exist $x_1\! < x_2$ and $\eta > 0$ such that ${g(x_1)} \, = \, {f(x_1)} - \eta$ , ${g(x_2)} \, = \, {f(x_2)} + \eta$ , and $ {g(z)} \, = \,{ f(z)}$ for all other values of z,

2. (b) G is obtained from F via a $\gamma$ -transfer if there exist $x_1\! < x_2 < x_3 < x_4$ and $\eta_1, \eta_2 > 0$ with $\eta_2(x_4 - x_3) = \gamma \eta_1 (x_2 - x_1)$ such that ${g(x_1)} \, = \, {f(x_1)} - \eta_1$ , ${g(x_2)} \, = \, {f(x_2)} + \eta_1$ , ${g(x_3)} \, = \, {f(x_3)} + \eta_2$ , ${g(x_4)} \, = \, {f(x_4)} - \eta_2$ , and ${g(z)} \, = \,{ f(z)}$ for all other values of z.

For notational convenience, for a given distribution F and $t \in \mathbb{R}$ , denote its right-censoring (or upper-tail truncation) version at level t by $F^r_t$ :

\begin{equation*}F^r_t(x)\,:\!=\,\begin{cases}F(x), & x < t,\\1 , & x \ge t.\end{cases}\end{equation*}

Observe that if $X\sim F$ , then the right-censored random variable $X\wedge t \,:\!=\, \min\{X,t\}$ has distribution function $F_t^r$ . Now it follows immediately from Definition 3 that for fixed $t\in\mathbb{R}$ and $\gamma\in[0, 1]$ ,

\begin{equation*}F \preceq_{(1+\gamma)_{t}^r\textrm{-SD}} G\quad \Longleftrightarrow\quad F_t^r \preceq_{(1+\gamma)\textrm{-SD}} G_t^r .\end{equation*}

Next we can state the following proposition.

3.1. FFSD: Definition and the integral condition

In Section 2 we extended fractional $(1+\gamma)$ -SD using an integral condition by replacing the constant parameter $\gamma$ with its functional analogue $\unicode{x1D6C4}$ . Due to the nature of the integral condition (1), this extension was relatively straightforward, as it involves associating the function $\unicode{x1D6C4}$ with a single point. However, the extension via the utility class introduces additional complexity, since the set condition for $\mathscr{U\,}^{*}_{\gamma}$ given in (2) involves four points. Therefore we require a new framework that enables a functional extension using a function $\unicode{x1D6C4}$ of a single variable. We construct this framework with the following requirements in mind.

1. (i) Fractional SD should be representable such that, for any $\gamma \in [0, 1]$ , the set condition given in (2) is both necessary and sufficient to verify whether a given function u belongs to the fractional SD utility space within the new framework (see Proposition 10).

2. (ii) The framework must be meaningful for decision analysis (see Section 4).

Based on the discussion above, we build our framework using the one-sided derivatives of utility functions. This approach not only serves our purposes but also provides a tractable and meaningful structure. Hence we introduce the following.

Definition 5. We say that a function u has the ${\boldsymbol{{D}}}_{\pm}$ property if u is two-sided differentiable, that is, the left-derivative $ u_-'(x) $ and the right-derivative $ u_+'(x) $ exist at each point $ x \in \mathbb{R}$ . For a given $\gamma \in [0, 1]$ , we define

\begin{equation*}\widetilde{\mathscr{U\,}}^{*}_{\gamma} \,:\!=\, \mathscr{U\,}^{*}_{\gamma} \cap \mathscr{U\,}({\boldsymbol{{D}}}_{\pm}),\end{equation*}

where

\begin{equation*} \mathscr{U\,}({\boldsymbol{{D}}}_{\pm})\,:\!=\, \{ u \colon \mathbb{R} \to \mathbb{R} \colon \textit{u}\text{ satisfies the $ {\boldsymbol{{D}}}_{\pm} $ property} \}.\end{equation*}

We present the following proposition, which establishes the representability of fractional SD.

Proposition 10. Let u be a function whose left- and right-derivatives exist at each point $x \in \mathbb{R}$ . Then, for a given $\gamma \in [0, 1]$ , we have $u\in \widetilde{\mathscr{U\,}}^{*}_{\gamma}$ if and only if

\begin{equation*}0\le\gamma u_+'(y) \le u_+'(x)\quad \textit{for all $ x \le y$.}\end{equation*}

A similar conclusion holds for the left-derivative $u'_-$ .

Building on our development so far, we provide the following derivative-relation property for the functional extension of $\widetilde{\mathscr{U\,}}^{*}_{\gamma}$ .

Definition 6. ( ${\boldsymbol{{D}}}_{\pm}(\unicode{x1D6C4})$ -property.) Let $\unicode{x1D6C4}\colon \mathbb{R}\to[0, 1]$ be an arbitrary non-decreasing function. We say that a function $u\colon \mathbb{R} \to \mathbb{R}$ satisfies the ${\boldsymbol{{D}}}_{\pm}(\unicode{x1D6C4})$ -property if it is differentiable from both sides (i.e. it has the ${\boldsymbol{{D}}}_{\pm}$ property) and, moreover, at least one of the following conditions hold:

(7) \begin{equation}{\rm (i)} \quad 0 \le \unicode{x1D6C4} (y) u_+'(y) \le u_+'(x) \quad \text{for all $x \le y$,}\quad {\rm (ii)} \quad 0 \le \unicode{x1D6C4}(y) u'_-(y) \le u'_{-}(x) \quad \text{for all $ x\le y$.}\end{equation}

Clearly, conditions (i) and (ii) are equivalent and simplify, at points where u is differentiable, to $0 \le \unicode{x1D6C4}(y)\,u'(y) \le u'(x)$ for all $ x \le y$ . We will revisit this relationship in Proposition 13. For a given non-decreasing function $\unicode{x1D6C4}\colon \mathbb{R}\to[0, 1]$ , we define the following utility space:

\begin{equation*} \widetilde{\mathscr{U\,}}^{\mathrm{FF}}_{\unicode{x1D6C4}}\,:\!=\, \{ u\colon \mathbb{R} \to \mathbb{R} \colon \textit{u} \text{ satisfies the $ {\boldsymbol{{D}}}_{\pm(\unicode{x1D6C4})}$-property} \}.\end{equation*}

Now we are ready to define functional fractional stochastic dominance.

Definition 7. (Functional fractional $(\boldsymbol{ \gamma})$ -SD.) Let $\unicode{x1D6C4}\colon \mathbb{R} \to [0, 1]$ be an arbitrary non-decreasing function, and let F and G be two distribution functions. We say that G dominates F in the sense of functional fractional $(\unicode{x1D6C4})$ -SD, denoted by $F \preceq^{\mathrm{FF}}_{\unicode{x1D6C4}\textrm{-SD}} G$ , if

\begin{equation*} \mathbb{E}_F[u] \le \mathbb{E}_G[u] \quad \text{for all $ u \in \widetilde{\mathscr{U\,}}^{\mathrm{FF}}_{\unicode{x1D6C4}}$,}\end{equation*}

provided that the expectations exist.

The following result establishes the relationship between MFSD and FFSD.

Next, we prove an ‘if and only if’ theorem for FFSD that establishes a connection with the integral condition:

(8) \begin{equation}\int_{-\infty}^t \dfrac{1}{\boldsymbol{ \gamma}(x)}(F(x)-G(x))_- \,{\mathrm{d}} x \le \int_{-\infty}^t (F(x)-G(x))_+ \,{\mathrm{d}} x \quad \text{for all $t \in \mathbb{R}$.}\end{equation}

But first we make the following remark concerning the well-definedness of the integral condition.

If the distributions never cross, then (8) holds trivially, and the statements follow directly from FSD. Therefore, for a given non-decreasing function $\unicode{x1D6C4}\colon \mathbb{R} \to (0,1]$ , by the monotonicity of $\unicode{x1D6C4}$ we have

\[\int_{-\infty}^{t}\dfrac{1}{\unicode{x1D6C4}(x)}(F(x)-G(x))_- \,{\mathrm{d}} x\le\int_{x^*_1}^{t}\dfrac{1}{\unicode{x1D6C4}(x^*_1)}(F(x)-G(x))_-\,{\mathrm{d}} x\quad \text{for all $ t \in \mathbb{R}$.}\]

Now we begin by establishing one direction of the ‘if and only if’ theorem.

Lemma 2. Let $\boldsymbol{ \gamma}\colon \mathbb{R} \rightarrow (0,1]$ be an arbitrary non-decreasing right-continuous function with finitely many jumps. Let F and G be two arbitrary distribution functions crossing finitely many times. For every $t \in \mathbb{R}$ , define the continuous function $u_{\mathrm{FF},t}$ via its right-derivative as

\begin{equation*} u_{\mathrm{FF},t}^{\prime}(x)=\begin{cases}1 & {if \, G(x) \le F(x)\, and \, x \leq t\,,} \\1/\boldsymbol{ \gamma}(x) & {if \, F(x) < G(x)\, and \, x \leq t\,,} \\0 & {if \, t < x\,.}\end{cases}\end{equation*}

Then $u_{\mathrm{FF},t} \in\widetilde{\mathscr{U\,}}^{\mathrm{FF}}_{\unicode{x1D6C4}}$ for every $t \in \mathbb{R}$ .

Proposition 12. Let $\boldsymbol{ \gamma}\colon \mathbb{R} \rightarrow (0,1]$ be an arbitrary non-decreasing right-continuous function with finitely many jumps. Let F and G be two arbitrary distribution functions with finite means and crossing finitely many times. Then the relation

\begin{equation*} \mathbb{E}_F[u_{\mathrm{FF},t}] \le \mathbb{E}_G [u_{\mathrm{FF},t}] \quad {for\, all \, t \in \mathbb{R}}\end{equation*}

implies that the integral condition (8) holds.

We are now prepared to present the main theorem.

Remark 9. The integral condition (8) admits a geometric interpretation as a balance of areas at every threshold t: positive gaps are integrated with respect to the Lebesgue measure ${\mathrm{d}} x$ , whereas negative gaps are effectively reweighted by the location-dependent factor ${\mathrm{d}} x/\unicode{x1D6C4}(x)$ . In view of the local $\unicode{x1D6C4}(t)$ -transfer interpretation of MFSD (Proposition 9), this motivates a transfer-based perspective for FFSD. Let $\unicode{x1D6C4}\colon \mathbb{R}\to(0,1]$ be a non-decreasing function. Fix $x_1\! < x_2\le x_3 < x_4$ and let $\eta_1,\eta_2>0$ . A $\unicode{x1D6C4}$ -weighted transfer consists of moving probability mass $\eta_1$ from $x_1$ to $x_2$ and simultaneously moving probability mass $\eta_2$ from $x_4$ to $x_3$ , subject to the balance constraint

\[\eta_2\int_{x_3}^{x_4}\dfrac{{\mathrm{d}} x}{\unicode{x1D6C4}(x)}=\eta_1\,(x_2-x_1).\]

Such weighted transfers, together with increasing transfers in the sense of [Reference Müller, Scarsini, Tsetlin and Winkler23], preserve the FFSD integral condition (8) uniformly over all censoring thresholds t. Under the regularity conditions of Theorem 3, one could potentially further develop an FFSD analogue of Proposition 9; however, we do not pursue these technical details here.

We conclude this section with the following remark.

3.2. FFSD: Examples

In this section we present examples that highlight the key differences between the MFSD and FFSD frameworks. Given the inclusion $\widetilde{\mathscr{U\,}}^{\mathrm{MF}}_{\unicode{x1D6C4}} \subseteq \widetilde{\mathscr{U\,}}^{\mathrm{FF}}_{\unicode{x1D6C4}}$ given in Proposition 11, it follows that all example utility functions presented in Section 2.3 for $\mathscr{U\,}^{\mathrm{MF}}_{\unicode{x1D6C4}}$ also belong to $\widetilde{\mathscr{U\,}}^{\mathrm{FF}}_{\unicode{x1D6C4}}$ , provided they are two-sided differentiable. There are also non-differentiable functions outside the MFSD utility class, but we focus on those that are both realistic and practically significant. Specifically, we concentrate on the family of piecewise continuously differentiable (PCD) functions, which are particularly relevant for modelling decision-making behaviour in applications.

Proposition 13(a) establishes that PCD functions indeed belong to $\widetilde{\mathscr{U\,}}^{\mathrm{FF}}_{\unicode{x1D6C4}}$ provided that they satisfy the derivative-relation property for an appropriately chosen function $\unicode{x1D6C4}$ . It is also possible to use mixed-side conditions, such as

\[0 \le \unicode{x1D6C4}(y) u'_+(y) \le u'_-(x)\quad \text{for all $ x\le y$.}\]

This condition is particularly useful in scenarios where the utility function exhibits upward jumps in its derivative, as discussed in Remark 7(iii). On the other hand, PCD functions are not necessarily included in $\widetilde{\mathscr{U\,}}^{\mathrm{MF}}_{\unicode{x1D6C4}}$ , since its definition requires u to be expressible as a linear combination of functions in $\widetilde{\mathscr{U\,}}^{\mathrm{Union}}_{\unicode{x1D6C4}}$ . To illustrate this distinction further, we provide the FFSD counterpart of Example 6. This example also highlights key structural differences between $\widetilde{\mathscr{U\,}}^{\mathrm{FF}}_{\unicode{x1D6C4}}$ and $\widetilde{\mathscr{U\,}}^*_{\gamma}$ .

Example 7. The purpose of this example is to provide a parametrized family $\mathscr{U\,}_{\Theta}$ such that the inclusion $\mathscr{U\,}_{\Theta} \subseteq \widetilde{\mathscr{U\,}}^*_{\gamma}$ can hold for a constant $\gamma \in [0, 1]$ only if $\gamma=0$ (hence corresponding to the utility class for FSD). However, one can choose a suitable non-decreasing function $\unicode{x1D6C4}$ such that

\begin{equation*}\mathscr{U\,}_{\Theta} \subseteq \widetilde{\mathscr{U\,}}^{\mathrm{FF}}_{\unicode{x1D6C4}} \neq \widetilde{\mathscr{U\,}}_{0} \quad \text{and more importantly} \quad \mathscr{U\,}_{\Theta} \not\subseteq \widetilde{\mathscr{U\,}}^{\mathrm{MF}}_{\unicode{x1D6C4}}.\end{equation*}

Let $\Theta=(0,1)$ and consider a function $\widetilde{\unicode{x1D6C4}} \colon \Theta \to (0,1)$ with $\lim_{\theta \to 0}\widetilde{\unicode{x1D6C4}}(\theta)=0$ . For every $\theta \in \Theta$ , we define the following PCD utility function $u_\theta$ via its right-derivative as

\begin{equation*}u'_\theta(x)=\begin{cases}u_{\theta,1}^{\prime}(x) & \text{if $ -\infty< x < \theta$,}\\x & \text{if $ \theta\le x < \sqrt{\theta} $,}\\u_{\theta,2}^{\prime}(x) & \text{if $ \sqrt{\theta}\le x < \infty$,}\\\end{cases}\end{equation*}

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

(9) \begin{equation}\sqrt{\theta} \ge \gamma\end{equation}

holds true. Let $\mathscr{U\,}_{\Theta } \,:\!=\, \{ u_\theta \colon \theta \in \Theta \}$ . Now observe that, for some constant $\gamma$ , the validity of the set inclusion $\mathscr{U\,}_{\Theta} \subseteq \mathscr{U\,}^*_{\gamma}$ dictates that necessarily $\gamma=0$ . This conclusion directly follows from (9) 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 $\widetilde{\mathscr{U\,}}_{0} = \widetilde{\mathscr{U\,}}^*_{\gamma=0}$ . Next, consider the continuous non-decreasing function

\begin{align*}\boldsymbol{ \gamma} (x) =\begin{cases}0 & \text{if $ x < 0 $,}\\x &\text{if $ 0 \le x \le 1$,}\\1 & \text{if $ x >1$.}\end{cases}\end{align*}

We first show that $\mathscr{U\,}_{\Theta} \subseteq \widetilde{\mathscr{U\,}}^{\mathrm{FF}}_{\unicode{x1D6C4}}$ . In order to do this, note that, for each $\theta \in \Theta$ , the condition

(10) \begin{equation}\theta \ge u'_{\theta}(y) \boldsymbol{ \gamma}(y) = y \times y \quad \text{for all $y \in [\theta, \sqrt{\theta})$}\end{equation}

is both necessary and sufficient for the validity of $u_\theta \in \widetilde{\mathscr{U\,}}^{\mathrm{FF}}_{\unicode{x1D6C4}}$ , which obviously holds true. Lastly, we show that $\mathscr{U\,}_{\Theta} \not\subseteq \widetilde{\mathscr{U\,}}^{\mathrm{MF}}_{\unicode{x1D6C4}}$ . Fix $\theta \in \Theta$ . First, when $y\approx\sqrt{\theta}$ (from left) and $x=\theta< y$ , it follows from (10) that

(11) \begin{equation}\dfrac{1}{\boldsymbol{ \gamma}(y)}=\dfrac{1}{\sqrt{\theta}}=\dfrac{u_\theta'(y)}{u_\theta'(x)}.\end{equation}

Next, we discuss that $u_\theta \notin \widetilde{\mathscr{U\,}}^{\mathrm{Union}}_{\unicode{x1D6C4}}$ . Note that $u_\theta$ is not a globally concave function and hence $u_\theta \notin \widetilde{\mathscr{U\,}}^{+\infty}_{{\stackrel{\rule{.6em}{1.5pt}}{\unicode{x1D6C4}}} =1}$ . Now, by contradiction, assume that there exists $t\in \mathbb{R} $ such that $u_\theta \in \widetilde{\mathscr{U\,}}^t_{\unicode{x1D6C4}(t)}$ . Therefore, based on the equality in (11), this is only possible when $t=\sqrt{\theta}$ . However, this leads to a contradiction, as the derivative of the subsequent concave segment, $u'_{2,\theta}$ of $u_\theta$ is not necessarily equal to 0 after $\sqrt{\theta}$ . Hence $u_\theta$ can only belong to $ \widetilde{\mathscr{U\,}}^{\mathrm{MF}}_{\unicode{x1D6C4}}$ if it can be represented as a convex combination of functions that belong to $\widetilde{\mathscr{U\,}}^{\mathrm{Union}}_{\unicode{x1D6C4}}$ . For the moment, assume that $u_\theta = \lambda_1 v_1 + \lambda_2 v_2$ where $v_1 \in \widetilde{\mathscr{U\,}}^{t = \sqrt{\theta}}_{\unicode{x1D6C4}(t) =\sqrt{\theta}}$ with ${{v_1'(\sqrt{\theta})}/{v_2'(\theta)}}={{1}/{\sqrt{\theta}}}$ and $v_2 \in \widetilde{\mathscr{U\,}}^*_{1}$ . Then, from the mediant inequality, we obtain

\begin{equation*}\dfrac{u_\theta'(\sqrt{\theta})}{u_\theta'(\theta)}=\dfrac{\lambda_1v_1(\sqrt{\theta})+\lambda_2v_2(\sqrt{\theta})}{\lambda_1v_1(\theta)+\lambda_2v_2(\theta)}=\dfrac{\lambda_1\sqrt{\theta}+\lambda_2v_2(\sqrt{\theta})}{\lambda_1\theta+\lambda_2v_2(\theta)}<\dfrac{1}{\sqrt{\theta}}\end{equation*}

since ${{v_2(\sqrt{\theta})}/{v_2(\theta)}}\le 1$ for any $v_2\in \widetilde{\mathscr{U\,}}^*_{1}$ by definition, which violates (11). The general case can be discussed in a similar way relying on mediant inequality and the fact that any linear combination should contain at least one piece from the utility space $\widetilde{\mathscr{U\,}}^{\sqrt{\theta}}_{\sqrt{\theta}}$ and another piece from $\widetilde{\mathscr{U\,}}^{*}_{1}$ .

Figure 3.

Plot of $u_\theta'$ .

We conclude with the following remark.

4.1. Partial greediness

We introduce the novel notion of partial greediness of a utility function at an arbitrary point $x\in \mathbb{R}$ that provides a meaning to the function $\boldsymbol{ \gamma}$ from a decision-making perspective in both the multi-fractional and functional fractional stochastic dominance frameworks. The notion of partial greediness at the location $x\in \mathbb{R}$ characterizes the greediness behaviour of decision makers over half-open intervals $(x,\infty)$ , rather than over the entire real line, as in [Reference Chateauneuf, Cohen and Meilijson5]. They define the index of global greediness (non-concavity) for a strictly increasing function $u\colon \mathbb{R}\to \mathbb{R}$ as follows:

(12) \begin{equation}G_u\,:\!=\, \sup\nolimits _{x_1 < x_2 \leq x_3 < x_4}\biggl(\dfrac{u(x_4)-u(x_3)}{x_4-x_3} \bigg/ \dfrac{u(x_2)-u(x_1)}{x_2-x_1}\biggr).\end{equation}

It is known that always $G_u \ge 1$ and $G_u =1$ if and only if u is a concave function. Moreover, when u is differentiable, it is true that $G_u = \sup_{x \le y} u'(y)/ u'(x)$ (see [Reference Chateauneuf, Cohen and Meilijson5]). 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. Upon closer examination of the set condition of the utility space $\mathscr{U\,}^*_{\gamma}$ , as provided in (2), it becomes evident that $\gamma$ serves as an upper bound for the level of greediness (non-concavity) $G_u$ for all $u \in \mathscr{U\,}^*_\gamma$ , established by the inequality $G_u \leq 1/\gamma$ . Inspired by [Reference Chateauneuf, Cohen and Meilijson5], we define a more local version of the greediness index as follows.

Definition 8. (Discrete partial greediness.) Let u be a strictly increasing function. The discrete partial greediness at location $x \in \mathbb{R}$ is defined as

\begin{equation*} G^d_{\mathrm{par}}(u;\,x)\,:\!=\, \sup\nolimits_{x < x_1 < x_2\le x_3 < x_4}\biggl(\dfrac{u(x_4)-u(x_3)}{x_4-x_3}\bigg/\dfrac{u(x_2)-u(x_1)}{x_2-x_1}\biggr).\end{equation*}

Although the discrete partial greediness $G^d_{\mathrm{par}}$ provides us with a measure of greediness attached to the location, it is not well suited for studying functions of interest in the space $\widetilde{\mathscr{U\,}}^{\mathrm{FF}}_{\unicode{x1D6C4}}$ . This is because functions in this space are defined via conditions on one-sided derivatives. Therefore, in line with the approach followed at the beginning of this section, we introduce a new local notion that is better suited to the structure of $\widetilde{\mathscr{U\,}}^{\mathrm{FF}}_{\unicode{x1D6C4}}$ structure. We present the following results in a more general setting, requiring continuity and one-sided differentiability rather than two-sided differentiability.

Definition 9. (Continuous partial greediness.) Let u be a strictly increasing function whose right-derivatives exist at each point $x \in \mathbb{R}$ . We define the continuous partial greediness at x as follows:

\begin{equation*} G^c_{\mathrm{par}}(u;\,x)\,:\!=\, \sup\nolimits_{x < y < z} \dfrac{u'_+(z)}{u'_+(y)}.\end{equation*}

The same definition applies to left-differentiable functions by replacing right-derivatives with left-derivatives.

Next we state the following proposition, which shows that the two notions of discrete and continuous partial greediness coincide in a more general framework.

Proposition 15. Let u be a continuous, strictly increasing function that has either left- or right-derivatives at each point. Then

\[G^d_{\mathrm{par}}(u,x)=G^c_{\mathrm{par}}(u,x) \quad {for\, all \, x \in \mathbb{R}.}\]

Hereafter, in our framework, Proposition 15 allows us to use the notation $G_{\mathrm{par}}(u;\cdot\!)$ interchangeably without any ambiguity.

Remark 14. In this remark we discuss both notions of partial greediness in the (utility) space $\widetilde{\mathscr{U\,}}^{\mathrm{FF}}_{\unicode{x1D6C4}}$ , where the assumption that utility functions are strictly increasing may fail in general. For instance, from the derivative-relation condition in (7),

\begin{equation*} 0\le \unicode{x1D6C4} (y) u_{-}' (y) \le u_{-}'(x) \quad \text{for all $ x\le y$,}\end{equation*}

it follows immediately that if $u_-'(x)=0$ then $u_-'(y)=0$ for all $y>x$ . Therefore, when considering the ratio $u_-'(y)/u_-'(x)$ , $ x\le y$ , the only potentially problematic case is $0/0$ . Adopting the convention $0/0=1$ removes this ambiguity. Moreover, this convention does not affect the measurement of non-concavity, since the lower bound 1 is attained if and only if the function is concave; in particular, constant functions are concave. In addition, under this convention, Proposition 16 remains valid for functions in $\widetilde{\mathscr{U\,}}^{\mathrm{FF}}_{\unicode{x1D6C4}}$ . We will use this on several occasions below in Section 4.2.

Corollary 1. Let $u \in \widetilde{\mathscr{U\,}}^{\mathrm{FF}}_{\unicode{x1D6C4}}$ . Then, for all $x \in \mathbb{R}$ , $G^d_{\mathrm{par}}(u;\,x)=G^c_{\mathrm{par}}(u;\,x)\,=\!:\, G_{\mathrm{par}}(u;\,x)$ . Moreover,

(13) \begin{equation}G_{\mathrm{par}} (u;\,x) \le \dfrac{1}{\unicode{x1D6C4}(x)}.\end{equation}

In the following remark, we highlight the decision-theoretic implications of the $\unicode{x1D6C4}$ function.

4.2. $\widetilde{\mathscr{U\,}}^{\mathrm{MF}}_{\unicode{x1D6C4}}$ and the partial greediness

In this section we provide several tools based on the partial greediness $G_{\mathrm{par}}(u;\cdot)$ to verify whether a given function $u \in \widetilde{\mathscr{U\,}}^{\mathrm{FF}}_{\unicode{x1D6C4}}$ belongs to the multi-fractional utility space $\widetilde{\mathscr{U\,}}^{\mathrm{MF}}_{\unicode{x1D6C4}}$ . The underlying principle relies on the observation in Remark 7: for a function $\unicode{x1D6C4}$ , the space $\widetilde{\mathscr{U\,}}^{\mathrm{FF}}_{\unicode{x1D6C4}}$ contains more non-concave functions than $\mathscr{U\,}^{\mathrm{MF}}_{\unicode{x1D6C4}}$ . Therefore, given a function u, we use partial greediness to assess its degree of non-concavity and, in turn, determine whether it belongs to $\widetilde{\mathscr{U\,}}^{\mathrm{MF}}_{\unicode{x1D6C4}}$ . First, we start with the following lemma.

Lemma 3. Let $\unicode{x1D6C4}\colon \mathbb{R} \to [0, 1]$ be an arbitrary non-decreasing function. Let $u=\sum_{k=1}^{n} \lambda_k u_k \in \widetilde{\mathscr{U\,}}^{\mathrm{MF}}_{\unicode{x1D6C4}}$ , $n\in \mathbb{N}$ , $\lambda_k \ge 0$ , and $u_k \in \widetilde{\mathscr{U\,}}^{\mathrm{Union}}_{\unicode{x1D6C4}}$ , $k=1,\dots,n$ . Assume that there exists $x_0 \in \mathbb{R}$ such that $\unicode{x1D6C4}(x_0) \neq 0$ and, moreover,

\begin{equation*} G_{\mathrm{par}}(u;\,x_0) = \dfrac{1}{\unicode{x1D6C4}(x_0)}.\end{equation*}

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

\begin{equation*}G_{\mathrm{par}}(u;\,x_0) = G_{\mathrm{par}}(u_j;\,x_0) =\max_{1 \le k\le n} G_{\mathrm{par}}(u_k;\,x_0) =\dfrac{1}{\unicode{x1D6C4}(x_0)}.\end{equation*}

The main message of Lemma 3 is that for $u \in \widetilde{\mathscr{U\,}}^{\mathrm{MF}}_{\unicode{x1D6C4}}$ , $G_{\mathrm{par}}(u;\,x)$ touches the upper bound given in (13) at $x_0$ if and only if u has an extreme component belonging to $\widetilde{\mathscr{U\,}}^{x_0}_{\unicode{x1D6C4}(x_0)} \subsetneq \widetilde{\mathscr{U\,}}^{\mathrm{Union}}_{\unicode{x1D6C4}}$ . We proceed with the following proposition, which asserts that for any $u \in \widetilde{\mathscr{U\,}}^{\mathrm{MF}}_{\unicode{x1D6C4}}$ , the upper bound for greediness (non-concavity) is reached only when $\boldsymbol{ \gamma}$ is constant in a neighbourhood of that point. The usefulness of this proposition becomes evident in the subsequent corollary, which states that if $G_{\mathrm{par}}(u;\,x)$ reaches the upper bound at some point and $\boldsymbol{ \gamma}$ is not constant around that point, then u cannot belong to $\widetilde{\mathscr{U\,}}^{\mathrm{MF}}_{\unicode{x1D6C4}}$ . It is important to note that such conditions are not required for functions in $\widetilde{\mathscr{U\,}}^{\mathrm{FF}}_{\unicode{x1D6C4}}$ .

Proposition 17. Let $\unicode{x1D6C4}\colon \mathbb{R} \to [0, 1]$ be an arbitrary non-decreasing function. Assume that $u \in \widetilde{\mathscr{U\,}}^{\mathrm{MF}}_{\unicode{x1D6C4}}$ and there exists $x_0 \in \mathbb{R}$ such that $\unicode{x1D6C4}(x_0) \neq 0$ and, moreover,

\begin{equation*} G_{\mathrm{par}}(u;\,x_0) = \dfrac{1}{\unicode{x1D6C4}(x_0)}.\end{equation*}

Then there exists an open interval $I=I_{x_0} = (x_0,x_0+\delta)$ for some $\delta >0$ such that $\unicode{x1D6C4}$ is a constant function on I.

Corollary 2. Let $\unicode{x1D6C4}\colon \mathbb{R} \rightarrow [0, 1]$ be an arbitrary non-decreasing function such that there exists an open subset $U \subseteq \mathbb{R}$ where $\unicode{x1D6C4}$ is strictly increasing on U. Assume that $u \in \widetilde{\mathscr{U\,}}^{\mathrm{FF}}_{\unicode{x1D6C4}}$ and there exists $x_0 \in U$ such that

\begin{equation*} G_{\mathrm{par}}(u;\, x_0)=\dfrac{1}{\boldsymbol{ \gamma}(x_0)}.\end{equation*}

Then $u \notin\widetilde{\mathscr{U\,}}^{\mathrm{MF}}_{\unicode{x1D6C4}}$ .

Next we present the main result of this section, which states that if, for a given $u \in \widetilde{\mathscr{U\,}}^{\mathrm{FF}}_{\unicode{x1D6C4}}$ , the partial greediness $G_{\mathrm{par}}(u;\, x)$ touches the upper bound for at least two distinct values of the function $\boldsymbol{ \gamma}$ , then $u \not\in \widetilde{\mathscr{U\,}}^{\mathrm{MF}}_{\unicode{x1D6C4}}$ . This follows because, for $u \in \widetilde{\mathscr{U\,}}^{\mathrm{MF}}_{\unicode{x1D6C4}}$ , once $G_{\mathrm{par}}(u;\, x)$ reaches its upper bound at a certain point x due to the extreme component (as per Lemma 3), u must remain constant thereafter. If not, the linear structure of $\widetilde{\mathscr{U\,}}^{\mathrm{MF}}_{\unicode{x1D6C4}}$ implies that combining this extreme component with other functions that are not constant after x – i.e. those unable to exhibit as steep non-concave segments – would detract the degree of non-concavity of u at x. Conversely, by the definition of the ${\boldsymbol{{D}}}_{\pm}(\unicode{x1D6C4})$ property, this does not apply to the functions within $\widetilde{\mathscr{U\,}}^{\mathrm{FF}}_{\unicode{x1D6C4}}$ .

Theorem 4. Let $\unicode{x1D6C4}\colon \mathbb{R} \to [0, 1]$ be an arbitrary non-decreasing function. Assume that $u \in \widetilde{\mathscr{U\,}}^{\mathrm{FF}}_{\unicode{x1D6C4}}$ and that there exist $x_0, x_1 \in \mathbb{R}$ such that $0 < \unicode{x1D6C4}(x_0) \neq \unicode{x1D6C4}(x_1)<1$ and

\begin{equation*} G_{\mathrm{par}}(u;\,x_k) = \dfrac{1}{\unicode{x1D6C4}(x_k)}, \quad k=0,1.\end{equation*}

Then $u \notin \widetilde{\mathscr{U\,}}^{\mathrm{MF}}_{\unicode{x1D6C4}}$ .

We conclude this section with the following corollary, which provides the last piece of the overall framework. From Proposition 17 we already know that for a given u, if $G_{\mathrm{par}}(u;\,x_0)$ reaches the upper bound around a point where $\boldsymbol{ \gamma}$ is strictly increasing, then u cannot belong to $\widetilde{\mathscr{U\,}}^{\mathrm{MF}}_{\unicode{x1D6C4}}$ . Furthermore, this corollary establishes that when $\boldsymbol{ \gamma}$ is a piecewise constant function, and $G_{\mathrm{par}}(u;\,x)$ attains the upper bound for more than one value of $\boldsymbol{ \gamma}$ , then u does not lie in $\widetilde{\mathscr{U\,}}^{\mathrm{MF}}_{\unicode{x1D6C4}}$ .

Corollary 3. Let $\unicode{x1D6C4}\colon \mathbb{R} \to [0, 1]$ be a piecewise step function of the form given in (6). Assume that $u \in \widetilde{\mathscr{U\,}}^{\mathrm{FF}}_{\unicode{x1D6C4}}$ and that there exist $x_i \in (t_{i-1},t_i)$ and $x_j \in (t_{j-1},t_j)$ , where $1 \le i \neq j \le n$ such that

\begin{equation*}G_{\mathrm{par}} (u;\,x_k) = \dfrac{1}{\unicode{x1D6C4}(x_k)}, \quad k=i,j.\end{equation*}

Then $u\notin \widetilde{\mathscr{U\,}}^{\mathrm{MF}}_{\unicode{x1D6C4}}$ .

4.3. Average risk aversion

In the preceding sections we explored the implications of the function $\boldsymbol{ \gamma}$ from an economics perspective, emphasizing non-differentiable utility functions. Now our focus shifts to differentiable utility functions, specifically strictly increasing twice-differentiable utility functions within $\widetilde{\mathscr{U\,}}^{\mathrm{FF}}_{\unicode{x1D6C4}}$ . In this context, the function $\boldsymbol{ \gamma}$ can be interpreted as a somewhat local lower bound on the average risk aversion (or an upper bound on risk loving behaviour) of a decision maker. Recall the well known Arrow–Pratt measure of absolute risk aversion (ARA) coefficient [Reference Arrow1, Reference Pratt26]:

\begin{equation*}r_u(x) = r(x)\,:\!=\, -\dfrac{u^{\prime\prime}(x)}{u^{\prime}(x)}, \quad x\in \mathbb{R}.\end{equation*}

Then, invoking the classical Arrow–Pratt log–marginal–utility identity, one obtains a local (interval–average) lower bound on the average absolute risk aversion coefficient:

(14) \begin{equation}\int_x^y -\dfrac{u^{\prime\prime}(z)}{u^{\prime}(z)}{\mathrm{d}} z= [\!-{\ln}(u^{\prime}(y))+\ln(u^{\prime}(x))]={\ln}\biggl(\dfrac{u^{\prime}(x)}{u^{\prime}(y)}\biggr) \geq \ln\!(\unicode{x1D6C4}(y)) \quad \text{for all $x\le y$.}\end{equation}

Meyer [Reference Meyer21] generalizes SSD by restricting attention to decision makers whose Arrow–Pratt absolute risk aversion lies between prescribed lower and upper (wealth-dependent) bounds for all x. In the same vein, fractional degree SD [Reference Huang, Tzeng and Zhao10] provides a continuum between FSD and SSD via a global, constant bound on risk aversion/loving. However, (14) shows that the function $\unicode{x1D6C4}$ imposes an interval-average lower bound on absolute risk aversion on the entire interval [x, y] rather than pointwise.

When $\unicode{x1D6C4}(y)=0$ , the estimate becomes obvious since $\ln\!(\unicode{x1D6C4}(y))=-\infty$ . If u is fully concave then $r_u(x)\ge 0$ and the left-hand side is non-negative, so the inequality is immediate whenever $\unicode{x1D6C4}(y)\in(0,1]$ . In the more interesting case where u allows local non-concavity, the bound (14) quantifies risk attitudes on an average basis over [x, y]. Because $\unicode{x1D6C4}$ is non-decreasing, lower wealth levels permit (on average) more risk loving behaviour in the admissible classes $\widetilde{\mathscr{U\,}}^{\mathrm{FF}}_{\unicode{x1D6C4}}$ and $\widetilde{\mathscr{U\,}}^{\mathrm{MF}}_{\unicode{x1D6C4}}$ . This is consistent with the non-monotone curvature patterns proposed by [Reference Friedman and Savage9] and [Reference Markowitz20], which explain the coexistence of gambling and insurance and a wealth-dependent willingness to bear risk.

Another interesting observation is that when the function $\unicode{x1D6C4}$ is continuous, and by a direct application of the mean value theorem, in addition, we can observe that, for every x and every y in a tiny neighbourhood of x,

\begin{equation*}\dfrac{\log \!( \unicode{x1D6C4} (x) ) }{y-x}\lessapprox r_u(x), \quad x \neq y.\end{equation*}

The latter heuristic observation interprets that the function $\unicode{x1D6C4}$ at each instance controls the non-concavity of the utility function u through the ARA coefficient. Similarly, one can discuss the relative absolute risk aversion coefficient $R_{u}(x)=-x({{u^{\prime\prime}(x)}/{u^{\prime}(x)}})$ as well.

In conclusion, it is evident that both the MFSD and FFSD methods utilize extra information regarding the position (wealth) of non-concave segments in the utility function u. Consequently, this approach enables a more comprehensive and realistic representation of decision makers’ behaviour, aligning with the expected utility theory.