Comparison of Risk Averse Utility Functions on Two-Dimensional Regions

Abstract

Weighted quasi-arithmetic means on two-dimensional regions are demonstrated, and risk averse conditions are discussed by the corresponding utility functions. For two utility functions on two-dimensional regions, we introduce a concept that decision making with one utility is more risk averse than decision making with the other utility. A necessary condition and a sufficient condition for the concept are demonstrated by their utility functions. Several examples are given to explain them.

1 Introduction

Weighted quasi-arithmetic means are important concept for mathematical theory such as the mean value theorems, and it is a fundamental tool for subjective estimation regarding information in management science, artificial intelligence and so on. Weighted quasi-arithmetic means of an interval are given mathematically by aggregation operations (Kolmogorov [4], Nagumo [6] and Aczél [1]). Bustince et al. [2] discussed aggregation operations on two-dimensional OWA operators, and Labreuche and Grabisch [5] demonstrated Choquet integral for aggregation in multicriteria decision making, and Torra and Godo [7] studied continuous WOWA operators for defuzzification. In micro-economics, subjective estimations with preference relations are formulated as utility functions (Fishburn [3]). From the view point of utility functions, Yoshida [8, 9] have studied the relations between weighted quasi-arithmetic means on an interval and decision maker’s behavior regarding risks. In one-dimensional cases, for twice continuously differentiable strictly increasing functions $\varphi , \psi : [a,b] \mapsto {\mathbb {R}}$ as decision makers’ utility functions and a continuous function $\omega : [a,b] \mapsto (0,\infty )$ as a weighting function, weighted quasi-arithmetic means $\mu$ and $\nu$ on a closed interval [a, b] are real numbers satisfying

$$\begin{aligned} \varphi (\mu ) \int _{a}^{b} \, \omega (x) \, dx = \int _{a}^{b} \varphi (x) \, \omega (x) \, dx, \end{aligned}$$

(1.1)

$$\begin{aligned} \psi (\nu ) \int _{a}^{b} \, \omega (x) \, dx = \int _{a}^{b} \psi (x) \, \omega (x) \, dx \end{aligned}$$

(1.2)

in the mean value theorem for integration. Then it is said that decision making with utility function $\varphi$ is more risk averse than decision making with utility function $\psi$ if $\mu \le \nu$ for all closed intervals [a, b]. Its equivalent condition is

$$\begin{aligned} \frac{\varphi ''}{\varphi '} \le \frac{\psi ''}{\psi '} \end{aligned}$$

(1.3)

on ${\mathbb {R}}$ (Yoshida [10, 11]).

Yoshida [12] introduced weighted quasi-arithmetic means on two-dimensional regions, which are related to multi-object decision making. In this paper, using decision makers’ utility functions we discuss relations between risk averse/risk neutral/risk loving conditions and the corresponding weighted quasi-arithmetic means on two-dimensional regions. In this paper we compare two decision makers’ behaviors regarding risks by the weighted quasi-arithmetic means on two-dimensional regions and we give a characterization by their utility functions.

In Sect. 2 we introduce weighted quasi-arithmetic means on two-dimensional regions and we discuss their risk averse conditions. For two utility functions f and g on two-dimensional regions, we introduce a concept that decision making with utility f is more risk averse than decision making with utility g. Further we derive a necessary condition where decision making with utility f is more risk averse than decision making with utility g on two-dimensional regions, and we investigate the condition by several examples. In Sect. 3 we give sufficient conditions for the results in Sect. 2 when utility functions are quadratic.

2 Weighted Quasi-arithmetic Means on Two-Dimensional Regions

Let ${\mathbb {R}}= (- \infty , \infty )$ and let a domain D be a non-empty open convex subset of ${\mathbb {R}}^2$, and let $\mathcal{R}(D)$ be a family of closed convex subsets of D. Denote by $\mathcal{L}$ a family of twice continuously differentiable functions $f : D \mapsto {\mathbb {R}}$ which is strictly increasing, i.e. $f_x>0$ and $f_y>0$ on D, and denote by $\mathcal{W}$ a family of continuous functions $w : D \mapsto (0,\infty )$. For a closed convex set $R \in \mathcal{R}(D)$, weighted quasi-arithmetic means on region R with utility $f \in \mathcal{L}$ and weighting $w \in \mathcal{W}$ are given by a subset $M^{f}_{w}(R)$ of region R as follows.

$$\begin{aligned} M^{f}_{w}(R) = \left\{ (\tilde{x},\tilde{y}) \in R \mid f(\tilde{x},\tilde{y}) \iint _{R} w(x,y) \, dx \, dy = \iint _{R} f(x,y) w(x,y) \, dx \, dy \right\} . \end{aligned}$$

(2.1)

Then we have $M^{f}_{w}(R) \ne \emptyset$ since f is continuous on R and

We introduce the following natural ordering on ${\mathbb {R}}^2$.

Definition 2.1

(A partial order $\preceq$ on ${\mathbb {R}}^2$).

• (i) For two points $(\underline{x},\underline{y}), (\overline{x},\overline{y}) (\in {\mathbb {R}}^2)$, an order $(\underline{x},\underline{y}) \preceq (\overline{x},\overline{y})$ implies $\underline{x} \le \overline{x} \ \text{ and } \ \underline{y} \le \overline{y}$.

• (ii) For two points $(\underline{x},\underline{y}), (\overline{x},\overline{y}) (\in {\mathbb {R}}^2)$, an order $(\underline{x},\underline{y}) \prec (\overline{x},\overline{y})$ implies $(\underline{x},\underline{y}) \preceq (\overline{x},\overline{y})$ and $(\underline{x},\underline{y}) \ne (\overline{x},\overline{y})$.

• (iii) For two sets $A,B (\subset {\mathbb {R}}^2)$, an order $A \preceq B$ implies the following (a) and (b):

  • (a) For any $(\underline{x},\underline{y}) \in A$ there exists $(\overline{x},\overline{y}) \in B$ satisfying $(\underline{x},\underline{y}) \preceq (\overline{x},\overline{y})$.

  • (b) For any $(\overline{x},\overline{y}) \in B$ there exists $(\underline{x},\underline{y}) \in A$ satisfying $(\underline{x},\underline{y}) \preceq (\overline{x},\overline{y})$.

Let a closed convex region $R \in \mathcal{R}(D)$ and let a weighting function $w \in \mathcal{W}$. We define a point $(\overline{x}_R,\overline{y}_R)$ on region R by the following weighted quasi-arithmetic means:

(2.2)

(2.3)

Hence, $(\overline{x}_R,\overline{y}_R)$ is called an invariant risk neutral point on R with weighting w (Yoshida [12]). We separate the space ${\mathbb {R}}^2$ as follows. Let $R_{w,-}^{(\overline{x}_R,\overline{y}_R)} = \{ (x,y) \in {\mathbb {R}}^2 \mid (x,y) \prec (\overline{x}_R,\overline{y}_R) \} = \{ (x,y) \in {\mathbb {R}}^2 \mid x \le \overline{x}_R, \ y \le \overline{y}_R, (x,y) \ne (\overline{x}_R,\overline{y}_R) \}$ and $R_{w,+}^{(\overline{x}_R,\overline{y}_R)} = \{ (x,y) \in {\mathbb {R}}^2 \mid (\overline{x}_R,\overline{y}_R) \prec (x,y) \} = \{ (x,y) \in {\mathbb {R}}^2 \mid x \ge \overline{x}_R, \ y \ge \overline{y}_R, (x,y) \ne (\overline{x}_R,\overline{y}_R) \}$. Then $R_{w,-}^{(\overline{x}_R,\overline{y}_R)}$ denotes a subregion of risk averse points and $R_{w,+}^{(\overline{x}_R,\overline{y}_R)}$ denotes a subregion of risk loving points. Let $R_w^{(\overline{x}_R,\overline{y}_R)} = R_{w,-}^{(\overline{x}_R,\overline{y}_R)} \cup R_{w,+}^{(\overline{x}_R,\overline{y}_R)} \cup \{ (\overline{x}_R,\overline{y}_R) \}$. Now we introduce the following relations between decision maker’s behavior and his utility.

Definition 2.2

Let a utility function $f \in \mathcal{L}$ and let a rectangle region $R \in \mathcal{R}(D)$.

• (i) Decision making with utility f is called risk neutral on R if

  $$\begin{aligned} f(\overline{x}_R,\overline{y}_R) \iint _{R} w(x,y) \, dx \, dy = \iint _{R} f(x,y) w(x,y) \, dx \, dy \end{aligned}$$

  (2.4)

  for all density functions w.

• (ii) Decision making with utility f is called risk averse on R if

  $$\begin{aligned} f(\overline{x}_R,\overline{y}_R) \iint _{R} w(x,y) \, dx \, dy \ge \iint _{R} f(x,y) w(x,y) \, dx \, dy \end{aligned}$$

  (2.5)

  for all density functions w.

• (iii) Decision making with utility f is called risk loving on R if

  $$\begin{aligned} f(\overline{x}_R,\overline{y}_R) \iint _{R} w(x,y) \, dx \, dy \le \iint _{R} f(x,y) w(x,y) \, dx \, dy \end{aligned}$$

  (2.6)

  for all density functions w.

Example 2.1

Let a domain $D=(-0.5,1.25)^2$ and a region $R=[0, 1]^2$, and let a weighting function $w(x,y)=1$ for $(x,y) \in D$. Then an invariant neutral point is $(\overline{x}_R,\overline{y}_R)=(0.5,0.5)$ and $R_{w,-}^{(\overline{x}_R,\overline{y}_R)} = [0,0.5]^2 \setminus \{ (0.5,0.5) \}$ and $R_{w,+}^{(\overline{x}_R,\overline{y}_R)} = [0.5,1]^2 \setminus \{ (0.5,0.5) \}$. Let us consider two utility functions $f(x,y)=-x^2-y^2+3x+3y$ and $g(x,y)=2x^2+2y^2-5x-5y$ for $(x,y) \in D$. Then by Yoshida [12, Example 3.1(i), Lemma 2.2] decision making with utility function f is called risk averse on R with weighting w, and decision making with utility function g is also called risk loving on R with weighting w. Hence the corresponding weighted quasi-arithmetic means $M^{f}_{w}(R)$ and $M^{g}_{w}(R)$ are ordered by the order $\preceq$ in a restricted subregion $R_{w}^{(\overline{x}_R,\overline{y}_R)} = R_{w,-}^{(\overline{x}_R,\overline{y}_R)} \cup R_{w,+}^{(\overline{x}_R,\overline{y}_R)} \cup \{ (\overline{x}_R,\overline{y}_R) \}$. However they can not be ordered on a subregion $R \setminus R_{w}^{(\overline{x}_R,\overline{y}_R)}$ (Fig. 1).

Fig. 1.

$M^{f}_{w}(R) \cap R_w^{(\overline{x}_R,\overline{y}_R)} \preceq M^{g}_{v}(R) \cap R_w^{(\overline{x}_R,\overline{y}_R)}$ ($f(x,y)=-x^2-y^2+3x+3y, g(x,y)=2x^2+2y^2-5x-5y$, $R=[0,1]^2$)

It is natural that the order $\preceq$ should be given between weighted quasi-arithmetic means $M^{f}_{w}(R)$ of risk averse utility f and weighted quasi-arithmetic means $M^{g}_{w}(R)$ of risk loving utility g in Example 3.1. Therefore when we compare weighted quasi-arithmetic means $M^{f}_{w}(R)$ and $M^{g}_{v}(R)$, we discuss it on the meaningful restricted subregion $R_w^{(\overline{x}_R,\overline{y}_R)}$. Hence we introduce the following definition regarding the comparison of utility functions.

Definition 2.3

Let $f,g \in \mathcal{L}$ be utility functions on D. Decision making with utility f is more risk averse than decision making with utility g if it holds that

$$\begin{aligned} M^{f}_{w}(R) \cap R_w^{(\overline{x}_R,\overline{y}_R)} \preceq M^{g}_{v}(R) \cap R_w^{(\overline{x}_R,\overline{y}_R)} \end{aligned}$$

(2.7)

for all weighting functions $w \in \mathcal{W}$ on D and all closed convex regions $R \in \mathcal{R}(D)$.

Example 2.2

Let a domain $D=(-0.5,1.25)^2$ and a region $R=[0, 1]^2$, and let a weighting function $w(x,y)=1$ for $(x,y) \in D$. Then an invariant neutral point is $(\overline{x}_R,\overline{y}_R)=(0.5,0.5)$ and $R_{w,-}^{(\overline{x}_R,\overline{y}_R)} = [0,0.5]^2 \setminus \{ (0.5,0.5) \}$ and $R_{w,+}^{(\overline{x}_R,\overline{y}_R)} = [0.5,1]^2 \setminus \{ (0.5,0.5) \}$. Let us consider two utility functions $f(x,y)=-x^2-y^2+3x+3y$ and $g(x,y)=-2x^2-2y^2+5x+5y$ for $(x,y) \in D$. Then decision making with utility f is more risk averse than decision making with utility g as we see the relation (2.7) in Fig. 2.

Fig. 2.

$M^{f}_{w}(R) \cap R_w^{(\overline{x}_R,\overline{y}_R)} \preceq M^{g}_{v}(R) \cap R_w^{(\overline{x}_R,\overline{y}_R)}$ ($f(x,y)=-x^2-y^2+3x+3y, g(x,y)=-2x^2-2y^2+5x+5y$, $R=[0, 1]^2$)

Now we give a necessary condition for (2.7), i.e. decision making with utility f is more risk averse than decision making with utility g.

Theorem 2.1

Let $f,g \in \mathcal{L}$ be utility functions on D. If decision making with utility f is more risk averse than decision making with utility g, then it holds that

$$\begin{aligned} \frac{ h^2 f_{xx} + 2r hk f_{xy} + k^2 f_{yy} }{ h f_x + k f_y } \le \frac{ h^2 g_{xx} + 2 rhk g_{xy} + k^2 g_{yy} }{ h g_x + k g_y } \end{aligned}$$

(2.8)

on D for all positive numbers h and k and all real numbers r satisfying $-1 \le r \le 1$.

From Theorem 2.1 we can easily obtain the following result, which is corresponding to [12, Theorem 3.1(i)].

Corollary 2.1

Let $f,g \in \mathcal{L}$ be utility functions on D. If decision making with utility f is more risk averse than decision making with utility g, then it holds that

$$\begin{aligned} \frac{f_{xx}}{f_x} \le \frac{g_{xx}}{g_x} \quad {and} \quad \frac{f_{yy}}{f_y} \le \frac{g_{yy}}{g_y} \quad {on} \ D. \end{aligned}$$

(2.9)

Equation (2.8) in Theorem 2.1 gives a detailed relation between f and g rather than (2.9). A parameter r in necessary condition (2.8) depends on the shapes of closed convex regions $R \in \mathcal{R}(D)$. Now we investigate several examples with different shapes of regions R.

Example 2.3

(Rectangle regions). Let h and k be positive numbers.Let rectangle regions

$$\begin{aligned} R^\mathrm{Rect}_{h,k}(a,b,t) = [a,a+ht] \times [b,b+kt] \end{aligned}$$

(2.10)

for $(a,b) \in D$ and $t>0$. Denote a family of rectangle regions by $\mathcal{R}^\mathrm{Rect}_{h,k}(D) = \{ R^\mathrm{Rect}_{h,k}(a,b,t) \mid R^\mathrm{Rect}_{h,k}(a,b,t) \subset D, \, (a,b) \in D, \ t>0 \} (\subset \mathcal{R}(D))$, (Fig. 3).

Corollary 2.2

If utility functions $f,g \in \mathcal{L}$ satisfy $M^{f}_{w}(R) \cap R_w^{(\overline{x}_R,\overline{y}_R)} \preceq M^{g}_{v}(R) \cap R_w^{(\overline{x}_R,\overline{y}_R)}$ for all weighting functions $w \in \mathcal{W}$ on D and all rectangle regions $R \in \mathcal{R}^\mathrm{Rect}_{h,k}(D)$, then it holds that

$$\begin{aligned} \frac{ h^2 f_{xx} + k^2 f_{yy} }{ h f_x + k f_y} \le \frac{ h^2 g_{xx} + k^2 g_{yy} }{ h g_x + k g_y} \end{aligned}$$

(2.11)

on D.

Example 2.4

(Oval regions). Let h and k be positive numbers. Let oval regions

$$\begin{aligned} R^\mathrm{Oval}_{h,k}(a,b,t) = \left\{ (x,y) \in {\mathbb {R}}^2 \left| \frac{(x-a)^2}{h^2} + \frac{(y-b)^2}{k^2} \le t^2 \right. \right\} \end{aligned}$$

(2.12)

for $(a,b) \in D$ and $t>0$. Denote a family of oval regions by $\mathcal{R}^\mathrm{Oval}_{h,k}(D) = \{ R^\mathrm{Oval}_{h,k}(a,b,t) \mid R^\mathrm{Oval}_{h,k}(a,b,t) \subset D, \, (a,b) \in D, \ t>0 \} (\subset \mathcal{R}(D))$, (Fig. 3).

Corollary 2.3

If utility functions $f,g \in \mathcal{L}$ satisfy $M^{f}_{w}(R) \cap R_w^{(\overline{x}_R,\overline{y}_R)} \preceq M^{g}_{v}(R) \cap R_w^{(\overline{x}_R,\overline{y}_R)}$ for all weighting functions $w \in \mathcal{W}$ on D and all oval regions $R \in \mathcal{R}^\mathrm{Oval}_{h,k}(D)$, then it holds that

$$\begin{aligned} \frac{ h^2 f_{xx} + k^2 f_{yy} }{ h f_x + k f_y} \le \frac{ h^2 g_{xx} + k^2 g_{yy} }{ h g_x + k g_y} \end{aligned}$$

(2.13)

on D.

Fig. 3.

Rectangle region $R^\mathrm{Rect}_{h,k}(a,b,t)$ and oval region $R^\mathrm{Oval}_{h,k}(a,b,t)$

Example 2.5

(Triangle regions). Let h and k be positive numbers. Let triangle regions

$$\begin{aligned} R^\mathrm{Tri}_{h,k}(a,b,t) = \left\{ (x,y) \in {\mathbb {R}}^2 \left| x \ge a, \, y \ge b, \, \frac{x-a}{h} + \frac{y-b}{k} \le t \right. \right\} \end{aligned}$$

(2.14)

for $(a,b) \in D$ and $t>0$. Denote a family of triangle regions by $\mathcal{R}^\mathrm{Tri}_{h,k}(D) = \{ R^\mathrm{Tri}_{h,k}(a,b,t) \mid R^\mathrm{Tri}_{h,k}(a,b,t) \subset D, \, (a,b) \in D, \ t>0 \} (\subset \mathcal{R}(D))$, (Fig. 4).

Corollary 2.4

If utility functions $f,g \in \mathcal{L}$ satisfy $M^{f}_{w}(R) \cap R_w^{(\overline{x}_R,\overline{y}_R)} \preceq M^{g}_{v}(R) \cap R_w^{(\overline{x}_R,\overline{y}_R)}$ for all weighting functions $w \in \mathcal{W}$ on D and all triangle regions $R \in \mathcal{R}^\mathrm{Tri}_{h,k}(D)$, then it holds that

$$\begin{aligned} \frac{ h^2 f_{xx} - hk f_{xy} + k^2 f_{yy} }{ h f_x + k f_y} \le \frac{ h^2 g_{xx} - hk g_{xy} + k^2 g_{yy} }{ h g_x + k g_y} \end{aligned}$$

(2.15)

on D.

Example 2.6

(Parallelogram regions). Let h and k be positive numbers. Let parallelogram regions

$$\begin{aligned} R^\mathrm{Para}_{h,k}(a,b,t) = \{ (x,y) \mid |k(x-a)-3h(y-b) | \le 4hkt, \, |3k(x-a)-h(y-b) | \le 4hkt \} \end{aligned}$$

(2.16)

for $(a,b) \in D$ and $t>0$. Denote a family of parallelogram regions by $\mathcal{R}^\mathrm{Para}_{h,k}(D) = \{ R^\mathrm{Para}_{h,k}(a,b,t) \mid R^\mathrm{Para}_{h,k}(a,b,t) \subset D, \, (a,b) \in D, \ t>0 \} (\subset \mathcal{R}(D))$, (Fig. 4).

Corollary 2.5

If utility functions $f,g \in \mathcal{L}$ satisfy $M^{f}_{w}(R) \cap R_w^{(\overline{x}_R,\overline{y}_R)} \preceq M^{g}_{v}(R) \cap R_w^{(\overline{x}_R,\overline{y}_R)}$ for all weighting functions $w \in \mathcal{W}$ on D and all parallelogram regions $R \in \mathcal{R}^\mathrm{Para}_{h,k}(D)$, then it holds that

$$\begin{aligned} \frac{ h^2 f_{xx} + \frac{3}{5} hk f_{xy} + k^2 f_{yy} }{ h f_x + k f_y} \le \frac{ h^2 g_{xx} + \frac{3}{5} hk g_{xy} + k^2 g_{yy} }{ h g_x + k g_y} \end{aligned}$$

(2.17)

on D.

Fig. 4.

Triangle region $R^\mathrm{Tri}_{h,k}(a,b,t)$ and parallelogram region $R^\mathrm{Para}_{h,k}(a,b,t)$

Example 2.3 (Rectangle regions) and Example 2.4 (Oval regions) are cases where $r=0$ in (2.8), and Example 2.5 (Triangle regions) and Example 2.6 (Parallelogram regions) are cases where $r=-\frac{1}{2}$ and $r=\frac{3}{10}$ respectively in (2.8).

3 A Sufficient Condition

Let $f,g \in \mathcal{L}$ be utility functions on an open convex domain D. Theorem 2.1 gives a necessary condition that decision making with utility f is more risk averse than decision making with utility g. In this section, we discuss its sufficient condition. For a utility function $f \in \mathcal{L}$, its Hessian matrix is written by

$$\begin{aligned} H^f(x,y) = \left( \begin{array}{cc} f_{xx}(x,y) &{} f_{xy}(x,y) \\ f_{yx}(x,y) &{} f_{yy}(x,y) \end{array} \right) \end{aligned}$$

(3.1)

for $(x,y) \in D$. The the following proposition gives a sufficient condition for (2.8) in Theorem 2.1.

Proposition 3.1

Let $f,g \in \mathcal{L}$ be utility functions on D. Then the following (i) and (ii) hold.

• (i) Matrices

  $$\begin{aligned} \frac{1}{f_x(x,y)} H^f(x,y) {-} \frac{1}{g_x(x,y)} H^g(x,y) \ {and} \ \frac{1}{f_y(x,y)} H^f(x,y) - \frac{1}{g_y(x,y)} H^g(x,y) \end{aligned}$$

  (3.2)

  are negative semi-definite for all $(x,y) \in D$ if and only if a matrix

  $$\begin{aligned} \frac{1}{h f_x(x,y) + k f_y(x,y)} H^f(x,y) - \frac{1}{h g_x(x,y) + k g_y(x,y)} H^g(x,y) \end{aligned}$$

  (3.3)

  is negative semi-definite for all $(x,y) \in D$ and all positive numbers h and k.

• (ii) If (3.2) are negative semi-definite at all $(x,y) \in D$, then (2.8) holds on D for all positive numbers h and k and all real numbers r satisfying $-1 \le r \le 1$.

From Proposition 3.1 implies that the condition (3.2) is stronger than the condition (2.8), however (3.2) is easier than (2.8) to check in actual cases. In this paper, utility functions $f (\in \mathcal{L})$ are called quadratic if the second derivatives $f_{xx}$, $f_{xy}$ and $f_{yy}$ are constant functions. When utility functions are quadratic, the following theorem gives a sufficient condition for what decision making with utility f is more risk averse than decision making with utility g.

Theorem 3.1

Let utility functions $f, g \in \mathcal{L}$ be quadratic on D. If

$$\begin{aligned} \frac{1}{f_x(x,y)} H^f(x,y) - \frac{1}{g_x(x,y)} H^g(x,y) \quad {and} \quad \frac{1}{f_y(x,y)} H^f(x,y) - \frac{1}{g_y(x,y)} H^g(x,y) \end{aligned}$$

(3.4)

are negative semi-definite at all $(x,y) \in D$, then decision making with utility f is more risk averse than decision making with utility g, i.e.

$$ M^{f}_{w}(R) \cap R_w^{(\overline{x}_R,\overline{y}_R)} \preceq M^{g}_{v}(R) \cap R_w^{(\overline{x}_R,\overline{y}_R)} $$

for all weighting functions $w \in \mathcal{W}$ and all closed convex regions $R \in \mathcal{R}(D)$.

Now we give an example for Theorem 3.1.

Example 3.1

(Quadratic utility functions). Let a domain $D=(-0.5,1.5)^2$ and a region $R= [0, 1]^2$, and let a weighting function $w(x,y)=1$ for $(x,y) \in D$. Then an invariant neutral point is $(\overline{x}_R,\overline{y}_R)=(0.5,0.5)$ and $R_{w,-}^{(\overline{x}_R,\overline{y}_R)} = [0,0.5]^2 \setminus \{ (0.5,0.5) \}$ and $R_{w,+}^{(\overline{x}_R,\overline{y}_R)} = [0.5,1]^2 \setminus \{ (0.5,0.5) \}$. Let us consider two quadratic utility functions $f(x,y)=-2x^2-2y^2+2xy+8x+8y$ and $g(x,y)=-x^2-y^2+xy+5x+5y$ for $(x,y) \in D$. Then f and g are increasing on D, i.e. $f_x(x,y)=-4x+2y+8>0$, $f_y(x,y)=2x-4y+8>0$, $g_x(x,y)=-2x+y+5>0$ and $g_y(x,y)=x-2y+5>0$ on D. Their Hessian matrices are

$$\begin{aligned} H^f(x,y) = \left( \begin{array}{rr} -4 &{} 2 \\ 2 &{} -4 \end{array} \right) \quad \text{ and } \quad H^g(x,y) = \left( \begin{array}{rr} -2 &{} 1 \\ 1 &{} -2 \end{array} \right) . \end{aligned}$$

(3.5)

Let A(x, y) and B(x, y) by $A(x,y)=\frac{1}{f_x(x,y)} H^f(x,y) - \frac{1}{g_x(x,y)} H^g(x,y)$ and $B(x,y)=\frac{1}{f_y(x,y)} H^f(x,y) - \frac{1}{g_y(x,y)} H^g(x,y)$ for $(x,y) \in D$, and then we have

$$\begin{aligned} A(x,y) = \frac{1}{-4x+2y+8} \left( \begin{array}{rr} -4 &{} 2 \\ 2 &{} -4 \end{array} \right) - \frac{1}{-2x+y+5} \left( \begin{array}{rr} -2 &{} 1 \\ 1 &{} -2 \end{array} \right) , \end{aligned}$$

(3.6)

$$\begin{aligned} B(x,y) = \frac{1}{2x-4y+8} \left( \begin{array}{rr} -4 &{} 2 \\ 2 &{} -4 \end{array} \right) - \frac{1}{x-2y+5} \left( \begin{array}{rr} -2 &{} 1 \\ 1 &{} -2 \end{array} \right) . \end{aligned}$$

(3.7)

We can easily check A(x, y) and B(x, y) are negative definite for all $(x,y) \in D$. From Theorem 3.1, decision making with utility f is more risk averse than decision making with utility g on R and it holds that $M^{f}_{w}(R) \cap R_w^{(\overline{x}_R,\overline{y}_R)} \preceq M^{g}_{v}(R) \cap R_w^{(\overline{x}_R,\overline{y}_R)}$ for all weighting functions $w \in \mathcal{W}$ (Fig. 5).

Fig. 5.

$M^{f}_{w}(R) \cap R_w^{(\overline{x}_R,\overline{y}_R)} \preceq M^{g}_{v}(R) \cap R_w^{(\overline{x}_R,\overline{y}_R)}$ ($f(x,y)=-2x^2-2y^2+2xy+8x+8y, g(x,y)=-x^2-y^2+xy+5x+5y$, $R=[0, 1]^2$)

Concluding Remark. When utility functions are quadratic, Theorem 3.1 gives a sufficient condition where decision making with utility f is more risk averse than decision making with utility g. It is an open problem whether (3.2) is a sufficient condition when utility functions are not quadratic but more general.