Financial Decisions Based on Zero-Sum Games: New Conceptual and Mathematical Outcomes

Abstract

All the n possible returns on a financial asset are the components of an element of a linear space over $\mathbb{R}$. This paper shows how to transfer all these n possible returns on a one-dimensional straight line. In this research work, two or more than two financial assets are studied. More than two financial assets are always studied in pairs, so they are treated inside the budget set of a given decision-maker. Two univariate financial assets give rise to a bivariate financial asset characterized by a bivariate (two-dimensional) distribution of probability. This research work shows how constrained choices being made by a given decision-maker under conditions of uncertainty and riskiness maximize his utility of an ordinal nature. For this reason, prevision bundles are dealt with. Furthermore, every choice identifies a zero-sum game. Since a specific kind of choice associated with two or more than two objects is investigated, new conceptual and mathematical outcomes related to financial decisions are shown.

1. Introduction

This paper studies decisions related to portfolios or sets of financial assets, where each financial asset is a random good that provides a monetary flow under conditions of uncertainty and riskiness. The monetary payments associated with a specific financial asset are all possible at present time. They can be treated as sampling units. The monetary payment that will happen in the imminent future is unknown at present time, so it has to be estimated. In this paper, two or more than two goods are always studied. They are the components of the portfolio of goods under consideration. More than two goods are steadily handled in pairs. Such goods are then studied inside the budget set of a given decision-maker. A specific kind of constrained choice based on zero-sum games is analyzed. This is a new matter compared to what has been investigated so far in the literature (see Chambers et al. (2017); Cassese et al. (2020); Li et al. (2020)). A given decision-maker has to estimate the expected monetary payment related to two or more than two financial assets based on the real data which may have been observed in the past. Specific indices of a statistical nature are used to obtain fair estimations. An infinite number of possible outcomes for each good into account can be considered. However, one can refer to a finite number of possible alternatives for each good under consideration. In particular, each good is bounded from above and below (see Echenique (2020)). The two starting goods are called marginal goods. They are univariate goods. They always give rise to a bivariate good. Since the latter is studied too, a further subdivision of outcomes of a random process takes place. The possible outcomes associated with two marginal goods and a bivariate good have to be summarized inside the budget set of a two-dimensional nature. All their coherent summaries are fair evaluations or estimations. They are barycentres of non-negative masses, where each mass is associated with a possible outcome. All their coherent summaries establish a closed convex set. This two-dimensional set is the budget set of a given decision-maker. Moreover, these coherent summaries also establish two closed convex sets, where each of them is of a one-dimensional nature. First of all, the whole of the formally admissible estimations related to an expected monetary payment associated with a specific good has to be studied. The budget set of a given decision-maker is characterized by the whole of the formally admissible estimations related to an expected monetary payment associated with a specific bivariate good. Every point of the budget set is a formally admissible estimation related to an expected monetary payment associated with a specific bivariate good. Secondly, the choice of one among such fair estimations is left to a judgment being made by a given individual based on a specific criterion of an empirical nature. A convergent process characterizes this choice. Bayes’ theorem applied to constrained choices says this (see Berti et al. (2020)). Also, this choice has to satisfy the optimization principle associated with the maximization of the notion of utility of an ordinal and cardinal nature (see Afriat (1967); Varian (1982); Angelini and Maturo (2022a)). For the purpose, prevision bundles satisfying the maximization of the notion of ordinal utility are treated inside the budget set of a given individual. Conversely, the maximization of the notion of cardinal utility takes place outside this framework based on data that are observed and estimated inside it. Given a probability space denoted by

$$(\Omega ,\mathcal{F},\mathbf{P}),$$

(1)

where $\Omega$ is the set of the possible outcomes of a random process and $\mathbf{P}$ is a prevision-function representing the opinion associated with a given decision-maker who is faced with a situation of uncertainty mathematically expressed by the possible values for a random magnitude, one observes the following $\sigma$-algebra

$$\mathcal{F}=\{\varnothing ,\Omega \}.$$

(2)

In this paper, $\Omega$ is always embedded in a linear space over $\mathbb{R}$ containing possible and impossible alternatives. Its nature is Euclidean. Only its finite dimension can be different.

The Objectives of This Paper

In Section 2, it is said that the conditions of certainty are ideal. Whenever two or more than two objects of decision-maker choice are treated, their real nature is multilinear. In particular, their real nature is bilinear. This is because more than two objects of decision-maker choice are always handled in pairs. In Section 3, a theorem showing how to transfer all the n possible returns on a financial asset on a one-dimensional straight line is proved. If $X_1$ and $X_2$ are two univariate financial assets, where each of them is assumed to have n possible returns, then the n possible returns on each financial asset into account are transferred on two one-dimensional straight lines, on which an origin, a same unit of length, and an orientation are established. Such lines are the two axes of the Cartesian plane. Section 4 studies the properties of $\mathbf{P}$, where $\mathbf{P}$ is a function of prevision defined as an expression of the opinion of a given decision-maker. Section 5 focuses on reductions of dimension characterizing the linear nature of the space of alternatives. Section 6 handles constrained choices focusing on specific objects called prevision bundles. Some axioms of revealed preference theory are used. Financial decisions based on zero-sum games are treated in Section 7, where a nonlinear analysis containing new conceptual and mathematical outcomes is shown. This analysis is of a multilinear nature. Finally, Section 8 provides conclusions and future perspectives.

2. The Real Nature of the Objects of Decision-Maker Choice under Claimed Conditions of Certainty

The conditions of certainty are not real, but they are ideal. For this reason, one has to speak about claimed conditions of certainty. What does this matter imply? First of all, this paper says that if two or more than two objects of decision-maker choice are studied, then the real nature of them is multilinear. In particular, their real nature is bilinear. This is because more than two objects of decision-maker choice are always handled in pairs. Given two ordinary goods having downward-sloping demand curves, $({\text{}}_1^{\text{}}{y}_{\text{}}^{\text{}},{\text{}}_2^{\text{}}{y}_{\text{}}^{\text{}})$ represents what is chosen for each of them by a given decision-maker inside his budget set (see Samuelson (1948); Varian (1983)). Nevertheless, this research work innovatively says that one can always write the following two objective functions of a linear nature given by

$${\text{}}_1^{\text{}}{y}_{\text{}}^{\text{}}={\text{}}_{\text{}}^{\text{}}{y}_1^1{\text{}}_{\text{}}^{\text{}}{p}_1^1+\dots +{\text{}}_{\text{}}^{\text{}}{y}_1^n{\text{}}_{\text{}}^{\text{}}{p}_1^n,$$

(3)

and

$${\text{}}_2^{\text{}}{y}_{\text{}}^{\text{}}={\text{}}_{\text{}}^{\text{}}{y}_2^1{\text{}}_{\text{}}^{\text{}}{p}_2^1+\dots +{\text{}}_{\text{}}^{\text{}}{y}_2^n{\text{}}_{\text{}}^{\text{}}{p}_2^n,$$

(4)

where $\left\{{\text{}}_{\text{}}^{\text{}}{p}_1^i\right\}$ and $\left\{{\text{}}_{\text{}}^{\text{}}{p}_2^j\right\}$ are two sets of n masses or weights summing to 1, with $0\le {\text{}}_{\text{}}^{\text{}}{p}_i^j\le 1$, $j=1,\dots ,n$, $i=1,2$. This means that average quantities are taken into account using the principles of the logic of prevision. In other words, ${\text{}}_1^{\text{}}{y}_{\text{}}^{\text{}}$ is not a raw datum only, but it is a barycentre of masses distributed over $\{{\text{}}_{\text{}}^{\text{}}{y}_1^1,\dots ,{\text{}}_{\text{}}^{\text{}}{y}_1^n\}$. Similarly, ${\text{}}_2^{\text{}}{y}_{\text{}}^{\text{}}$ is not a raw datum only, but it is a barycentre of masses distributed over $\{{\text{}}_{\text{}}^{\text{}}{y}_2^1,\dots ,{\text{}}_{\text{}}^{\text{}}{y}_2^n\}$. The possible quantities that can be demanded for good 1 and good 2 are estimated together with their masses. The rules of the logic of prevision always hold inside the budget set of a given decision-maker, so studying $({\text{}}_1^{\text{}}{y}_{\text{}}^{\text{}},{\text{}}_2^{\text{}}{y}_{\text{}}^{\text{}})$ by itself is not sufficient. In fact, it is necessary to estimate two objective functions of a linear nature given by (3) and (4). These functions derive from an objective function of a bilinear nature. For this reason, the Cartesian product given by $\{{\text{}}_{\text{}}^{\text{}}{y}_1^1,\dots ,{\text{}}_{\text{}}^{\text{}}{y}_1^n\}\times \{{\text{}}_{\text{}}^{\text{}}{y}_2^1,\dots ,{\text{}}_{\text{}}^{\text{}}{y}_2^n\}$ is also treated together with $n^2$ non-negative masses. Each of them is associated with each pair of $\{{\text{}}_{\text{}}^{\text{}}{y}_1^1,\dots ,{\text{}}_{\text{}}^{\text{}}{y}_1^n\}\times \{{\text{}}_{\text{}}^{\text{}}{y}_2^1,\dots ,{\text{}}_{\text{}}^{\text{}}{y}_2^n\}$. Given $({\text{}}_1^{\text{}}{y}_{\text{}}^{\text{}},{\text{}}_2^{\text{}}{y}_{\text{}}^{\text{}})$, the weighted average of $n^2$ possible quantities that can be demanded for good 1 and good 2 is a summarized element of the Fréchet class, where the Fréchet class contains all the bivariate distributions of mass established by $n^2$ non-negative masses. In particular, with $n\ge 10$, masses or weights can be chosen in such a way that one out of three paradigmatic cases is handled. They coincide with three specific values, $-1$, 0, and 1, that can be taken by the correlation coefficient. Given $({\text{}}_1^{\text{}}{y}_{\text{}}^{\text{}},{\text{}}_2^{\text{}}{y}_{\text{}}^{\text{}})$, a closed neighborhood of ${\text{}}_1^{\text{}}{y}_{\text{}}^{\text{}}$ is represented by $[{\text{}}_1^{\text{}}{y}_{\text{}}^{\text{}}-ϵ;{\text{}}_1^{\text{}}{y}_{\text{}}^{\text{}}+{ϵ}^{\prime }]$ on the horizontal axis, and a closed neighborhood of ${\text{}}_2^{\text{}}{y}_{\text{}}^{\text{}}$ is expressed by $[{\text{}}_2^{\text{}}{y}_{\text{}}^{\text{}}-ϵ;{\text{}}_2^{\text{}}{y}_{\text{}}^{\text{}}+{ϵ}^{\prime }]$ on the vertical one, where both $ϵ$ and ${ϵ}^{\prime }$ are two small positive quantities. The possible quantities that can be demanded for good 1 belong to $[{\text{}}_1^{\text{}}{y}_{\text{}}^{\text{}}-ϵ;{\text{}}_1^{\text{}}{y}_{\text{}}^{\text{}}+{ϵ}^{\prime }]$, and the possible quantities that can be demanded for good 2 belong to $[{\text{}}_2^{\text{}}{y}_{\text{}}^{\text{}}-ϵ;{\text{}}_2^{\text{}}{y}_{\text{}}^{\text{}}+{ϵ}^{\prime }]$. It is not necessary that one out of n possible alternatives coincides with ${\text{}}_1^{\text{}}{y}_{\text{}}^{\text{}}$. The same is true with respect to ${\text{}}_2^{\text{}}{y}_{\text{}}^{\text{}}$. $({\text{}}_1^{\text{}}{y}_{\text{}}^{\text{}},{\text{}}_2^{\text{}}{y}_{\text{}}^{\text{}})$ belongs to a subset of $\mathbb{R}\times \mathbb{R}$, where $\mathbb{R}\times \mathbb{R}$ is the direct product of $\mathbb{R}$ and $\mathbb{R}$. Two half-lines give rise to the budget set of a given decision-maker. Each of them extends indefinitely in a positive direction from zero before being restricted. The budget set is a right triangle belonging to the first quadrant of the Cartesian plane. The vertex of the right angle is given by $(0,0)$. The budget line identifying the budget set is a hyperplane embedded in the Cartesian plane. This line is the side of the right triangle opposite the right angle. The negative slope of the budget line is given by

$$-\frac{b_1}{b_2}.$$

(5)

It depends on the known prices of the two goods into account. The budget constraint of a given decision-maker is written in the form

$$b_1\left({\text{}}_1^{\text{}}{y}_{\text{}}^{\text{}}\right)+b_2\left({\text{}}_2^{\text{}}{y}_{\text{}}^{\text{}}\right)\le b,$$

(6)

where the prices of good 1 and good 2 are given by $(b_1,b_2)$, whereas the amount of money he has to spend is given by b. The horizontal intercept of the budget line is given by

$$\frac{b}{b_1}.$$

(7)

Its vertical intercept is expressed by

$$\frac{b}{b_2}.$$

(8)

The two choice functions are given by

$${\text{}}_1^{\text{}}{y}_{\text{}}^{\text{}}={\text{}}_1^{\text{}}{y}_{\text{}}^{\text{}}\left(b_1,b_2,b\right),$$

(9)

and

$${\text{}}_2^{\text{}}{y}_{\text{}}^{\text{}}={\text{}}_2^{\text{}}{y}_{\text{}}^{\text{}}\left(b_1,b_2,b\right).$$

(10)

They are two objective functions of a linear nature deriving from an objective function of a bilinear nature. The elements identifying the decision-maker’s budget are represented by $(b_1,b_2,b)$. They are all objective. Nevertheless, this paper says that b is assumed to be an uncertain element at the time of choice. As a consequence, the role of a finite number of possible alternatives is essential for each objective function. This observation answers the question posed at the beginning of this section.

Two Different Notions: Prevision and Prediction

Another point of view about choice functions is associated with problems contemplating infinite alternatives in number. Problems contemplating infinite alternatives in number are precious whenever one wants to obtain a unique result without any error. This result can be called a sure prediction. Hence, to make a prediction means to try to guess, among the possible alternatives, the one that will happen in the imminent future. With respect to the previous discussion, there are infinite points between 0 and (7), as well as between 0 and (8). If one finds oneself in a state of uncertainty, then such points represent infinite alternatives in number. They are all possible at present time. This means that each of them could be either true or false at the right time. The right time is when uncertainty ceases. In particular, if infinite alternatives between 0 and (7) denoted by $[a,b]=[0,\frac{b}{b_1}]$ are supposed to be the support of the probability density function f of the continuous uniform distribution with parameters 0 and $\frac{b}{b_1}$ characterizing $X_1$, where $X_1$ is a random variable, then one writes

$$\mathbb{E}\left(X_1\right)={\int }_a^{b}xf\left(x\right)dx=\frac{1}{2}(a+b)=\frac{1}{2}\frac{b}{b_1},$$

(11)

with $\frac{1}{2}\frac{b}{b_1}\in [0,\frac{b}{b_1}]$. Among infinite alternatives, the one that is chosen is given by $\frac{1}{2}\frac{b}{b_1}$. Similarly, one writes

$$\mathbb{E}\left(X_2\right)={\int }_a^{b}xf\left(x\right)dx=\frac{1}{2}(a+b)=\frac{1}{2}\frac{b}{b_2}$$

(12)

with respect to the mathematical expectation of $X_2$, where $X_2$ is another random variable. The support of the probability density function f of the continuous uniform distribution with parameters 0 and $\frac{b}{b_2}$ characterizing $X_2$ is denoted by $[a,b]=[0,\frac{b}{b_2}]$. The expression given by (12) is passed off as a sure prediction like (11). Among infinite alternatives denoted by $[a,b]=[0,\frac{b}{b_2}]$, the one that is chosen is expressed by $\frac{1}{2}\frac{b}{b_2}\in [0,\frac{b}{b_2}]$. Consequently, a sure prediction takes infinite alternatives in number as a necessary prerequisite together with a probability distribution characterized by specific parameters. Infinite alternatives in number are uniquely summarized using a specific mathematical tool (integral). There also exists a more immediate way that can be used to make a prediction. Thus, a prediction reduces to the arbitrary choice of any point in a given set containing possible alternatives. A given individual assumes himself capable of guessing it without using mathematics. A prediction reduces to the arbitrary choice of any point in $[0,\frac{b}{b_1}]$ and in $[0,\frac{b}{b_2}]$, so it is a prophecy. In this paper, constrained choices which are studied inside the budget set of a given decision-maker do not obey what is passed off as a sure prediction. They obey the rules of the logic of prevision. Hence, to make a prevision means to exchange $X_1$ for $\mathbf{P}\left(X_1\right)$, where $\mathbf{P}\left(X_1\right)\in [0,\frac{b}{b_1}]$. The uncountable set denoted by $[0,\frac{b}{b_1}]$ identifies infinite previsions in the first stage. This is because there are infinitely many possible opinions about the evaluations of probability. They are all admissible in the first stage. The possible values for $X_1$ belonging to $I\left(X_1\right)$ are finite in number (see Example 1 later on). They are contained in $[0,\frac{b}{b_1}]$. One observes $\inf I\left(X_1\right)=0$ and $\sup I\left(X_1\right)=\frac{b}{b_1}$. Every prevision contained in $[0,\frac{b}{b_1}]$ is a weighted average. Nonparametric distributions of probability are used. Information and knowledge associated with a given individual always affect the exchange of $X_1$ for $\mathbf{P}\left(X_1\right)$ in the second stage. The exchange of $X_1$ for $\mathbf{P}\left(X_1\right)$ in the second stage is not a prophecy, but it is a fair estimation based on a specific state of information and knowledge associated with a given individual (see Bassan et al. (2003)). This state is intrinsically variable. The same is true for $X_2$ and $\mathbf{P}\left(X_2\right)\in [0,\frac{b}{b_2}]$. A prevision never transforms the uncertainty into an artificial certainty. Mathematically, one writes $\mathbf{P}=\mathbb{E}$. Conceptually, while $\mathbf{P}$ is a fair estimation admitting positive and negative errors, $\mathbb{E}$ is the result of a calculation, when it is possible, obtained without any error. In this paper, $\mathbf{P}$ is a linear functional (scalar or inner product) and zero-sum games with incomplete information (see De Meyer et al. (2010)) are based on it. On the other hand, linear functionals are used in machine learning too, where it is known that artificial neural networks are models influenced by the structure and function of biological neural networks in animal brains (see Awasthi et al. (2022)). In particular, scalar products underlie a classification algorithm that makes its estimations combining a set of weights with the feature vector.

3. A Random Good and Its Representation in a Linear Form

Let X be a random good with n possible alternatives. Let ${\mathcal{B}}_n^{\perp }=\left\{{\mathbf{e}}_i\right\}$, $i=1,\dots ,n$, be an orthonormal basis of $E^n$, where $E^n$ is a linear space over $\mathbb{R}$. If $\mathbf{x}\in E^n$, then it is possible to write

$$\mathbf{x}=x^i{\mathbf{e}}_i$$

(13)

using the Einstein summation convention. The set of all contravariant components of $\mathbf{x}$ is denoted by $\left\{x^i\right\}$. They are uniquely determined with respect to ${\mathcal{B}}_n^{\perp }$. Given ${\mathcal{B}}_n^{\perp }$, they uniquely identify $\mathbf{x}$. Since one writes

$$\mathbf{x}=(x^1,x^2,\dots ,x^n),$$

(14)

(14) is a vector belonging to $E^n$, whose contravariant components represent the elements of a finite partition of events concerning X. These events are implicitly and “en masse” considered. Each contravariant component of $\mathbf{x}$ is a possible value for X (see Gilio and Sanfilippo (2014)). The set of all possible values for X is denoted by $I\left(X\right)=\{x^1,\dots ,x^n\}$, where one can observe $x^1<\dots <x^n$. If X is a financial asset, then $I\left(X\right)=\{x^1,\dots ,x^n\}$ is the set of possible returns on it. Studying constrained choices, it is appropriate to treat bounded random quantities, whose possible values are contained in a specific interval after their geometric transfer on a one-dimensional straight line. Let $E_i$, $i=1,\dots ,n$, be the generic element of a finite partition of events. One writes

$$X=x^1\left|E_1\right|+x^2\left|E_2\right|+\dots +x^n\left|E_n\right|,$$

(15)

where one has

$$\left|E_i\right|=\left\{\begin{array}{cc}1,\hfill & \mathrm{i}\mathrm{f} E_i \mathrm{i}\mathrm{s} \mathrm{t}\mathrm{r}\mathrm{u}\mathrm{e}\hfill \\ 0,\hfill & \mathrm{i}\mathrm{f} E_i \mathrm{i}\mathrm{s} \mathrm{f}\mathrm{a}\mathrm{l}\mathrm{s}\mathrm{e}\hfill \end{array}\right.$$

(16)

for every $i=1,\dots ,n$. X is therefore linearly dependent on n random elements. Linear dependence is a special case of logical dependence. Linear dependence is more restrictive than logical one. For instance, logical dependence could also be of a quadratic nature. Logical dependence of one random entity on others has the same meaning that it has in mathematical analysis with respect to a one-valued function of several variables. X is “a posteriori” logically dependent on $\left|E_i\right|$, $i=1,\dots ,n$. All possible linear combinations expressed by (15) give rise to random goods which are different because their possible values can be different. They identify a linear space denoted by $E^{n*}$. It is dual to $E^n$. Such dual spaces are two in number and they are superposed by means of a quadratic metric introduced by considering an orthonormal basis of $E^n$. In fact, the notion of scalar or inner product is used to say that an orthonormal basis of $E^n$ consists of unit vectors and pairwise orthogonal. All possible values for X are “a priori” logically independent and the additive form of X given by (15) derives from this independence.

3.1. Two n-Dimensional Linear Spaces That Are Superposed

Let ${\Phi }^j$ be a linear functional such that it is possible to write

$${\Phi }^j:E^n\to \mathbb{R}.$$

(17)

If $\mathbf{t}\in E^n$, then one writes

$$\mathbf{t}=t^j{\mathbf{e}}_j.$$

(18)

Hence, one observes

$${\Phi }^j\left(\mathbf{t}\right)=t^j,j=1,\dots ,n.$$

(19)

It follows that one obtains

$${\Phi }^j\left({\mathbf{e}}_i\right)={\delta }_i^j,$$

(20)

where ${\delta }_i^j$ is the Kronecker delta, so one has ${\delta }_i^j=1$ if $i=j$, and ${\delta }_i^j=0$ if $i\ne j$. Thus, it is known that $\left\{{\Phi }^j\right\}$, $j=1,\dots ,n$, is a basis of $E^{n*}$, so one writes

$$\Phi =u_1{\Phi }^1+u_2{\Phi }^2+\dots +u_n{\Phi }^n,$$

(21)

with $\Phi \in E^{n*}$. Let F be a linear functional such that one writes

$$F:E^n\to \mathbb{R}.$$

(22)

One consequently obtains

$$F\left(\mathbf{x}\right)=F\left(x^j{\mathbf{e}}_j\right)=x^{j}F\left({\mathbf{e}}_j\right),\mathbf{x}\in E^n.$$

(23)

Given (19), one observes

$$F\left(\mathbf{x}\right)=F\left({\mathbf{e}}_j\right){\Phi }^j\left(\mathbf{x}\right)=\left[F\left({\mathbf{e}}_j\right){\Phi }^j\right]\left(\mathbf{x}\right),\mathbf{x}\in E^n,$$

(24)

where (24) is valid for every $\mathbf{x}\in E^n$. It follows that F and $F\left({\mathbf{e}}_j\right){\Phi }^j$ are linear functionals that coincide, so one writes

$$F=F\left({\mathbf{e}}_j\right){\Phi }^j,$$

(25)

where one observes $F\left({\mathbf{e}}_j\right)=u_j\in \mathbb{R}$. Hence, the elements of $\left\{{\Phi }^j\right\}$, $j=1,\dots ,n$, give rise to $E^{n*}$. One writes

$$\Phi \left(\mathbf{x}\right)=u_1{\Phi }^1\left(\mathbf{x}\right)+u_2{\Phi }^2\left(\mathbf{x}\right)+\dots +u_n{\Phi }^n\left(\mathbf{x}\right)=u_1{x}^1+u_2{x}^2+\dots +u_n{x}^n,\mathbf{x}\in E^n,$$

(26)

with $\Phi \in E^{n*}$, where (26) denotes the scalar or inner product of two n-dimensional vectors. The former is denoted by $\mathbf{x}=(x^1,x^2,\dots ,x^n)\in E^n$, whereas the latter is denoted by $\mathbf{u}=(u_1,u_2,\dots ,u_n)\in E^{n*}$. It is known that ${\Phi }^1,{\Phi }^2,\dots ,{\Phi }^n$ are linearly independent. Suppose that it is possible to write

$${\alpha }_j{\Phi }^j=0,$$

(27)

where one has ${\alpha }_j\in \mathbb{R}$. From (27), it follows that one can write

$${\alpha }_j{\Phi }^j\left(\mathbf{x}\right)=0,\forall \mathbf{x}\in E^n.$$

(28)

In particular, if one chooses $\mathbf{x}={\mathbf{e}}_k$, then one writes

$${\alpha }_j{\Phi }^j\left({\mathbf{e}}_k\right)={\alpha }_k=0.$$

(29)

This is because the expression given by (20) holds. Since (28) is valid for every $\mathbf{x}\in E^n$, (29) is true for every ${\mathbf{e}}_j\in {\mathcal{B}}_n^{\perp }$, $j=1,\dots ,n$. This means that one obtains

$${\alpha }_1={\alpha }_2=\dots ={\alpha }_n=0,$$

(30)

so all linear functionals of the set $\left\{{\Phi }^j\right\}$ are linearly independent. They represent a basis of $E^{n*}$. One then observes

$$dim{E}^n=dim{E}^{n*}=n.$$

(31)

3.2. How to Obtain Mathematically the Possible Alternatives for a Random Good

In this subsection, the mathematical origin of the possible values for X is treated. They belong to a closed structure coinciding with a linear space over $\mathbb{R}$. It is the space of alternatives (see von Neumann (1936)). If X is a financial asset, then the mathematical origin of the possible returns on X is treated. Let ${\mathcal{B}}_n^{\perp }$ be an orthonormal basis of $E^n$. If there is no uncertainty, then the real coefficients of each linear combination of n basis vectors belonging to ${\mathcal{B}}_n^{\perp }$ coincide with 0 or 1 only (see Coletti et al. (2016)). Let $\mathcal{A}\subset E^n$ be the set of elements denoted by $\mathbf{a}$. They are n-dimensional vectors. Their contravariant components are all equal to 0 or 1 only. One writes

$$\mathbf{a}=a^j{\mathbf{e}}_j,$$

(32)

where if $a^j=1$ and $a^i=0$, $\forall i\ne j$, then $E_j$ corresponding to ${\mathbf{e}}_j$ is true, being false all others. Given n events generically denoted by $E_1,E_2,\dots ,E_n$, $\mathcal{A}\subset E^n$ contains all the constituents of $E_1,E_2,\dots ,E_n$. Their number is at most equal to $k=2^n$. In general, each constituent of $E_1,E_2,\dots ,E_n$ is an event obtained through the logical product involving $E_1$ or its negation denoted by ${\overline{E}}_1$, $E_2$ or its negation denoted by ${\overline{E}}_2$, …, and $E_n$ or its negation denoted by ${\overline{E}}_n$. This product takes n factors into account. For instance,

$$E_1{E}_2\dots E_n,$$

(33)

$${\overline{E}}_1{\overline{E}}_2\dots {\overline{E}}_n,$$

(34)

and

$${\overline{E}}_1{E}_2\dots E_n$$

(35)

are logical products identifying some constituents of $E_1,E_2,\dots ,E_n$. All possible constituents of $E_1,E_2,\dots ,E_n$, whose number is equal to $k\le 2^n$, give rise to a finite partition of events. In particular, if $E_1,E_2,\dots ,E_n$ identify incompatible and exhaustive values for X, then one observes $k=n$. Hence, with respect to the elements of $\mathcal{A}\subset E^n$, it is possible to define the following linear functional

$$F\left(\mathbf{a}\right)=u_j{\Phi }^j\left(\mathbf{a}\right)=u_1{a}^1+u_2{a}^2+\dots +u_n{a}^n,$$

(36)

where one has $F\left(\mathbf{a}\right)\in E^{n*}$. This functional represents the possible return on X which is associated with $\mathbf{a}$. In particular, $u_j$ is the possible return on X which is associated with ${\mathbf{e}}_j$ to which $\mathbf{a}$ reduces when and only when $E_j$ is true. From (36) and (20), it follows that it is possible to write

$$F\left({\mathbf{e}}_k\right)=u_j{\Phi }^j\left({\mathbf{e}}_k\right)=u_j{\delta }_k^j=u_k,$$

(37)

with $u_k\in \mathbb{R}$. On the other hand, F is a linear map, so it is determined whenever its value on basis elements is known. Hence, $u_j$ is a real number depending on a specific state of information and knowledge associated with a given decision-maker. Since $\mathbf{a}\in \mathcal{A}$ is a random vector, $F\left(\mathbf{a}\right)$ is a scalar or inner product representing all possible returns on X. The two dual spaces denoted by $E^n$ and $E^{n*}$ are superposed, so $F\left(\mathbf{a}\right)$ identifies homogeneous linear combinations. The number of the possible values for $F\left(\mathbf{a}\right)$, $\mathbf{a}\in \mathcal{A}$, is overall equal to n. Thus, one observes

$$u_1\ne u_2\ne \dots \ne u_n.$$

(38)

They can be denoted by $b_1,b_2,\dots ,b_n$, so one writes

$$(u_1=b_1)\ne (u_2=b_2)\ne \dots \ne (u_n=b_n).$$

(39)

They exactly correspond to $x^1,x^2,\dots ,x^n$, so one observes $b_i=x^i$, $i=1,\dots ,n$. If an orthonormal basis of $E^n$ is considered, then the contravariant and covariant components of a same vector of $E^n$ coincide. Thus, one writes

$${\mathcal{S}}_r=\{\mathbf{a}\in \mathcal{A}|F\left(\mathbf{a}\right)=b_r\},$$

(40)

so one has

$$\mathcal{A}=\bigcup _{r=1}^n{\mathcal{S}}_r,$$

(41)

as well as

$${\mathcal{S}}_r\cap {\mathcal{S}}_t=\varnothing ,r\ne t.$$

(42)

If $\mathcal{A}$ is extended to $E^n$, then (36) is an expression of a hyperplane embedded in $E^n$. One rewrites it in the following form

$$u_1{a}^1+u_2{a}^2+\dots +u_n{a}^n,$$

(43)

so different n-dimensional vectors denoted by $\mathbf{a}\in E^n$ give rise to different values for it denoted by $b_1,b_2,\dots ,b_n$.

3.3. Convex Combinations of Possible Alternatives

This research work says that the set of n possible returns on X is embedded in an n-dimensional linear space over $\mathbb{R}$ provided with a quadratic metric (see Angelini and Maturo (2023)). All possible returns on X can be expressed by means of two of them in the form of a convex combination. This happens after transferring all possible returns on X on a one-dimensional straight line, on which an origin, a unit of length, and an orientation are chosen. In this paper, a convex combination is also used in the field of the logic of prevision. This is because the role of convex sets is essential. Such sets contain the whole of the formally admissible estimations related to an expected monetary payment associated with a specific financial asset. How to transfer all the n possible returns on X on a one-dimensional straight line is proved by the following:

Theorem 1.

Let ${\mathcal{B}}_n^{\perp }=\{{\mathbf{e}}_1,\dots ,{\mathbf{e}}_n\}$ be an orthonormal basis of $E^n$ and let $b_1,b_2,\dots ,b_n$ be the possible values for X. If each possible value for X is obtained by means of a homogeneous linear combination of n possible alternatives, then each possible value for X is expressed as a convex combination of two possible values for X visualized on a one-dimensional straight line.

Proof. 

An n-dimensional located vector at the origin of $E^n$ is fully determined by its endpoint. It is then possible to call an ordered n-tuple of real numbers either a vector of $E^n$ or a point of ${\mathcal{E}}^n$, where ${\mathcal{E}}^n$ is an affine space. This means that $E^n$ and ${\mathcal{E}}^n$ are isomorphic. There exists a one-to-one correspondence between the vectors of $E^n$ and the points of ${\mathcal{E}}^n$. If one writes

$$F\left(\mathbf{x}\right)=u_j{\Phi }^j\left(\mathbf{x}\right)=b_r,\mathbf{x}\in E^n,$$

(44)

then one has $F\left(\mathbf{x}\right)\in E^{n*}$. This means that (44) is a hyperplane embedded in $E^n$. It is also a hyperplane embedded in ${\mathcal{E}}^n$. On the other hand, if one writes

$$F\left(\mathbf{a}\right)=u_j{\Phi }^j\left(\mathbf{a}\right)=b_r,\mathbf{a}\in \mathcal{A},$$

(45)

then one observes $F\left(\mathbf{a}\right)\in E^{n*}$. This means that (45) is a hyperplane embedded in $E^n$. It is also a hyperplane embedded in ${\mathcal{E}}^n$. (44) and (45) are characterized by the same possible value for X. Let

$$\mathbf{u}=u^j{\mathbf{e}}_j\in E^n$$

(46)

be an n-dimensional vector and let ${\rho }_0\in {\mathcal{E}}^n$ be a straight line. If one writes

$$\left\{\lambda \mathbf{u}\right|\forall \lambda \in \mathbb{R}\},$$

(47)

then the straight line given by (47) is orthogonal to all hyperplanes established by (44). They are obtained as $b_r$ varies, $r=1,\dots ,n$. In particular, (47) is orthogonal to the hyperplane given by $F\left(\mathbf{x}\right)=0$. The latter passes through the point of ${\mathcal{E}}^n$ with coordinates which are all equal to 0. It follows that (44) identifies a sheaf of parallel hyperplanes. Given two vectors of $E^n$ expressed by

$${\mathbf{x}}^{\prime }={\lambda }^{\prime }\mathbf{u}+{\mathbf{x}}_0^{\prime },$$

(48)

and

$${\mathbf{x}}^{″}={\lambda }^{″}\mathbf{u}+{\mathbf{x}}_0^{″},$$

(49)

the parallel components to the vector $\mathbf{u}\in E^n$ of the vectors ${\mathbf{x}}^{\prime }$ and ${\mathbf{x}}^{″}$ are respectively ${\lambda }^{\prime }\mathbf{u}$ and ${\lambda }^{″}\mathbf{u}$, whereas ${\mathbf{x}}_0^{\prime }$ and ${\mathbf{x}}_0^{″}$ are the orthogonal components to $\mathbf{u}\in E^n$ of ${\mathbf{x}}^{\prime }$ and ${\mathbf{x}}^{″}$. (48) and (49) can be viewed as two n-dimensional located vectors at the origin of $E^n$, whose endpoints belong to two hyperplanes. Each of them is expressed by (44). Their value is given by $b_{r^{\prime }}$ and $b_{r^{″}}$, respectively, so one has

$$F\left({\mathbf{x}}^{\prime }\right)=〈\mathbf{u},{\mathbf{x}}^{\prime }〉=b_{r^{\prime }},$$

(50)

as well as

$$F\left({\mathbf{x}}^{″}\right)=〈\mathbf{u},{\mathbf{x}}^{″}〉=b_{r^{″}}.$$

(51)

One has evidently

$$F\left({\mathbf{x}}^{\prime }\right)=〈\mathbf{u},{\mathbf{x}}^{\prime }〉=〈\mathbf{u},{\lambda }^{\prime }\mathbf{u}+{\mathbf{x}}_0^{\prime }〉={\lambda }^{\prime }{\parallel \mathbf{u}\parallel }^2+〈\mathbf{u},{\mathbf{x}}_0^{\prime }〉=b_{r^{\prime }},$$

(52)

as well as

$$F\left({\mathbf{x}}^{″}\right)=〈\mathbf{u},{\mathbf{x}}^{″}〉=〈\mathbf{u},{\lambda }^{″}\mathbf{u}+{\mathbf{x}}_0^{″}〉={\lambda }^{″}{\parallel \mathbf{u}\parallel }^2+〈\mathbf{u},{\mathbf{x}}_0^{″}〉=b_{r^{″}}.$$

(53)

Since one observes $〈\mathbf{u},{\mathbf{x}}_0^{\prime }〉=0$ and $〈\mathbf{u},{\mathbf{x}}_0^{″}〉=0$, it is possible to write

$${\lambda }^{\prime }{\parallel \mathbf{u}\parallel }^2=b_{r^{\prime }},$$

(54)

and

$${\lambda }^{″}{\parallel \mathbf{u}\parallel }^2=b_{r^{″}}.$$

(55)

In general, all points of a hyperplane characterized by the same value denoted by $b_r$ can be summarized using the intersection of it with the straight line denoted by ${\rho }_0\in {\mathcal{E}}^n$. Such an intersection coincides with the real number given by

$$\lambda =\frac{b_r}{{\parallel \mathbf{u}\parallel }^2}.$$

(56)

This means that the orthogonal component of the vectors $\mathbf{x}\in E^n$ and $\mathbf{a}\in \mathcal{A}$ is insignificant with respect to the straight line denoted by ${\rho }_0\in {\mathcal{E}}^n$. Hence, one refers oneself to such a line instead of different hyperplanes. Every point belonging to ${\rho }_0\in {\mathcal{E}}^n$ can be expressed as a convex combination of two different points belonging to it. Given three possible values for X denoted by $b_{r^{\prime }}$, $b_{r^{″}}$, and $b_{r^{\prime \prime \prime }}$, one writes

$${\mathbf{x}}_1^{\prime }={\lambda }^{\prime }\mathbf{u},$$

(57)

$${\mathbf{x}}_1^{″}={\lambda }^{″}\mathbf{u},$$

(58)

and

$${\mathbf{x}}_1^{\prime \prime \prime }={\lambda }^{\prime \prime \prime }\mathbf{u}.$$

(59)

One observes $F\left({\mathbf{x}}_1^{\prime }\right)=b_{r^{\prime }}$, $F\left({\mathbf{x}}_1^{″}\right)=b_{r^{″}}$, and $F\left({\mathbf{x}}_1^{\prime \prime \prime }\right)=b_{r^{\prime \prime \prime }}$. The following expression is therefore obtained

$${\lambda }^{″}\mathbf{u}=t{\lambda }^{\prime }\mathbf{u}+(1-t){\lambda }^{\prime \prime \prime }\mathbf{u},$$

(60)

with $0\le t\le 1$. (56) is taken into account, so one writes

$$b_{r^{″}}\mathbf{u}=t{b}_{r^{\prime }}\mathbf{u}+(1-t)b_{r^{\prime \prime \prime }}\mathbf{u}$$

(61)

after multiplying by ${\parallel \mathbf{u}\parallel }^2$ both sides of (60). After dividing by $\mathbf{u}$ both sides of (61), one writes

$$b_{r^{″}}=t{b}_{r^{\prime }}+(1-t)b_{r^{\prime \prime \prime }},$$

(62)

so one has

$$t=\frac{b_{r^{\prime \prime }}-b_{r^{\prime \prime \prime }}}{b_{r^{\prime }}-b_{r^{\prime \prime \prime }}}.$$

(63)

The stated property related to the one-dimensional points identifying the possible values for X is shown by (62). □

In general, given two one-dimensional points denoted by A and B belonging to a one-dimensional straight line and identifying two different values for X, every point P belonging to the same one-dimensional straight line and identifying another possible value for X is expressed by

$$P=tA+(1-t)B,$$

(64)

where one observes $0\le t\le 1$. A convex combination appears, so one has

$$t+(1-t)=1.$$

(65)

It is possible to note the following:

Remark 1.

There exists a one-to-one correspondence between the elements of a sheaf of parallel hyperplanes and the points of intersection of them with a straight line denoted by ${\rho }_0\in {\mathcal{E}}^n$. Since a given decision-maker does not know which possible value for X belonging to $I\left(X\right)$ will be true at the right time, he focuses on one out of n axes of an n-dimensional Cartesian coordinate system. They are pairwise orthogonal. He considers all collinear vectors with respect to one out of n basis vectors. Such collinear vectors give rise to ${\rho }_0\in {\mathcal{E}}^n$, where ${\rho }_0$ is orthogonal to all parallel hyperplanes into account. The points of intersection of all these parallel hyperplanes with a straight line denoted by ${\rho }_0\in {\mathcal{E}}^n$ are real numbers transferred on a one-dimensional straight line, on which an origin, a unit of length, and an orientation are established. They coincide with all possible values for X belonging to $I\left(X\right)$. After focusing on another axis of an n-dimensional Cartesian coordinate system, if the same decision-maker considers all collinear vectors with respect to another basis vector, then the same possible values for X belonging to $I\left(X\right)$ are obtained.

Remark 2.

$E^{n*}$ contains all those random goods obtained by considering all homogeneous linear combinations of n events generically denoted by $E_1,E_2,\dots ,E_n$. If one writes

$$X=b_1|E_1|+b_2|E_2|+\dots +b_n\left|E_n\right|,$$

then X is that random good such that its possible values coincide with $I\left(X\right)=\{b_1,b_2,\dots ,b_n\}$. They are found on distinct hyperplanes expressed by

$$b_i{a}^i=constant,$$

where $b_1,b_2,\dots ,b_n$ are coordinates of points of $E^{n*}$, whereas $a^1,a^2,\dots ,a^n$ are components of vectors of $E^n$. The values characterizing each $a^i$, $i=1,\dots ,n$, coincide with 0 or 1 only.

Remark 3.

Since $E^n$ is a linear space over $\mathbb{R}$ of a Euclidean nature, $E^n$ and $E^{n*}$ coincide. Points of $E^{n*}$ and vectors of $E^n$ can be identified because a specific n-tuple denoted by $(0,0,\dots ,0)$ has meaning in both of them. It is their origin. One observes $u_i=b_i$, $i=1,\dots ,n$, so one writes $b_i=x^i$, $i=1,\dots ,n$. This means that all possible values for X are given by $I\left(X\right)=\{b_1,\dots ,b_n\}=\{x^1,\dots ,x^n\}$.

Another basic result for what will be shown in the next sections is the following. If an infinite number of possible outcomes for each random good into account is treated, then one can always refer to a finite number of possible alternatives. This implies that finite random quantities are handled. Check the following:

Example 1.

Let $X=$ “the future percentage of return on a financial asset” be a random quantity, whose possible values belonging to $I\left(X\right)$ are all real numbers between 0 and 1. X is then bounded from above and below, so one writes $\mathit{inf}I\left(X\right)=0$ and $\mathit{sup}I\left(X\right)=1$. The atomic events $E_1=(X=x^1=0)$, $E_2=(X=x^2=0.05)$, $E_3=(X=x^3=0.08)$, $E_4=(X=x^4=0.12)$, and $E_5=(X=x^5=0.15)$ are incompatible, but they are not exhaustive. It is then necessary to consider the event $E_6=(X=x^6=1-(E_1+E_2+\dots +E_5)=0.6)$. The latter is both an atomic event and an event belonging to an infinite set of events. For this reason, one writes $E_6\in E_x=(X=x,\mathit{with}0<x<1,\mathit{and}x\ne 0,0.05,0.08,0.12,0.15,1)$. Distinguishing between atomic events and events belonging to an infinite set of events is meaningless according to the approach followed by this research work. A finite partition of events is formed from $E_1$, $E_2$, $E_3$, $E_4$, $E_5$, and $E_6$, together with the atomic event $E_7=(X=x^7=1)$. An element of a finite-dimensional linear space over $\mathbb{R}$ is therefore $(0,0.05,0.08,0.12,0.15,0.6,1)$. With respect to the future percentage of return on X, the real data are then $(0,0.05,0.08,0.12,0.15,0.6,1)$. $\mathbf{P}\left(X\right)$ is consequently expressed as a function of the probabilities of all possible values for X. These values are estimated to be finite in number. $\mathbf{P}\left(X\right)$ is a linear functional (scalar or inner product).

A random quantity (also known as a random variable in other formulations) X is a function from $\Omega$ into the set $\mathbb{R}$ of real numbers such that the pre-image of an interval of $\mathbb{R}$ corresponding to a singleton or one-point set is an event in $\Omega$.

4. Choice Functions and Their Properties

In the decision-making problems treated in this research work, uncertainty does not cease (see Camerer and Weber (1992); Bossaerts et al. (2010)). Thus, probabilities associated with the range of possibility have to be chosen (see Viscusi and Evans (2006)). A nonparametric distribution of mass is written in the form of a finite sequence expressed by

$$(x^1,p_1),(x^2,p_2),\dots ,(x^n,p_n).$$

(66)

The scalar or inner product given by

$$\mathbf{P}\left(X\right)=〈\left(\begin{array}{c}{x}^1\\ x^2\\ ⋮\\ x^n\end{array}\right),\left(\begin{array}{c}{p}_1\\ p_2\\ ⋮\\ p_n\end{array}\right)〉$$

(67)

has an intrinsic bilinear nature. $\mathbf{P}\left(X\right)$ is a function of prevision of a linear nature. While

$$\left(\begin{array}{c}{x}^1\\ x^2\\ ⋮\\ x^n\end{array}\right)$$

(68)

does not change, the following vector expressed by

$$\mathbf{p}=\left(\begin{array}{c}{p}_1\\ p_2\\ ⋮\\ p_n\end{array}\right)$$

(69)

can change. For this reason, one writes

$$\left(\begin{array}{c}0\le p_1\le 1\\ 0\le p_2\le 1\\ ⋮\\ 0\le p_n\le 1\end{array}\right).$$

(70)

This means that one can make ${\infty }^{n-1}$ possible choices of probabilities in the first stage. The only restriction is that all the probabilities into account sum to 1. Whenever the possible values for X denoted by $I\left(X\right)=\{x^1,x^2,\dots ,x^n\}$ are found on a one-dimensional straight line, a closed line segment including both its endpoints is observed (see Maturo and Angelini (2023)). This line segment contains an infinite number of coherent previsions of X. The rules of linearity hold. It is possible to extend and, symmetrically, to restrict the set of possible values for X (see de Finetti (1989)). This is because such a set always depends on a specific state of information and knowledge associated with a given individual. This state is intrinsically variable. Whenever two or more than two random goods are studied, ordered pairs of them are treated. Thus, the rules of linearity no longer hold on their own. The rules of multilinearity hold too. Such rules are actually preeminent. This is a new issue compared to what has been investigated so far in the literature. For instance, given $X_1$ and $X_2$, it is possible to treat not only $(X_1,X_2)$, but also three other ordered pairs. It is then possible to treat the following pairs: $(X_1,X_1)$, $(X_1,X_2)$, $(X_2,X_1)$, and $(X_2,X_2)$. Similarly, given $X_1$, $X_2$, and $X_3$, it is possible to treat the following pairs: $(X_1,X_1)$, $(X_1,X_2)$, $(X_1,X_3)$, $(X_2,X_1)$, $(X_2,X_2)$, $(X_2,X_3)$, $(X_3,X_1)$, $(X_3,X_2)$, and $(X_3,X_3)$. Let $X_1$ and $X_2$ be two random goods. Since it is always possible to write

$$X_1=Y_1-Z_1,$$

(71)

and

$$X_2=Y_2-Z_2,$$

(72)

where one has $Y_1=X_1(X_1\ge 0)$ and $Z_1=-X_1(X_1\le 0)$, as well as $Y_2=X_2(X_2\ge 0)$ and $Z_2=-X_2(X_2\le 0)$, the possible values for $Y_1$ and $Z_1$ are always non-negative. The same is true for $Y_2$ and $Z_2$. This implies that all possible values for $X_1$ and $X_2$ can be studied as if they are steadily non-negative. It is then sufficient to consider two half-lines. They meet each other at $(0,0)$ (see Markowitz (1956)). The space within which fair estimations related to $X_1$ and $X_2$ are made by a given individual is therefore a right triangle belonging to the first quadrant of the Cartesian plane (see Cherchye et al. (2018)). Its hypotenuse is a hyperplane embedded in the Cartesian plane. The vertex of the right angle of this triangle is given by $(0,0)$. One writes

$$\mathbf{P}\left(a{X}_1\right)=a\mathbf{P}\left(X_1\right),$$

(73)

and

$$\mathbf{P}\left(a{X}_2\right)=a\mathbf{P}\left(X_2\right)$$

(74)

for every real number denoted by a. More generally, one writes

$$\mathbf{P}(a{X}_1^{\prime }+b{X}_1^{″}+c{X}_1^{\prime \prime \prime }+\dots )=a\mathbf{P}\left(X_1^{\prime }\right)+b\mathbf{P}\left(X_1^{″}\right)+c\mathbf{P}\left(X_1^{\prime \prime \prime }\right)+\dots$$

(75)

for any finite number of random goods $X_1^{\prime }$, $X_1^{\prime \prime }$, $X_1^{\prime \prime \prime }$, …that are considered on the horizontal axis, and

$$\mathbf{P}(a{X}_2^{\prime }+b{X}_2^{″}+c{X}_2^{\prime \prime \prime }+\dots )=a\mathbf{P}\left(X_2^{\prime }\right)+b\mathbf{P}\left(X_2^{″}\right)+c\mathbf{P}\left(X_2^{\prime \prime \prime }\right)+\dots$$

(76)

for any finite number of summands $X_2^{\prime }$, $X_2^{\prime \prime }$, $X_2^{\prime \prime \prime }$, …that are considered on the vertical one, with a, b, c, …any real numbers. With respect to (73), if a is a real number, then a has to lie between $a^{\prime }$ and $a^{\prime \prime }$. This implies that $a{X}_1$ has to lie between $a^{\prime }{X}_1$ and $a^{″}{X}_1$. This is true because all possible values for $X_1$ are considered to be non-negative. Otherwise, this is false. The same is true with respect to (74). One writes

$$\mathbf{P}(Y_1-Z_1)=\mathbf{P}\left(Y_1\right)-\mathbf{P}\left(Z_1\right),$$

(77)

and

$$\mathbf{P}(Y_2-Z_2)=\mathbf{P}\left(Y_2\right)-\mathbf{P}\left(Z_2\right).$$

(78)

If $\mathbf{P}(Y_1-Z_1)$ is decomposed into $\mathbf{P}\left(Y_1\right)$ and $\mathbf{P}\left(Z_1\right)$, then (77) works when and only when $\mathbf{P}(Y_1-Z_1)$, $\mathbf{P}\left(Y_1\right)$, and $\mathbf{P}\left(Z_1\right)$ have the same masses expressed by the same vector $\mathbf{p}$. Differently, (77) does not work. If $\mathbf{P}(Y_2-Z_2)$ is decomposed into $\mathbf{P}\left(Y_2\right)$ and $\mathbf{P}\left(Z_2\right)$, then (78) works when and only when $\mathbf{P}(Y_2-Z_2)$, $\mathbf{P}\left(Y_2\right)$, and $\mathbf{P}\left(Z_2\right)$ have the same masses expressed by the same vector $\mathbf{p}$. Differently, (78) does not work. One of the properties of the scalar or inner product shows just this. It follows that the number of the possible values for $Y_1$ has to be the same as the one for $Z_1$, and the number of the possible values for $Y_2$ has to be the same as the one for $Z_2$. One observes

$$\inf I(Y_1-Z_1)\le \mathbf{P}(Y_1-Z_1)\le \sup I(Y_1-Z_1),$$

(79)

with $Y_1-Z_1=X_1$, and

$$\inf I(Y_2-Z_2)\le \mathbf{P}(Y_2-Z_2)\le \sup I(Y_2-Z_2),$$

(80)

with $Y_2-Z_2=X_2$. This is because $\mathbf{P}$ is convex. Such properties of $\mathbf{P}$ are essential for coherence (see Chudjakow and Riedel (2013)). Any transgression of them leads to choices which are not of a rational nature (see Choi et al. (2014)). The notion of prevision is unique. In the case of single events, it is also called probability. The same symbol $\mathbf{P}$ is used in both cases.

A Two-Dimensional Probability Distribution: A Projection of a Bilinear Measure onto Two Mutually Orthogonal Axes

From two marginal random goods $X_1$ and $X_2$, a bivariate random good denoted by $X_1{X}_2$ arises. If $I\left(X_1\right)$ contains n elements and $I\left(X_2\right)$ contains n elements too, then $I\left(X_1\right)\times I\left(X_2\right)$ contains $n^2$ elements. Two marginal probability distributions associated with two marginal random goods give rise to a bivariate probability distribution associated with a bivariate random good. A bivariate probability distribution is of a two-dimensional nature. A function of prevision of a bilinear nature is denoted by $\mathbf{P}\left(X_1{X}_2\right)$. The set of all coherent previsions of $X_1{X}_2$ is expressed by $\mathcal{P}$. It is an uncountable subset of $\mathbb{R}\times \mathbb{R}$. All the pairs of real numbers denoted by $(\mathbf{P}\left(X_1\right),\mathbf{P}\left(X_2\right))$ are the Cartesian coordinates of all the points of this subset. Projecting $(\mathbf{P}\left(X_1\right),\mathbf{P}\left(X_2\right))$ onto the two mutually orthogonal axes of the Cartesian plane means that $\mathbf{P}\left(X_1{X}_2\right)$ is decomposed into two linear indices, $\mathbf{P}\left(X_1\right)$ and $\mathbf{P}\left(X_2\right)$ (see Pompilj (1957); Ahn et al. (2014)). A linear inequality given by

$$c_1{X}_1+c_2{X}_2\le c,$$

(81)

where $c_1$, $c_2$, and c are positive real numbers, is analytically considered at first. This inequality must also be satisfied by the corresponding marginal previsions $\mathbf{P}\left(X_1\right)$ and $\mathbf{P}\left(X_2\right)$, so one has

$$c_1\mathbf{P}\left(X_1\right)+c_2\mathbf{P}\left(X_2\right)\le c.$$

(82)

The space within which fair estimations of $X_1$, $X_2$, and $X_1{X}_2$ are made by a given individual is his budget set. It obeys the rules of the logic of prevision (see Markowitz (1952)). The expression given by

$$c_1\mathbf{P}\left(X_1\right)+c_2\mathbf{P}\left(X_2\right)=c$$

(83)

identifies a straight line expressed in an implicit form, whose slope is given by −$\frac{c_1}{c_2}$. Its horizontal intercept is given by $\frac{c}{c_1}$. Its vertical intercept is expressed by $\frac{c}{c_2}$. By definition, the line given by (83) does not separate any point $\mathbf{P}$ of $\mathcal{P}$ from $I\left(X_1\right)\cup \left\{\frac{c}{c_1}\right\}$, $I\left(X_2\right)\cup \left\{\frac{c}{c_2}\right\}$, nor from $[I\left(X_1\right)\cup \left\{\frac{c}{c_1}\right\}]\times [I\left(X_2\right)\cup \left\{\frac{c}{c_2}\right\}]$. Not all points of $[I\left(X_1\right)\cup \left\{\frac{c}{c_1}\right\}]\times [I\left(X_2\right)\cup \left\{\frac{c}{c_2}\right\}]$ are possible. The point denoted by

$$\left(\sup I\left(X_1\right),\sup I\left(X_2\right)\right)$$

(84)

always belongs to (83). It is possible to establish the following:

Definition 1.

After decomposing $\mathbf{P}\left(X_1{X}_2\right)$ inside $\mathbb{R}\times \mathbb{R}$, the decision-maker’s choice functions are expressed by

$$\mathbf{P}\left(X_1\right)=\left\{\mathbf{P}\left(X_1\right)\left[(c_1,c_2,c)\right]\right\},$$

(85)

and

$$\mathbf{P}\left(X_2\right)=\left\{\mathbf{P}\left(X_2\right)\left[(c_1,c_2,c)\right]\right\}.$$

(86)

They are two objective functions of a linear nature deriving from an objective function of a bilinear nature.

It is possible to note the following:

Remark 4.

A given individual estimates both marginal masses associated with $X_1$ and $X_2$ and the joint ones associated with $X_1{X}_2$. Given $(\mathbf{P}\left(X_1\right),\mathbf{P}\left(X_2\right))$, $\mathbf{P}\left(X_1{X}_2\right)$ is a summarized element of the Fréchet class being chosen by a given individual. This summarized element corresponds to each point of the budget set. With $n\ge 10$, a given individual can choose $\mathbf{P}\left(X_1{X}_2\right)$ in such a way that there is no linear correlation between $X_1$ and $X_2$. He could also choose $\mathbf{P}\left(X_1{X}_2\right)$ in such a way that there is an inverse or direct linear relationship between $X_1$ and $X_2$. This implies that the joint masses of the two-dimensional probability distribution are chosen in such a way that the correlation coefficient is, respectively, equal to 0, or $-1$, or 1.

If $X_1$ and $X_2$ are two financial assets, then all expected returns on $X_1$ and $X_2$ can be expressed by

$$\frac{c_1}{c_1+c_2}\mathbf{P}\left(X_1\right)+\frac{c_2}{c_1+c_2}\mathbf{P}\left(X_2\right)\le \frac{c}{c_1+c_2}.$$

(87)

A given individual divides his relative monetary wealth given by

$$\frac{c_1}{c_1+c_2},$$

(88)

and

$$\frac{c_2}{c_1+c_2}$$

(89)

between the two financial assets into account, where one has

$$\frac{c_1}{c_1+c_2}+\frac{c_2}{c_1+c_2}=1.$$

(90)

The budget set of a given decision-maker is firstly established by the budget constraint given by (82). It does not change when all objective prices and income are secondly multiplied by a positive number. The best rational choice being made by a given individual inside his budget set does not change either. The decision-maker’s choice functions are then expressed by

$$\mathbf{P}\left(X_1\right)=\left\{\mathbf{P}\left(X_1\right)\left[\left(\frac{c_1}{c_1+c_2},\frac{c_2}{c_1+c_2},\frac{c}{c_1+c_2}\right)\right]\right\},$$

(91)

and

$$\mathbf{P}\left(X_2\right)=\left\{\mathbf{P}\left(X_2\right)\left[\left(\frac{c_1}{c_1+c_2},\frac{c_2}{c_1+c_2},\frac{c}{c_1+c_2}\right)\right]\right\}.$$

(92)

They are two objective functions of a linear nature deriving from an objective function of a bilinear nature.

5. Reductions of Dimension Characterizing the Budget Set of a Given Decision-Maker

5.1. Contravariant and Covariant Components of Vectors and Tensors

If $X_1$ and $X_2$ are two marginal random goods, where the number of the possible values for each of them is equal to $n+1$, then each marginal probability distribution is individuated by two vectors of $E^{n+1}$. One writes $dim{E}^{n+1}=n+1$. Two marginal probability distributions identify a bivariate probability distribution. The latter is individuated by an affine tensor of order 2 belonging to $E^{n+1}\otimes E^{n+1}$ and representing the joint masses of the two-dimensional probability distribution. One writes $dim(E^{n+1}\otimes E^{n+1})={(n+1)}^2$. One observes

$$dim{E}^{n+1}\ne dim(E^{n+1}\otimes E^{n+1})$$

(93)

whenever $n+1\ge 2$ is an integer. Given an orthonormal basis of $E^{n+1}$ expressed by $\left\{{\mathbf{e}}_i\right\}$, $i=1,\dots ,n+1$, the contravariant components of ${\text{}}_{\left(1\right)}^{\text{}}{\mathbf{x}}_{\text{}}^{\text{}}$ identifying the set $\{{\text{}}_{\left(1\right)}^{\text{}}{x}_{\text{}}^1,\dots ,{\text{}}_{\left(1\right)}^{\text{}}{x}_{\text{}}^{n+1}\}$ represent the possible values for $X_1$. They represent the possible returns on $X_1$ whenever $X_1$ is a financial asset. One writes

$${\text{}}_{\left(1\right)}^{\text{}}{\mathbf{x}}_{\text{}}^{\text{}}={\text{}}_{\left(1\right)}^{\text{}}{x}^i{\text{}}_{\text{}}^{\text{}}{\mathbf{e}}_i^{\text{}}.$$

(94)

Even though a contravariant notation is used for the possible returns on $X_1$, the contravariant and covariant components of a same vector of $E^{n+1}$ coincide whenever an orthonormal basis of $E^{n+1}$ is treated. The same is true with respect to the contravariant components of ${\text{}}_{\left(2\right)}^{\text{}}{\mathbf{x}}_{\text{}}^{\text{}}$. One writes

$${\text{}}_{\left(2\right)}^{\text{}}{\mathbf{x}}_{\text{}}^{\text{}}={\text{}}_{\left(2\right)}^{\text{}}{x}_{\text{}}^j{\text{}}_{\text{}}^{\text{}}{\mathbf{e}}_j^{\text{}}.$$

(95)

Given

$${\text{}}_{\left(1\right)}^{\text{}}{\mathbf{p}}_{\text{}}^{\text{}}={\text{}}_{\left(1\right)}^{\text{}}{p}_i^{\text{}}{\text{}}_{\text{}}^{\text{}}{\mathbf{e}}^i,$$

(96)

where the covariant components of ${\text{}}_{\left(1\right)}^{\text{}}{\mathbf{p}}_{\text{}}^{\text{}}$ identify all the masses associated with all possible returns on $X_1$, one writes

$$\mathbf{P}\left(X_1\right)={\text{}}_{\left(1\right)}^{\text{}}{x}^i{\text{}}_{\text{}}^{\text{}}{\text{}}_{\left(1\right)}^{\text{}}{p}_i^{\text{}}{\text{}}_{\text{}}^{\text{}}.$$

(97)

In the first stage, each mass associated with a possible return on $X_1$ can take all values between 0 and 1, endpoints included, into account. Even though a covariant notation is used for the masses, the contravariant and covariant components of a same vector of $E^{n+1}$ coincide whenever an orthonormal basis of $E^{n+1}$ is treated. Similarly, given

$${\text{}}_{\left(2\right)}^{\text{}}{\mathbf{p}}_{\text{}}^{\text{}}={\text{}}_{\left(2\right)}^{\text{}}{p}_j^{\text{}}{\text{}}_{\text{}}^{\text{}}{\mathbf{e}}_{\text{}}^j,$$

(98)

one writes

$$\mathbf{P}\left(X_2\right)={\text{}}_{\left(2\right)}^{\text{}}{x}^i{\text{}}_{\text{}}^{\text{}}{\text{}}_{\left(2\right)}^{\text{}}{p}_i^{\text{}}{\text{}}_{\text{}}^{\text{}}.$$

(99)

$\mathbf{P}\left(X_1\right)$ and $\mathbf{P}\left(X_2\right)$ are two scalar or inner products of two vectors belonging to the same linear space over $\mathbb{R}$. Given the two affine tensors of order 2 expressed by

$$T={\text{}}_{\left(1\right)}^{\text{}}{x}^i{\text{}}_{\left(2\right)}^{\text{}}{x}_{\text{}}^j{\text{}}_{\text{}}^{\text{}}{\mathbf{e}}_i^{\text{}}\otimes {\text{}}_{\text{}}^{\text{}}{\mathbf{e}}_j^{\text{}},$$

(100)

and

$$P=p_{ij}{\text{}}_{\text{}}^{\text{}}{\mathbf{e}}^i\otimes {\text{}}_{\text{}}^{\text{}}{\mathbf{e}}_{\text{}}^j,$$

(101)

where each of them has ${(n+1)}^2$ components, one writes

$$\mathbf{P}\left(X_1{X}_2\right)={\text{}}_{\left(1\right)}^{\text{}}{x}^i{\text{}}_{\left(2\right)}^{\text{}}{x}_{\text{}}^j{\text{}}_{\text{}}^{\text{}}{p}_{ij}.$$

(102)

In the first stage, each mass associated with a possible return on $X_1{X}_2$ belonging to $[I\left(X_1\right)\cup \left\{\frac{c}{c_1}\right\}]\times [I\left(X_2\right)\cup \left\{\frac{c}{c_2}\right\}]$ can take all values between 0 and 1, endpoints included, into account. Given an orthonormal basis of $E^{n+1}$, the contravariant and covariant components of a same affine tensor of order 2 coincide whenever a basis of $E^{n+1}\otimes E^{n+1}$ is treated. If a covariant notation is used, then the components of T are expressed by the same numbers. If a contravariant notation is used, then the components of P are expressed by the same numbers.

5.2. The Metric Notion of $\alpha$-Product

If the notion of $\alpha$-product between ${\text{}}_{\left(1\right)}^{\text{}}{\mathbf{x}}_{\text{}}^{\text{}}$ and ${\text{}}_{\left(2\right)}^{\text{}}{\mathbf{x}}_{\text{}}^{\text{}}$ is used, then it is a scalar or inner product obtained using the joint masses denoted by $p_{ij}$ of the bivariate distribution of $X_1$ and $X_2$ together with the contravariant components of ${\text{}}_{\left(1\right)}^{\text{}}{\mathbf{x}}_{\text{}}^{\text{}}$ and ${\text{}}_{\left(2\right)}^{\text{}}{\mathbf{x}}_{\text{}}^{\text{}}$. Check the following:

Example 2.

From the following

Table 1. Data of a two-dimensional probability distribution.

	Random Good 2	0	9	10	Sum
Random Good 1
0		0	0	0	0
7		0	0.15	0.25	0.4
8		0	0.3	0.3	0.6
Sum		0	0.45	0.55	1

it follows that one has $\mathbf{P}\left(X_1{X}_2\right)=72.55$. Given the contravariant components of ${\text{}}_{\left(2\right)}^{\text{}}{\mathbf{x}}_{\text{}}^{\text{}}$ identifying the following column vector

$$\left(\begin{array}{c}0\\ 9\\ 10\end{array}\right),$$

its covariant components are expressed by

$$0·0+9·0+10·0=0,$$

$$0·0+9·0.15+10·0.25=3.85,$$

and

$$0·0+9·0.3+10·0.3=5.7,$$

so it is possible to write the following result

$$〈\left(\begin{array}{c}0\\ 7\\ 8\end{array}\right),\left(\begin{array}{c}0\\ 3.85\\ 5.7\end{array}\right)〉={\langle {\text{}}_{\left(1\right)}^{\text{}}{\mathbf{x}}_{\text{}}^{\text{}},{\text{}}_{\left(2\right)}^{\text{}}{\mathbf{x}}_{\text{}}^{\text{}}\rangle }_{\alpha }=\mathbf{P}\left(X_1{X}_2\right)=72.55.$$

On the other hand, after calculating the covariant components of ${\text{}}_{\left(1\right)}^{\text{}}{\mathbf{x}}_{\text{}}^{\text{}}$ in a similar way, one writes

$$〈\left(\begin{array}{c}0\\ 3.45\\ 4.15\end{array}\right),\left(\begin{array}{c}0\\ 9\\ 10\end{array}\right)〉={\langle {\text{}}_{\left(1\right)}^{\text{}}{\mathbf{x}}_{\text{}}^{\text{}},{\text{}}_{\left(2\right)}^{\text{}}{\mathbf{x}}_{\text{}}^{\text{}}\rangle }_{\alpha }=\mathbf{P}\left(X_1{X}_2\right)=72.55.$$

It is possible to write

$$\left(\mathbf{P}\left(X_1\right),\mathbf{P}\left(X_2\right)\right)$$

in order to identify $\mathbf{P}\left(X_1{X}_2\right)$. This is because $\mathbf{P}\left(X_1{X}_2\right)$ is always decomposed into two linear previsions. A bivariate probability distribution is of a two-dimensional nature. The notion of α-norm is a particular α-product. From the following

Table 2. Data of a particular $\alpha$-product.

	Random Good 2	0	7	8	Sum
Random Good 1
0		0	0	0	0
7		0	0.4	0	0.4
8		0	0	0.6	0.6
Sum		0	0.4	0.6	1

it follows that one has ${\parallel {\text{}}_{\left(1\right)}^{\text{}}{\mathbf{x}}_{\text{}}^{\text{}}\parallel }_{\alpha }^2=\mathbf{P}\left(X_1{X}_1\right)=58$, whereas from the following

Table 3. Data of a particular $\alpha$-product.

	Random Good 2	0	9	10	Sum
Random Good 1
0		0	0	0	0
9		0	0.45	0	0.45
10		0	0	0.55	0.55
Sum		0	0.45	0.55	1

it follows that it is possible to write ${\parallel {\text{}}_{\left(2\right)}^{\text{}}{\mathbf{x}}_{\text{}}^{\text{}}\parallel }_{\alpha }^2=\mathbf{P}\left(X_2{X}_2\right)=91.45$.

5.3. Nonparametric Distributions of Mass Transferred on Straight Lines and Their Indices

Theorem 1 says that all the $n+1$ possible returns on a financial asset identifying an element of a linear space over $\mathbb{R}$ of dimension equal to $n+1$ are transferred on a one-dimensional straight line, on which an origin, a unit of length, and an orientation are chosen. A reduction of dimension is firstly observed. Given $X_1$ and $X_2$, where each of them has $n+1$ possible values, these values are transferred on two one-dimensional straight lines, on which an origin, a same unit of length, and an orientation are established. Such lines are linearly independent. They are the two axes of the Cartesian plane. The space within which a given decision-maker chooses is his budget set. It contains an infinite number of coherent bilinear previsions of $X_1{X}_2$, and an infinite number of coherent linear previsions of $X_1$ and $X_2$. A given decision-maker chooses $\mathbf{P}\left(X_1{X}_2\right)$ among infinite coherent bilinear previsions. He chooses a prevision bundle identified with $\mathbf{P}\left(X_1{X}_2\right)$. This choice has to satisfy the optimization principle associated with the maximization of the notion of utility of an ordinal nature. Since $\mathbf{P}\left(X_1{X}_2\right)$ belongs to a two-dimensional convex set, it is expressed in the form given by $\left(\mathbf{P}\left(X_1\right),\mathbf{P}\left(X_2\right)\right)$. A given decision-maker also chooses $\mathbf{P}\left(X_1\right)$ and $\mathbf{P}\left(X_2\right)$. A reduction of dimension is secondly observed. This is because it is possible to pass from $\mathbf{P}\left(X_1{X}_2\right)$ to $\mathbf{P}\left(X_1\right)$ and $\mathbf{P}\left(X_2\right)$, respectively.

5.4. The Direct Product of $\mathbb{R}$ and $\mathbb{R}$

The direct product of $\mathbb{R}$ and $\mathbb{R}$ is denoted by $\mathbb{R}\times \mathbb{R}$, so $\mathbb{R}\times \mathbb{R}$ is a two-dimensional linear space over $\mathbb{R}$. The budget set of a given decision-maker is an uncountable subset of $\mathbb{R}\times \mathbb{R}$. Two half-lines are firstly considered instead of two one-dimensional straight lines. Each of them extends indefinitely in a positive direction from zero before being restricted. Two line segments belonging to these two half-lines are obtained whenever all coherent previsions of two univariate financial assets are taken into account. Formally, the set of all the pairs is denoted by $({\mathbf{P}}_0^{+}\left(X_1\right),{\mathbf{P}}_0^{+}\left(X_2\right))$. Its first component is given by ${\mathbf{P}}_0^{+}\left(X_1\right)\in {\mathbb{R}}_0^{+}$. Its second component is given by ${\mathbf{P}}_0^{+}\left(X_2\right)\in {\mathbb{R}}_0^{+}$. One has ${\mathbf{P}}_0^{+}\left(X_1\right)\ge 0$, as well as ${\mathbf{P}}_0^{+}\left(X_2\right)\ge 0$. The addition of such pairs works componentwise. If $({\mathbf{P}}_0^{+}\left(X{\text{}}_1^{\prime }\right),{\mathbf{P}}_0^{+}\left(X{\text{}}_2^{\prime }\right))\in {\mathbb{R}}_0^{+}\times {\mathbb{R}}_0^{+}$ and $({\mathbf{P}}_0^{+}\left(X{\text{}}_1^{″}\right),{\mathbf{P}}_0^{+}\left(X{\text{}}_2^{″}\right))\in {\mathbb{R}}_0^{+}\times {\mathbb{R}}_0^{+}$, then it is possible to write

$$[({\mathbf{P}}_0^{+}\left(X{\text{}}_1^{\prime }\right),{\mathbf{P}}_0^{+}\left(X{\text{}}_2^{\prime }\right))+({\mathbf{P}}_0^{+}\left(X{\text{}}_1^{\prime \prime }\right),{\mathbf{P}}_0^{+}\left(X{\text{}}_2^{\prime \prime }\right))]=\left[({\mathbf{P}}_0^{+}\left(X{\text{}}_1^{\prime }\right)+{\mathbf{P}}_0^{+}\left(X{\text{}}_1^{\prime \prime }\right),{\mathbf{P}}_0^{+}\left(X{\text{}}_2^{\prime }\right)+{\mathbf{P}}_0^{+}\left(X{\text{}}_2^{\prime \prime }\right))\right].$$

(103)

If $k\in \mathbb{R}$, then the product given by $k({\mathbf{P}}_0^{+}\left(X{\text{}}_1^{\prime }\right),{\mathbf{P}}_0^{+}\left(X{\text{}}_2^{\prime }\right))$ is written in the following form expressed by

$$k({\mathbf{P}}_0^{+}\left(X{\text{}}_1^{\prime }\right),{\mathbf{P}}_0^{+}\left(X{\text{}}_2^{\prime }\right))=(k{\mathbf{P}}_0^{+}\left(X{\text{}}_1^{\prime }\right),k{\mathbf{P}}_0^{+}\left(X{\text{}}_2^{\prime }\right)).$$

(104)

6. Decision-Making Problems under Conditions of Uncertainty and Riskiness

6.1. Financial Assets and Utility of an Ordinal Nature: Prevision Bundles

Some axioms of revealed preference theory applied to financial assets are treated in this section. Revealed preference theory gives empirical meaning to the neoclassical economic hypothesis according to which the best rational choice being made by a given decision-maker inside his budget set has to be the one maximizing his utility of an ordinal nature (see Nishimura et al. (2017)). In this paper, the best rational choice being made by a given decision-maker inside his budget set is related to expected returns on financial assets (see Matzkin and Richter (1991)). This is a new issue compared to what has been investigated so far in the literature. The space of alternatives is firstly denoted by $E^{n+1}$. Two one-dimensional linear subspaces of $E^{n+1}$ are transferred on two one-dimensional straight lines, on which an origin, a same unit of length, and an orientation are chosen. Two half-lines are handled. A subset of $E^2$ is secondly treated. It coincides with a subset of $\mathbb{R}\times \mathbb{R}$. This is because $n+1$ non-negative and finitely additive masses are studied with respect to each univariate financial asset. The set of all $\mathbf{x}\in E^2$, with $x_1=\mathbf{P}\left(X_1\right)\ge 0$ and $x_2=\mathbf{P}\left(X_2\right)\ge 0$, is denoted by $E_{+}^2$. The set of all $\mathbf{x}\in E^2$, with $x_1=\mathbf{P}\left(X_1\right)>0$ and $x_2=\mathbf{P}\left(X_2\right)>0$, is denoted by $E_{++}^2$. All decision-maker’s fair previsions concerning bivariate financial assets identify a finite sequence belonging to $E^2$ and denoted by

$$\left\{{\mathbf{x}}^k\right|k=1,\dots ,K\}.$$

(105)

The space within which a given decision-maker chooses is expressed by $E_{+}^2$. It coincides with the first quadrant of the Cartesian plane. A collection denoted by $\mathcal{U}$ of utility functions written in the form

$$U:E_{+}^2\to \mathbb{R}$$

(106)

can be considered. Each decision-maker’s fair prevision concerning a bivariate financial asset is a vector $\mathbf{x}\in E_{+}^2$ obtained from his budget set denoted by

$$B(\mathbf{c},c)=\{\mathbf{x}\in E_{+}^2|\mathbf{c}·\mathbf{x}\le c\},$$

(107)

where $\mathbf{c}=(c_1,c_2)$ is a price vector, whereas c is the amount of money a given individual has to spend. A generic pair denoted by

$$(\mathbf{x},\mathbf{c})\in E_{+}^2\times E_{++}^2$$

(108)

shows that a fair estimation related to a financial asset is made by a given individual inside his budget set. A finite collection of pairs written in the form

$$\{({\mathbf{x}}^1,{\mathbf{c}}^1),\dots ,({\mathbf{x}}^K,{\mathbf{c}}^K)\}$$

(109)

expresses a dataset (see Crawford and De Rock (2014)). All datasets that are coherent with revealed preference theory represent its empirical content (see Blundell et al. (2003)). Given a collection $\mathcal{U}$ of utility functions, it is possible to say that a dataset expressed by (109) is $\mathcal{U}$-rational if there exists $U\in \mathcal{U}$ such that one writes, for each k,

$${\mathbf{x}}^k\in \mathrm{argmax}\left\{U\left(\mathbf{x}\right)\right|\mathbf{x}\in B({\mathbf{c}}^k,{\mathbf{c}}^k·{\mathbf{x}}^k)\}.$$

(110)

If one wants to represent $U\left(\mathbf{x}\right)$, then it is necessary to consider three axes. It is then necessary to go away from the space within which a given individual chooses.

6.2. General Utilities Whose Arguments Are Fair Estimations

Let $({\mathbf{x}}^k,{\mathbf{c}}^k){\text{}}_{k=1}^K$ be a dataset. It is possible to define two binary relations on $E_{+}^2$. It is possible to establish the following:

Definition 2.

$\mathbf{x}$ is revealed preferred to $\mathbf{y}$, that is $\mathbf{x}{R}^P\mathbf{y}$, if there exists k such that one observes $\mathbf{x}={\mathbf{x}}^k$ as well as ${\mathbf{c}}^k·\mathbf{y}\le {\mathbf{c}}^k·{\mathbf{x}}^k$.

Definition 3.

$\mathbf{x}$ is strictly revealed preferred to $\mathbf{y}$, that is $\mathbf{x}{P}^P\mathbf{y}$, if there exists k such that one observes $\mathbf{x}={\mathbf{x}}^k$ as well as ${\mathbf{c}}^k·\mathbf{y}<{\mathbf{c}}^k·{\mathbf{x}}^k$.

A dataset expressed by $({\mathbf{x}}^k,{\mathbf{c}}^k){\text{}}_{k=1}^K$ satisfies the Weak Axiom of Revealed Preference (WARP) if there is no pair of observations k and $k^{\prime }$ such that it is possible to observe ${\mathbf{x}}^k{R}^P{\mathbf{x}}^{k^{\prime }}$, whereas one has also ${\mathbf{x}}^{k^{\prime }}{P}^P{\mathbf{x}}^k$. If $({\mathbf{x}}^k,{\mathbf{c}}^k){\text{}}_{k=1}^K$ does not satisfy the WARP, then it cannot be ${\mathcal{U}}_{LNS}$-rational, where ${\mathcal{U}}_{LNS}$ denotes the set of locally non-satiated utility functions. A dataset expressed by $({\mathbf{x}}^k,{\mathbf{c}}^k){\text{}}_{k=1}^K$ satisfies the Generalized Axiom of Revealed Preference (GARP) when, for any finite sequence $\left(k_i\right){\text{}}_{i=1}^M$ in $\{1,\dots ,K\}$, if one observes ${\mathbf{x}}^{k_i}{R}^P{\mathbf{x}}^{k_{i+1}}$, with $i=1,\dots ,M-1$, then it is false to observe ${\mathbf{x}}^{k_M}{P}^P{\mathbf{x}}^{k_1}$. Hence, the empirical content of the rational behavior of a locally non-satiated decision-maker maximizing his utility of an ordinal nature associated with a prevision bundle is the same as the one of a decision-maker with a strictly increasing and concave utility function. Since a given decision-maker can be modeled as being a consumer, the set of strictly increasing and concave utility functions is denoted by ${\mathcal{U}}_{MC}$. Also, those datasets that are coherent with revealed preference theory are those satisfying the GARP (see Diewert (1973)).

6.3. Additive Separability of Utility of Prevision Bundles

Let $\mathbf{P}\left(X_1{X}_2\right)=(\mathbf{P}\left(X_1\right),\mathbf{P}\left(X_2\right))$ be a generic prevision bundle. Let ${\mathcal{U}}_{AS}$ be the set of utility functions denoted by $U:E_{+}^2\to \mathbb{R}$ for which there exists a concave and strictly increasing utility function denoted by $u:{\mathbb{R}}_{+}\to \mathbb{R}$. Hence, it is possible to write $U\left(\mathbf{r}\right)\ge U\left(\mathbf{s}\right)$ if and only if one observes

$$u\left(r_1\right)+u\left(r_2\right)\ge u\left(s_1\right)+u\left(s_2\right),$$

(111)

where one has $r_1=\mathbf{P}\left(X_1\right)$, $r_2=\mathbf{P}\left(X_2\right)$, as well as $s_1=\mathbf{P}\left(X_1\right)$, $s_2=\mathbf{P}\left(X_2\right)$, with $r_1\ne s_1$ and $r_2\ne s_2$. A given decision-maker is consequently faced with the problem expressed by

$$\max \left\{u\left(r_1\right)+u\left(r_2\right)|\mathbf{r}\in B(\mathbf{c},c)\right\}.$$

(112)

If u is smooth, then the first-order conditions of the maximization problem, where an interior solution is assumed to exist, require that one has

$$\frac{u^{\prime }\left(r_1\right)}{u^{\prime }\left(r_2\right)}=\frac{c_1}{c_2}.$$

(113)

The first-order conditions, together with the concavity of u, involve that whenever one observes $r_1>r_2$, it has then to be the case that $\frac{c_1}{c_2}\le 1$. This means that $\mathbf{P}\left(X_1{X}_2\right)$ is decomposed into $\mathbf{P}\left(X_1\right)$ and $\mathbf{P}\left(X_2\right)$ inside an uncountable subset of $\mathbb{R}\times \mathbb{R}$ in order that a larger estimation related to $X_1$ expressed by $\mathbf{P}\left(X_1\right)$ is only possible when it is cheaper than $\mathbf{P}\left(X_2\right)$. In other words, demand slopes down. Suppose that one is faced with a budget set where $\mathbf{P}\left(X_1\right)$ is cheaper than $\mathbf{P}\left(X_2\right)$. This means that the budget set below the 45-degree line is larger than the one above. If a dataset includes a fair prevision of $X_1{X}_2$ that is found on the budget line above the 45-degree line, then it violates downward-sloping demand property (see Echenique and Saito (2015)). It is possible to say that this dataset is not ${\mathcal{U}}_{AS}$-rational. If a dataset conversely includes a coherent prevision of $X_1{X}_2$ that is found on the budget line below the 45-degree line, then it does not violate downward-sloping demand property. Such a prevision is then compatible with a ${\mathcal{U}}_{AS}$-rational decision-maker (see Halevy et al. (2018)). To fix ideas, let $({\mathbf{r}}^k,{\mathbf{c}}^k){\text{}}_{k=1}^3$ be a dataset with three observations. If they violate the WARP, then such a dataset cannot be rationalized by any utility function. It cannot even be rationalized by an element belonging to ${\mathcal{U}}_{AS}$. Given a balanced sequence of pairs denoted by

$$(r_1^1,r_2^1),(r_1^2,r_2^2),(r_1^3,r_2^3),$$

(114)

it is possible to say that (114) has the downward-sloping demand property if

$$r_1^1>r_2^1,r_1^2>r_2^2,\mathrm{and}{r}_1^3>r_2^3\mathrm{imply that}\frac{c_1^1}{c_2^1}·\frac{c_1^2}{c_2^2}·\frac{c_1^3}{c_2^3}\le 1.$$

(115)

If any balanced sequence of pairs has the downward-sloping demand property, then the Strong Axiom of Revealed Additively Separable Utility is satisfied. Such an axiom is a test for whether a dataset is ${\mathcal{U}}_{AS}$-rational (see Chambers and Echenique (2009)). The WARP, the GARP, and the Strong Axiom of Revealed Additively Separable Utility basically say that an increase in price will unambiguously reduce the demand for a given good. The demand for a good under conditions of uncertainty and riskiness coincides with its coherent prevision.

6.4. Datasets Associated with Choices Being Made by a Given Individual Who Is Not Averse to Risk

The notion of risk is intrinsically of a subjective nature (see Angelini and Maturo (2022b)). For this reason, the Fréchet class can be treated. Let $({\mathbf{x}}^k,{\mathbf{c}}^k){\text{}}_{k=1}^K$ be a dataset. Given a collection ${\mathcal{U}}^{\prime }$ of strictly increasing and convex utility functions, a dataset expressed by (109) is ${\mathcal{U}}^{\prime }$-rational if there exists $U^{\prime }\in {\mathcal{U}}^{\prime }$ such that one writes, for each k,

$${\mathbf{x}}^k\in \mathrm{argmax}\left\{U^{\prime }\left(\mathbf{x}\right)\right|\mathbf{x}\in B({\mathbf{c}}^k,{\mathbf{c}}^k·{\mathbf{x}}^k)\}.$$

(116)

If a given decision-maker is conversely risk-neutral, then his linear utility function coincides with the 45-degree line. A dataset expressed by (109) is ${\mathcal{U}}^{″}$-rational if there exists one and one only $U^{″}\in {\mathcal{U}}^{″}$ such that one writes, for each k,

$${\mathbf{x}}^k\in \mathrm{argmax}\left\{U^{″}\left(\mathbf{x}\right)\right|\mathbf{x}\in B({\mathbf{c}}^k,{\mathbf{c}}^k·{\mathbf{x}}^k)\}.$$

(117)

It is not necessary to consider three axes to depict $U^{″}\left(\mathbf{x}\right)$.

7. Financial Decisions Based on Zero-Sum Games

7.1. A Decomposition of Fair Bets

Given $X_1$, $X_2$, and $X_1{X}_2$, in the first stage three convex sets containing fair estimations of $X_1$, $X_2$, and $X_1{X}_2$ are established by a given decision-maker. In the second stage, the same decision-maker chooses $\mathbf{P}\left(X_1{X}_2\right)\in \mathcal{P}$ in such a way his utility is the highest. With respect to the choice function $\mathbf{P}\left(X_1{X}_2\right)$, a zero-sum game steadily takes place. Thus, a given decision-maker has systematically and endlessly to accept any bet whatsoever after choosing $\mathbf{P}\left(X_1{X}_2\right)$ as if he is a bookmaker (see Levitt (2004); Turnbull (1987)). The gain obtained by a competitor is given by

$$c\left(X_1{X}_2-\mathbf{P}\left(X_1{X}_2\right)\right),$$

(118)

with c that is a positive or negative number (betting amount). This number is chosen by the same competitor. Since $\mathbf{P}\left(X_1{X}_2\right)$ is decomposed into $\mathbf{P}\left(X_1\right)$ and $\mathbf{P}\left(X_2\right)$, the same decision-maker automatically accepts two bets. He accepts a bet, whose gain obtained by the same competitor is expressed by

$$\frac{c}{2}\left(X_1-\mathbf{P}\left(X_1\right)\right),$$

(119)

and another one, whose gain obtained by him is denoted by

$$\frac{c}{2}\left(X_2-\mathbf{P}\left(X_2\right)\right),$$

(120)

where one has $\frac{c}{2}+\frac{c}{2}=c$. $\mathbf{P}\left(X_1{X}_2\right)$ represents an opinion expressed by a given individual based on a further hypothesis of an empirical nature. A utility function of an ordinal nature is a way of assigning a number to every prevision bundle $\mathbf{P}\left(X_1{X}_2\right)$ of $\mathcal{P}$ such that more-preferred bundles get assigned larger numbers than less-preferred ones, so a further hypothesis of an empirical nature is that $\mathbf{P}\left(X_1{X}_2\right)$ coincides with a specific prevision bundle of $\mathcal{P}$ with a larger utility than any other one. In this paper, a larger utility coincides with a higher distance of $\mathbf{P}\left(X_1{X}_2\right)$ from $(0,0)$ measured along the 45-degree line (see Angelini (2023)). The latter intersects different indifference curves in a point of them only. Each indifference curve has a negative slope. $\mathbf{P}\left(X_1{X}_2\right)$ is always based on probabilities that are subjectively chosen. Their value is, intrinsically, of a psychological nature. A given individual does not want to set up a bet which will with certainty result in a loss for him, so $\mathbf{P}\left(X_1{X}_2\right)$ has to be such that the differences or deviations between $X_1{X}_2$ and $\mathbf{P}\left(X_1{X}_2\right)$ are not real numbers with the same sign (see Crawford (1974)). Hence, real numbers with the same plus sign never appear in order to avoid a sure loss. Similarly, real numbers with the same minus sign never appear in order to avoid a sure loss. Conversely, it is possible that real numbers with the same plus sign appear together with zero, and that real numbers with the same minus sign appear together with zero. Consequently, in the latter cases there is no sure loss. After making the choice of $\mathbf{P}\left(X_1{X}_2\right)$, the differences or deviations between $X_1{X}_2$ and $\mathbf{P}\left(X_1{X}_2\right)$ are the possible values for a specific random quantity. They are the possible values for a random gain meant in an algebraic sense. A bet being set up by a given individual is a zero-sum game taking place under incomplete information, so it is said to be a fair bet when and only when the scalar or inner product of two vectors is equal to zero. This product is visualized on a one-dimensional straight line. The former vector is given by the differences between $X_1{X}_2$ and $\mathbf{P}\left(X_1{X}_2\right)$, whereas the latter one is given by the same probabilities used to obtain $\mathbf{P}\left(X_1{X}_2\right)$. Check the following:

Example 3.

With respect to Example 2, a given individual chooses $\mathbf{P}\left(X_1{X}_2\right)=72.55$. Hence, the following scalar or inner product given by

$$〈\left(\begin{array}{c}-9.55\\ -2.55\\ -0.55\\ 7.45\end{array}\right),\left(\begin{array}{c}0.15\\ 0.25\\ 0.30\\ 0.30\end{array}\right)〉=0$$

holds in the case that a given competitor chooses $c=1$. Conversely, the following scalar or inner product given by

$$〈\left(\begin{array}{c}9.55\\ 2.55\\ 0.55\\ -7.45\end{array}\right),\left(\begin{array}{c}0.15\\ 0.25\\ 0.30\\ 0.30\end{array}\right)〉=0$$

holds in the case that the same competitor chooses $c=-1$. A reduction of dimension caught by Theorem 1 appears. The situation related to a bivariate (two-dimensional) probability distribution can also be visualized by considering a one-dimensional straight line as the space of alternatives and on it the finite set of the only values which are possible. Such values are given by

$$\left(\begin{array}{c}\left(\right[7·9]-72.55)=-9.55\\ \left(\right[7·10]-72.55)=-2.55\\ \left(\right[8·9]-72.55)=-0.55\\ \left(\right[8·10]-72.55)=7.45\end{array}\right),$$

or by

$$\left(\begin{array}{c}-\left(\right[7·9]-72.55)=9.55\\ -\left(\right[7·10]-72.55)=2.55\\ -\left(\right[8·9]-72.55)=0.55\\ -\left(\right[8·10]-72.55)=-7.45\end{array}\right).$$

Since $\mathbf{P}\left(X_1{X}_2\right)$ is decomposed into $\mathbf{P}\left(X_1\right)$ and $\mathbf{P}\left(X_2\right)$, the same decision-maker automatically accepts two bets. He accepts a bet for which the following scalar or inner product given by

$$\frac{1}{2}〈\left(\begin{array}{c}-0.55\\ 0.45\end{array}\right),\left(\begin{array}{c}0.45\\ 0.55\end{array}\right)〉=0$$

holds, and a bet for which the following scalar or inner product given by

$$\frac{1}{2}〈\left(\begin{array}{c}-0.6\\ 0.4\end{array}\right),\left(\begin{array}{c}0.4\\ 0.6\end{array}\right)〉=0$$

holds in the case that $c=\frac{1}{2}+\frac{1}{2}=1$ is chosen. Conversely, if $c=-\frac{1}{2}-\frac{1}{2}=-1$ is chosen, then one writes

$$-\frac{1}{2}〈\left(\begin{array}{c}0.55\\ -0.45\end{array}\right),\left(\begin{array}{c}0.45\\ 0.55\end{array}\right)〉=0,$$

and

$$-\frac{1}{2}〈\left(\begin{array}{c}0.6\\ -0.4\end{array}\right),\left(\begin{array}{c}0.4\\ 0.6\end{array}\right)〉=0.$$

7.2. Strategic Interactions among Rational Agents

There exists a strategic interaction between a bookmaker and his competitor. Check the following:

Example 4.

With respect to Example 2, if the choice function is $\mathbf{P}\left(X_1{X}_2\right)=72.55$, then two bets are indifferent for a given bookmaker. Two bets are indifferent for a given bookmaker if and only if the scalar or inner product given by

$$〈\left(\begin{array}{c}-9.55\\ -2.55\\ -0.55\\ 7.45\end{array}\right),\left(\begin{array}{c}0.15\\ 0.25\\ 0.30\\ 0.30\end{array}\right)〉=0$$

holds in the case that a competitor chooses $c=1$, and

$$〈\left(\begin{array}{c}9.55\\ 2.55\\ 0.55\\ -7.45\end{array}\right),\left(\begin{array}{c}0.15\\ 0.25\\ 0.30\\ 0.30\end{array}\right)〉=0$$

holds in the case that the same competitor chooses $c=-1$. Conversely, if $\mathbf{P}\left(X_1{X}_2\right)=60$ is chosen by the same bookmaker together with $(0.15,0.25,0.30,0.30)$, then the same competitor can choose $c=1$ in such a way that one has

$$〈\left(\begin{array}{c}3\\ 10\\ 12\\ 20\end{array}\right),\left(\begin{array}{c}0.15\\ 0.25\\ 0.30\\ 0.30\end{array}\right)〉\ne 0.$$

Hence, if $\mathbf{P}\left(X_1{X}_2\right)=60$ is chosen by a given bookmaker together with $(0.15,0.25,0.30,0.30)$, then a sure loss appears for him. It follows that a sure gain appears for a given competitor. There exists an infinite number of rational choices. Theorem 1 says that they can be visualized on a one-dimensional straight line after making a reduction of dimension. There also exists an infinite number of choices leading to a sure loss.

A strategic interaction between a bookmaker and his competitor can be interpreted in more intuitive terms. Thus, two individuals are in conditions of strategic interaction whenever one of them has to divide a given cake into two parts and the other person will choose the larger part of the cake. To prevent the other person from gaining an advantage, the individual who divides the cake into two parts takes care to cut it in such a way that the two parts are judged to be equal by him.

7.3. A Nonlinear Analysis: New Conceptual and Mathematical Outcomes

It is possible to consider the following nonlinear index

$$\mathbf{P}\left(X_{12}\right)=\left|\begin{array}{cc}\mathbf{P}\left(X_1{X}_1\right)& \mathbf{P}\left(X_1{X}_2\right)\\ \mathbf{P}\left(X_2{X}_1\right)& \mathbf{P}\left(X_2{X}_2\right)\end{array}\right|.$$

(121)

Its intrinsic nature is multilinear. One writes

$$X_{12}=\{X_1,X_2\}$$

(122)

to denote a multilinear relationship between the two components, $X_1$ and $X_2$, of $X_{12}$, where $X_{12}$ is a multiple random good of order 2. Within this context, a multiple random good of order 2 is a portfolio or set of two financial assets. $\mathbf{P}\left(X_{12}\right)$ is a real number. $\mathbf{P}\left(X_1{X}_1\right)$, $\mathbf{P}\left(X_1{X}_2\right)$, $\mathbf{P}\left(X_2{X}_1\right)$, and $\mathbf{P}\left(X_2{X}_2\right)$ are ordered pairs of real numbers. $\mathbf{P}\left(X_1{X}_1\right)$, $\mathbf{P}\left(X_1{X}_2\right)$, $\mathbf{P}\left(X_2{X}_1\right)$, and $\mathbf{P}\left(X_2{X}_2\right)$ identify four evaluations giving rise to four bivariate bets. $\mathbf{P}\left(X_{12}\right)$ identifies an aggregate evaluation of an aggregate bet. If one exchanges $X_1$ for $\mathbf{P}\left(X_1\right)$ and $X_2$ for $\mathbf{P}\left(X_2\right)$, then one has to exchange $X_{12}$ for $\mathbf{P}\left(X_{12}\right)$. It is not absolutely appropriate to exchange $X_1{X}_2$ for $\mathbf{P}\left(X_1{X}_2\right)$. This is because $\mathbf{P}\left(X_1{X}_2\right)$ summarizes a two-dimensional probability distribution, so $\mathbf{P}\left(X_1{X}_2\right)=(\mathbf{P}\left(X_1\right),\mathbf{P}\left(X_2\right))$ is properly an ordered pair of real numbers. This is a new issue compared to what has been investigated so far in the literature. Check the following:

Example 5.

With respect to Example 2, one writes

$$\mathbf{P}\left(X_{12}\right)=\left|\begin{array}{cc}\mathbf{P}\left(X_1{X}_1\right)=58& \mathbf{P}\left(X_1{X}_2\right)=72.55\\ \mathbf{P}\left(X_2{X}_1\right)=72.55& \mathbf{P}\left(X_2{X}_2\right)=91.45\end{array}\right|=40.5975.$$

A given individual also chooses $\mathbf{P}\left(X_1{X}_1\right)=58$, so the following scalar or inner product given by

$$〈\left(\begin{array}{c}-9\\ 6\end{array}\right),\left(\begin{array}{c}0.4\\ 0.6\end{array}\right)〉=0$$

holds in the case that a given competitor chooses $c=1$. Conversely, the following scalar or inner product given by

$$〈\left(\begin{array}{c}9\\ -6\end{array}\right),\left(\begin{array}{c}0.4\\ 0.6\end{array}\right)〉=0$$

holds in the case that the same competitor chooses $c=-1$. The same individual also chooses $\mathbf{P}\left(X_2{X}_2\right)=91.45$, so the following scalar or inner product given by

$$〈\left(\begin{array}{c}-10.45\\ 8.55\end{array}\right),\left(\begin{array}{c}0.45\\ 0.55\end{array}\right)〉=0$$

holds in the case that a given competitor chooses $c=1$. Conversely, the following scalar or inner product given by

$$〈\left(\begin{array}{c}10.45\\ -8.55\end{array}\right),\left(\begin{array}{c}0.45\\ 0.55\end{array}\right)〉=0$$

holds in the case that the same competitor chooses $c=-1$. Example 3 shows that $\mathbf{P}\left(X_1{X}_2\right)=\mathbf{P}\left(X_2{X}_1\right)=72.55$ are two evaluations of two fair bets of a bivariate nature, so one writes

$$\left|\begin{array}{cc}0& 0\\ 0& 0\end{array}\right|=0.$$

It follows that $\mathbf{P}\left(X_{12}\right)$ is an aggregate measure of a multiple fair bet of order 2.

If one writes

$$X_{12\dots m}=\{X_1,X_2,\dots ,X_m\},$$

(123)

then a multilinear measure given by

$$\mathbf{P}\left(X_{12\dots m}\right)=\left|\begin{array}{cccc}\mathbf{P}\left(X_1{X}_1\right)& \mathbf{P}\left(X_1{X}_2\right)& \dots & \mathbf{P}\left(X_1{X}_m\right)\\ \mathbf{P}\left(X_2{X}_1\right)& \mathbf{P}\left(X_2{X}_2\right)& \dots & \mathbf{P}\left(X_2{X}_m\right)\\ ⋮& ⋮& \ddots & ⋮\\ \mathbf{P}\left(X_m{X}_1\right)& \mathbf{P}\left(X_m{X}_2\right)& \dots & \mathbf{P}\left(X_m{X}_m\right)\end{array}\right|$$

(124)

is associated with a multiple fair bet of order m. It is the determinant of a square matrix of order m, so it is a real number. Given $X_1,X_2,\dots ,X_m$, the possible values for each of them are firstly the components of vectors such that each vector is a linear combination of basis vectors. An orthonormal basis of a linear space over $\mathbb{R}$ is arbitrarily chosen. $\mathbf{P}\left(X_1\right)$, $\mathbf{P}\left(X_2\right)$, …, $\mathbf{P}\left(X_m\right)$ are scalar or inner products such that they are always independent of any coordinate system, so only the affine properties are really meaningful (see Angelini (2024)). Even $\mathbf{P}\left(X_1{X}_1\right)$, $\mathbf{P}\left(X_1{X}_2\right)$, …, $\mathbf{P}\left(X_1{X}_m\right)$, $\mathbf{P}\left(X_2{X}_1\right)$, $\mathbf{P}\left(X_2{X}_2\right)$, …, $\mathbf{P}\left(X_2{X}_m\right)$, …, $\mathbf{P}\left(X_m{X}_1\right)$, $\mathbf{P}\left(X_m{X}_2\right)$, …, $\mathbf{P}\left(X_m{X}_m\right)$ are scalar or inner products such that they are always independent of any coordinate system. With respect to the expected utility function visualized in a subset of the Cartesian plane, suppose that it is convex. It is convex because the masses of the two-dimensional probability distributions are chosen in such a way that the correlation coefficient is equal to 1. Hence, a multiple random gain of order m denoted by $X_{12\dots m}$ is preferred to a multiple certain gain denoted by $x_{12\dots m}$. One observes $x_{12\dots m}>\mathbf{P}\left(X_{12\dots m}\right)$ on the x-axis, where both $x_{12\dots m}$ and $\mathbf{P}\left(X_{12\dots m}\right)$ are two real numbers. Conversely, $X_{12\dots m}$ is not preferred to a multiple certain gain denoted by $x_{12\dots m}$ if one observes $x_{12\dots m}<\mathbf{P}\left(X_{12\dots m}\right)$ on the x-axis. This means that the expected utility function is concave. The masses of the two-dimensional probability distributions are chosen in such a way that the correlation coefficient is equal to $-1$. Finally, if one observes $x_{12\dots m}=\mathbf{P}\left(X_{12\dots m}\right)$ on the x-axis, then the expected utility function coincides with the 45-degree line. The masses of the two-dimensional probability distributions are chosen in such a way that the correlation coefficient is equal to 0. With respect to the expected utility function visualized in a subset of the Cartesian plane, one goes away from the budget set of a given decision-maker in order to process data that are observed and estimated inside it. The expected utility function is of a cardinal nature. The cardinal utility can be defined based on ideas put forward by D. Bernoulli and developed by various authors later on (J. von Neumann, O. Morgenstern, L. J. Savage, B. de Finetti, and some others). The expected utility function denoted by $y=u\left(x\right)$ is a manifold embedded in a subset of the Cartesian plane. The graph of $y=u\left(x\right)$ contains $(0,0)$. The barycentre of masses that are found on the graph of $y=u\left(x\right)$ need not itself belong to the graph of $y=u\left(x\right)$. If the expected utility function is convex or concave, then the barycentre of masses that are found on the graph of $y=u\left(x\right)$ belongs to a convex region outside the graph of $y=u\left(x\right)$. Conversely, if $y=u\left(x\right)$ is the 45-degree line, then the barycentre of masses that are found on the graph of $y=u\left(x\right)$ belongs to the graph of $y=u\left(x\right)$ itself. A multiple certain gain denoted by $x_{12\dots m}$ on the x-axis corresponds to $u\left(x_{12\dots m}\right)$ on the y-axis. The maximization of the expected utility is at $u\left(x_{12\dots m}\right)$. The latter point belongs to the union of convex sets on the y-axis and it derives from the calculation of the determinant of a square matrix of order m. The result of the multiplication of two numbers is considered on the x-axis together with the corresponding mass. The form and extent of the convexity or concavity of $y=u\left(x\right)$ will depend upon the temperament, or the current mood, or some other circumstances associated with a given individual. The degree of preferability of a multiple random gain is inserted into the scale of the multiple certain gains. This scale is of a cardinal nature, so the increments of utility or distances on the y-axis, $u\left(x_{i+1}\right)-u\left(x_i\right)$, are equal when, and only when, a given individual is indifferent between the corresponding increments of gain on the x-axis, $x_{i+1}-x_i$. It is possible to note the following:

Remark 5.

With respect to Example 2, a given decision-maker chooses $\mathbf{P}\left(X_1{X}_2\right)=72.55$. Thus, he suffers the following penalty given by

$$〈\left(\begin{array}{c}{(-9.55)}^2\\ {(-2.55)}^2\\ {(-0.55)}^2\\ {\left(7.45\right)}^2\end{array}\right),\left(\begin{array}{c}0.15\\ 0.25\\ 0.30\\ 0.30\end{array}\right)〉=32.0475.$$

Conversely, if $\mathbf{P}\left(X_1{X}_2\right)=74$ is incorrectly chosen by the same decision-maker, then he suffers a greater penalty given by

$$〈\left(\begin{array}{c}{(-11)}^2\\ {(-4)}^2\\ {(-2)}^2\\ {\left(6\right)}^2\end{array}\right),\left(\begin{array}{c}0.15\\ 0.25\\ 0.30\\ 0.30\end{array}\right)〉=34.15.$$

This is because $\mathbf{P}\left(X_1{X}_2\right)=74$ cannot be chosen together with $(0.15,0.25,0.30,0.30)$. The same is true if $\mathbf{P}\left(X_1{X}_2\right)=69$ is incorrectly chosen.

8. Conclusions and Future Perspectives

In this research work, constrained choices are reinterpreted by considering zero-sum games characterized by an incomplete state of information and knowledge associated with a given individual. An extension of zero-sum games is provided. The set $\Omega$ of the possible outcomes of a random process is embedded in a linear space over $\mathbb{R}$ of a Euclidean nature. Only its finite dimension can be different. The set $\Omega$ does not emphasize measures having a special status according to other formalistic formulations, but it emphasizes linear functionals within which the masses can be put in whatever way a given individual feels. Whenever two or more than two objects of decision-maker choice are studied, the rules of linearity no longer hold on their own. The rules of multilinearity hold too. Such rules are actually preeminent. This is a new issue compared to what has been investigated so far in the literature. The rules of multilinearity extend the properties of the barycentre given by a stable equilibrium and minimum of the moment of inertia. To separate what, in a decision-making problem, is logical from what is of an empirical value is essential according to the approach followed by this research work. This implies that closed convex sets are handled. The budget set of a given individual is a closed convex set. The choice of any point belonging to a given closed convex set depends on the underlying hypothesis related to a specific state of information and knowledge associated with a given decision-maker. This state is intrinsically variable. Given a closed convex set, it is not at all correct to say that one always chooses the same point of it regardless of the underlying hypothesis related to a specific state of information and knowledge associated with a given individual. In this research work, probability and utility of an ordinal nature are jointly handled using the notion of prevision bundle. In this paper, a given decision-maker innovatively maximizes his subjective utility of an ordinal nature associated with expected returns on portfolios of financial assets. Hence, he maximizes the notion of distance which is treated inside a subset of a linear space over $\mathbb{R}$ of a Euclidean nature. Financial assets are studied inside the budget set of a given decision-maker. This is a new matter compared to what has been investigated so far in the literature. Although the arguments of $U\left(\mathbf{x}\right)$ and $U^{\prime }\left(\mathbf{x}\right)$ are observed inside the budget set of a given individual, it is necessary to go away from it in order to represent both $U\left(\mathbf{x}\right)$ and $U^{\prime }\left(\mathbf{x}\right)$. It is not absolutely unusual to go away from the budget set of a given decision-maker in order to process data that are observed and estimated inside it. With respect to the expected utility function of a cardinal nature visualized in a subset of the Cartesian plane, one goes away from the budget set of a given decision-maker to process data that are observed and estimated inside it. In this case, multiple random quantities of order m, where $m\ge 2$ is an integer, appear. In this case, an index measuring the expected return on a portfolio of m financial assets takes place. What is said in this research work is useful with respect to the study of estimation problems faced in the field of statistical inference. A reinterpretation of the central limit theorem can be provided in order to handle estimation problems based on the Bayesian interpretation of probability. It is possible to focus on a mathematical model of multilinear regression. It is possible to consider infinite translations identifying repeated sampling, where the mean and variance of a specially defined deviation-variable do not change.

• This study was not funded.
• The authors declare that they have no conflicts of interest
• This study does not contain any studies with human participants or animals performed by any of the authors
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