The Standard Model of Rational Risky Decision-Making

Abstract

Expected utility theory (EUT) is currently the standard framework which formally defines rational decision-making under risky conditions. EUT uses a theoretical device called von Neumann–Morgenstern utility function, where concepts of function and random variable are employed in their pre-set-theoretic senses. Any von Neumann–Morgenstern utility function thus derived is claimed to transform a non-degenerate random variable into its certainty equivalent. However, there can be no certainty equivalent for a non-degenerate random variable by the set-theoretic definition of a random variable, whilst the continuity axiom of EUT implies the existence of such a certainty equivalent. This paper also demonstrates that rational behaviour under utility theory is incompatible with scarcity of resources, making behaviour consistent with EUT irrational and justifying persistent external inconsistencies of EUT. A brief description of a new paradigm which can resolve the problems of the standard paradigm is presented. These include resolutions of such anomalies as instant endowment effect, asymmetric valuation of gains and losses, intransitivity of preferences, profit puzzle as well as the St. Petersburg paradox.

1. Introduction

It is generally accepted that science is provisional by nature. To confirm, improve or reject it, it must be open to impartial and rigorous re-examination. It is this openness to challenge which distinguishes science from dogma and mythology. Without such scrutiny, resolutions of existing scientific anomalies cannot emerge, hence the need for them. However, any rejection of the extant views as a result of such a re-examination is a bitter pill to swallow for unsuspecting holders of those views, and hence their understandable surprise and resistance to a change of views, a resistance which must be overcome for science to progress. In this context, this paper aims to re-examine expected utility theory (EUT) by first setting out the claims of EUT and the problems that they generate, and later by identifying the generic cause of these problems and finally indicating how they can be resolved. This article peruses the issues that EUT raises, with the care and respect that they deserve, i.e., rigorously and thoroughly from the perspectives of the disciplines of mathematics/statistics and economics/finance, hence the length of it.

Assuming the axioms of EUT hold for all gambles, the utility of each gamble from the set of all gambles available to a rational decision-maker is claimed to be the statistical expectation of the utility of its outcomes (e.g., Jehle and Reny 2011, pp. 97–118). Let random variable $X$ be a function from the set of all possible outcomes of a specific gamble, denoted by $\underset{X}{G}$, to the set of real numbers. Let each value of function $X$ be the quantity of an object (e.g., money) to be paid or received by the individual who plays this gamble. When $X$ takes only discrete real values $a_1,a_2,a_3\dots$ with probabilities $p_1,p_2,p_3\dots$, respectively, this gamble can also be denoted by $\underset{X}{G}=p_1{a}_1\oplus p_2{a}_2\oplus p_3{a}_3\oplus \dots$ (Varian 1992, pp. 173–76). The current literature claims that under the axioms of EUT, for a rational decision-maker, a function $U$ (in a set-theoretic sense) always exists such that $U(\underset{X}{G})=E[U(X)]$, where $E$ is the expectation operator. Function $U$ is claimed to be such that for any two gambles $\underset{X}{G}\mathrm{and}\underset{Y}{G}$ which satisfy the axioms of EUT, one can write $\underset{X}{G}\succ \underset{Y}{G}\iff U(\underset{X}{G})>U(\underset{Y}{G})$ and $\underset{X}{G~}\underset{Y}{G}\iff U(\underset{X}{G})=U(\underset{Y}{G})$ (note: symbol $\succ$ denotes preference and symbol ∼ denotes indifference). As such, $U$ is called the von Neumann–Morgenstern (vN-M) utility function following the latter’s 1953 book Theory of Games and Economic Behavior, which purports to prove EUT axiomatically in its appendix, whilst relying on its main text for certain definitions and explanations.

The decision-maker can be an individual, a firm or a State. The psychological processes that the decision-maker at an individual or collective level goes through to come to a decision is outside the scope of EUT. The axioms of EUT can be viewed as to what a rational decision-maker either does or should do; thus, EUT can be seen as a descriptive or a normative theory of behaviour. EUT was originally devised for gambles with objective probability; Savage (1954) extended it to gambles with subjective probability. Therefore, EUT is currently claimed to be applicable to many real-world risky or uncertain situations, including interactive decision-making scenarios envisioned in game theory, in relation to which EUT is of foundational significance. EUT is claimed to predict or explain rational behaviour, based on the decision-makers’ assessments of their prospects, given all the readily available free information to them.

The vN-M book Theory of Games and Economic Behavior was first published in 1944 without the axiomatic proof of EUT, as von Neumann and Morgenstern had doubts on the validity of their proof. In the light of Gödel’s incompleteness theorems (Gödel 1931), it is not possible to prove the internal consistency of EUT. Nonetheless, it is possible to prove the internal inconsistency of certain theories, and in particular EUT, as EUT is founded on the vN-M definition of a function (Von Neumann and Morgenstern 1953, p. 88), which has significant shortcomings by current standards of rigour, and generates the following hereto overlooked internal inconsistencies (note: this is not an exhaustive list):

(a)

If the probability distribution of $X$ has no mean, which can occur as the axioms of EUT do not rule out any probability distribution of $X$, no $U(\underset{X}{G})$ can exist.

(b)

If $X$ has a mean, there is no guarantee that $U(\underset{X}{G})$ can exist.

(c)

Leaving out Cases (1) and (2), if $U(\underset{X}{G})=E[U(X)]$, a certainty equivalent $E(X)-\mathsf{\Pi }$ for the random variable $X$ (where $\mathsf{\Pi }$ is the risk premium of $X$) is claimed to exit such that the decision-maker will be indifferent in replacing this random variable with its certainty equivalent or vice versa, i.e., $U(\underset{X}{G})=E[U(X)]=E\{U[E(X)-\mathsf{\Pi }]\}=U(\underset{E(X)-\mathsf{\Pi }}{G})=U[E(X)-\mathsf{\Pi }]$. However, no certainty equivalent for any non-degenerate random variable can exist by the set-theoretic definition of the concept of a random variable in statistics.

On the other hand, EUT defines a risk-seeker and a risk-averter in such a way (see Section 5.1 for the definition of these terms) that it becomes impossible for a rational decision-maker to be both a strict risk-seeker and a strict risk-averter concurrently, whilst everyday behaviour in the real-world often requires taking risk and seeking protection from it concurrently. For instance, one may choose to have a mortgage for buying a house with a volatile price, aiming to repay the mortgage with one’s income, and concurrently seek cover against loss of income by way of having a mortgage payment protection insurance policy. However, under EUT, such behaviour is deemed irrational, revealing the external inconsistency of EUT. Bizarrely, EUT deems the behaviour of this house-buyer rational if he/she seeks no insurance policy. Analogously, under EUT, anyone who chooses to bear the dangers of a new pandemic such as COVID-19, and refuses any protection for his/her health, is regarded as rational. On the other hand, whilst at the same time that such an individual accepts to bear these dangers, if he/she seeks any protection against this pandemic, he/she is said to be irrational!

The structure of the rest of this article is as follows. Section 2 describes the methodological approach of this paper. Section 3 provides simple cases of the internal inconsistency of EUT to stimulate research interest in this area for the unsuspecting believers in EUT. Section 4 highlights the practical rejection of EUT in risk management in the finance industry. Section 5 generalizes the findings of Section 3 on the internal inconsistency of EUT. Section 6 draws attention to the errors in the existing proofs of the claims of EUT. Section 7 is on the limitations of this paper. Section 8 presents the results of this research and discusses their implications, including an indication of how the problems of EUT can be resolved in a new paradigm. Section 9 discusses the broad implications of this paper. Section 10 concludes. New concepts are defined at the point of their first use.

A critical literature review under five headings is presented in the appendices. Appendix A studies recognized theoretical, empirical and experimental failures of EUT and considers their implications for economic theory. Appendix B addresses the deep problems which utility theory generates for economic theory, by articulating the axiom of scarcity of resources accurately in a formal sense and noting that rational behaviour under utility theory is incompatible with this fundamental axiom; it also points to how these foundational problems in economics/finance can be resolved in a newly emerging paradigm. Appendix C critiques the vN-M definition of a mathematical function on which EUT is founded, and studies ordinal and vN-M utility functions from the Zermelo–Fraenkel set-theoretic perspective. Appendix D brings to light the hidden errors in the current textbook proofs of the existence of a vN-M utility function. Appendix E reviews the notations used for the operation of addition versus the operation of mixture in the proofs of Von Neumann and Morgenstern (1953) and Herstein and Milnor (1953) and demonstrates the contradictions that they generate for these proofs.

2. Methodology

Utility theory, including EUT, rests at the foundation of standard economic theory currently. However, it has generated a huge number of puzzles and anomalies (see Appendix A). Whilst ad hoc reasons for these developments have been advanced, the generic cause of these problems has only recently come to light through a rigorous examination of its underlying axiomatic and mathematical foundations (Falahati 2019a). Consistent with the latter approach, the methodology of this paper which focuses on EUT is to scrutinize the underlying axiomatic and mathematical foundations of EUT. This brings to light hereto unaddressed issues on which EUT is founded. In particular, this paper proves that EUT is inconsistent with axiomatic set theory, notes the incompatibility of the axioms of EUT with scarcity of resources and proposes that these are the reasons why EUT is rejected in practice (see Section 4) and in experiments (see Appendix A).

The criteria for the validity of any theory, as a logically constructed representation of part of the real-world, are its internal consistency and the external consistency of its results. For a reviewer of a theory, the examination of its internal consistency normally takes precedence over the examination of its external consistency; however, surprisingly, whilst historically there has been much research on the latter aspect of EUT, highlighting its empirical and experimental failures, virtually no in-depth research has been carried out on its internal consistency. Perhaps, this reflects the widespread belief that EUT is valid as a theory for the perfectly rational specimen; and the observed failures of EUT are attributable to the bounded rationality of the ordinary human being (Simon 1955). A view which is reinforced in the light of the high esteem in the existing literature for John von Neumann (1903–1957), appearing as an unerring mathematical genius, on account of his foundational contributions to various mathematical fields (Macrae 1992), one of which was axiomatic set theory. Incongruously, in the latter area, von Neumann’s contribution was seriously deficient, in particular in defining the concept of function, and left Paul Bernays (1888–1977) and Kurt Gödel (1906–1978) to correct his oversight, leading to von Neumann–Bernays–Gödel set theory (Hamilton 1982, pp. 115–33, 145–55).

Alas, von Neumann used the same deficient definition of function in his work on EUT (Von Neumann and Morgenstern 1953, p. 88). However, the impact of this error on the proof of the existence of vN-M utility functions has never been explored hereto. This represents a gaping hole in the history of research on EUT. Therefore, the primary focus of this paper is on the examination of the mathematical integrity of the expected utility theorem leading to $U(\underset{X}{G})=E[U(X)]$ in the light of the set-theoretic definitions of the concepts of function and random variable in mathematics and statistics. This paper demonstrates that these rigorous definitions are not followed in any proof of this theorem, with fatal consequences for EUT.

3. Unnoticed Cases of Internal Inconsistency of EUT

In the following three cases, for all the gambles studied, the random variables generating the outcomes of gambles lead to monetary payments or receipts by individuals who play the gambles, and whilst the axioms of EUT hold by assumption, yet EUT fails.

Case 1.

Where for gamble$\underset{X}{G}$, the probability distribution of$X$does not have a mean. This makes it impossible for any vN-M utility function$U(\underset{X}{G})$to exist, contrary to the prediction of EUT. For instance, let the decision-maker’s utility function be an exponential one:$U(X)=(1-e^{-aX})/a$, as used in standard textbooks, where$e$is the Euler’s number, and$a=1$,$a=0$and$a=-1$represent risk-averter, risk-neutral and risk-seeker behaviour, respectively, under EUT. The axioms of EUT put no restriction on the characteristics of the random variable$X$generating the outcomes of gamble$\underset{X}{G}$; hence,$X$can have a Cauchy probability distribution which has no defined mean, variance or higher moment. Given that no$E(X)$exists in this case, no$E[U(X)]$and no vN-M utility function$U(\underset{X}{G})=E[U(X)]$can exist either.

Case 2.

Where random variable$X$has a continuous probability density function$f$with a finite mean,$a$and$b$are finite real numbers, and for gamble$\underset{X}{G}$,$U(\underset{X}{G})$is an improper integral in the form$U(\underset{X}{G})={\displaystyle {\int }_a^{\infty }U(X)f(X)dX}$,$U(\underset{X}{G})={\displaystyle {\int }_{-\infty }^{b}U(X)f(X)dX}$or$U(\underset{X}{G})={\displaystyle {\int }_{-\infty }^{\infty }U(X)f(X)dX}$such that$U(\underset{X}{G})$does not exist. Geweke (2001) and Yoon (2004) provide examples of such failures of vN-M utility functions; however, they fail to note that these represent internal contradictions for EUT; and they do not explain the cause of these contradictions or offer any remedy for them.

Case 3.

Assuming the axioms of EUT lead to the existence of a multivariate vN-M utility function$u$, where the probability distribution of each random variable$x_i$has a finite mean for$i=1,2,3,\dots$, a certainty equivalent$E(x_i)-{\pi }_i$for$x_i$with risk premium${\pi }_i$is claimed to exist (Kihlstrom and Mirman 1974) such that$E[u(x_1,x_2,x_3,\dots )]=u[E(x_1)-{\pi }_1,E(x_2)-{\pi }_2,E(x_3)-{\pi }_3,\dots )]$. Paroush (1975, p. 283) and Duncan (1977, p. 896) recognize that this situation leads to a multiplicity of certainty equivalents for each random element of multivariate wealth derived from the same utility function for the same individual. The existing literature fails to notice that this generates a multiplicity of preference-orderings of the same elements of wealth, each of which can contradict the other, contrary to EUT, which implicitly assumes one and only one preference-ordering of the elements of wealth, hence an internal contradiction for EUT.

For example, let the decision-maker’s vN-M utility function be $u$ of the Cobb–Douglas form, as in Duncan (1977, p. 896): $u(x_1,x_2)=x_1{x}_2$, where $x_1$ and $x_2$ are random monetary amounts, each with a finite mean, hence $E[u(x_1,x_2)]=E(x_1{x}_2)$. Let the certainty equivalents of $x_1$ and $x_2$ be $E(x_1)-{\pi }_1$ and $E(x_2)-{\pi }_2$, respectively, such that $E[u(x_1,x_2)]=u\{[E(x_1)-{\pi }_1],[E(x_2)-{\pi }_2]\}=[(E(x_1)-{\pi }_1][E(x_2)-{\pi }_2)]$. Given $[E(x_1)-{\pi }_1][E(x_2)-{\pi }_2)]=E(x_1)E(x_2)+{\pi }_1{\pi }_2-{\pi }_{1}E(x_2)-{\pi }_{2}E(x_1)$, and $E(x_1{x}_2)=[(E(x_1)-{\pi }_1][E(x_2)-{\pi }_2)]\Rightarrow E(x_1{x}_2)-E(x_1)E(x_2)={\pi }_1{\pi }_2-{\pi }_{1}E(x_2)-{\pi }_{2}E(x_1)$, by letting ${\sigma }_{12}$ to be the covariance of $x_1$ and $x_2$, and noting that ${\sigma }_{12}=E(x_1{x}_2)-E(x_1)E(x_2)$, one obtains ${\sigma }_{12}={\pi }_1{\pi }_2-{\pi }_{1}E(x_2)-{\pi }_{2}E(x_1)$. Duncan (1977, p. 896) misstates the latter equality in the form ${\sigma }_{12}={\pi }_1{\pi }_2-{\pi }_1{x}_2-{\pi }_2{x}_1$!

Assuming $E(x_1)$, $E(x_2)$ and ${\sigma }_{12}$ exist and are finite and known, the latter equality is one equation for two unknown risk premia, i.e., ${\pi }_1$ and ${\pi }_2$; thus, there can be no unique certainty equivalent for either $x_1$ or $x_2$.

This generates an internal contradiction for EUT. To see this, let ${\pi }_1=p$ and ${\pi }_2=q$ such that ${\sigma }_{12}=pq-pE(x_2)-qE(x_1)$ and $p-q<E(x_1)-E(x_2)$, hence $E(x_2)-q<E(x_1)-p$. It follows that the certainty equivalent of $x_1$ is greater than the certainty equivalent $x_2$, and thus for gambles $\underset{x_1}{G}$ and $\underset{x_2}{G}$, one can write $\underset{x_1}{G}\succ \underset{x_2}{G}$. On the other hand, one can let ${\pi }_1=m$ and ${\pi }_2=n$ such that ${\sigma }_{12}=mn-mE(x_2)-nE(x_1)$ and $m-n>E(x_1)-E(x_2)$, hence $E(x_2)-n>E(x_1)-m$.

It follows that the certainty equivalent of $x_1$ is less than the certainty equivalent $x_2$, and thus for gambles $\underset{x_1}{G}$ and $\underset{x_2}{G}$, one can write $\underset{x_2}{G}\succ \underset{x_1}{G}$, hence a contradiction for EUT. Let us note that any single random variable $x$ can be thought of as a product of two random variables such that $x=x_1{x}_2$; hence, this internal contradiction can arise for univariate and multivariate random wealth.

4. EUT versus Risk Management in Practice in the Finance Industry

Friedman and Savage (1948) observe that contrary to EUT, the same decision-maker can be a risk-seeker and a risk-averter at two different levels of wealth (see Appendix A). However, they do not realize that a decision-maker can also be a strict risk-seeker and a strict risk-averter concurrently, i.e., at the same level of wealth. This is easily seen in the finance industry, as the following explains.

For example, when a bank engages in covered interest rate arbitrage to exploit the interest differential on deposits in two different currencies, it acts as a strict risk-seeker (by exposing the bank to currency fluctuations) and a strict risk-averter (by using forward contracts to protect the bank from currency fluctuations); however, such behaviour is irrational under EUT.

The empirical failure of EUT can be seen quite clearly in relation to the management of risk by insurers, an industry which Bernoulli (1954, p. 30) refers to, in justifying his belief in an early version of EUT. For, under EUT, a rational decision-making firm cannot be a specialist insurer (e.g., a marine insurer) as a strict risk-seeker and also obtain reinsurance as a strict risk-averter against claims it may not be able to afford, contrary to everyday practice in the insurance/reinsurance industry. The rationale for this industry practice is that specialization gives the insurer a competitive advantage against non-specialist insurers, and reinsurance can curtail the cost of policyholders’ claims to a limit which can help the specialist insurer avoid insolvency. Moreover, under EUT, it is not rational for this specialist insurer (as a strict risk-seeker) to invest (as a strict risk-averter) the premia it receives from its policyholders in risk-free bonds rather than risky assets (e.g., equities) to cover its cost of expected claims, contrary to the normal investment practice in this industry. The rationale for this industry practice is that investing in risk-free bonds rather than equities reduces the insolvency risk of the specialist insurer, e.g., when a marine insurer concurrently faces a tsunami of claims and the collapse of equity markets as a result of a sudden war in the seas.

5. Irreparably of EUT

This section demonstrates the self-contradictory implications of rational behaviour under EUT in the disciplines of economics/finance and mathematics/statistics.

5.1. Implications of EUT for the Standard Paradigm in Economics/Finance

Let us note that Cases 1 and 2 of Section 3 cannot be amended by making the vN-M utility function bounded as no vN-M utility function with a defined mean can exist in these cases. On the other hand, the axioms of EUT do not make the vN-M utility function bounded. To avoid the problems encountered in these cases, one needs to add to the axioms of EUT a new axiom: the size of the actual or expected outcomes of the gambles, whether simple or compound, must never approach positive or negative infinity, when measured in non-infinitesimal units. The justification in economics and finance for this restriction is the axiom of scarcity of resources (see Appendix B), which requires that at any date, the total quantity of each scarce resource is finite, and the total number of all the different types of distinct scarce resources is also finite. This axiom requires that as a valid claim against scarce resources, the quantity of money, whether in commodity form (e.g., gold) or in fiat form (e.g., $), must be finite at any date, as otherwise it will cause unbounded inflation. Therefore, one cannot pay or receive an actual or expected infinite monetary outcome of any gamble, and thus such a gamble is never played. Incidentally, this new axiom for EUT removes the St. Petersburg paradox, the resolution of which was the historical motivation for the development of an early version of a vN-M utility function (Bernoulli 1954).

For supporters of EUT, this new axiom is unwelcome, as it eliminates applications of EUT to gambles with unbounded probability distributions of their outcomes, e.g., the normal distribution, which are relied on in financial economics. Supporters of EUT argue that if the utility function is bounded, this objection does not hold; however, they ignore the fact that in no economy which admits scarcity of resources, an infinite quantity of money (in commodity or fiat form) can ever exist; hence, the St. Petersburg paradox will not be offered in such an economy.

Let us presume that this new axiom is complied with, and the axioms of EUT (Jehle and Reny 2011, pp. 111–13) hold for gamble $\underset{Z}{G}$ and lead to the existence of $U$ as a strictly increasing vN-M utility function for a decision-maker with sure monetary wealth $M$, where $Z$ is the random monetary outcome of $\underset{Z}{G}$ with a known probability distribution. Then, this decision-maker is claimed to be a strict risk-averter if for him/her $U\left[E(M+Z)\right]>E[U(M+Z)]$, and he/she is claimed to be a strict risk-seeker if for him/her $U\left[E(M+Z)\right]<E[U(M+Z)]$, and he/she is claimed to be risk neutral if for him/her $U\left[E(M+Z)\right]=E[U(M+Z)]$. Thus, for strict risk-averters, $U$ is strictly concave, and for strict risk-seekers, $U$ is strictly convex, whilst for risk neutral decision-makers, $U$ is strictly linear. This is where EUT does not define the concept of risk per se such that the meaning of attraction towards it i.e., risk-seeking, or repulsion from it i.e., risk-aversion or indifference towards it i.e., risk-neutrality, can be unambiguously derived from the definition of risk. This is where the decision-maker is implicitly assumed to be indifferent towards gambling per se, as gambling per se generates no utility or disutility for decision-makers under EUT (Von Neumann and Morgenstern 1953, pp. 629–30)!

Supporters of EUT seek to justify the trading of a lottery (e.g., gamble $\underset{Z}{G}$ with random monetary prize $Z$) at a fixed price by deriving an equivalence relationship between the random prize of the lottery and the certainty equivalent of this random prize as follows: Consider a decision-maker with a strictly increasing vN-M function $U$ with respect to each of its variables. Following the Jensen inequality and Pratt (1964), if this decision-maker has the sure monetary wealth $M$ and the non-degenerate random monetary outcome $Z$ with mean $E(Z)$, arising from his/her right to play gamble $\underset{Z}{G}$, one can define risk premium ${}_{M}h{}_Z$ such that

$$U(\underset{M+Z}{G})=E\left[U(M+Z)\right]=E\{U[M+E(Z)-{}_{M}h{}_Z]\}=U(M+E(Z)-{}_{M}h{}_Z),$$

(1)

or

$$U(\underset{M+Z}{G})=E\left[U(M+Z)\right]=E\{U[M+E(Z)-{}_{M}h{}_Z]\}=U(\underset{M+E(Z)-{}_{M}h{}_Z}{G}).$$

(2)

The sure amount $E(Z)-{}_{M}h{}_Z$ is claimed to be the certainty equivalent of the random amount $Z$ when the remainder of the decision-maker’s wealth is the sure amount $M$, in the sense that he/she will be indifferent in replacing the random amount $Z$ with the certain amount $E(Z)-{}_{M}h{}_Z$. $E(Z)-{}_{M}h{}_Z$ is thus claimed to be the minimum selling (or maximum buying) price that the decision-maker requires to give up (or acquire) the right to play $\underset{Z}{G}$. If the decision-maker was a strict risk-averter, $U$ would be concave and ${}_{M}h{}_Z>0$, and if the decision-maker was a strict risk-seeker, $U$ would be strictly convex and ${}_{M}h{}_Z<0$, and if the decision-maker was risk-neutral, then $U$ would be linear and ${}_{M}h{}_Z=0$. Nonetheless, the foregoing new axiom does not remove the objections to EUT that arise under Case 3 in Section 3, which has a much wider scope than Cases 1 and 2. Indeed, as the following proposition proves, rational behaviour under EUT is incompatible with the axiom of scarcity of resources (defined in Appendix B), and in particular, it is incompatible with the standard paradigm in economics and finance.

Assumptions of the proposition:

In any competitive, efficient and frictionless economy (CEFE), by definition, free lunches are fully exploited. In the standard CEFE, by assumption, a spot transaction takes zero length of time, i.e., it occurs timelessly at a date of zero length on a timeline. In contrast, in the CEFE of the new paradigm (see Appendix B), by assumption, a spot transaction takes a positive length of time to occur, however small that length may be.

Consider a standard CEFE in which the axioms of EUT hold for all decision-makers in respect of all gambles, including gamble $\underset{Z}{G}$, where $Z$ is a positive non-infinitesimal random amount of money (in the fiat form, e.g., $, or in the commodity form, e.g., gold). In addition, by assumption, there are always strict risk-averter, strict risk-seeker and risk-neutral gamblers in this CEFE, and strict risk-averters have the property right to play gambles, whilst strict risk-seekers wish to acquire this right. Let us assume that at date $d$ for strict risk-averter A, the certainty equivalent of $Z$ is $E(Z)-h$, and for strict risk-seeker B, the certainty equivalent of $Z$ is $E(Z)+k$, and for risk-neutral gambler C, the certainty equivalent of $Z$ is $E(Z)$, where $h>0$ and $k>0$ are sure amounts of money. Further, by assumption, strict risk-averter A has the property right to play gamble $\underset{Z}{G}$, and strict risk-seeker B wishes to acquire this right.

Proposition. 

In a CEFE of the standard paradigm where spot transactions take place timelessly and by assumption all risk attitudes always exist, rational behaviour as defined by EUT generates arbitrage opportunities, leading to a free lunch and a money pump.

Proof. 

Let us note that at date $d$, risk-neutral gambler C has an arbitrage opportunity in terms of the standard paradigm in economics and finance, as he/she can pay A the amount $E(Z)-h$ to acquire the right to play gamble $\underset{Z}{G}$ from A and receive $E(Z)+k$ from B for transferring his/her right to play gamble $\underset{Z}{G}$ to B. Consequently, these transactions occur without any loss of utility for A and B and with C ending up with a free lunch of $k+h>0$ at date $d$. Clearly, within the standard paradigm of economics and finance in this CEFE, where spot transactions take place timelessly, such arbitrage transactions can be repeated at an infinite number of dates within a finite period to generate a money pump or an infinite quantity of gold for C. These trades violate the axiom of scarcity of resources, whilst they reflect rational decision-making under EUT! This is despite the fact that these certainty equivalents, which were derived earlier in this subsection, were based on the presumption that the new axiom, which was introduced to ensure consistency with the axiom of scarcity of resources, could hold under EUT. It is now clear that this new axiom cannot hold under EUT, which means that EUT is incompatible with the axiom of scarcity of resources! □

5.2. Implications of EUT for Mathematics/Statistics

It is the continuity axiom of EUT (e.g., Ingersoll 1987, p. 10; Varian 1992, p. 174; Jehle and Reny 2011, p. 100) which leads to the concept of certainty equivalent of a random variable. For, according to this axiom, if $a,b\&c$ are objects of choice for which $a\succ b\succ c$, then there will be a unique $\mu$ such that $0\le \mu \le 1$ and $b~\mu a\oplus (1-\mu )c$. This is where, by letting random variable $X$ of gamble $\underset{X}{G}$ take values $a$ and $c$ with probabilities $\mu$ and $1-\mu$, respectively, one obtains $\underset{X}{G}=\mu a\oplus (1-\mu )c$ by definition; and by letting $a,b\&c$ be sure monetary amounts, it follows from $b~\mu a\oplus (1-\mu )c\Rightarrow \underset{b}{G}~\underset{X}{G}$. Thus $b$ becomes the certainty equivalent of random variable $X$.

However, this is contrary to the definition of a random variable (Doob 1996, p. 590; Fristedt and Gray 1997, p. 11) as a measurable function which assigns a single numerical value to each possible outcome of an experiment with a well-defined set of outcomes with no predictable order of realization of each one of the individual outcomes when they may not be the same outcomes. Thus, a non-degenerate random variable cannot have a unique specific value, but it will have a set of values. On the contrary, the existence of a certainty equivalent for a non-degenerate random variable leads to a unique specific value for the random variable and implicitly requires the existence of an outcome which is equivalent to each one of the different outcomes of such an experiment. This can be deduced formally from the definition of the certainty equivalent as in the following:

Consider gamble $\underset{Z}{G}$ in Section 5.1, where by the definition of the concept of certainty equivalent, the decision-maker must be indifferent in replacing $Z$ with $E(Z)-{}_{M}h{}_Z$ in his/her utility function $U$, and he/she must be also indifferent in replacing $E(Z)-{}_{M}h{}_Z$ with $Z$ in $U$. It follows from Equation (1) that whatever the decision-maker’s risk attitude may be $E\left[U(M+Z)\right]=U\left[M+E(Z)-{}_{M}h{}_Z\right]$. Replacing $Z$ with $E(Z)-{}_{M}h{}_Z$ on the left side of the latter equality and replacing $E(Z)-{}_{M}h{}_Z$ with $Z$ on the right side of it yields $E\left[U(M+E(Z)-{}_{M}h{}_Z)\right]=U\left[M+Z\right]$ or $U[M+E(Z)-{}_{M}h{}_Z]=U\left[M+Z\right]$. Given that $U$ is a one-to-one function, then $E(Z)-{}_{M}h{}_Z=Z$, which is absurd! If every random variable had a certainty equivalent, there would be no need for the discipline of statistics.

Supporters of EUT presume that since a lottery ticket can be traded, like any other commodity, at a fixed price, therefore its random prize must have a certainty equivalent, such that both traders in such an exchange will be indifferent between the fixed price and the random prize of the lottery. Let us note that this view considers the only source of value of a gamble to traders is its random prize, and it overlooks any positive or negative value that gambling per se may have. Moreover, let us note that trade takes place between two parties, who exchange one object with another, when both parties think that they will be better-off, rather than being indifferent, in doing so. For, if the parties are indifferent, then the exchange will be reversible and can be annulled. Hence, it is not possible to conclude from this assumed indifference that such trade must occur between these two parties. Therefore, there is no compelling logic for equating the price of a lottery ticket to such a derived certainty equivalent of its random prize.

6. Why Existing Proofs of EUT Are Fallacious

The previous sections raise the question as to what is wrong with the existing proofs of existence of vN-M utility functions. I have touched on some of these reasons already. I address them thoroughly in Appendix C, Appendix D and Appendix E, an overview of which follows here.

Appendix C provides, by current standard of rigour in mathematics which employs the concept of function in its set-theoretic sense, the criteria to judge the mathematical integrity of any claim to the proof of the existence of any utility function. In doing so, it disproves ordinal and expected utility theories without challenging their axioms, and only by relying on the set-theoretic definition of a function.

The proofs of existence of vN-M utility functions in current textbooks (e.g., Varian 1992, pp. 172–76; Jehle and Reny 2011, pp. 97–118) overlook the errors which Appendix C brings to light, and are proved to be self-contradictory by a theorem in Appendix D.

Von Neumann and Morgenstern (1953) and Herstein and Milnor (1953) use the same notation for the operations of addition and mixture, Appendix E brings to light the self-contradictory implications of this conflation and thus rejects both these proofs. Herstein and Milnor (1953) try not to contradict explicitly either the vN-M or the post-set-theoretic concept of a function; thus they do not define what they mean by a mathematical function, which is fundamental to their proof; hence their analysis is deficient in this crucial respect. Moreover, the axioms which Herstein and Milnor (1953, pp. 292–95) impose on their mixture sets generate their Theorem 6. The latter supports, and indeed upholds, the vN-M axiom of continuity, which, as discussed in Section 5.2, leads to EUT becoming self-contradictory.

Remarkably, neither Herstein and Milnor (1953) nor Von Neumann and Morgenstern (1953) show any awareness of the failures of EUT in such simple cases as those in Section 3, and thus they make no attempt to remedy these failures. In contrast, to justify the need for the axiom of continuity in ordinal utility theory, to which all these authors subscribe, they are perfectly content with just one counterexample, i.e., the well-known case of lexicographic preferences. This paper points out that EUT cannot be valid with or without the axiom of continuity, as EUT violates the axiom of scarcity of resources and it lacks mathematical integrity, highlighted in Section 5 and Appendix B, Appendix C, Appendix D and Appendix E.

7. Limitations

The description of the newly emerging paradigm as a new overarching economic theory which resolves long-standing puzzles and accords with the real-world, discussed in Appendix B, is quite condensed. To engage in an expanded description would make this article too long. Nonetheless, there is a need to elucidate the new paradigm further and show in a separate article in the future the extent it can go to resolve the extant anomalies of the currently dominant paradigm.

8. Results and Discussion

This article notes that contrary to the definition of a random variable, a vN-M utility function is claimed to transform a non-degenerate random variable into its certainty equivalent, and brings to light the internal contradictions that this implies for EUT. On the other hand, it explains that there is no compelling logic for equating the price of a lottery ticket to such a certainty equivalent. This paper articulates precisely for the first time the axiom of scarcity of resources in its primary form, and it points out that the axioms of utility theories, including EUT, are not consistent with scarcity of resources (e.g., Section 5.1); hence they cannot be rational. It highlights why certain persistent observed individual behaviour in the face of risk and certain persistent observed behaviour in risk management in the finance industry, including the insurance industry, are economically justifiable, contrary to EUT which regards such behaviour irrational. The findings of this paper are relevant to the development of any alternative theory to EUT and explain why attempts to construct new functions which can describe risky/uncertain economic behaviour in non-EUT research studies have not achieved sweeping resolutions of existing anomalies. In contrast, a condensed description of a new paradigm is provided where very many major behavioural puzzles are resolved without such a function. These include long-standing puzzles such as St. Petersburg paradox, instant endowment effect, asymmetric valuation of gains and losses, intransitivity of preferences, the reason for the existence of the firm and the explanation on how the profit of the firm emerges and it is financed in the CEFE of the new paradigm.

9. Broad Implications of This Paper

The history of mathematics, like all other intellectual disciplines, illustrates that it is possible for a new generation to identify errors in the works of an earlier generation. The reason why such polymaths as Daniel Bernoulli and John von Neumann could not see the internal inconsistencies of EUT disclosed here is that in common with many mathematicians of their eras, for a very long time, mathematicians did not employ rigorous definitions of the mathematical concepts of function and random variable. The evidence for the latter facts is seen in Kleiner (1989, p. 284) and in Von Neumann and Morgenstern (1953, p. 88). It is in this context that Doob (1996, p. 586) notes, ‘The mathematization of probability required new ideas, and in particular required a new approach to the idea of … a function’ (emphasis added).

Doob (1996, p. 588) also notes the observation of Max Planck (1858–1947) on the power of received wisdom, where Planck states, ‘A new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die, and a new generation grows up with it.’ This can be emphatically so when the new idea challenges the validity of received wisdom. I trust that scientists have learnt from history and that this gloomy prediction of Planck will not be applicable to new scientific truths found in modern times. However, it is hard to be optimistic, as the sad and the striking finding of this research is that too many intelligent people and their followers, over many generations, can be mesmerized by what appears to them as an elegant theory, such as EUT, based on a non-rigorous justification. The aesthetic attraction of such a theory and halos of its promotors in the eyes of its followers seem to motivate repeated propagation of it, regardless of its internal and external inconsistencies. Hence, such a myth, dressed up as science, is spread widely, and its repetition to the naive makes it believable to them, as long as their views are left unchallenged.

The problem becomes quite serious when this theory is routinely taught in educational institutions as part of a standard curriculum, and thus mythology masqueraded as science dominates scientific discourse. This is where any rejection of the extant theory by the avant-garde, however well substantiated, will be opposed by the laggard, given their vested interests in the status quo, as the latter fear losing respectability for their research and teaching materials. This gives rise to a duel between the mythology supported by the dominant paradigm and the challenges of the avant-garde to it. The shift to a new paradigm for science can come if a new generation is able to sift through independently and freely all relevant received wisdom carefully, impartially and thoroughly and verify the new scientific truth.

10. Conclusions

This paper demonstrates that EUT is founded on pre-set-theoretic understandings of what a mathematical function and a random variable is and that it is incompatible with modern mathematics/statistics and it is also incompatible with the foundational concept of scarcity in economics/finance. For, EUT generates internal contradictions in mathematics/statistics and economics/finance. A new paradigm which overcomes these problems and resolves very many anomalies of the standard paradigm in microeconomics, financial economics and macroeconomics is presented. To help this new paradigm be more widely recognized, there is a need to elucidate it more and to expand it by demonstrating its potential for resolution of any more remaining anomalies in the future.