Quantification of emotions in decision making

V.I. Yukalov

Bogolubov Laboratory of Theoretical Physics,
Joint Institute for Nuclear Research, Dubna 141980, Russia
and
Instituto de Fisica de São Carlos, Universidade de São Paulo,
CP 369, São Carlos 13560-970, São Paulo, Brazil

e-mail<EMAIL_ADDRESS>

###### Abstract

The problem of quantification of emotions in the choice between alternatives
is considered. The alternatives are evaluated in a dual manner. From one side,
they are characterized by rational features defining the utility of each
alternative. From the other side, the choice is affected by emotions labeling
the alternatives as attractive or repulsive, pleasant or unpleasant. A
decision maker needs to make a choice taking into account both these features,
the utility of alternatives and their attractiveness. The notion of utility is
based on rational grounds, while the notion of attractiveness is vague and
rather is based on irrational feelings. A general method, allowing for the
quantification of the choice combining rational and emotional features is
described. Despite that emotions seem to avoid precise quantification, their
quantitative evaluation is possible at the aggregate level. The analysis of a
series of empirical data demonstrates the efficiency of the approach,
including the realistic behavioral problems that cannot be treated by the
standard expected utility theory.

Keywords: emotions in decision making, quantification of emotions, behavioral
probability, dual choice, affective computing, artificial intelligence

Declarations Funding: Not applicable, Conflicts of interest: Not applicable,
Availability of data and material: Not applicable, Code availability: Not
applicable

ORCID: 0000-0003-4833-0175

##  1 Introduction

The problem of making a choice between alternatives is a core of decision
theory and its numerous applications in economics, finances, and the operation
of intelligence, whether artificial or human. The most well developed
procedure of decision making is based on the expected utility theory
formalized by von Neumann and Morgenstern (1953). However, as is well known,
it is rather a rare occasion when decisions are made on the basis of purely
rational grounds estimating the alternative utility. Almost always the choice
is essentially affected by emotions, and humans do not strictly follow the
prescriptions of the expected utility theory, which results in numerous
paradoxes and often does not allow even for qualitative predictions. To take
into account behavioral effects related to the influence of emotions and other
subjective biases, various so-called non-expected utility theories were
suggested by replacing the expected utility with specially constructed
functionals invented for the purpose of a posteriori interpretation of one or
just a few phenomena. A list of such non-expected utility theories can be
found in the review by Machina (2008).

However, non-expected utility theories are descriptive requiring fitting of
several parameters from particular experimental data. In addition, spoiling
the structure of the expected utility leads to the appearance of
inconsistences and new paradoxes producing more problems than it resolves
(Safra and Segal 2008; Birnbaum 2008; Al Najjar and Weinstein 2009, 2009).

The major problem in describing real-life decision making is caused by the
difficulty of quantifying such behavioral phenomena as emotions. This is
because subjective emotions are not precisely defined in explicit mathematical
terms, contrary to such a crisp notion as utility that can be evaluated on
rational grounds. Therefore emotions increase the uncertainty that always
exists in any choice, when decision makers evaluate the features of the given
alternatives (Scherer and Moors 2019). When analyzing alternatives, decision
makers experience different feelings, emotions, and subconscious intuitive
movements (Kahneman 1982; Picard 1997; Minsky 2006; Plessner et al 2008). This
is why, even choosing between seemingly well formulated lotteries, humans
often do not obey the normative prescriptions of utility theory, but make
decisions qualitatively contradicting the latter (Kahneman and Tversky 1979).

It is important to differentiate two sides in the problem of emotion
quantification. One side is the assessment of emotions experienced by a
subject as reactions on external events, e.g. hearing voice or looking at
pictures. The arising emotions can include happiness, anger, pleasure,
disgust, fear, sadness, astonishment, pain, and so on. The severity or
intensity of such emotions can be estimated by studying the expressive forms
manifesting themselves in motor reactions, such as facial expressions,
pantomime, and general motor activity, and by measuring physiological
reactions, such as the activity of the sympathetic and parasympathetic parts
of the autonomic nervous system, as well as the activity of the endocrine
glands. Vegetative manifestations of emotions can be noticed by studying
changes in the electrical resistance of the skin, the frequency and strength
of heart contractions, blood pressure, skin temperature, hormonal and chemical
composition of the blood, and like that. Several methods of appraising
particular emotions in separate setups have been considered (Amjadzadeh and
Ansari-Asl 2017; Vartanov and Vartanova 2018; Scherer and Moors 2019; Vartanov
et al. 2020, Wang et al. 2021).

The other, principally different, side of emotion evaluation concerns the
study of the influence of emotions on taking decisions by subjects. It is
generally accepted that human decisions are not purely rational, but emotions
do play a great role in decision making. However the quantitative influence of
emotions on the process of decision making remains yet an unsolved problem.

The present paper studies the second problem: How it would be possible to
assess the influence of emotions on decisions taken by humans? We do not
consider the somatic or physiological effects produced by emotions, but we aim
at analyzing how subjective emotions, arising in the process of decision
making, influence the resulting decisions.

Subjectivity in decision making arises because of uncertainty in the suggested
choice. This uncertainty can be of dual nature. From one side, there is the
usual probabilistic uncertainty based on deliberations related to the
alternative utility. From the other side, there is an uncertainty in the
choice due to the subjective feelings that are not regulated by rational
rules. Emotions can be separated into three classes. One class contains,
loosely speaking, positive, features, such as ”good”, ”pleasant”, ”attractive”
and like that. The second class is composed of negative characterizations,
such as ”bad”, ”unpleasant”, ”repulsive” etc. And the third class is
intermediate, comprising neutral definitions expressing indifference with
respect to the alternatives under consideration.

Despite the subjectiveness of emotions, their influence in the choice between
alternatives sometimes can be quantified. Of course, this looks to be
impossible for a particular decision maker and for each separate choice
procedure. Yet, it turns out that quantification is admissible at the
aggregate level for a typical decision maker representing the average
characteristics of a large group of decision makers.

The formulation of explicit mathematical rules allowing for the selection of
an optimal alternative under vague uncertainty due to the influence of
emotions, is not merely useful for characterizing human decision making, but
it is compulsory for the realization of affective computing (Picard 1997) and
for overcoming the challenge of creating artificial intelligence (Russel and
Norvig 2016; Poole and Mackworth 2017; Neapolitan and Jiang 2018). The
achievement of human-level machine intelligence is a principal goal of
artificial intelligence since its inception.

The process of decision making, actually, consists of two sides that can
conditionally be named rational and irrational. The rational side describes
the comparative usefulness of the considered alternatives, while the
irrational side is due to emotions making the decision process less
predictable. It is the rational-irrational duality that makes the
quantification of the decision-making process so difficult.

The distinction between rational and irrational has been extensively discussed
in the literature on dual processes (Sun 2002; Paivio 2007; Evans 2007;
Stanovich 2011; Kahneman 2011) according to which the procedure of taking
decisions in human brains can be treated as a result of two different
processes that can be called rational (logical, controlled, regulated,
deterministic, slow, defined by clear rules) and irrational (intuitive,
uncontrolled, emotional, random, fast, defined in a fuzzy manner). These
processes may proceed in parallel or in turn, but in any case they act
differently (Milner and Goodale 2008; Kahneman 2011).

It is important to stress that the differentiation of mental processes onto
rational and irrational has the meaning for the moment of taking a decision
(Ariely 2008). It is a psychological distinction but not a philosophical one.
It is clear that giving a specially invented philosophical definition it is
straightforward to include afterwards all intuitive and emotional effects into
the rank of rational just giving a definition that rational is all what leads
to the desired goal. Then illogical uncontrolled feelings that occasionally
lead to the goal should be termed rational, and vice versa logical conclusions
that occasionally miss the goal should be named irrational (Searle 2001; Julmi
2019). The philosophical definition of rational has the meaning only
afterwards, when the goal has been reached. Only then it becomes clear what
was leading to the goal and what was not.

Moreover, the philosophical definition of rational as what leads to the goal
is ambiguous. For instance, assume that your goal is to become rich. The
easiest way to become rich is to steal. Hence to steal is rational. But then
you are caught by police and put into jail. To be jailed was not your goal.
Hence to steal is not rational. So it is not clear, is it rational or not,
while from the psychological point of view there is no ambiguity. An action
that is logically and explicitly formulated is psychologically rational. The
psychological definition is based on real physiological processes in the
brain, while the philosophical definition is not uniquely defined and depends
on interpretations.

In what follows, we distinguish rational from irrational as it is accepted in
decision making, where rational is what can be explicitly formulated, based on
clear rules, deterministic, logical, prescriptive, normative, while irrational
is the opposite to rational (Ariely 2008; Zafirovski 2012), being intuitive,
uncontrolled, emotional, random, defined in a fuzzy way.

The dual nature of decision making, comprising the rational-irrational
duality, suggests that this duality could be mathematically represented by a
theory that naturally includes some kind of duality in its basis. The proper
candidate for this could be quantum theory, with its particle-wave duality. A
consistent approach realizing this analogy, by treating decision making as the
procedure of quantum measurements, is the recently developed Quantum Decision
Theory (Yukalov and Sornette 2008, 2009, 2011, 2014, 2016, 2018; Yukalov 2020,
2021).

However, mathematical techniques of quantum theory are not customary for the
majority of people. Therefore it would be desirable to develop a theory that
could incorporate the achievements of quantum decision theory at the same time
avoiding mathematical complications of quantum techniques and the language of
quantum theory so unfamiliar for the majority of researchers. The development
of such an approach and its farther elaboration is the goal of the present
paper. Specifically, the new results of the present article are as follows.

(i) The axiomatic formulation of dual decision theory, taking account of
cognition-emotion duality, that is rational-irrational duality in decision
making, and comprising the main points of quantum decision theory, but without
involving any quantum formulae.

(ii) The derivation, without appealing to quantum theory, of the non-
informative prior estimate for attraction factor measuring the typical
influence of emotions on the process of decision making.

(iii) Illustration of a simple rule for distinguishing emotionally attractive
and repulsive characteristics of the considered alternatives in the case of
highly uncertain lotteries of the Kahneman-Tversky type.

(iv) Analysis of empirical data confirming that the typical influence of
emotions in decision making composes $25\%$.

The plan of the paper is as follows. The approach to be formulated possesses
two major features. First, it is probabilistic, which requires to define the
corresponding probability measure. Second, it is dual, aiming at taking into
consideration rational as well as irrational characteristics of alternatives.
In Sec. 2, the rules for defining the rational probabilistic choice,
describing the utility of the alternatives, is formulated. In Sec. 3, it is
shown how it is possible to characterize the emotional attractiveness of
alternatives. In Sec. 4, the behavioral probability is defined, combining the
rational utility measure and the irrational emotional characteristic, called
attraction factor. The properties of the attraction factor are considered in
Sec. 5, where the typical value of the attraction factor is found to be $1/4$,
which is called the quarter law. This value allows for the estimation of non-
informative priors for the attraction factors describing the influence of
emotions at the aggregate level. Section 6 shows how the attraction factors
for multiple alternatives can be estimated. Section 7, by the example of
binary decision tasks, illustrates that the definition of attraction factors,
generally speaking, is contextual. This is because the attraction factors can
contain parameters whose values need to be adjusted for describing a
particular set of decision problems, which limits their use for other sets of
decision tasks. In Sec. 8, the method of defining the attraction factor
structure in the case of two alternatives with equal or very close utilities
is described. Section 9 considers difficult choice tasks in the case of the
Kahneman-Tversky lotteries, whose utilities are either exactly equal or very
close to each other, so that the standard utility theory is not applicable. A
method is suggested estimating the quality of the lotteries and their
attractiveness and giving good quantitative predictions at the aggregate
level. In Sec. 10, the analysis of a large set of lotteries is given
demonstrating the validity of the quarter law at the aggregate level. Section
11 concludes.

##  2 Probabilistic uncertainty

The main task of decision theory is to describe the process of choice between
a given set of alternatives

${\cal A}=\\{A_{n}:\;n=1,2,\ldots,N_{A}\\}\;.$ (1)

Each alternative can be characterized from two sides, from the rational point
of view of its usefulness and, from the other side, following irrational
feelings and emotions.

In this and the following sections, we describe a new approach to decision
making, taking into account the rational reasoning by estimating the utility
of the considered alternatives as well as the presence of irrational emotions
accompanying the choice.

Even when there exist rational logical arguments explaining the utility of the
given alternatives, not all subjects incline to prefer a single alternative,
but always an alternative $A_{n}$, with a clearly defined utility, is selected
only by a fraction $f(A_{n})$ of decision makers (Slovic and Tversky 1974),
which can be termed rational fraction.

In the present section, the definition of the rational fraction is formulated
and its properties are described.

Definition 1. A rational fraction $f(A_{n})$ is the fraction of decision
makers that would choose the alternative $A_{n}$ provided their decisions
would be based solely on rational grounds. The rational fraction is semi-
positive and normalized,

$\sum_{n=1}^{N_{A}}f(A_{n})=1\;,\qquad 0\leq f(A_{n})\leq 1\;.$ (2)

The rational fraction represents the classical probability, with its standard
properties, including the additivity with respect to mutually exclusive
alternatives,

$f\left(\bigcup_{n}A_{n}\right)=\sum_{n}f(A_{n})\;.$ (3)

Definition 2. An alternative $A_{1}$ is called more useful than $A_{2}$ if and
only if

$f(A_{1})>f(A_{2})\;.$ (4)

Two alternatives, $A_{1}$ and $A_{2}$, are equally useful if and only if

$f(A_{1})=f(A_{2})\;.$ (5)

The rational fraction $f(A_{n})$ shows how useful the alternative is, because
of which it can be called the utility fraction.

The ideas of decision theory ate typically illustrated by the choice between
lotteries. Let the alternatives be represented by the lotteries

$A_{n}=\\{x_{i},\;p_{n}(x_{i}):~{}i=1,2,\ldots,N_{n}\\}\;,$ (6)

which are the probability distributions over payoffs $x_{i}$, with
$p_{n}(x_{i})$ being the payoff probabilities that can be either objective
(von Neumannn and Morgenstern 1953) or subjective (Savage 1954). The lottery
utility is quantified by the expected utility

$U(A_{n})=\sum_{i}u(x_{i})p_{n}(x_{i})\;,$ (7)

where $u(x)$ is a utility function. More generally, it is possible to
introduce a utility functional

$U(A_{n})=\sum_{i}u(x_{i})w(p_{n}(x_{i}))\;,$

with $w(p_{n}(x_{i}))$ being a postulated weighting function (Kahneman and
Tversky 1979).

The rational fraction, associated with the expected utility, should satisfy
the limiting conditions

$f(A_{n})\rightarrow 1\;,\qquad U(A_{n})\rightarrow\infty\;,$
$f(A_{n})\rightarrow 0\;,\qquad U(A_{n})\rightarrow-\infty\;,$ (8)

whose meaning is clear. An explicit form of the rational fraction can be done
by the Luce rule (Luce 1959; Luce and Raiffa 1989) according to which, if an
alternative $A_{n}$ is characterized by an attribute $a_{n}$, then the weight
of this alternative can be defined as

$f(A_{n})=\frac{a_{n}}{\sum_{n=1}^{N_{A}}a_{n}}\qquad(a_{n}\geq 0)\;.$ (9)

When the expected utilities of all lotteries are semi-positive, the attribute
values can be defined by these utilities

$a_{n}=U(A_{n})\;,\qquad U(A_{n})\geq 0\;,$ (10)

while when the expected utilities are negative, the attribute values are
defined by the inverse quantities

$a_{n}=\frac{1}{|\;U(A_{n})\;|}\;,\qquad U(A_{n})<0\;.$ (11)

In the case of mixed utility signs, it is straightforward to shift the
utilities by a minimal available wealth making these utilities semi-positive.

Definition 3. When alternatives are represented by lotteries, the rational
fractions can be defined as

$f(A_{n})=\frac{U(A_{n})}{\sum_{n=1}^{N_{A}}U(A_{n})}\qquad(U(A_{n})\geq 0)$
(12)

for semi-positive utilities and as

$f(A_{n})=\frac{|\;U(A_{n})\;|^{-1}}{\sum_{n=1}^{N_{A}}|\;U(A_{n})\;|^{-1}}\qquad(U(A_{n})<0)$
(13)

for negative utilities.

Generally speaking, as the utility $U(A_{n})$, one can imply either the
standard expected utility (von Neumannn and Morgenstern 1953), or a utility
functional, for instance as is used in the prospect theory (Kahneman and
Tversky 1979). It is also possible to define the rational fraction as the
minimizer of an information functional, as has been done for resolving the St.
Petersburg paradox (Yukalov 2021).

The so-defined rational fraction quantifies the fraction of decision makers
choosing an alternative being based only on rational arguments of the
alternative utility. In other words, it is the probability that an alternative
would be chosen by decision makers, provided they are purely rational.

##  3 Emotional uncertainty

In addition to the probabilistic uncertainty that can be quantified by the
rational fraction, there exists an emotional uncertainty ascribing to the
alternatives such vague emotional characteristics that do not seem to allow
for a quantification. In the simplest case, these characteristics can be
separated into three classes of different quality. One quality class includes
such specifications as ”positive”, ”good”, ”pleasant”, and ”attractive”, while
the other is composed of such depictions as ”negative”, ”bad”, ”unpleasant”,
and ”repulsive”. The third, intermediate class qualifies the related
alternatives as ”neutral” or ”indifferent” with respect to their
attractiveness. The principal question is how it would be possible to describe
in mathematical terms and quantify these classes of emotional uncertainty?

We shall denote the set of alternatives pertaining to the positive quality
class as $\mathcal{A}_{+}$, while the set of alternatives pertaining to the
negative quality class, as $\mathcal{A}_{-}$. The set of alternatives from the
neutral quality class is denoted by $\mathcal{A}_{0}$. Let the emotional
attractiveness of an alternative $A_{n}$ be represented by an attraction
factor $q(A_{n})$. For a positive, attractive alternative, the attraction
factor is positive, for a negative, repulsive alternative, it is negative, and
for a neutral alternative, it is zero. The absolute value of the attraction
factor is limited by one.

Definition 4. The attraction factor pertaining to a positive, negative or
neutral quality class, respectively, varies in the intervals

$0<q(A_{n})\leq 1\qquad(A_{n}\in{\cal A}_{+})\;,$ $-1\leq
q(A_{n})<0\qquad(A_{n}\in{\cal A}_{-})\;,$ $q(A_{n})=0\qquad(A_{n}\in{\cal
A}_{0})\;.$ (14)

Definition 5. An alternative $A_{1}$ is more attractive than $A_{2}$ if and
only if

$q(A_{1})>q(A_{2})\;.$ (15)

Conversely, an alternative $A_{2}$ is more repulsive than $A_{1}$. Two
alternatives are said to be equally attractive, or equally repulsive, if and
only if

$q(A_{1})=q(A_{2})\;.$ (16)

Recall that quality, or attractiveness, is a vague subjective notion which can
be interpreted as that the attraction factor is a random quantity varying in
the frame of its quality class. The qualities ”attractive” or ”repulsive” are
subjective, being associated with concrete decision makers. Moreover, they can
change for the same decision maker taking decisions at different moments of
time or under different circumstances, hence they are contextual (Helland
2018).

##  4 Behavioral probability

In real life, humans make decisions taking into account rational arguments, at
the same time being influenced by irrational feelings and emotions. This
implies that both quantities, the rational fraction and attraction factor
define the probability $p(A_{n})$ of choosing alternatives by decision makers.
Thus the probability $p(A_{n})$ of choosing an alternative $A_{n}$ embodies
both a rational evaluation of the alternative utility as well as reflects the
emotional attitude of decision makers towards the considered alternatives.
This rational-irrational duality is typical for the behavior of real-life
decision makers, because of which the probability $p(A_{n})$ can be called
behavioral probability.

When looking for the form of this probability, it is necessary to keep in mind
that the rational decision making has to be a particular case of the more
general process encompassing both rational and irrational sides of decision
making. That is, when irrational effects become not important, the choice
becomes purely rational. This requirement can be written as a limiting
condition.

Correspondence principle. Rational decision making is a particular case of
behavioral decision making, when irrational effects play no role:

$p(A_{n})\rightarrow f(A_{n})\;,\qquad q(A_{n})\rightarrow 0\;.$ (17)

The behavior of decision makers reflects the superposition of rational and
irrational sides of consciousness. In other words, the real-life behavior is a
superposition of cognition and emotions. This suggests the following axiom.

Axiom 1. Behavioral probability is the sum of a rational fraction and of an
attraction factor:

$p(A_{n})=f(A_{n})+q(A_{n})\;,$ (18)

with $p(A_{n})$ being semi-positive and normalized,

$\sum_{n=1}^{N_{A}}p(A_{n})=1\;,\qquad 0\leq p(A_{n})\leq 1\;.$ (19)

The condition of additivity is not required, so that, in general, the
probability measure $\\{p(A_{n})\\}$ is not necessarily additive.

The rational part of the behavioral probability is explicitly defined by the
rational fraction, while the irrational part is characterized by a quantity
represented by the attraction factor. The behavioral probability, being a
superposition of two terms, reflects the existence in life of rational-
irrational duality, or cognition-emotion duality, or utility-attractiveness
duality. When dealing with empirical data, the probability $p(A_{n})$
describes the total fraction of decision makers preferring the given
alternative.

The alternatives $A_{n}$ from the set $\mathcal{A}$ acquire the following
properties understood as the corresponding relations between their
probabilities.

  1. (i)

Ordering: For any two alternatives $A_{1}$ and $A_{2}$, one of the relations
necessarily holds: either $A_{1}\prec A_{2}$, in the sense that
$p(A_{1})<p(A_{2})$, or $A_{1}\preceq A_{2}$, when $p(A_{1})\leq p(A_{2})$, or
$A_{1}\succ A_{2}$, if $p(A_{1})>p(A_{2})$, or $A_{1}\succeq A_{2}$, when
$p(A_{1})\geq p(A_{2})$, or $A_{1}\sim A_{2}$, if $p(A_{1})=p(A_{2})$.

  2. (ii)

Linearity: The relation $A_{1}\preceq A_{2}$, implying $p(A_{1})\leq
p(A_{2})$, means that $A_{2}\succeq A_{1}$, in the sense that $p(A_{2})\geq
p(A_{1})$.

  3. (iii)

Transitivity: For any three alternatives, such that $A_{1}\preceq A_{2}$, with
$p(A_{1})\leq p(A_{2})$, and $A_{2}\preceq A_{3}$, when $p(A_{2})\leq
p(A_{3})$, it follows that $A_{1}\preceq A_{3}$, in the sense that
$p(A_{1})\leq p(A_{3})$.

  4. (iv)

Completeness: The set of alternatives ${\cal A}$ contains a minimal $A_{min}$
and a maximal $A_{max}$ elements, for which $p(A_{min})=\min_{n}p(A_{n})$ and,
respectively, $p(A_{max})=\max_{n}p(A_{n})$. The ordered set of these
alternatives is called a complete lattice.

Relations between behavioral probabilities determine preference relations
between the alternatives.

Definition 6. An alternative $A_{1}$ is called preferable to $A_{2}$ if and
only if

$p(A_{1})>p(A_{2})\qquad(A_{1}\succ A_{2})\;.$ (20)

Two alternatives $A_{1}$ and $A_{2}$ are indifferent if and only if

$p(A_{1})=p(A_{2})\qquad(A_{1}\sim A_{2})\;.$ (21)

Definition 7. The alternative $A_{opt}$ is called optimal if and only if it
corresponds to the maximal behavioral probability,

$p(A_{opt})=\max_{n}p(A_{n})\;.$ (22)

It is clear that an alternative can be more useful but not preferable, since
its behavioral probability consists of a rational fraction and an irrational
attraction factor. An alternative $A_{1}$ is preferable to $A_{2}$, implying
that $p(A_{1})>p(A_{2})$, then and only then when

$f(A_{1})-f(A_{2})>q(A_{2})-q(A_{1})\;.$ (23)

Both quantities, the rational fraction and attraction factor are important in
the process of taking decisions.

##  5 Attraction factor

Although the attraction factor is a random quantity, it possesses, on average,
some general properties that, because of their importance, are formulated as
theorems.

Theorem 1. The attraction factor $q(A_{n})$ varies in the interval

$-f(A_{n})\leq q(A_{n})\leq 1-f(A_{n})$ (24)

and satisfies the alternation law

$\sum_{n=1}^{N_{A}}q(A_{n})=0\;.$ (25)

Proof. These properties follow directly from the definition of the behavioral
probability (18), its semi-definiteness and normalization (19), and from the
semi-definiteness and normalization of the rational fraction (2). $\square$

Irrational feelings and emotions, playing a very important role in decision
making, are characterized by the attraction factor. Strictly speaking, the
attraction factor $q(A_{n})$ is a random quantity that varies for different
people and different conditions. Despite that it is random, it enjoys some
specific features that can be used for estimating the non-informative priors
quantifying this factor.

Recall that being random does not prevent the quantity from possessing well
defined properties on average. In decision making this means that, although
the attraction factor is difficult to define for a single decision maker and a
single choice, but it may enjoy quite explicit properties at the aggregate
level as an average for a large group of decision makers and over several
choices. Such averages, playing the role of non-informative priors, could
allow us to evaluate typical attraction factors, even having no detailed
information on each of the separate decision makers.

Definition 8. If a quantity $y$ is given on an interval $[a,b]$, the average
of $y$ is defined as the arithmetic average

$\overline{y}\equiv\frac{a+b}{2}\;.$ (26)

In the case when the boundaries $a$ and $b$ themselves are the quantities
given on the intervals $[a_{1},a_{2}]$ and, respectively, $[b_{1},b_{2}]$, the
non-informative prior for $y$ is the arithmetic average

$\overline{y}\equiv\frac{\overline{a}+\overline{b}}{2}\qquad\left(\overline{a}\equiv\frac{a_{1}+a_{2}}{2}\;,~{}\overline{b}\equiv\frac{b_{1}+b_{2}}{2}\right)\;.$
(27)

Theorem 2. Quarter Law. The average value for the attraction factor $q(A_{n})$
in the positive quality class is

$\overline{q}(A_{n})=\frac{1}{4}\qquad(A_{n}\in{\cal A}_{+})$ (28)

and in the negative quality class, it is

$\overline{q}(A_{n})=-\;\frac{1}{4}\qquad(A_{n}\in{\cal A}_{-})\;.$ (29)

Proof. According to Theorem 1, the attraction factor is defined on the
interval $[-f(A_{n}),1-f(A_{n})]$. Hence for the positive quality class it is
given on the interval $[0,1-f(A_{n})]$ and for the negative quality class, on
the interval $[-f(A_{n}),0]$. By the definition of the averages, we have for
the positive quality class

$\overline{q}(A_{n})=\frac{1-\overline{f}(A_{n})}{2}\qquad(A_{n}\in{\cal
A}_{+})\;,$ (30)

while for the negative quality class

$\overline{q}(A_{n})=-\;\frac{\overline{f}(A_{n})}{2}\qquad(A_{n}\in{\cal
A}_{-})\;.$ (31)

Since $f(A_{n})$ is defined on the interval $[0,1]$, its non-informative prior
is the average $\bar{f}(A_{n})=1/2$. This gives the averages (28) and (29).
$\square$

In this way, the average behavioral probability of choosing an alternative
$A_{n}$ by a group of decision makers can be estimated by the expression

$p(A_{n})=f(A_{n})\pm 0.25\;,$ (32)

provided the probability properties (18) are preserved.

The attraction factor, being a random quantity, varies for different agents as
well as for the same decision maker at different moments of time. Therefore
the same decision task, with the same utility factors, even for the same pool
of subjects may be accompanied by different behavioral probabilities (Murphy
and Fu 2018), hence different attraction factors. Such variations is a kind of
random noise. Numerous empirical data show that these variations lead to
statistical errors of about $0.1$ (Murphy and ten Brincke 2018). This implies
that if the difference $p(A_{n})-f(A_{n})$ is smaller than $0.1$, then the
attractiveness of the alternative pertains to the neutral class and the
attraction factor can be set to zero.

##  6 Multiple alternatives

The estimate for the non-informative prior of the attraction factor, derived
above, is especially useful for the case of choosing between two alternatives.
In the case, where there are many alternatives in the set $\mathcal{A}$, it is
possible to more precisely estimate typical attraction factors, playing the
role of non-informative priors.

Suppose $N_{A}$ alternatives can be classified according to the level of their
attractiveness, so that

$q(A_{n})>q(A_{n+1})\qquad(n=1,2,\ldots,N_{A}-1)\;.$ (33)

Let the nearest to each other attraction factors $q(A_{n})$ and $q(A_{n+1})$
be separated by a typical gap

$\Delta\equiv q(A_{n})-q(A_{n+1})\;.$ (34)

And let us accept that the average over the set $\mathcal{A}$ absolute value
of the attraction factor can be estimated by the non-informative prior
$\overline{q}=1/4$, so that

$\overline{q}\equiv\frac{1}{N_{A}}\sum_{n=1}^{N_{A}}|\;q(A_{n})\;|=\frac{1}{4}\;.$
(35)

Theorem 3. For a set $\mathcal{A}$ of $N_{A}$ alternatives, under conditions
(33), (34), and (35), the non-informative priors for the attraction factors
are

$q(A_{n})=\frac{N_{A}-2n+1}{2N_{A}}\qquad(N_{A}~{}even)\;,$
$q(A_{n})=\frac{N_{A}(N_{A}-2n+1)}{2(N_{A}^{2}-1)}\qquad(N_{A}~{}odd)\;,$ (36)

depending on whether $N_{A}$ is even or odd.

Proof. In accordance with conditions (33), (34), and (35), we can write

$q(A_{n})=q(A_{1})-(n-1)\Delta\;.$ (37)

From the alternation law (25), it follows

$q(A_{1})=\frac{N_{A}-1}{2}\;\Delta\;.$ (38)

Using the definition of the average $\overline{q}$ in equation (35) gives the
gap

$\displaystyle\Delta=\left\\{\begin{array}[]{ll}4\overline{q}/N_{A},{}&N_{A}~{}even\\\
4\overline{q}N_{A}/(N_{A}^{2}-1),{}&N_{A}~{}odd\end{array}\right.\;,$ (41)

depending on whether the number of alternatives $N_{A}$ is even or odd. Then
expression (38) becomes

$\displaystyle
q(A_{1})=\left\\{\begin{array}[]{ll}2\overline{q}(N_{A}-1)/N_{A},{}&N_{A}~{}even\\\
2\overline{q}N_{A}/(N_{A}+1),{}&N_{A}~{}odd\end{array}\right.\;.$ (44)

Using (37), we get

$\displaystyle
q(A_{n})=\left\\{\begin{array}[]{ll}2\overline{q}(N_{A}+1-2n)/N_{A},{}&N_{A}~{}even\\\
2\overline{q}N_{A}(N_{A}+1-2n)/(N_{A}^{2}-1),{}&N_{A}~{}odd\end{array}\right.\;,$
(47)

In view of equality (35), we have $\bar{q}=1/4$. Then expression (44) leads to

$\displaystyle
q(A_{1})=\left\\{\begin{array}[]{ll}(N_{A}-1)/2N_{A},{}&N_{A}~{}even\\\
N_{A}/2(N_{A}+1),{}&N_{A}~{}odd\end{array}\right.\;.$ (50)

And finally, equation (47) results in the answer (36). $\square$

As applications of the above theorem, let us give some examples. Thus for a
set of two alternatives, we have the already known values of the non-
informative priors for the attraction factors

$\\{q(A_{n}):~{}n=1,2\\}=\left\\{\frac{1}{4}\;,\;-\;\frac{1}{4}\right\\}\;.$
(51)

For three alternatives, we find

$\\{q(A_{n}):~{}n=1,2,3\\}=\left\\{\frac{3}{8}\;,\;0\;,\;-\;\frac{3}{8}\right\\}\;.$
(52)

Respectively, for the set of four alternatives, we obtain the quality factors

$\\{q(A_{n}):~{}n=1,2,3,4\\}=\left\\{\frac{3}{8}\;,\;\frac{1}{8}\;,\;-\;\frac{1}{8}\;,\;-\;\frac{3}{8}\right\\}\;.$
(53)

##  7 Binary alternatives

The case of binary alternatives is, probably, the most often considered in
applications, being a typical choice problem. In previous sections, a method
of evaluating the average value of the attraction factors is described. As is
shown, the typical attraction factor can be estimated, despite that, in
general, attractiveness seems to be a vague notion. The natural question
arises whether it would be feasible to give a more detailed assessment of
attractiveness. This problem has been discussed for the case of two
alternatives, when it has been necessary to choose between two lotteries
(Favre et al. 2016; Vincent et al. 2016; Ferro et al. 2021, Zhang and
Kjellström 2021).

Let us consider the choice between two lotteries, $A$ and $B$, whose utilities
are $U(A)$ and $U(B)$, respectively. These quantities can represent either the
standard expected utility (von Neumann and Morgenstern 1953) or other utility
functionals employed in decision theory, e.g. the utility functional of
prospect theory (Kahneman and Tversky 1979; Tversky and Kahneman 1992). The
attraction factor in the form

$q(A)=\min\\{\varphi(A),\;\varphi(B)\\}\tanh\\{a[\;U(A)-U(B)\;]\\}\;,\qquad
q(B)=-q(A)\;,$ (54)

has been considered (Vincent et al. 2016; Ferro et al. 2021), where

$\varphi(A)=\frac{1}{Z}\;e^{\beta
U(A)}\;,\qquad\varphi(B)=\frac{1}{Z}\;e^{\beta U(B)}\;,\qquad Z=e^{\beta
U(A)}+e^{\beta U(B)}\;.$

It has been used for characterizing the series of $91$ binary decision tasks
(lotteries), where the choice is made by the pool of $142$ subjects (Murphy
and ten Brincke 2018). The parameters of the attraction factor are fitted so
that to optimally agree with the given experimental data. It is shown (Vincent
et al. 2016; Ferro et al. 2021) that the decision theory with this attraction
factor better describes the empirical data than the stochastic cumulative
prospect theory (Tversky and Kahneman 1992) and than the stochastic rank-
dependent utility theory (Quiggin 1982).

As is clear, the parameters calibrated so that to optimally describe a given
set of lotteries may be not appropriate for another set of lotteries. In that
sense, each combination of parameters is contextual, being suitable for a
particular set of decision tasks, but not necessarily adequate for other
groups of decision problems. This can be easily understood noticing that the
attraction factor (54) becomes zero, when the utilities $U(A)$ and $U(B)$
coincide, or it becomes negligible when these utilities are close to each
other. At the same time very close, or equal utilities often correspond to
high uncertainty in the choice, which results in large attraction factors. A
typical example of such a situation has been illustrated by Kahneman and
Tversky (1979) for a set of lotteries with close or coinciding expected
utilities.

The choice between two alternatives with equal or close utilities is a kind of
the ”Buridan’s donkey problem” (Kane 2005) that can be solved only considering
emotions.

##  8 Buridan’s donkey problem

Emotions are characterized by attraction factors. Hence to quantify emotions
means the necessity of evaluating the values of the attraction factors for the
considered alternatives. As is explained in Sec. 5, the first step in this
estimation is the formulation of the quarter law stating that the non-
informative prior for the magnitude of the attraction factor for either
positive or negative quality classes is $\pm 0.25$. However we need to define
how the actual classification could be realized, so that each alternative
would be associated with the corresponding quality class, either positive or
negative. This problem is typical for the research area known as soft
computing aspiring to find methods that tolerate imprecision and uncertainty
of fuzzy notions to achieve tractability and robustness allowing for
quantitative conclusions (Clocksin 2003; de Silva 2003; Jamshidi 2003).

Below we suggest an algorithm that is applicable to that situation of close
utilities of lotteries, whether with gains or with losses. This algorithm can
be justified on the basis of studies in experimental neuroscience, which have
discovered that, when making a choice, the main and foremost attention of
decision makers is directed towards the payoff probabilities (Kim, Seligman
and Kable 2012). This implies that subjects evaluate higher the probabilities
than the related payoffs (Yukalov and Sornette 2014, 2018). In mathematical
terms, this can be formulated as the existence of different types of scaling
for the alternative quality with respect to payoff utility and payoff
probability. Say, the payoff utility is scaled linearly, while the payoff
probability, exponentially.

To be explicit, let us consider a simple case of two alternatives, one lottery

$A_{1}=\\{u,p\;|\;0,1-p\\}\;,$ (55)

with a payoff utility $u$ and a related probability $p$, and the other lottery

$A_{2}=\left\\{\lambda
u,\frac{p}{\lambda}\;|\;0,1-\;\frac{p}{\lambda}\right\\}\;,$ (56)

whose payoff utility and probability are scaled in such a way that the
expected utilities of both lotteries are equal,

$U(A_{1})=U(A_{2})=up\;.$

Then the corresponding rational fractions coincide, $f(A_{1})=f(A_{2})=1/2$,
and one cannot choose a preferable alternative being based on rational
arguments. This is a typical example of a series of lotteries considered by
Kahneman and Tversky (1979). However, empirical studies show that subjects do
make clear preferences between the lotteries, depending on their payoffs and
probabilities. This implies that subjects are able to intuitively classify the
lotteries into positive (attractive) or negative (repulsive).

Since ”quality” or ”attractiveness” are vague notions, it would be tempting to
accomplish the quality classification by means of words. For instance, we
could accept that between two lotteries that one is of better quality, or more
attractive, that yields a more certain gain or less certain loss. Often this
is a reasonable way of classification, although not always.

Suppose the lottery $A_{1}$ is quite certain, which implies that the payoff
probability is in the interval $1/2<p\leq 1$. Hence the average probability is
$p=3/4$, which is appreciated by people as highly certain (Hillson 2003,
2019). Let the scaling with $\lambda>1$ be such that the payoff utility
increases, while its probability diminishes. When $\lambda$ is not large,
subjects do prefer the more certain lottery $A_{1}$. However strongly
increasing the payoff utility may attract more people, despite a small payoff
probability, as has been confirmed by real-life lotteries (Rabin 2000).

In order to describe the method of classification of alternatives into
positive (attractive) or negative (repulsive) quality classes, let us
introduce the quality functional $Q(A_{n})$. The fact that decision makers in
their choice pay the main and foremost attention to the payoff probabilities
(Kim, Seligman and Kable 2012) is formalized by a linear dependence of the
quality functional with respect to the payoff utility and by an exponential
dependence with respect to the payoff probability. For the case of the
lotteries (55) and (56), this implies the quality functionals

$Q(A_{1})=ub^{p}\;,\qquad Q(A_{2})=\lambda ub^{p/\lambda}\;.$

When the scaling $\lambda$ is of order one and $A_{1}$ is more certain,
subjects consider the more certain lottery as more attractive, which means
that $Q(A_{1})$ is larger than $Q(A_{2})$. But if the payoff probability is
diminished by an order, which assumes $\lambda=10$, while the payoff utility
increases by an order, then the lottery $A_{1}$ can become less attractive
than $A_{2}$, so that $Q(A_{1})$ becomes smaller than $Q(A_{2})$. The change
of attractiveness occurs where $Q(A_{1})=Q(A_{2})$. The latter equality gives
the expression for the base $b$ that for $p=3/4$ and $\lambda=10$ yields

$b=\lambda^{\lambda/(\lambda-1)p}=30\;.$ (57)

The above arguments give a clue allowing us to define the quality functional
for any lottery.

Definition 9. The quality functional of an arbitrary lottery is

$Q(A_{n})=\sum_{i}u(x_{i})30^{p_{n}(x_{i})}\;.$ (58)

Comparing the quality functionals of different lotteries, we can meet the
case, where these functionals are equal, but the lotteries differ from each
other by the gain-loss number difference

$N(A_{n})=N_{+}(A_{n})-N_{-}(A_{n})$ (59)

between the number of admissible gains $N_{+}(A_{n})$ and the number of
possible losses $N_{-}(A_{n})$. A typical example is the comparison of the
lottery $A_{1}$ defined in (55) and the lottery

$A_{3}=\\{u_{1},p\;|\;u_{2},p\;|\;0,1-2p\\}\;,$ (60)

in which $u_{1}+u_{2}=u$. Then the related quality functionals are equal

$Q(A_{3})=u_{1}b^{p}+u_{2}b^{p}=ub^{p}=Q(A_{1})\;,$

where $b=30$. If $u_{n}>0$, then the lottery $A_{1}$ possesses only one
admissible gain and no losses, while the lottery $A_{3}$, two gains and also
no losses. Hence $N(A_{1})=1$ and $N(A_{3})=2$. Since $N(A_{3})$ is larger
than $N(A_{1})$, the lottery $A_{3}$ is treated as more attractive. Similarly,
for the lotteries with losses, where $u_{n}<0$, and quality functionals are
equal, the gain-loss number difference $N(A_{1})=-1$ is larger than
$N(A_{3})=-2$, so that the lottery $A_{1}$ with a smaller number of losses is
more attractive.

Definition 10. A lottery $A_{1}$ is of better quality, or more attractive,
than $A_{2}$, so that $q(A_{1})>q(A_{2})$, if either

$Q(A_{1})>Q(A_{2})\;,$ (61)

or if

$Q(A_{1})=Q(A_{2})\;,\qquad N(A_{1})>N(A_{2})\;.$ (62)

If some lotteries $A_{1}$ and $A_{2}$ cannot be classified as more or less
attractive, they are said to be of equal quality, or equally attractive, so
that $q(A_{1})=q(A_{2})$. If there are only two of these lotteries, then the
alternation law $q(A_{1})+q(A_{2})=0$ implies $q(A_{1})=q(A_{2})=0$. In that
case, the lotteries are in the neutral quality class.

If the alternative $A_{1}$ is more attractive than $A_{2}$, then the related
behavioral probabilities can be estimated as

$p(A_{1})=f(A_{1})+0.25\;,\qquad p(A_{2})=f(A_{2})-0.25\;,$ (63)

where $f(A_{n})$ are rational fractions. Here the inequality $0\leq
p(A_{n})\leq 1$ is assumed, which can be formalized by the definition

$p(A_{n})={\rm Ret}_{[0,1]}\\{f(A_{n})\pm 0.25\\}\;,$

where the retract function is defined as

$\displaystyle{\rm Ret}_{[0,1]}z=\left\\{\begin{array}[]{ll}0,{}&~{}z<0\\\
z,{}&~{}0\leq z\leq 1\\\ 1,{}&~{}z>1\end{array}\right..$ (67)

Recall that the above expressions estimate the aggregate fractions of decision
makers averaged over many subjects and a set of choices. For a single decision
maker, the attraction factor is a random quantity. However, the average
attraction factor and, respectively, the average behavioral probability can be
estimated according to rules (63). The rational fraction $f(A_{n})$ shows the
fraction (frequentist probability) of decision makers that would choose the
corresponding alternative on the basis of only rational rules. While the
behavioral probability $p(A_{n})$ defines the real total fraction of decision
makers actually choosing $A_{n}$, taking into account both the rational
utility as well as the irrational emotional attractiveness of the
alternatives.

##  9 Kahneman-Tversky lotteries

By a number of examples, Kahneman and Tversky (1979) have shown that the
expected utility theory in many cases does not work at all, so that decision
makers do not decide according to utility theory, because of the very close
lottery utilities, and even choose the alternatives that should be neglected
according to the utility theory prescriptions. In their experiments, the
number of participants was about $100$. The typical statistical error was
close to $\pm 0.1$. Payoffs below are given in monetary units, whose measures
are of no importance when using dimensionless rational fractions.

Below, we show that the method described above correctly predicts the
aggregate choice, giving good quantitative estimates for behavioral
probabilities. Rational fractions are calculated by formulas of Sec. 2. For
simplicity, the linear utility function $u(x)$ is accepted. The attraction
factor is represented by its non-informative prior, with the sign prescribed
by the lottery quality functional defined in Sec. 8. For brevity, we use the
notation $Q(A_{n})\equiv Q_{n}$.

For the convenience of the reader, we summarize the formulae that are used
below in characterizing the lotteries. The rational utility fraction is
calculated according to the definition in Sec. 2 as

$f(L_{n})=\frac{U(L_{n})}{\sum_{n}U(L_{n})}\qquad(U(L_{n})\geq 0)$

for semi-positive expected utilities and as

$f(L_{n})=\frac{|\;U(L_{n})\;|^{-1}}{\sum_{n}|\;U(L_{n})\;|^{-1}}\qquad(U(L_{n})<0)$

for negative expected utilities, where the latter are given by the expression

$U(L_{n})=\sum_{i}x_{i}p_{n}(x_{i})\;.$

The quality functional $Q_{n}=Q(L_{n})$ is defined in (58).

Choice 1. Consider two lotteries

$L_{1}=\\{2.5,\;0.33\;|\;2.4,\;0.66\;|\;0,\;0.01\\}\;,\qquad
L_{2}=\\{2.4,\;1\\}\;.$

The rational fractions are $f(L_{1})=0.501$ and $f(L_{2})=0.499$, so that the
first lottery should be chosen on the rational grounds. However, the lottery
quality functionals $Q_{1}=30.3$ and $Q_{2}=72$ show that the second lottery,
being more certain, is more attractive, since $Q_{2}>Q_{1}$. Hence
$q(L_{2})>q(L_{1})$, and involving the non-informative prior, we have
$q(L_{1})=-0.25$, while $q(L_{2})=0.25$. This gives the behavioral
probabilities

$p(L_{1})=0.25\;,\qquad p(L_{2})=0.75\;,$

according to which the second lottery is optimal. This is in agreement with
the empirical results

$p_{exp}(L_{1})=0.18\;,\qquad p_{exp}(L_{2})=0.82\;.$

The more certain, but less useful lottery is chosen.

Choice 2. One chooses between the lotteries

$L_{1}=\\{2.5,\;0.33\;|\;0,\;0.67\\}\;,\qquad
L_{2}=\\{2.4,\;0.34\;|\;0,\;0.66\\}\;.$

The rational fractions are close to each other, $f(L_{1})=0.503$ and
$f(L_{2})=0.497$. At the first glance, it is difficult to say which of the
lotteries is more attractive, since the first lottery has a slightly larger
payoff, while the second is a little more certain. But the lottery qualities
$Q_{1}=7.68$ and $Q_{2}=7.63$ show that the first lottery is a bit more
attractive. Hence $q(L_{1})=0.25$ and $q(L_{2})=-0.25$. Then the behavioral
probabilities are

$p(L_{1})=0.75\;,\qquad p(L_{2})=0.25$

which is well comparable with the experimental data

$p_{exp}(L_{1})=0.83\;,\qquad p_{exp}(L_{2})=0.17\;.$

This is an example, where the majority prefer a less certain, but more useful
lottery.

Choice 3. Considering the lotteries

$L_{1}=\\{4,\;0.8\;|\;0,\;0.2\\}\;,\qquad L_{2}=\\{3,\;1\\}\;,$

one sees that the first lottery, although being less certain, is more useful,
having a larger rational fraction $f(L_{1})=0.516$ while $f(L_{2})=0.484$. But
its quality is lower than that of the second lottery, $Q_{1}=60.8$, while
$Q_{2}=90$. This means that the second lottery is more attractive, because of
which $q(L_{1})=-0.25$ and $q(L_{2})=0.25$. As a result, the behavioral
probabilities are

$p(L_{1})=0.27\;,\qquad p(L_{2})=0.73\;,$

being close to the experimentally observed

$p_{exp}(L_{1})=0.20\;,\qquad p_{exp}(L_{2})=0.80\;.$

Here the more certain, although less useful lottery is chosen.

Choice 4. For the lotteries

$L_{1}=\\{4,\;0.20\;|\;0,\;0.80\\}\;,\qquad
L_{2}=\\{3,\;0.25\;|\;0,\;0.75\\}\;,$

the rational fractions are again close to each other, as in the previous case,
$f(L_{1})=0.516$ and $f(L_{2})=0.484$. But the quality of the first lottery is
higher than that of the second, $Q_{1}=7.9$, but $Q_{2}=7.02$. This makes the
first lottery more attractive, with $q(L_{1})=0.25$ and $q(L_{2})=-0.25$. And
the choice reverses, as compared to the previous case,

$p(L_{1})=0.77\;,\qquad p(L_{2})=0.23\;,$

in agreement with the empirical results

$p_{exp}(L_{1})=0.65\;,\qquad p_{exp}(L_{2})=0.35\;.$

Again, a less certain, although more useful, lottery is chosen.

Choice 5. Between the lotteries

$L_{1}=\\{6,\;0.45\;|\;0,\;0.55\\}\;,\qquad
L_{2}=\\{3,\;0.9\;|\;0,\;0.10\\}\;,$

it is difficult to choose which is better. The first lottery suggests a twice
larger payoff, while the second, twice higher payoff probability. The utility
of both the lotteries is the same, with the same rational fractions
$f(L_{1})=f(L_{2})=0.5$. However, the lottery qualities are different,
$Q_{1}=27.7$, while $Q_{2}=64.1$, showing that the second lottery is more
attractive, which gives $q(L_{1})=-0.25$ and $q(L_{2})=0.25$. Therefore the
behavioral probabilities become

$p(L_{1})=0.25\;,\qquad p(L_{2})=0.75\;.$

And the empirical data are

$p_{exp}(L_{1})=0.14\;,\qquad p_{exp}(L_{2})=0.86\;.$

More certain lottery is chosen.

Choice 6. The lotteries

$L_{1}=\\{6,\;0.001\;|\;0,\;0.999\\}\;,\qquad
L_{2}=\\{3,\;0.002\;|\;0,\;0.998\\}\;,$

have the same rational fractions $f(L_{1})=f(L_{2})=0.5$. But the quality of
the first lottery is higher than that of the second, $Q_{1}=6.02$, while
$Q_{2}=3.02$. That is, the first lottery is more attractive, so that
$q(L_{1})=0.25$ and $q(L_{2})=-0.25$. This yields the behavioral probabilities

$p(L_{1})=0.75\;,\qquad p(L_{2})=0.25\;,$

practically coinciding with the experimental data

$p_{exp}(L_{1})=0.73\;,\qquad p_{exp}(L_{2})=0.27\;.$

Between two equally useful lotteries, the less certain is chosen.

Choice 7. For the lotteries

$L_{1}=\\{6,\;0.25\;|\;0,\;0.75\\}\;,\qquad
L_{2}=\\{4,\;0.25\;|\;2,\;0.25\;|\;0,\;0.5\\}\;,$

the rational fractions are again the same, which does not make it possible to
choose on the basis of utility, $f(L_{1})=f(L_{2})=0.5$. Although the lottery
qualities are equal, $Q_{1}=Q_{2}=14$, but the second lottery suggests a
larger choice of gains, $N(L_{2})=2>N(L_{1})=1$, which makes it more
attractive, with $q(L_{1})=-0.25$ and $q(L_{2})=0.25$. As a result, the
behavioral probabilities read as

$p(L_{1})=0.25\;,\qquad p(L_{2})=0.75\;.$

The empirical data are

$p_{exp}(L_{1})=0.18\;,\qquad p_{exp}(L_{2})=0.82\;.$

Among seemingly equally useful lotteries, the choice is made under the
influence of the attraction factor.

Choice 8. Considering the lotteries

$L_{1}=\\{5,\;0.001\;|\;0,\;0.999\\}\;,\qquad L_{2}=\\{0.005,\;1\\}\;,$

we see that they are of equal utility, with the rational fractions
$f(L_{1})=f(L_{2})=0.5$. But the lottery qualities are essentially different,
$Q_{1}=5.02$ and $Q_{2}=0.15$, which defines the attraction factors
$q(L_{1})=0.25$ and $q(L_{2})=-0.25$. Then the behavioral probabilities are

$p(L_{1})=0.75\;,\qquad p(L_{2})=0.25\;.$

This is very close to the empirical data

$p_{exp}(L_{1})=0.72\;,\qquad p_{exp}(L_{2})=0.28\;.$

Again, this is an example, when a less certain lottery is chosen among two
equally useful lotteries.

Choice 9. The lotteries

$L_{1}=\\{10,\;0.5\;|\;0,\;0.5\\}\;,\qquad L_{2}=\\{5,\;1\\}$

have equal utilities, with the rational fractions $f(L_{1})=f(L_{2})=0.5$. The
lottery qualities $Q_{1}=54.8$ and $Q_{2}=150$ show that the second lottery is
more attractive, hence $q(L_{1})=-0.25$ and $q(L_{2})=0.25$. Therefore the
behavioral probabilities become

$p(L_{1})=0.25\;,\qquad p(L_{2})=0.75\;.$

The empirical probabilities are

$p_{exp}(L_{1})=0.16\;,\qquad p_{exp}(L_{2})=0.84\;.$

The second lottery is more certain, although has a smaller payoff.

Choice 10. For the lotteries

$L_{1}=\\{2,\;0.5\;|\;1,\;0.5\\}\;,\qquad L_{2}=\\{1.5,\;1\\}\;,$

rational fractions are equal, $f(L_{1})=f(L_{2})=0.5$. The lottery qualities
are $Q_{1}=16.4$ and $Q_{2}=45$. Hence the second lottery is more attractive,
which means that $q(L_{1})=-0.25$ and $q(L_{2})=0.25$. Then the behavioral
probabilities are

$p(L_{1})=0.25\;,\qquad p(L_{2})=0.75\;.$

This is very close to the experimentally found probabilities

$p_{exp}(L_{1})=0.20\;,\qquad p_{exp}(L_{2})=0.80\;,$

actually coinciding with them within the accuracy of experiments.

Choice 11. The previous lotteries dealt with gains. Now we shall treat the
lotteries with losses, which implies that the subject has to pay, that is to
loose, the amount of monetary units marked as negative. Consider the lotteries

$L_{1}=\\{-4,\;0.8\;|\;0,\;0.2\\}\;,\qquad L_{2}=\\{-3,\;1\\}\;.$

The rational fraction of the second lottery is larger, $f(L_{1})=0.484$, while
$f(L_{2})=0.516$. However, the first lottery is more attractive, since its
quality is higher, $Q_{1}=-60.8$, while $Q_{2}=-90$. This tells us that
$q(L_{1})=0.25$ and $q(L_{2})=-0.25$, which leads to the behavioral
probabilities

$p(L_{1})=0.73\;,\qquad p(L_{2})=0.27\;.$

In experiments, the majority also choose the first lottery,

$p_{exp}(L_{1})=0.92\;,\qquad p_{exp}(L_{2})=0.08\;.$

The situation is opposite to the case of gains. Now a lottery with a less
certain loss is preferable.

Choice 12. For the lotteries

$L_{1}=\\{-4,\;0.2\;|\;0,\;0.8\\}\;,\qquad
L_{2}=\\{-3,\;0.25\;|\;0,\;0.75\\}\;,$

the rational fractions are $f(L_{1})=0.484$ and $f(L_{2})=0.516$. The related
lottery qualities read as $Q_{1}=-7.9$ and $Q_{2}=-7.02$, showing that the
second lottery is more attractive, with $q(L_{1})=-0.25$ and $q(L_{2})=0.25$.
Then we find the behavioral probabilities

$p(L_{1})=0.23\;,\qquad p(L_{2})=0.77\;.$

Now the majority of decision makers choose the second lottery,

$p_{exp}(L_{1})=0.42\;,\qquad p_{exp}(L_{2})=0.58\;,$

although it suggests a more certain loss.

Choice 13. The lotteries

$L_{1}=\\{-3,\;0.9\;|\;0,\;0.1\\}\;,\qquad L_{2}=\\{-6,\;0.45\;|\;0,\;0.55\\}$

possess equal utility, hence equal rational fractions $f(L_{1})=f(L_{2})=0.5$.
But the second lottery is more attractive, since its quality is higher,
$Q_{1}=-64.1$, while $Q_{2}=-27.7$. Therefore $q(L_{1})=-0.25$ and
$q(L_{2})=0.25$. The behavioral probabilities

$p(L_{1})=0.25\;,\qquad p(L_{2})=0.75$

show that the second lottery is optimal, in agreement with the empirical
observations,

$p_{exp}(L_{1})=0.08\;,\qquad p_{exp}(L_{2})=0.92\;.$

The second lottery is preferred, although its loss is higher, but the loss is
less certain.

Choice 14. For the lotteries

$L_{1}=\\{-3,\;0.002\;|\;0,\;0.998\\}\;,\qquad
L_{2}=\\{-6,\;0.001\;|\;0,\;0.999\\}$

rational fractions are equal, $f(L_{1})=f(L_{2})=0.5$. However, the lottery
qualities $Q_{1}=-3.02$ and $Q_{2}=-6.02$ demonstrate that the first lottery
is more attractive, so that $q(L_{1})=0.25$ and $q(L_{2})=-0.25$. This results
in the behavioral probabilities

$p(L_{1})=0.75\;,\qquad p(L_{2})=0.25$

that are very close to the experimentally found,

$p_{exp}(L_{1})=0.70\;,\qquad p_{exp}(L_{2})=0.30\;.$

Now, between two equally useful lotteries, the lottery suggesting a more
certain loss is chosen.

Choice 15. The lotteries

$L_{1}=\\{-1,\;0.5\;|\;0,\;0.5\\}\;,\qquad L_{2}=\\{-0.5,\;1\\}$

also have equal rational fractions, $f(L_{1})=f(L_{2})=0.5$. But the quality
of the first lottery is higher, $Q_{1}=-5.48$, while $Q_{2}=-15$. Hence the
first lottery is more attractive, with $q(L_{1})=0.25$, but $q(L_{2})=-0.25$.
The resulting behavioral probabilities

$p(L_{1})=0.75\;,\qquad p(L_{2})=0.25$

are in good agreement with the experimental data

$p_{exp}(L_{1})=0.69\;,\qquad p_{exp}(L_{2})=0.31\;.$

The first lottery is preferred, although its loss is larger.

Choice 16. Among the lotteries

$L_{1}=\\{-6,\;0.25\;|\;0,\;0.75\\}\;,\qquad
L_{2}=\\{-4,\;0.25\;|\;-2,\;0.25\;|\;0,\;0.5\\}\;,$

that look similar, having the same rational fractions $f(L_{1})=f(L_{2})=0.5$,
and equal qualities $Q_{1}=Q_{2}=-14$, the second is less attractive,
exhibiting a larger number of losses, $N(L_{1})=-1>N(L_{2})=-2$. Therefore
$q(L_{1})=0.25$ and $q(L_{2})=-0.25$. This yields the behavioral probabilities

$p(L_{1})=0.75\;,\qquad p(L_{2})=0.25\;,$

practically coinciding with the empirical data

$p_{exp}(L_{1})=0.70\;,\qquad p_{exp}(L_{2})=0.30\;,$

within the accuracy of experiments.

Choice 17. The lotteries

$L_{1}=\\{-5,\;0.001\;|\;0,\;0.999\\}\;,\qquad L_{2}=\\{-0.005,\;1\\}$

have the same utility, with equal rational fractions $f(L_{1})=f(L_{2})=0.5$.
But their qualities $Q_{1}=-5.02$ and $Q_{2}=-0.15$ show that the second
lottery is more attractive, having a much larger quality. Hence
$q(L_{1})=-0.25$ and $q(L_{2})=0.25$. The behavioral probabilities are

$p(L_{1})=0.25\;,\qquad p(L_{2})=0.75\;,$

as compared with the experimental data

$p_{exp}(L_{1})=0.17\;,\qquad p_{exp}(L_{2})=0.83\;.$

Surprisingly, the lottery with certain loss is chosen, which is explained by
its higher quality.

Choice 18. For the lotteries

$L_{1}=\\{-10,\;0.5\;|\;0,\;0.5\\}\;,\qquad L_{2}=\\{-5,\;1\\}\;,$

the rational fractions are equal, $f(L_{1})=f(L_{2})=0.5$. But for the lottery
qualities, we have $Q_{1}=-54.8$ and $Q_{2}=-150$. Thus the first lottery is
more attractive, hence $q(L_{1})=0.25$ and $q(L_{2})=-0.25$. This gives the
behavioral probabilities

$p(L_{1})=0.75\;,\qquad p(L_{2})=0.25\;,$

in agreement with empirical data

$p_{exp}(L_{1})=0.69\;,\qquad p_{exp}(L_{2})=0.31\;.$

Now the lottery with a larger, but less certain loss is chosen.

The results for all $18$ choices between the Kahneman-Tversky lotteries are
summarized in Table 1 showing which of the lotteries is optimal, that is,
having the largest predicted behavioral probability

$p(L_{opt})\equiv\max_{n}p(L_{n})$ (68)

over the given lattice of alternatives. Also shown are the rational fractions
for the optimal lottery, $f(L_{opt})$, experimental probabilities of the
optimal lottery, defined as the fractions of decision makers choosing the
optimal lottery $p_{exp}(L_{opt})$, and the related empirical attraction
factors

$q_{exp}(L_{opt})=p_{exp}(L_{opt})-f(L_{opt})\;.$ (69)

The results, corresponding to the non-optimal lotteries, can be easily found
from the normalization conditions

$p(L_{1})+p(L_{2})=1\;,\qquad f(L_{1})+f(L_{2})=1\;,\qquad
q(L_{1})+q(L_{2})=0\;.$ (70)

At the bottom of Table 1, the average values over all $18$ cases are given for
the rational fraction $\overline{f}(L_{opt})=0.5$, predicted behavioral
probability $\overline{p}(L_{opt})=0.75$, experimentally observed probability
$\overline{p}_{exp}(L_{opt})=0.77$, and the experimentally observed average
absolute value of the attraction factor

$\overline{q}_{exp}=\overline{p}_{exp}(L_{opt})-\overline{f}(L_{opt})=0.27\;.$
(71)

Within the accuracy of the experiment, the predicted average behavioral
probability of choosing an optimal lottery, $0.75$, equals the empirical
average fraction of decision makers $0.77$, and the average attraction factor
$0.27$ practically coincides with the theoretical estimate of $0.25$.

As the analysis of this set of choices demonstrates, it is not possible to
predict the behavioral decision making of humans by considering separately
either lottery utilities, payoffs, or payoff probabilities. But reliable
predictions can be made by defining behavioral probabilities, including the
estimates of both, rational fractions as well as attraction factors. On the
aggregate level, such predictions are not merely qualitative, but provide good
quantitative agreement with empirical data, involving no fitting parameters.

At the same time, the expected utility theory is not applicable to the
Kahneman-Tversky lotteries, since the lottery with a higher utility is
preferred only twice among $18$ lotteries, that is only in the $1/9$ part of
the lotteries. Also, it is important to notice that the formula (54) here is
not valid, as far as for the coinciding utilities it gives zero attraction
factor, while the aggregate experimental data give for the attraction factor
$0.27$.

##  10 Quarter law

In the previous sections, it has been shown that the average influence of
emotions in decision making can be quantified by the typical value of
attraction factor, which turns out to be close to $0.25$, which is termed
quarter law and which follows from the non-informative prior estimate of Sec.
5. Thus in the set of Kahneman-Tversky lotteries of Sec. 9 the experimentally
measured average attraction factor is $0.27$, which, within the typical
statistical error of $0.1$, coincides with the predicted attraction factor
$0.25$.

In the present section, we verify the quarter law on the basis of a large set
of binary lotteries studied recently (Murphy and ten Brincke 2018). In the
analyzed experiment, $142$ subjects were suggested a set of binary decision
tasks (lotteries). The same experiment was repeated after two weeks, with
randomly changing the order of the pairs of lotteries. The experiments at
these two different times are referred as session $1$ and session $2$. There
are three types of lotteries: lotteries containing only gains (all payoffs are
positive), lotteries with only losses (all payoffs are negative), and mixed
lotteries containing gains as well as losses. As usual, a loss implies the
necessity to pay the designed amount of money. Keeping in mind the estimation
of attraction factors in positive and negative quality classes, we consider
the related lotteries, where the difference between the rational utility
factors and the empirical choice probabilities, at least in one of the
sessions, are larger than the value of the typical statistical error of $0.1$
corresponding to random noise. On the basis of these lotteries, we calculate
the quantities of interest and summarize the results in several tables.

Table 2 presents the results for the optimal lotteries with only gains and
Table 3 shows the results for the optimal lotteries with only losses. Recall
that a lottery $L_{1}$ is called optimal, as compared to a lottery $L_{2}$ if
and only if the corresponding probability $p(L_{1})$ is larger than
$p(L_{2})$. In both the cases, of either the lotteries with only gains or the
lotteries with only losses, an optimal lottery is always a lottery from the
positive quality class, in which $q(L_{opt})>0$. The situation can be
different for the mixed lotteries, containing gains as well as losses. In
these cases, an optimal lottery can occasionally pertain to a negative quality
class. Table 4 summarizes the results for the mixed lotteries containing both
gains and losses. Among these lotteries, the first sixteen examples in Table 4
are the lotteries from the positive quality class, which at the same time are
the optimal lotteries. The last five cases are the lotteries that are not
optimal, however being from the positive quality class.

As is seen, the value of the attraction factor in the positive or negative
quality classes is $\pm 0.22$, which is in very good agreement with the
predicted non-informative priors $\pm 0.25$. Thus the quarter law provides a
rather accurate estimate of the attraction factor at the aggregate level.

##  11 Conclusion

An approach is developed allowing for the quantification of emotions in
decision making. The approach takes into account the duality of decision
making, including both rational and irrational sides of decision process. The
rational evaluation of alternatives is based on logical clearly prescribed
rules defining a rational fraction representing the probability of choosing
alternatives on the basis of rational principles.

The irrational side of decision processes is due to subconscious feelings,
emotions, and intuition that cannot be exactly measured for a given subject at
a given moment of time, thus inducing emotional uncertainty in the process of
decision making. Irrational processes are superimposed on the rational
evaluation of the considered alternatives and define for each alternative a
correction term called attraction factor. Since irrational processes cannot be
exactly quantified, the attraction factor is a random quantity. The attraction
factor can be described by linguistic characteristics that can be classified
into three quality classes, briefly speaking, positive, negative, and neutral.
The positive quality class includes such specifications as attractive,
pleasant, good and like that. The negative quality class comprises the
features like repulsive, unpleasant, bad, and so on. The neutral quality class
is intermediate, being neither positive nor negative.

The attraction factor is a variable randomly varying for different decision
makers and even for the same decision maker at different moments of time.
Nevertheless, being random, does not preclude this quantity to have a well
defined average value inside each of the quality classes. A theorem is proved
defining the average values of the quality factor inside the positive class as
$1/4$ and inside the negative class as $-1/4$. For alternatives represented by
lotteries with equal or close utilities, a method is suggested ascribing each
lottery to the appropriate quality class.

Being able to determine the belonging of alternatives to the related quality
classes and knowing the average values of attraction factors allows us to find
the average behavioral probabilities associated with the typical fractions of
decision makers choosing this or that alternative.

The method is illustrated by a series of lotteries with a difficult choice,
when the standard expected utility theory is not applicable, or its
prescriptions contradict the choice of real humans. The empirical data confirm
that the non-informative prior for attraction factors provides an accurate
quantification of emotions at the aggregate level.

Summarizing, the main points of the suggested approach can be formulated as
follows.

  1. (i)

Decision making is treated as a probabilistic process that can be
characterized by behavioral probabilities defining the portions of decision
makers choosing this or that alternative from the given set of alternatives.

  2. (ii)

The behavioral probability, taking into account the rational-irrational or
cognition-emotion duality of decision processes, describes decision making
affected by emotions. The superposition of utility and attractiveness is
represented as a sum of two terms, a rational fraction and an attraction
factor.

  3. (iii)

The rational fraction, having the properties of the standard additive
probability, describes the fraction of decision makers that would make their
choice being based solely on rational grounds, following prescribed rational
rules. The rational fraction quantifies the utility of the choice.

  4. (iv)

The attraction factor takes into account irrational effects influencing the
choice, such as feelings, emotions, and biases. The attraction factor
characterizes subconscious attractiveness of the considered alternatives,
because of which it is called attraction factor. The attraction factor is a
random quantity, varying for different subjects, different choices, and
different times.

  5. (v)

Despite being random, the attraction factor possesses well defined average
features. The average values of the attraction factor for positive or negative
quality classes can be defined by non-informative priors.

  6. (vi)

The approach makes it possible to give quantitative predictions in the choice
between the lotteries with emotional uncertainty, where the expected utility
theory does not work. The aggregate predictions, averaged over decision makers
and choices, are in good quantitative agreement with empirical data.

  7. (vii)

Empirical data confirm the quarter law providing, at the aggregate level, an
accurate evaluation of typical influence of emotions in decision making.

  8. (viii)

The appealing feature of the approach is its straightforward axiomatic
formulation employing rather simple mathematics. Although the structure of the
approach is implicitly influenced by quantum theory, but it completely avoids
borrowed from physics complicated quantum techniques.

## Acknowledgment

The author is grateful for helpful advise and useful discussions to D.
Sornette and E.P. Yukalova.

This research did not receive any specific grant from funding agencies in the
public, commercial, or not-for-profit sector.

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Table 1: Optimal lotteries $L_{opt}$ from the Kahneman-Tversky set, the
rational fractions for the optimal lotteries, $f(L_{opt})$, predicted
behavioral probabilities $p(L_{opt})$, experimentally observed probabilities
$p_{exp}(L_{opt})$, defined as the fractions of the participants choosing the
optimal lottery $L_{opt}$, and the experimental attraction factors
$q_{exp}(L_{opt})$ corresponding to the optimal lotteries. At the bottom, the
average values are shown.
| $L_{opt}$ | $f(L_{opt})$ | $p(L_{opt})$ | $p_{exp}(L_{opt})$ | $q_{exp}(L_{opt})$
---|---|---|---|---|---
1 | $L_{2}$ | 0.50 | 0.75 | 0.82 | 0.32
2 | $L_{1}$ | 0.50 | 0.75 | 0.83 | 0.33
3 | $L_{2}$ | 0.48 | 0.73 | 0.80 | 0.32
4 | $L_{1}$ | 0.52 | 0.77 | 0.65 | 0.13
5 | $L_{2}$ | 0.50 | 0.75 | 0.86 | 0.36
6 | $L_{1}$ | 0.50 | 0.75 | 0.73 | 0.23
7 | $L_{2}$ | 0.50 | 0.75 | 0.82 | 0.32
8 | $L_{1}$ | 0.50 | 0.75 | 0.72 | 0.22
9 | $L_{2}$ | 0.50 | 0.75 | 0.84 | 0.34
10 | $L_{2}$ | 0.50 | 0.75 | 0.80 | 0.30
11 | $L_{1}$ | 0.48 | 0.73 | 0.92 | 0.44
12 | $L_{2}$ | 0.52 | 0.77 | 0.58 | 0.06
13 | $L_{2}$ | 0.50 | 0.75 | 0.92 | 0.42
14 | $L_{1}$ | 0.50 | 0.75 | 0.70 | 0.20
15 | $L_{1}$ | 0.50 | 0.75 | 0.69 | 0.19
16 | $L_{1}$ | 0.50 | 0.75 | 0.70 | 0.20
17 | $L_{2}$ | 0.50 | 0.75 | 0.83 | 0.33
18 | $L_{1}$ | 0.50 | 0.75 | 0.69 | 0.19
|  | 0.50 | 0.75 | 0.77 | 0.27
Table 2: Optimal lotteries with gains. The rational fraction $f(L_{opt})$ of
the optimal lottery, fractions of subjects (frequentist probabilities)
$p_{i}(L_{opt})$ choosing the optimal lottery in the session $i=1,2$, and the
attraction factors $q_{i}(L_{opt})$ of the optimal lottery in the session $i$.
At the bottom, the average values for the related quantities.
| $f(L_{opt})$ | $p_{1}(L_{opt})$ | $p_{2}(L_{opt})$ | $q_{1}(L_{opt})$ | $q_{2}(L_{opt})$
---|---|---|---|---|---
1 | 0.55 | 0.86 | 0.89 | 0.31 | 0.34
2 | 0.48 | 0.66 | 0.69 | 0.18 | 0.21
3 | 0.51 | 0.68 | 0.62 | 0.17 | 0.11
4 | 0.59 | 0.80 | 0.75 | 0.22 | 0.17
5 | 0.63 | 0.89 | 0.90 | 0.26 | 0.27
6 | 0.66 | 0.96 | 0.95 | 0.30 | 0.29
7 | 0.51 | 0.79 | 0.81 | 0.28 | 0.30
8 | 0.48 | 0.60 | 0.63 | 0.12 | 0.15
9 | 0.63 | 0.88 | 0.92 | 0.26 | 0.30
10 | 0.56 | 0.89 | 0.82 | 0.33 | 0.26
11 | 0.63 | 0.77 | 0.73 | 0.14 | 0.10
12 | 0.51 | 0.72 | 0.73 | 0.21 | 0.21
13 | 0.61 | 0.87 | 0.85 | 0.26 | 0.24
14 | 0.63 | 0.93 | 0.93 | 0.30 | 0.30
15 | 0.64 | 0.85 | 0.87 | 0.21 | 0.23
16 | 0.64 | 0.80 | 0.80 | 0.16 | 0.16
17 | 0.64 | 0.89 | 0.89 | 0.25 | 0.25
18 | 0.48 | 0.65 | 0.70 | 0.17 | 0.22
19 | 0.65 | 0.87 | 0.93 | 0.22 | 0.28
20 | 0.66 | 0.86 | 0.82 | 0.20 | 0.16
21 | 0.58 | 0.84 | 0.80 | 0.26 | 0.22
22 | 0.52 | 0.75 | 0.74 | 0.23 | 0.22
23 | 0.48 | 0.64 | 0.65 | 0.16 | 0.17
24 | 0.44 | 0.60 | 0.53 | 0.16 | 0.10
25 | 0.62 | 0.73 | 0.79 | 0.11 | 0.17
26 | 0.64 | 0.81 | 0.90 | 0.17 | 0.26
27 | 0.66 | 0.93 | 0.96 | 0.27 | 0.30
| 0.58 | 0.80 | 0.80 | 0.22 | 0.22

Table 3: Optimal lotteries with losses. The rational fraction $f(L_{opt})$ of
the optimal lottery, fractions of subjects (frequentist probabilities)
$p_{i}(L_{opt})$ choosing the optimal lottery in the session $i=1,2$, and the
attraction factors $q_{i}(L_{opt})$ of the optimal lottery in the session $i$.
At the bottom, the average values for the related quantities.
| $f(L_{opt})$ | $p_{1}(L_{opt})$ | $p_{2}(L_{opt})$ | $q_{1}(L_{opt})$ | $q_{2}(L_{opt})$
---|---|---|---|---|---
1 | 0.52 | 0.77 | 0.75 | 0.25 | 0.23
2 | 0.60 | 0.85 | 0.83 | 0.25 | 0.23
3 | 0.53 | 0.72 | 0.71 | 0.19 | 0.18
4 | 0.64 | 0.96 | 0.92 | 0.32 | 0.28
5 | 0.55 | 0.70 | 0.68 | 0.15 | 0.13
6 | 0.54 | 0.73 | 0.72 | 0.20 | 0.19
7 | 0.63 | 0.79 | 0.84 | 0.16 | 0.21
8 | 0.54 | 0.66 | 0.63 | 0.12 | 0.09
9 | 0.56 | 0.80 | 0.89 | 0.24 | 0.33
10 | 0.58 | 0.89 | 0.92 | 0.31 | 0.34
11 | 0.49 | 0.66 | 0.71 | 0.17 | 0.22
12 | 0.62 | 0.87 | 0.93 | 0.25 | 0.31
13 | 0.55 | 0.79 | 0.74 | 0.24 | 0.19
14 | 0.54 | 0.82 | 0.77 | 0.29 | 0.24
15 | 0.53 | 0.65 | 0.70 | 0.12 | 0.17
16 | 0.51 | 0.59 | 0.62 | 0.08 | 0.11
17 | 0.56 | 0.79 | 0.86 | 0.23 | 0.30
18 | 0.58 | 0.89 | 0.90 | 0.31 | 0.32
19 | 0.61 | 0.76 | 0.74 | 0.15 | 0.13
| 0.56 | 0.77 | 0.78 | 0.21 | 0.22

Table 4: Mixed lotteries, containing gains and losses, from the positive
quality class. The rational fraction $f(L_{+})$ of the lottery, fractions of
subjects $p_{i}(L_{+})$ choosing the corresponding lottery in the session
$i=1,2$, and the attraction factors $q_{i}(L_{+})$ of the lottery in that
session $i$. At the bottom, the average values of the related quantities.
| $f(L_{+})$ | $p_{1}(L_{+})$ | $p_{2}(L_{+})$ | $q_{1}(L_{+})$ | $q_{2}(L_{+})$
---|---|---|---|---|---
1 | 0.40 | 0.69 | 0.66 | 0.29 | 0.26
2 | 0.62 | 0.85 | 0.85 | 0.23 | 0.23
3 | 0.67 | 0.87 | 0.82 | 0.20 | 0.15
4 | 0.44 | 0.62 | 0.61 | 0.18 | 0.17
5 | 0.50 | 0.64 | 0.54 | 0.15 | 0.05
6 | 0.59 | 0.71 | 0.65 | 0.12 | 0.06
7 | 0.54 | 0.69 | 0.63 | 0.16 | 0.10
8 | 0.49 | 0.66 | 0.60 | 0.18 | 0.16
9 | 0.57 | 0.87 | 0.85 | 0.30 | 0.28
10 | 0.65 | 0.75 | 0.77 | 0.10 | 0.12
11 | 0.52 | 0.77 | 0.70 | 0.26 | 0.19
12 | 0.49 | 0.58 | 0.63 | 0.09 | 0.14
13 | 0.55 | 0.87 | 0.92 | 0.32 | 0.37
14 | 0.52 | 0.61 | 0.67 | 0.09 | 0.15
15 | 0.53 | 0.80 | 0.83 | 0.27 | 0.30
16 | 0.56 | 0.67 | 0.63 | 0.11 | 0.07
17 | 0.00 | 0.27 | 0.27 | 0.27 | 0.27
18 | 0.00 | 0.29 | 0.36 | 0.29 | 0.36
19 | 0.00 | 0.30 | 0.45 | 0.30 | 0.45
20 | 0.00 | 0.39 | 0.38 | 0.39 | 0.38
21 | 0.00 | 0.37 | 0.35 | 0.37 | 0.35
| 0.41 | 0.63 | 0.63 | 0.22 | 0.22