imitate the actions of others, which is called herding effect. Herd behaviour
occurs in animal herds, packs, bird flocks, fish schools and so on, and it is
very widely spread in humans. The herding effect is well known and studied for
many years [286, 287, 288, 289, 290, 291, 292]. The mathematical description
of herding in evolution equations of social and biological systems is
represented by the replication term [293], which we take in the form

$h_{j}(A_{n},t)=\varepsilon_{j}\left\\{\frac{1}{N-1}\sum_{i(\neq
j)}^{N}[\;f_{i}(A_{n})+q_{i}(A_{n},t)\;]-[\;f_{j}(A_{n})+q_{j}(A_{n},t)\;]\right\\}\;.$
(5.18)

The strength of the herding behaviour is given by the parameters
$\varepsilon_{j}$. Because of the normalization conditions (5.1), (5.2),
(5.3), and (5.4), these parameters are in the interval

$0\leq\varepsilon_{j}\leq 1\qquad(j=1,2,\ldots,N)\;.$ (5.19)

Substituting the herding term into equation (5.5) and measuring time in units
of $\tau$ yields the delayed evolution equations

$p_{j}(A_{n},t+1)=(1-\varepsilon_{j})[\;f_{j}(A_{n})+q_{j}(A_{n},t)\;]+\frac{\varepsilon_{j}}{N-1}\sum_{i(\neq
j)}^{N}[\;f_{i}(A_{n})+q_{i}(A_{n},t)\;]$ (5.20)

defining the probability that the $j$-th agent at time $t+1$ chooses the
alternative $A_{n}$.

For negative times, the utility factor is fixed,
$f_{j}(A_{n},t<0)=f_{j}(A_{n})$, there is no yet any memory, $M_{j}(t<0)=0$,
hence the attraction factor is $q_{j}(A_{n},t<0)=q_{j}(A_{n})$. Therefore, at
zero time, the initial condition for the probability is

$p_{j}(A_{n},0)=(1-\varepsilon_{j})[\;f_{j}(A_{n})+q_{j}(A_{n})\;]+\frac{\varepsilon_{j}}{N-1}\sum_{i(\neq
j)}^{N}[\;f_{i}(A_{n})+q_{i}(A_{n})\;]\;.$ (5.21)

At the first step, equation (5.20) gives $p_{j}(A_{n},1)$, in which
$q_{j}(A_{n},0)=q_{j}(A_{n})\exp\\{-M_{j}(0)\\}$, etc.

The evolution equations describe the dynamics of collective decision making
developing in time due to the interaction of agents through information
exchange. The dynamics of opinions in choosing among alternatives depends on
the utility of alternatives, their emotional attractiveness, and herding
tendency of the agents.

###  5.3 Networks with uniform memory

Under the fixed type of interactions, the society agents can differ in their
initial conditions $f_{j}(A_{n})$ and $q_{j}(A_{n})$, in the strength of the
herding effect, and in the kind of the agent memory, whether long-term or
short-term. There can exist networks, where all agents enjoy the same type of
memory, for instance all agents have long-term memory or all agents possess
short-term memory, but the agents at the initial moment of time are
characterized by different initial conditions. This case of agents with the
same type of memory but different initial conditions is studied in [279, 280].
A network, where all agents have the same type of memory, can be called
uniform-memory network. Despite that all agents in the uniform-memory society
have the same type of memory, they can display nontrivial dynamics due to
different initial conditions. The case of two groups of agents making a choice
between two alternatives has been investigated [279, 280]. The groups differ
by their initial conditions. Two situations have been studied.

(i) Network with uniform long-term memory. This network always displays the
convergence of behavioural probabilities to fixed points. Depending on the
initial conditions and herding strength, the dynamics can be monotonic or with
intermediate oscillations, the fixed points can coincide or not, the
probability trajectories can intersect, but always the probabilities tend to
their fixed points. Increasing the herding parameters $\varepsilon_{j}$ makes
the probability trajectories closer to each other, and sufficiently strong
herding forces the probabilities to converge to a consensual limit.

(ii) Network with uniform short-term memory. This network displays two kinds
of behaviour. One kind is the convergence of the behavioural probabilities to
fixed points that, under week herding effect, do no coincide, but start
coinciding for sufficiently strong herding parameters. The other kind of
behaviour of such short-term memory networks is the existence of permanent
oscillations of both probabilities. These oscillations become suppressed when
the herding parameters are increased.

###  5.4 Network with mixed memory

The most interesting and rich case is when the society contains the groups of
agents with different types of memory and where all three features, utility,
attractiveness, and herding are taken into account [281]. This case also looks
as more realistic, since in any society there always exist agents with
different types of memory. A network, where a part of agents has long-term
memory, while the other part, short-term memory, can be named mixed-memory
network.

For concreteness, let us consider the often met situation of choice between
two alternatives, $A_{1}$ and $A_{2}$. The overall society is composed of a
group of $N_{1}$ agents with long-term memory and $N_{2}$ agents with short-
term memory. The total number of agents is the sum of the agents in each of
the groups, $N=N_{1}+N_{2}$. The number of agents in a $j$-th group preferring
an alternative $A_{n}$ is $N_{j}(A_{n},t)$ depending on time and being
normalized,

$N_{j}(A_{1},t)+N_{j}(A_{2},t)=N_{j}\qquad(j=1,2)\;.$ (5.22)

Then

$p_{j}(A_{n},t)=\frac{N_{j}(A_{n},t)}{N_{j}}\qquad(j=1,2)$ (5.23)

is the frequentist probability of choosing an alternative $A_{n}$ by a member
of the $j$-th group at time $t$.

For the sake of convenience, it is possible to use the simplified notations
for the probabilities

$p_{j}(A_{1},t)\equiv p_{j}(t)\;,\qquad p_{j}(A_{2},t)\equiv 1-p_{j}(t)\;,$
(5.24)

utility factors

$f_{j}(A_{1})\equiv f_{j}\;,\qquad f_{j}(A_{2})\equiv 1-f_{j}\;,$ (5.25)

attraction factors

$q_{j}(A_{1},t)\equiv q_{j}(t)\;,\qquad q_{j}(A_{2},t)\equiv-q_{j}(t)\;,$
(5.26)

and herding factors

$h_{j}(A_{1},t)=h_{j}(t)\;,\qquad h_{j}(A_{2},t)\equiv-h_{j}(t)\;.$ (5.27)

The evolution equations become

$p_{j}(t+1)=f_{j}+q_{j}(t)+h_{j}(t)\qquad(j=1,2)\;,$ (5.28)

with the attraction factors

$q_{j}(t)=q_{j}\exp\\{-M_{j}(t)\\}\qquad(j=1,2)$ (5.29)

and herding factors

$h_{j}(t)=\varepsilon_{j}[\;f_{i}+q_{i}(t)-f_{j}-q_{j}(t)\;]\qquad(i\neq
j)\;,$ (5.30)

where $q_{j}$ is the attraction factor before receiving any information.

The first group, by assumption, consists of agents with long-term memory of
the type

$M_{1}(t)=\sum_{t^{\prime}=0}^{t}\mu_{12}(t^{\prime})\;,$ (5.31)

while the second group has short-term memory with

$M_{2}(t)=\mu_{21}(t)\;.$ (5.32)

The information gain reads as

$\mu_{ij}(t)=p_{i}(t)\ln\;\frac{p_{i}(t)}{p_{j}(t)}+[\;1-p_{i}(t)\;]\ln\;\frac{1-p_{i}(t)}{1-p_{j}(t)}\;,$
(5.33)

where $i\neq j$.

The evolution equations take the form

$p_{1}(t+1)=(1-\varepsilon_{1})[\;f_{1}+q_{1}(t)\;]+\varepsilon_{1}[\;f_{2}+q_{2}(t)\;]\;,$
$p_{2}(t+1)=(1-\varepsilon_{2})[\;f_{2}+q_{2}(t)\;]+\varepsilon_{2}[\;f_{1}+q_{1}(t)\;]\;,$
(5.34)

with the initial conditions

$p_{1}(0)=(1-\varepsilon_{1})(\;f_{1}+q_{1}\;)+\varepsilon_{1}(\;f_{2}+q_{2}\;)\;,$
$p_{2}(0)=(1-\varepsilon_{2})(\;f_{2}+q_{2}\;)+\varepsilon_{2}(\;f_{1}+q_{1}\;)\;.$
(5.35)

Recall that $p_{1}(t)$ describes the fraction of agents with long-term memory
choosing the alternative $A_{1}$ at time $t$, while $p_{2}(t)$ characterizes
the fraction of agents with short-term memory preferring, at time $t$, the
same alternative $A_{1}$.

###  5.5 Dynamic regimes of preferences

A detailed analysis of equations (5.34) has been done in [281]. Depending on
the values of the initial conditions and herding parameters, the following
dynamic regimes are found for the probabilities.

1\. Both probabilities tend to fixed points as in Fig. 1. The tendency can be
smooth or with intermediate oscillations. The fixed points can coincide or
not.

Figure 1: Time dependence of the probabilities $p_{1}(t)$ (solid line) and
$p_{2}(t)$ (dash-dotted line), for the initial conditions $f_{1}=0.8$,
$q_{1}=0.19$ and $f_{2}=0.9$, $q_{2}=-0.8$, under weak herding effect, with
the herding parameters $\varepsilon_{1}=0.05$ and $\varepsilon_{2}=0$. The
probabilities tend to the fixed points $p_{1}^{*}=0.778$ and
$p_{2}^{*}=0.362$.

2\. Under sufficiently strong herding effect the dynamics is smooth and the
probabilities are getting closer to each other, as in Fig. 2, where the
herding parameters are larger, under the same initial conditions as in Fig. 1.

Figure 2: Time dependence of the probabilities $p_{1}(t)$ (solid line) and
$p_{2}(t)$ (dash-dotted line), for the same initial conditions, as in Fig.1,
with $f_{1}=0.8$, $q_{1}=0.19$ and $f_{2}=0.9$, $q_{2}=-0.8$, but under
stronger herding effect, with the herding parameters $\varepsilon_{1}=0.1$ and
$\varepsilon_{2}=0.85$. The probabilities tend to the fixed points
$p_{1}^{*}=0.730$ and $p_{2}^{*}=0.695$.

3\. One of the probabilities tends to a fixed point, while the other at some
moment of time starts periodically oscillating and continues oscillating
forever, as in Fig. 3.

Figure 3: Time dependence of the probabilities $p_{1}(t)$ (solid line) and
$p_{2}(t)$ (dash-dotted line), for the initial conditions $f_{1}=0.3$,
$q_{1}=0.699$ and $f_{2}=0$, $q_{2}=0.98$, in the absence of herding, with
$\varepsilon_{1}=\varepsilon_{2}=0$. The probability $p_{1}(t)$ tends to the
fixed points $p_{1}^{*}=0.3$ while $p_{2}(t)$, after a period of smooth
behaviour, starts everlasting oscillations.

4\. Both probabilities exhibit everlasting periodic oscillations, as in Fig.
4.

Figure 4: Time dependence of the probabilities $p_{1}(t)$ (solid line) and
$p_{2}(t)$ (dash-dotted line), for the initial conditions $f_{1}=0.6$,
$q_{1}=0.39$ and $f_{2}=1$, $q_{2}=-0.9$, under strong herding, with the
parameters $\varepsilon_{1}=1$ and $\varepsilon_{2}=0.9$. Both probabilities
perpetually oscillate.

5\. One of the probabilities tends to a fixed point, but the other starts
fluctuating chaotically and continues fluctuating for ever, as in Fig. 5.

Figure 5: Time dependence of the probabilities $p_{1}(t)$ (solid line) and
$p_{2}(t)$ (dash-dotted line), for the initial conditions $f_{1}=0.3$,
$q_{1}=0.699$ and $f_{2}=0$, $q_{2}=0.99$, with the herding parameters
$\varepsilon_{1}=\varepsilon_{2}=1$. The probability $p_{1}(t)$, after a
period of smooth behaviour, suddenly begins chaotically oscillate, while
$p_{2}(t)$ monotonically tends to the fixed point $p_{2}^{*}=0.3$.

6\. Both probabilities at some moment of time suddenly begin fluctuating
chaotically, and keep fluctuating for ever, as in Fig. 6.

Figure 6: Time dependence of the probabilities $p_{1}(t)$ (solid line) and
$p_{2}(t)$ (dash-dotted line), for the initial conditions $f_{1}=0.6$,
$q_{1}=0.3$ and $f_{2}=1$, $q_{2}=-0.999$, under the herding parameters
$\varepsilon_{1}=1$ and $\varepsilon_{2}=0.8$. Both probabilities exhibit
perpetual chaotic oscillations.

The sudden appearance of self-excited periodic and chaotic oscillations is an
interesting effect arising in a closed system. The herding behaviour smoothes
the trajectories and suppresses both kinds of oscillations, periodic as well
as chaotic. The appearance of large oscillations and chaotic fluctuations in
decision making can be understood as representing unstable decision making of
mentally ill individuals.

###  5.6 Attenuation of emotion influence

The existence of behavioural paradoxes, considered above, is caused by the
presence of emotions of decision makers. The paradoxes arise when subjects
make decisions without exchanging information with each other. However, when
they are allowed to discuss the problem and to exchange their thoughts, the
deviation of the behavioural probabilities from the utility factors
diminishes, hence the attraction factors decrease. The paradox attenuation has
been confirmed in numerous experimental studies [294, 295, 296, 297, 298, 299,
300, 301, 302, 303, 304].

Thus a series of experiments [304] were accomplished, designed to test whether
and to what extent individuals succumb to the conjunction fallacy after
discussing the problem with each other. The experimental design of Tversky and
Kahneman [255] was used. It was found that, when subjects are allowed to
consult between themselves, the proportion of individuals who violate the
conjunction principle falls dramatically, especially when the size of the
group rises. When individuals are forced to think, they recover in their minds
additional information that has been forgotten or shadowed by emotions. The
amount of received information increases with the size of the group. As a
result, there is a substantially larger drop in the error rate when the group
size is increased from two to three than when it is increased from one to two
[300, 304]. These experiments confirm the earlier results [298] that there is
only a marginal difference between the choices of single individuals and two-
person groups, but a significant difference between the choices of two-person
and four-person groups in a game. In the experiment by Charness et al. [304]
with groups of three members, the fraction of individuals giving the answers
deviating from the utility factor dropped to $0.17$. The latter quantity,
actually, is the measure of the attraction factor. Thus, after consultations
providing additional information, that is after the repeated exchange of
information, the attraction factor falls from about $0.25$ to $0.17$. It seems
to be natural that, if the discussions and information exchange would
continue, the attraction factor would drop further, but it is clear that it
should not disappear completely, since the amount of received information is
actually never infinite.

The effect of the paradox attenuation, or in other words, the attenuation of
the attraction factors after repeated exchange of information, under the
conditions discussed above, is illustrated in Fig. 7 and Fig. 8 demonstrating
the disjunction effect attenuation. Here the agents of both groups enjoy long-
term memory, share the same utility factor, taken as in the experiments of
Tversky and Shafir [254] $f_{1}=f_{2}=0.64$, but differ by the initial
conditions for the attraction factors. The probabilities $p_{1}(t)$ and
$p_{2}(t)$ tend to the consensual limit equal to the utility factor, while the
attraction factor, characterizing the paradox, tends to zero, in agreement
with the paradox attenuation effect [294, 295, 296, 297, 298, 299, 300, 301,
302, 303, 304]. The tendency of the attraction factors to zero is preserved
under different herding parameters.

Figure 7: Dynamic disjunction effect. (a) Time dependence of the probabilities
$p_{1}(t)$ (solid line) and $p_{2}(t)$ (dash-dotted line) for the initial
conditions $f_{1}=f_{2}=0.64$, and $q_{1}=0.3$, $q_{2}=-0.2$, in the absence
of herding $\varepsilon_{1}=\varepsilon_{2}=0$. Both probabilities tend to the
consensual limit $p_{1}^{*}=p_{2}^{*}=0.64$; (b) Time dependence of the
attraction factors $q_{1}(t)$ (solid line) and $q_{2}(t)$ (dash-dotted line)
for the same conditions as in Fig.7a. Both attraction factors tend to
$q_{1}^{*}=q_{2}^{*}=0$, demonstrating disjunction-effect attenuation.

Figure 8: Dynamic disjunction effect. (a) Time dependence of the probabilities
$p_{1}(t)$ (solid line) and $p_{2}(t)$ (dash-dotted line) for the initial
conditions $f_{1}=f_{2}=0.64$, and $q_{1}=0.35$, $q_{2}=0.15$, for the herding
parameters $\varepsilon_{1}=0.1$ and $\varepsilon_{2}=0.05$. Both
probabilities tend to the consensual limit $p_{1}^{*}=p_{2}^{*}=0.64$; (b)
Time dependence of the attraction factors $q_{1}(t)$ (solid line) and
$q_{2}(t)$ (dash-dotted line) for the same conditions as in Fig.8a. Both
attraction factors tend to $q_{1}^{*}=q_{2}^{*}=0$, demonstrating disjunction-
effect attenuation.

###  5.7 Continuous decision making

The evolution equations describing the dynamics of preferences deal with
multistep decision making, which implies that decisions are taken in discrete
moments of time. The temporal interval between subsequent decisions is denoted
by $\tau$. If the time of observing the process of taking decisions by
decision makers $t$ is much longer than $\tau$, then it seems admissible to
expand the probability $p_{j}(A_{n},t+\tau)$ over the small ratio $\tau/t\ll
1$. To first order, this gives

$p_{j}(A_{n},t+\tau)\simeq p_{j}(A_{n},t)+\frac{\partial
p_{j}(A_{n},t)}{\partial t}\;\tau\;.$ (5.36)

Measuring time in units of $\tau$, we have

$p_{j}(A_{n},t+1)-p_{j}(A_{n},t)=\frac{\partial p_{j}(A_{n},t)}{\partial
t}\;.$ (5.37)

Then equation (5.5) becomes a differential equation

$\frac{\partial p_{j}(A_{n},t)}{\partial
t}=f_{j}(A_{n},t)+q_{j}(A_{n},t)+h_{j}(A_{n},t)-p_{j}(A_{n},t)\;.$ (5.38)

Equation (5.38) describes the imaginary process of continuous decision making,
assuming that agents can take decisions continuously in time. Of course, such
a continuous process sounds not realistic. However, equation (5.38), as it
seems, is derived under a reasonable assumption that the observation time is
much longer than the time of taking a single decision. It is therefore
interesting whether the description of multistep decision making can be done
by employing the continuous approximation, that is whether the solutions of
the discrete in time equations, characterizing multistep decision making,
could be approximated by solutions of continuous in time equation.

The case of two groups deciding on two alternatives, characterized by Eq.
(5.34), in the continuous approximation reduces to the equation

$\frac{dp_{j}(t)}{dt}=f_{j}+q_{j}(t)+h_{j}(t)-p_{j}(t)\;.$ (5.39)

Substituting here the herding factors (5.30) yields the evolution equations

$\frac{dp_{1}(t)}{dt}=(1-\varepsilon_{1})[\;f_{1}+q_{1}(t)\;]+\varepsilon_{1}[\;f_{2}+q_{2}(t)\;]-p_{1}(t)\;,$
$\frac{dp_{2}(t)}{dt}=(1-\varepsilon_{2})[\;f_{2}+q_{2}(t)\;]+\varepsilon_{2}[\;f_{1}+q_{1}(t)\;]-p_{2}(t)\;,$
(5.40)

with the attraction factors (5.29). Initial conditions are given by (5.35). If
we keep in mind the case of mixed memories, then one group possesses long-term
memory (5.31) that, using the Euler-Maclaurin formula, translates into

$M_{1}(t)=\int_{0}^{t}\mu_{12}(t^{\prime})\;dt^{\prime}+\frac{\mu_{12}(0)+\mu_{12}(t)}{2}\;,$
(5.41)

while the other group has short-term memory (5.32). For $t=0$, we have

$M_{1}(0)=\mu_{12}(0)\;,\;M_{2}(0)=\mu_{21}(0)\;.$ (5.42)

The problem of whether the solutions of time-discrete equations are close to
the solutions of their time-continuous approximation is interesting by its
own. It is also important for the understanding which of the variants more
correctly characterizes real multistep decision making.

###  5.8 Discrete versus continuous processes

Comparing the solutions to discrete and continuous in time equations (5.34)
and (5.40), respectively, it is instructive to compare the solutions for the
same parameters and the same initial conditions. As is well known, for two-
dimensional dynamical systems, continuous differential equations usually
produce more smooth solutions than those of discrete difference equations
[305]. This is confirmed by the results of numerical solution of the
differential equations (5.40) and their comparison with the solutions of
discrete equations (5.34).

Figure 9 demonstrates that the solution to the continuous equation is smooth,
and does not display oscillations as the solution to the discrete equation.
When the solution to the discrete problem is smooth, then the solution to the
continuous problem is also smooth and the related probabilities practically
coincide, as it happens for the parameters of Fig. 2, when the solutions of
discrete and continuous equations are not distinguishable, because of which we
do not repeat them here.

Figure 9: Comparison of the solutions for the probabilities $p_{1}(t)$ and
$p_{2}(t)$ for the discrete and continuous decision making. The initial
conditions $f_{1}=0.8$, $q_{1}=0.19$ and $f_{2}=0.9$, $q_{2}=-0.8$, under weak
herding, with the parameters $\varepsilon_{1}=0.05$ and $\varepsilon_{2}=0$.
The probabilities tend to the fixed points $p^{*}_{1}=0.778$ and
$p_{2}^{*}=0.362$. (a) Solution $p_{1}(t)$ to continuous equations (solid
line), as compared to $p_{1}(t)$ for discrete equations (dash-dotted line);
(b) Solution $p_{2}(t)$ to continuous equations (solid line) as compared to
$p_{2}(t)$ for discrete equations (dash-dotted line).

If the evolution equations possess fixed points, they are defined by the
equations

$p_{1}^{*}=(1-\varepsilon_{1})(f_{1}+q_{1}^{*})+\varepsilon_{1}(f_{2}+q_{2}^{*})\;,\qquad
p_{2}^{*}=(1-\varepsilon_{2})(f_{2}+q_{2}^{*})+\varepsilon_{2}(f_{1}+q_{1}^{*})\;,$

where $q_{j}^{*}\equiv q_{j}(\infty)$.

Figures 10 and 11 show that the solutions to the continuous equations do not
display oscillatory behaviour that exists for discrete equations. And Fig. 12
evidences that the solutions to the continuous equations, as it should be
expected, do not possess chaotic behaviour, demonstrated by the discrete
equations.

Figure 10: Comparison of the solutions for the probabilities $p_{1}(t)$ and
$p_{2}(t)$ for the discrete and continuous decision making. The initial
conditions are $f_{1}=0.3$, $q_{1}=0.699$ and $f_{2}=0$, $q_{2}=0.98$, in the
absence of herding, with $\varepsilon_{1}=\varepsilon_{2}=0$. (a) Solution
$p_{1}(t)$ to continuous equations (solid line), as compared to $p_{1}(t)$ for
discrete equations (dash-dotted line). Both probabilities $p_{1}(t)$ tend to
the same fixed point $p_{1}^{*}=0.3$; (b) Solution $p_{2}(t)$ to continuous
equations (solid line), as compared to $p_{2}(t)$ for discrete equations
(dash-dotted line). The continuous solution $p_{2}(t)$ tends to the fixed
point $p_{2}^{*}=0.699$, whereas the discrete $p_{2}(t)$, after a period of
smooth behaviour, starts periodically oscillating.

Figure 11: Comparison of the solutions for the probabilities $p_{1}(t)$ and
$p_{2}(t)$ for the discrete and continuous decision making. The initial
conditions are $f_{1}=0.6$, $q_{1}=0.39$, and $f_{2}=1$, $q_{2}=-0.9$, under
strong herding, with the parameters $\varepsilon_{1}=1$ and
$\varepsilon_{2}=0.9$. (a) Solution $p_{1}(t)$ to the continuous equations
(solid line), as compared to $p_{1}(t)$ for the discrete equations (dash-
dotted line). The continuous probability tends to the fixed point
$p_{1}^{*}=0.265$, while the discrete probability perpetually oscillates; (b)
Solution $p_{2}(t)$ to continuous equations (solid line), as compared to
$p_{2}(t)$ for discrete equations (dash-dotted line). The continuous solution
$p_{2}(t)$ tends to the fixed point $p_{2}^{*}=0.566$, while the discrete
solution $p_{2}(t)$ permanently oscillates.

Figure 12: Comparison of the solutions for the probabilities $p_{1}(t)$ and
$p_{2}(t)$ for the discrete and continuous decision making. The initial
conditions are $f_{1}=0.6$, $q_{1}=0.3$ and $f_{2}=1$, $q_{2}=-0.999$, under
the herding parameters $\varepsilon_{1}=1$ and $\varepsilon_{2}=0.8$. (a)
Solution $p_{1}(t)$ to the continuous equations (solid line), as compared to
$p_{1}(t)$ for the discrete equations (dash-dotted line). The continuous
probability tends to the fixed point $p_{1}^{*}=0.210$, while the discrete
probability chaotically oscillates; (b) Solution $p_{2}(t)$ to the continuous
equations (solid line), as compared to $p_{2}(t)$ for the discrete equations
(dash-dotted line). The continuous solution $p_{2}(t)$ tends to the fixed
point $p_{2}^{*}=0.522$, while the discrete solution $p_{2}(t)$ chaotically
oscillates.

A special case is illustrated in Fig. 13. The solutions $p_{1}(t)$ and
$p_{2}(t)$ to the discrete equations sometimes can intersect, as is shown in
Fig. 13b. However, the solutions to the continuous equations do not intersect,
instead they glue to each other and tend to a common limit, while in the case
of the discrete problem, one of the probabilities chaotically oscillates, as
is shown in Fig. 14.

Figure 13: Comparison of the solutions for the probabilities $p_{1}(t)$ (solid
line) and $p_{2}(t)$ (dash-dotted line) for the discrete and continuous
decision making. The initial conditions are $f_{1}=0.3$, $q_{1}=0.699$ and
$f_{2}=0$, $q_{2}=0.99$, with the herding parameters
$\varepsilon_{1}=\varepsilon_{2}=1$. (a) Continuous solutions $p_{1}(t)$
(solid line) and $p_{2}(t)$ (dash-dotted line) at the beginning of the process
for $t\in[0,30]$; (b) Discrete solutions $p_{1}(t)$ (solid line) and
$p_{2}(t)$ (dash-dotted line) at the beginning of the process for
$t\in[0,30]$.

Figure 14: Comparison of the solutions for the probabilities $p_{1}(t)$ and
$p_{2}(t)$ for the discrete and continuous decision making. The initial
conditions are $f_{1}=0.3$, $q_{1}=0.699$ and $f_{2}=0$, $q_{2}=0.99$, with
the herding parameters $\varepsilon_{1}=\varepsilon_{2}=1$. (a) Solution
$p_{1}(t)$ to the continuous equations (solid line), as compared to $p_{1}(t)$
for the discrete equations (dash-dotted line). The continuous probability
tends to the fixed point $p_{1}^{*}=f_{2}+q_{2}=0.99$, while the discrete
probability chaotically oscillates; (b) Solution $p_{2}(t)$ to the continuous
equations (solid line), as compared to $p_{2}(t)$ for the discrete equations
(dash-dotted line). The continuous probability tends to the fixed point
$p_{2}^{*}=f_{2}+q_{2}=0.99$, while the discrete probability tends to
$p_{2}^{*}=f_{1}=0.3$.

Thus, the solutions to the continuous equations are always smooth and exhibit
neither periodic oscillations nor chaotic fluctuations that exist for the
discrete equations. The considered equations describe the process of decision
making in a network of intelligent agents. The discrete equations correspond
to multistep decision making, as it can be done in real life. The continuous
equations serve as an approximation to the studied problem. This approximation
is rather accurate when the discrete solutions are smooth. However, the
continuous approximation does not catch such interesting cases as periodic
oscillations and chaotic fluctuations. These oscillations and fluctuations can
arise in realistic problems, where there are the groups of agents with
different features, for instance when there occurs the co-existence of agents
with different types of memory, say long-term and short-term memory.

###  5.9 Time discounting of utility

In the dynamical processes of preferences, considered above, the utility
factor was treated as a constant quantity. This is a reasonable assumption
based on the fact that the period of time when utility does not change is
usually much longer than the time of information exchange between the agents
of the network. For instance, prices often are not changed during years or at
least months, while the information exchange discussing these prices can occur
several times a day.

However, in order to have general understanding, let us consider how the
utility factors could vary when taking account of time discounting of utility.
Then we need to deal with discounted expected utility [282, 306]. In that
case, the utility function $u(x)$ for a payoff $x$ is discounted at time $t$
as $D(t)u(x)$. Here $D(t)$ is a discount function with the properties

$0<D(t)<1\qquad(0<t<\infty)\;,$ $D(0)=1\;,\qquad D(\infty)=0\;.$ (5.43)

A typical example of a discount function is $D(t)=1/(1+r)^{t}$, where $r$ is a
discount rate and $t$ is a discrete time counted in some temporal units,
usually in units of an year. With the discounted utility functions, the
expected utility of a lottery $L_{n}$ becomes discounted as $D(t)U(L_{n})$.
Denoting the discounted expected utility of an alternative $A_{n}$ as

$U(A_{n},t)=U(A_{n})D(t)\;,$ (5.44)

for the discounted utility factor (3.25), we obtain

$f(A_{n},t)=\frac{U(A_{n},t)\exp\\{\beta
U(A_{n},t)\\}}{\sum_{n}U(A_{n},t)\exp\\{\beta U(A_{n},t)\\}}\;.$ (5.45)

Using here the discounted utility (5.44) gives

$f(A_{n},t)=\frac{U(A_{n})\exp\\{\beta(t)U(A_{n})\\}}{\sum_{n}U(A_{n})\exp\\{\beta(t)U(A_{n})\\}}\;,$
(5.46)

with the initial expected utilities and the discounted belief function

$\beta(t)=\beta D(t)$ (5.47)

enjoying the properties

$0<\beta(t)<\beta\qquad(0<t<\infty)\;,$
$\beta(0)=\beta\;,\qquad\beta(\infty)=0\;.$ (5.48)

As we see, the utility factor can be considered to be constant if $t\ll 1$,
when $\beta(t)\simeq\beta$, if $t\gg 1$, when $\beta(t)\simeq 0$, and also if
$\beta\ll 1$, when $\beta(t)\simeq 0$.

In that way, the expected utilities are discounted with time, however they
enter the utility factors so that the latter are almost constant for wide time
periods, and they do not vary at all for the neutral beliefs of decision
makers, when $\beta=0$. Thus the assumption that the utility factor is
practically constant has the wide range of applicability.

###  5.10 Collective network operation

The functioning of a network can be characterized in two ways, by considering
the actions of its separate parts and by studying the operation of the network
as a whole. Under a network, we can imply a society with the population
structured into several different groups or an artificial intelligence
consisting of several groups of nodes with different properties. That is, it
is possible to study the decision making of parts of a society or the
collective decision making by the whole society. Otherwise, it is possible to
concentrate on the actions of different parts of a network or on the
collective functioning of the whole network.

Suppose, we consider a society (network) of $N$ agents (nodes). The society is
structured into $G$ groups (parts) enumerated by $j=1,2,\ldots,G$. Each group
is characterized by its specific properties, say its type of memory and
herding parameters. The group $j$ consists of $N_{j}$ agents, so that the
group $j$ composition fraction is

$c_{j}\equiv\frac{N_{j}}{N}\qquad(j=1,2,\ldots,G)\;.$ (5.49)

Hence the normalization conditions are

$\sum_{j=1}^{G}N_{j}=N\;,\qquad\sum_{j=1}^{G}c_{j}=1\;.$ (5.50)

As early, the number of agents in a group $j$ choosing an alternative $A_{n}$
at time $t$ is $N_{j}(A_{n},t)$. Since each agent of a group chooses one of
the alternatives,

$\sum_{n=1}^{N_{A}}N_{j}(A_{n},t)=N_{j}\;.$ (5.51)

The probability that an agent of a group $j$ chooses an alternative $A_{n}$ at
time $t$ is

$p_{j}(A_{n},t)\equiv\frac{N_{j}(A_{n},t)}{N_{j}}\;.$ (5.52)

The number of agents in the whole society choosing the alternative $A_{n}$
reads as

$N(A_{n},t)=\sum_{j=1}^{G}N_{j}(A_{n},t)\;.$ (5.53)

Then the probability that an agent of the whole society chooses an alternative
$A_{n}$ at time $t$ is defined as

$p(A_{n},t)\equiv\frac{N(A_{n},t)}{N}\;.$ (5.54)

The related normalization condition is

$\sum_{n=1}^{N_{A}}p(A_{n},t)=1\;.$ (5.55)

In the above subsections, the temporal behaviour of the group probabilities
(5.52) has been investigated. The collective decision of a society, or the
collective action of a network is described by the probability (5.54). The
group probabilities define the total network probability

$p(A_{n},t)=\sum_{j=1}^{G}c_{j}p_{j}(A_{n},t)\;,$ (5.56)

where the structure of the society (network) is prescribed by the given
composition fractions $c_{j}$. The collective probability (5.56) characterizes
the behaviour of the society (network) as a whole.

##  6 Conclusion

The review discusses the principal points of how the operation of artificial
intelligence has to be organized in order to simulate the so-called human-
level artificial intelligence that is able to make decisions taking into
account the utility of the considered alternatives as well as emotions
accompanying the choice. In order to formalize the operation of such an
affective intelligence, it is necessary to characterize the basic rules of
human decision making.

First of all, it is required to understand the general structure of affective
decision making. The pivotal idea suggesting the main principles of this
structure is the formal similarity between decision making under emotions and
quantum measurements under intrinsic noise. Cognition-emotion duality in
decision making is formally analogous to observable-noise duality in quantum
measurements.

It is in no way assumed that the brain is a quantum object or consciousness is
in any sense quantum. The analogies between quantum measurements and decision
making are purely nominal. It is possible to adduce a number of examples where
physically very different processes are characterized by very close
mathematics. For instance, many processes of self-organization in complex
systems can be interpreted as decision making [201]. As an example of two
different in nature phenomena characterized by very similar mathematical
description, it is possible to mention that the development and arising
distribution of cities over their size can be described as the process of
Bose-Einstein condensation [307]. Although the growth of cities and Bose-
Einstein condensation, by their nature, are absolutely different processes,
they can be described by similar mathematics. In the same vein one has to
accept the analogies between quantum measurements and decision making. These
analogies turned out to be rather productive suggesting the principal points
governing the description of affective decision making.

To be concrete, Sec. 2 presents the general theory of quantum measurements
under intrinsic noise. The goal is to analyze what ideas and methods could be
accepted for the design of affective decision theory. However, there is no
need to involve all that quantum machinery for decision theory because of the
following:

(a) From one side, quantum theory is overloaded by numerous notions having no
direct meaning in decision theory. Thus, it is not defined what in decision
making would be the form of Hamiltonians, of statistical operators, and so on.

(b) From the other side, the notions of quantum theory are not sufficient for
suggesting quantitative theory of decision making. This is why the use of
quantum techniques for decision making has produced practically no
quantitative description and no quantitative predictions. In order to develop
a quantitative theory of decision making, one has to complement quantum
formulas by quite a number of assumptions of purely classical origin and
introduce several fitting parameters.

(c) In addition, involving rather complicated techniques of quantum theory,
complementing them by a number of classical assumptions, and equipping with
fitting parameters seems to be excessive if a simpler classical description is
admissible.

Therefore, borrowing some general ideas from the theory of quantum
measurements under intrinsic noise, it looks beneficial to formulate an
axiomatic approach to affective decision making in purely classical terms,
given in Sec. 3. The ideas of quantum measurement theory serve rather as
Ariadne’s thread for formulating affective decision theory that can be used by
artificial intelligence. It is in that sense the term “quantum operation of
artificial intelligence” is used. The intelligence is not quantum, but its
operation imitates some rules specific for quantum theory. The main points
borrowed from the theory of noisy measurements are:

  1. (i)

Decision making has to be treated as a probabilistic process that needs to be
characterized by behavioural probabilities defining the fractions of decision
makers choosing the related alternatives from a given set.

  2. (ii)

The role of emotions in decision making is similar to the role of intrinsic
noise under quantum measurements.

  3. (iii)

Behavioural probability, taking account of cognition-emotion duality, is
represented by the superposition of two terms, a rational utility factor,
measuring the utility of the alternatives, and an attraction factor,
characterizing the role of emotions in the process of decision making. In
particular cases, the attraction factor can be split into several terms
responsible for different types of emotions.

  4. (iv)

Alternatives and accompanying emotions are entangled in the sense that they
are connected with each other and both of them influence the choice.

  5. (v)

Different types of emotions interfere with each other producing in the
behavioural probability the term named attraction factor.

  6. (vi)

It is necessary to distinguish the joint probability of two consecutive events
occurring at different moments of time and the joint probability of two events
defined as a synchronous probability of two events at different locations.
This latter probability, similarly to the probability of a single event, is
represented by a sum of utility factor and attraction factor, if emotions are
taken into account.

  7. (vii)

Temporal dependence of behavioural probabilities can be modeled by the form
arising from the consideration of time dependence of the averages
corresponding to observable quantities under non-destructive repeated quantum
measurements.

To have predictive power, these principal items can be complemented by non-
informative priors characterizing the behaviour of a typical decision maker.
All elaboration of affective decision theory can be given in classical
terminology, with no quantum formulas explicitly implicated. The behaviour of
real decision makers on the aggregate level is quantitatively well described
by the formulated affective decision theory, using no fitting parameters. This
is confirmed by treating a set of lotteries in Sec. 3.

The famous behavioural paradoxes plaguing classical decision making find their
natural resolution in the frame of the affective decision theory taking
account of emotions. The paradoxes are not merely qualitatively explained but
are in good quantitative agreement with empirical data on the aggregate level,
without involving any fitting parameters.

A family of affective intelligent agents composes a network where the agents
interact through information exchange. The intelligent agents can differ by
their initial conditions and by the type of memory, in the ultimate cases
being long-term or short-term memory. There can exist the networks of agents
with uniform memory, that is having the same type of memory, but different
initial conditions. The most interesting and rich is the behaviour of a mixed
network, where a part of agents enjoys long-term memory, while the other part
has short-term memory. In these networks there can arise self-excited waves of
preferences demonstrating either periodic or chaotic dynamics. Chaotic
decision making reminds us the behaviour of patients suffering from mental
diseases.

Discussing the multi-step decision making of society agents, we can keep in
mind either a human society composed of different groups or a network
consisting of several parts of nodes, or the brain composed of neurons.

The described approach, from one side, helps understanding the behaviour of
real humans making a choice between alternatives by evaluating their rational
utility at the same time being subject to accompanying irrational emotions.
From the other side, the formalized rules of the choice followed by a typical
decision maker can help for implicating the related algorithms in the
operation of a complex Affective Artificial Intelligence.

## Acknowledgment

I am very grateful for many useful discussions to D. Sornette and E.P.
Yukalova. A part of the material composing this review has been presented in
my lectures I have been giving for several years in Swiss Federal Institute of
Technology (ETH Zurich), Zurich, Switzerland.

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

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