# Discrete versus Continuous Algorithms in Dynamics of Affective Decision Making
## 3 Discrete Dynamics in Affective Decision Making  
The approach to affective decision making, considered in the present paper, is based on the probabilistic theory [9; 10; 11] characterized by probabilities of choosing an alternative among the set of given alternatives, taking account of utility as well as emotions. In studying dynamic equations, one has to define initial conditions, that is, the utility factors and attraction factors at time \(t=0\). At the initial time, the decisions are taken by agents independently, since they have no time for exchanging information and observing the behavior of their neighbors. Thus, the initial behavioral probabilities define the required initial conditions for the following dynamics. A society, or a network, is considered to consist of many agents. For each member of a society, the other members play the role of surrounding. The agents of a society interact with each other through the exchange of information and by imitating the actions of others. The probability dynamics is due to these features [35; 36; 37]. Let us consider \(N_{A}\) alternatives between which one needs to make a choice. The alternatives are enumerated by the index \(n=1,2,\ldots,N_{A}\). A society of \(N_{tot}\) agents is making a choice among the available alternatives. The overall society is structured into \(N\) groups enumerated by the index \(j=1,2,\ldots,N\). Each group differs from other groups by its specific features, such as its type of memory and the inclination to replicate the actions of others, which is termed _herding_. The herding effect is well known and has been studied in voluminous literature [38; 39; 40; 41; 42; 43; 44; 45; 46]. The number of agents in a group \(j\) is \(N_{j}\) so that the summation over all groups gives the total number of agents,  
\[\sum_{j=1}^{N}N_{j}\ =\ N_{tot}. \tag{10}\]  
The number of agents in a group \(j\) choosing an alternative \(A_{n}\) at time \(t\) is \(N_{j}(A_{n},t)\). Since each member of a group \(j\) chooses one alternative, then  
\[\sum_{n=1}^{N_{A}}N_{j}(A_{n},t)\ =\ N_{j}. \tag{11}\]  
The probability that a member of a group \(j\) chooses an alternative \(A_{n}\) at time \(t\) is  
\[p_{j}(A_{n},t)\ \equiv\ \frac{N_{j}(A_{n},t)}{N_{j}}, \tag{12}\]  
which satisfies the normalization condition  
\[\sum_{n=1}^{N_{A}}p_{j}(A_{n},t)\ =\ 1\,\qquad 0\ \leq\ p_{j}(A_{n},t)\ \leq\ 1. \tag{13}\]  
Probability (12) is a functional of the utility factor \(f_{j}(A_{n},t)\) and the attraction factor \(q_{j}(A_{n},t)\). The utility factor characterizes the utility of an alternative \(A_{n}\) at time \(t\) and obeys the normalization condition  
\[\sum_{n=1}^{N_{A}}f_{j}(A_{n},t)\ =\ 1\,\qquad 0\ \leq\ f_{j}(A_{n},t)\ \leq\ 1. \tag{14}\]  
The attraction factor quantifies the influence of emotions when selecting an alternative \(A_{n}\) at time \(t\) and satisfies the normalization condition  
\[\sum_{n=1}^{N_{A}}q_{j}(A_{n},t)\ =\ 0\,\qquad-1\ \leq\ q_{j}(A_{n},t)\ \leq\ 1. \tag{15}\]  
At the initial moment of time \(t=0\), the functional dependence of the probability on the utility and attraction factors has the form  
\[p_{j}(A_{n},0)\ =\ f_{j}(A_{n},0)+q_{j}(A_{n},0), \tag{16}\]  
where the initial utility factor and attraction factor can be calculated following the rules explained in detail in earlier works [9; 10; 11; 46; 47; 48]. The tendency of agents of a group \(j\) to replicate the actions of the members of other groups is described by the _herding parameters_\(\varepsilon_{j}\), which lay in the interval  
\[0\ \leq\ \varepsilon_{j}\ \leq\ 1\qquad(j=1,2,\ldots,N). \tag{17}\]  
The other meaning of these parameters is the level of tendency for acting as others, which in the present setup models the agents' cooperation. Generally, the value \(\epsilon_{j}\) can vary in time. However, this variation is usually very slow so that the herding parameters can be treated as constants characterizing the members of the related groups.